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Derivatives and Risk Management: Introduction Marta Wisniewska
Module Outline
Literature
Grading
RISK
MANAGEMENT DERIVATIVES
PRICING OF DERIVATIVES
USE OF DERIVATIVES & RISK MANAGEMENT
Derivatives and Risk Management: Introduction Marta Wisniewska
PV of Cash Flows the asset is producingValue of an asset
Value of a derivative At what price there is no arbitrage?
PV Valuation: e.g. DCF, DDM
No Arbitrage Valuation
Derivatives and Risk Management: Introduction Marta Wisniewska
The purpose of this module is to discuss how derivaties can
be used to manage financial risk.
We will be analyzing:
the impact of using derivaties
and the pricing of derivaties.
Derivatives and Risk Management: Introduction Marta Wisniewska
Risk…
…something different than expected
Derivatives…
…financial assets whose value depend of value of
underlying asset (the value of an asset is derived from
value of another asset)
Derivatives and Risk Management: Introduction Marta Wisniewska
DATE TITLE TEXTBOOK
15.09
Introduction to Risk Management
Bonds, Interest Rates and Swaps ch 1, 4, 7
16.09
Futures and Forwards
Introduction to Options ch 3, 5-6, 9-11
6.10 Option Valuation ch 12, 14
7.10
The Greeks & Hedging
Value at Risk
American Options & Dividends
Real Optionsch 10, 12, 18
34
4.11
Project presentations
TEST
Derivatives and Risk Management: Introduction Marta Wisniewska
Hull, J. C. (2012): Options,
Futures and Other
Derivatives, 8th Edition,
Pearson
or earlier eddition
Available in the liberary and online:
www.witor.biz/drm
Derivatives and Risk Management: Introduction Marta Wisniewska
LECTURE NOTES
Grading
TEST: 70% Not multiple choice
You will need perform calculations
…and interpret the numbers (e.g. hedging)
Section 1:
Option Question
60%
Section 2:
Answer 2 out of 3 questions
40%
Derivatives and Risk Management: Introduction Marta Wisniewska
Project: 30% → Trading Game
see sample exam paper
http://fxtrade.oanda.com/forex_trading/fxtrade/
Trading Game
Stock Indexes
Currencies
Commodities
goal: experience the impact of leverage on trading
Derivatives and Risk Management: Introduction Marta Wisniewska
Derivatives and Risk Management: Introduction Marta Wisniewska
UK
Derivatives and Risk Management: Introduction Marta Wisniewska
Derivatives and Risk Management: Introduction Marta Wisniewska
Derivatives and Risk Management: Introduction Marta Wisniewska
Derivatives and Risk Management: Introduction Marta Wisniewska
Derivatives and Risk Management: Introduction Marta Wisniewska
1.5 → 0.02 0.02/1.5
= 0.013 (1.3%)
Derivatives and Risk Management: Introduction Marta Wisniewska
Derivatives and Risk Management: Introduction Marta Wisniewska
RULES OF THE GAME 1
1. CREATE A DEMO ACCOUNT asapif you have any technical problems I send me an email
2. GAME TIME: 17 Sept - 2 Nov
3. PRESENTATIONS OF RESULTS: 4th Nov (PPT)- deadline for the report: 3rd Nov (Word/PDF file)
4. You need to activelly manage your portfoliothat means you can not: make no trades at all or buy 1 EUR and hold it till 4 Dec
Derivatives and Risk Management: Introduction Marta Wisniewska
RULES OF THE GAME 2
5. You decide on the strategy:
You decide what currency/ccommodity you trade /
focus on
You decide what position you take: long, short
You decide for how long you hold the position
(exit strategy)
6. You follow your strategy
7. In case your strategy fails during the game, you
can change the strategy and follow the new one
Derivatives and Risk Management: Introduction Marta Wisniewska
RULES OF THE GAME : Final PRESENTATION
Each presentation should include:
Brief summary of what you have been doing for the
last 1.5 months
Presentation of your strategy (Why this strategy?
Have you changed it? Why?)
History of your trades (and your final results)
Evaluation of your strategy including answer to
question of what went wrong (if anything) and why
Evaluation of impact of leverage on your trading
REMEMBER YOU TRADE ON A MARGIN!
see Trading Game Report Template
Derivatives and Risk Management: Introduction Marta Wisniewska
?
Derivatives and Risk Management: Introduction Marta Wisniewska
1.Introduction to
Risk Management
2.Bonds, Interest
Rates and Swaps
Lecture 1:
1. Introduction to
Risk Management
Introduction to Risk Management
Financial Markets
Financial markets facilitate the transfer of funds between
borrowers and lenders
To trade time & risk
FUNDSLENDERS BORROWERSFUNDS
HouseholdsBusinessesGovernmentsForeigners
HouseholdsBusinessesGovernmentsForeigners
DEBT
EQUITY
Money
Market
Capital
Market
Hedgers
Speculators
Arbitrageurs Arbitrage: a trading strategy that allows making profits without any risk of loss
Possible when instruments generating the same cash flow are sold at 2 different prices at 2 markets
All current methods of pricing derivatives utilize the notion of arbitrage.
Arbitrage pricing methods derive the prices of derivatives from conditions that preclude arbitrage opportunities.
Participants of the financial markets
reduce risk exposure
increase risk exposure
Introduction to Risk Management
Risk and exposure
Why manage risk?
Types of risk
Risk management process
Risk management instruments
Misuse of derivatives
Introduction to Risk Management
Risk and exposure
What investors care about when making the investments?
Return
Risk
Risk-return trade off: the higher the risk the higher expected rate of return
Introduction to Risk Management
Risk and exposure
What is return (R)?
𝒐𝒓 𝑹 = 𝐥𝐧𝑫𝒕 + 𝑷𝒕𝑷𝒕−𝟏
𝑹 =𝑫𝒕 + 𝑷𝒕 − 𝑷𝒕−𝟏
𝑷𝒕−𝟏
simple return
logarithmic return
additive properties
Income received (Dt) on an investment plus any change in the
market price (Pt – Pt-1), usually expressed as a percent of the
beginning market price of the investment.
Introduction to Risk Management
Risk and exposureIntroduction to Risk Management
Risk and exposure
What is risk?
𝜎 =
𝑖=1
𝑛
𝑅𝑖 − 𝑅 2𝑝𝑖
In common language risk is viewed as something ‘negative’, as an “exposure to danger or hazard”.
The Chinese symbols for risk (危機) combines danger and opportunity.
In Finance Risk is something different than expected:Traditionally risk measured by standard deviation of returns (σ), and is referred to as volatility
Introduction to Risk Management
What is risk?
Expected Return
High Variance Investment
Low Variance Investment
Probability
NO RISK
Risk and exposure
Normal distribution68, 96, 99.7 rule
+/- 1 σ , +/- 2 σ , +/- 3 σ
Introduction to Risk Management
Risk and exposure
What is risk?
Is volatility the best measure of risk?
STD
DEV
OF
PO
RTF
OLI
O R
ETU
RN
NUMBER OF SECURITIES IN THE PORTFOLIO
TotalRisk
Unsystematic risk (Unique risk)
Systematic risk(Market risk)
Factors such as changes
in nation’s economy, tax
reforms, or a change in
the world situation.
Factors unique to a particular company
or industry. For example, the death of a
key executive or loss of a governmental
defense contract.
𝑅𝑖 = 𝑅𝐹 + 𝛽𝑖 𝑅𝑀 − 𝑅𝐹 𝛽𝑖 =𝐶𝑜𝑣 𝑅𝑀, 𝑅𝑖
𝜎𝑀2 𝜎𝑖
2 = 𝛽𝑖2𝜎𝑀
2 + 𝜎𝑟𝑒𝑠𝑖𝑑2
𝑅𝑝 =
𝑖=1
𝑛
𝑅𝑖𝑤𝑖
𝜎𝑝2 =
𝑖=1
𝑛
𝜎𝑖2𝑤𝑖
2 + 2
1≤𝑖<𝑗≤𝑛
𝑤𝑖𝑤𝑗𝜎𝑖𝜎𝑗𝜌𝑖𝑗
Capital Asset pricing model (CAPM) assumes that investors hold well diversified portfolios, thus they care only about exposure towards market risk and not the total risk.
This exposure is measure by beta (β)
Introduction to Risk Management
Risk and exposure
. regress rbar beta sig, robust
Linear regression Number of obs = 101
F( 2, 98) = 2.54
Prob > F = 0.0838
R-squared = 0.1040
Root MSE = .00047
------------------------------------------------------------------------------
| Robust
rbar | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
beta | -.0005175 .0002296 -2.25 0.026 -.0009732 -.0000618
sig | .0327825 .0169871 1.93 0.057 -.000928 .0664929
_cons | .0002485 .0001712 1.45 0.150 -.0000912 .0005881
------------------------------------------------------------------------------
no idiosyncratic risk
Testing CAPM
Does CAPM hold?
𝑅𝑖 = 𝑅𝐹 + 𝛽𝑖 𝑅𝑀 − 𝑅𝐹
Introduction to Risk Management
Risk and exposure
Testing CAPM
. regress rbar beta beta2, robust
Linear regression Number of obs = 101
F( 2, 98) = 2.28
Prob > F = 0.1081
R-squared = 0.0763
Root MSE = .00048
------------------------------------------------------------------------------
| Robust
rbar | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
beta | .0009477 .0004908 1.93 0.056 -.0000264 .0019217
beta2 | -.0004873 .0002336 -2.09 0.040 -.0009509 -.0000236
_cons | .0000256 .0002206 0.12 0.908 -.0004122 .0004634
------------------------------------------------------------------------------
quadratic relation
Does CAPM hold?
NO
Introduction to Risk Management
Risk and exposure
What is risk?
Is volatility the best measure of risk?
p = (100 - X)% = 3%
VaR is a 3rd percentile of
the distribution of gain in
the value of the portfolio
in the next 5 days.
Value at Risk (VaR)
‘We are X percent sure there will not be a loss of more than VaR in the next N days’.
Focus on downside.
Example: If N = 5 and X = 97 what is the VaR?
Introduction to Risk Management
time t0 t1 t2 … t10
Machine -350 000 0
Today you can buy a machine for £350 000. Thanks to this machine, in the next 10 years, you can produce t-shirts.Assume that you know the following about cash flows in years t0 - t10:
Sales volume: 20 000 units per year Sales price: £ 8.50 per unit Variable cost: £ 3.50 per unit Fixed costs: £ 24 875 per year
Cash Flows CF1 CF2 … CF10
Exposure Example:
Investment Appraisal: Sensitivity Analysis
Risk and exposure
Exposure: state of having no protection from
something risky
Should you buy the machine?
Is NPV positive?
Under what assumptions
Introduction to Risk Management
time t0 t1 t2 … t10Cost of capital 15%
Machine -350 000 0
Cash Flows in years t1-t10: Sales volume: 20 000 units per year Sales price: £ 8.50 per unit Variable cost: £ 3.50 per unit Fixed costs: £ 24 875 per year
Cash Flows CF1 CF2 … CF10
Exposure Example:
Investment Appraisal: Sensitivity Analysis
Risk and exposure
Since every year we receive constant cash flow (=CF):
Therefore to calculate the NPV the annuity formula can be used:
1st derivative of NPV with respect to chosen factor (exposure) tells us how
much NPV will change if the value of chosen factor changes by 1 unit.
𝐶𝐹 = 20 000 × 8.5 − 3.5 − 24 875 = 75 125
NPV = −350 000 +75 125
0.151 −
1
1 + 0.15 10 What happens to NPV if there is change in production costs?
𝜕𝑁𝑃𝑉
𝜕𝑉𝐶=−20 000
0.151 −
1
1 + 0.15 10
= −133 333 × 0.752815
= −100 375.37
= 27 035
If variable cost increases by £1,the NPV decreases by £100375.37 to £-73 340, 37 (= 27 035 -100 375.37).
Approx. 12%change in VCleads tonegative NPV
Introduction to Risk Management
time t0 t1 t2 … t10
Machine -350 000 0
Today you can buy a machine for £350 000. Thanks to this machine, in the next 10 years, you can produce t-shirts.Assume that you know the following about cash flows in years t0 - t10:
Sales volume: 20 000 units per year Sales price: £ 8.50 per unit Variable cost: £ 3.50 per unit Fixed costs: £ 24 875 per year
Cash Flows CF1 CF2 … CF10
Exposure Example:
Investment Appraisal: Sensitivity Analysis
Risk and exposure
What are the sources of Exposure?
time
Cost Factors Income
price of cotton (in USD)USD/GBP exchange rate…
Introduction to Risk Management
Why manage risk?
Cotton price exposure
Introduction to Risk Management
Why manage risk?
FX exposureCurrency Risk
Introduction to Risk Management
Why manage risk?
Interest Rate exposureInterest Rate Risk
Introduction to Risk Management
Why manage risk?
Interest Rate exposureInterest Rate Risk
Introduction to Risk Management
Why manage risk?
Market exposureMarket Risk
Introduction to Risk Management
Why manage risk?
Oil Prices exposureOil Prices Risk
Introduction to Risk Management
Why manage risk?
Why manage risk?
Decision not to manage an expose is a speculation
Exposure to risk factors can affect:The valuation of the projectThe valuation of the assetThe value of the whole company
Yet such speculation is not necessary a bad thing!
Always consider your situation vs. your competitors and the industry standard
Introduction to Risk Management
Types of risk
2. Event driven definition of risk
1. Diversifiability Systematic Unsystematic
Classification
Business risk Non-business risks
Financial risk Other risks
Reputational
risk
Jorion P.,(2007) Value at Risk
Regulatory,
political risk
Market risk, liquidity
risk
Credit risk
Operational risk
Business decisions
Business environment
Strategic risk
Product, marketing,
organization
Macroeconomics
Competition,
technology
These classifications are somewhere arbitrary(some risks overlap)
One company faces various types of risks, some more important
than othersWhat can he do?
Introduction to Risk Management
Expected Return
High Variance Investment
Low Variance Investment
Probability
NO RISK
Risk management process
What is risk management?
Assuring an outcome
How to assure outcome?
Introduction to Risk Management
Risk management process
Reduce Reduce the probability that the event will occur
Reduce the impact if the event does occur
Transfer Transfer the cost of an undesirable outcome to someone else
Avoid Completely avoid potential events thus provide a zero probability that
they will occur
Do Nothing Let the risk happen and be ready to bear the consequences
How to assure outcome?
Introduction to Risk Management
Price Risk
What Happens to the Distribution When You Have a Floor Price at $40?
100 14040
Reduce
Risk management process
Oil producer
Introduction to Risk Management
Price Risk
What Happens to the Distribution When you Establish a Short Fence from $100 to $140 ?
Via use of derivativesHow to
achieve this?
Reduce
Risk management processIntroduction to Risk Management
Production Risk
Risk: of a poor weather event causing the undesirable outcome of lower than expected yields
160 19595 wheat yields
Reduce the cost of the risk via spatial location, multiple variety selection, and other cropping practices.
Reduce
Risk management processIntroduction to Risk Management
160 19595Transfer the cost of the risk via
crop insurance
Transfer
wheat yields
Production Risk
Risk: of a poor weather event causing the undesirable outcome of lower than expected yields
Risk management processIntroduction to Risk Management
Financial Risk
Risk: higher interest rates causing the undesirable outcome of lower than expected cash flow
Cash FlowTransfer the risk
via fixed rate loans
Transfer
Risk management processIntroduction to Risk Management
Reduce the cost of the negative impact via
lower debt financing
Financial Risk
Risk: higher interest rates causing the undesirable outcome of lower than expected cash flow
Cash Flow
Reduce
Risk management processIntroduction to Risk Management
Risk management process
Risk management (RM) is the process by which various risk exposures are
identified,
measured, and
controlled.
Introduction to Risk Management
Phases risk management process:
1. Identify a company’s current risk profile and set a target risk profile.
2. Achieve the target risk profile by coordinating resources and executing transactions (i.e. reduce, transfer, avoid, do nothing, or some combination)
3. Evaluate the altered risk profile.
Risk management processIntroduction to Risk Management
RM process – phase 1
Decompose corporate assets and liabilities into risk pools: interest rate, foreign exchange, crude oil, etc.
Develop market scenarios and test the impact of these on the values of the risk pools and on the value of the company as a whole. This determines the company’s “value at risk”.
Develop a target risk profile, which may or may not include a complete elimination of risk.
Risk management processIntroduction to Risk Management
This is the implementation phase.
Many companies centralize their risk management activities.
This allows for coordination and avoids unnecessary transactions.
Division 2
Exposed short to Polish interest
rates.
Has floating rate loan in zloty.
Division 1
Exposed long to Polish interest
rates.
Has bank account in zloty.
Net
exposure
RM process – phase 2
Risk management processIntroduction to Risk Management
This is the evaluation phase.
Key questions to consider:
Has the firm’s risk profile changed?
Is the current risk profile still appropriate?
What new economic and market scenarios should be considered in the next iteration?
RM process – phase 3
Risk management processIntroduction to Risk Management
Risk prioritization matrix
Probability of happening
Potential impact
Act if cost effective
No action required
Immediate action
Action required
Small Catastrophic
High
Low
Risk management process
Proximity
Introduction to Risk Management
Risk management is an individual decision
No one "right" decision
The "right" decision depends on the characteristics of the operation and
individual decision-maker
Risk
Revenue
1
2
3
Risk management process
M&M
Introduction to Risk Management
Risk management instruments
Approaches/ Actions Instruments
Eliminate/ Avoid
Transfer
Absorb/ Manage
Hedge/ Sell
Diversify
Insure
Set policy
Hold capital
derivatives
Introduction to Risk Management
Forward/ Futures contracts A forward/futures contract is an agreement between two parties, a buyer and a seller, to exchange
an asset at a later date for a price (delivery/ futures price) agreed to in advance, when the contract is first entered into
Forwards OTC
Futures exchange traded
Options An option gives the buyer the right, but not the obligation, to buy/sell the underlying at a later date
for a price (strike or exercise price) agreed to in advance, when the contract is first entered into.
Call option: an option to buy the underlying at the strike price
Put option: an option to sell the underlying at the strike price
The option buyer pays the seller a sum of money called the option price or premium.
OTC and exchange traded.
Swaps Swap is an over-the-counter agreement to exchange cash flows in the future.
Risk management instruments
A derivative is a financial instrument whose value derives from (depends on) the value of
something else (underlying asset).
Introduction to Risk Management
Consider a British fund manager with a portfolio of U.S. equities.
If he buys IBM shares, he is exposed to three risks:
Prices in the U.S. equity market generally.
The price of IBM stock specifically.
The dollar/sterling exchange rate.
Risk management instruments
He is bearish about:
The dollar’s medium-term prospects.
The overall U.S. stock market.
To hedge the currency risk, he could sell dollars under the terms of a forward contract
To hedge the market risk, he could short futures contracts on the S&P 500 index.
He would be left with exposure to IBM’s share price only.
EXPOSURE
Believes
Action
EXAMPLE:
Introduction to Risk Management
Derivatives allow firms to:
Separate out the financial risks that they face.
Remove or neutralize the risk exposures they do not want.
Retain or possibly increase the risk exposures they want.
Using derivatives, firms can transfer, for a price, any undesirable risk to other parties
who either have risks that offset or want to assume that risk.
Risk management instrumentsIntroduction to Risk Management
Risk management instruments
Derivative markets have a long history.
Futures markets: date back to the Middle Ages.
Options markets: date back to 17th century Holland.
Last 35 years: extraordinary growth worldwide:
Increased market volatility.
Deregulation of markets.
Globalization of business
Derivative markets:The over-the-counter (OTC) marketThe exchanges
Derivatives and Financial Risk Management, Spring 2016 67/54M. Wisniewska
Introduction to Risk Management
Leverage
Leverage is the ability to control large amounts of an underlying asset with a
comparatively small amount of capital.
As a result, small price changes can lead to large gains or losses.
Leverage makes derivatives:
Powerful and efficient
Potentially dangerous
Risk management instrumentsIntroduction to Risk Management
EXAMPLE: LEVERAGE WITH OPTIONS
It is May.
The price of XYZ stock is £28.30.
A December call option on XYZ stock with a £29 strike price is selling for £2.80.
A speculator thinks the stock price will rise.
To make a profit, the speculator might:
Buy, say, 100 shares of XYZ stock for £2,830.
Buy 1,000 options (10 option contracts on 100 shares each) for £2,800,
(roughly the same amount of money).
Risk management instrumentsIntroduction to Risk Management
Suppose the speculator is right. The stock price rises to £33 by December.
EXAMPLE: LEVERAGE WITH OPTIONS Cont
Risk management instruments
Strategy Profit
Buy the stock
Buy options
£33 − £28.3 × 100 = £470
£33 − £29 × 1000 − £2800= £1200
Introduction to Risk Management
Suppose the speculator is wrong. The stock price falls to £27 by December.
EXAMPLE: LEVERAGE WITH OPTIONS Cont
Risk management instruments
Strategy Loss
Buy the stock
Buy options
£28.3 − £27 × 100 = £130
£2800
Introduction to Risk Management
Misuse of derivatives
Entity Date Instrument Loss ($million)
Orange County, California Dec.1994 Reverse repos 1 810
margin call
Local government fund Bob Citron (county treasurer)
$7.5 million own money + $12.5 borrowed (via reverse repos) Invest money in agency notes with average maturity of 4 years Short-term funding of mind-term investment Works if rates are falling
Since Feb 1994 rates started to hike Margin calls Dec 1994 investors tried to pull out money Fund defaulted on margin payments Orange County declared bankruptcy
Introduction to Risk Management
Misuse of derivatives
Entity Date Instrument Loss ($million)
Metallgesellschaft, Germany Jan. 1994 Oil futures 1 580
margin call
Germany’s 14th largest industrial group 58 000 employees
US subsidiary (MG Refinig & Marketing) offered long-term contracts for oil products By 1993 180 million barrels of oil sold to be supplied over a period of 10 years Short-term futures & rolling hedge Long term exposure hedged via series of short-term contracts (3months maturity)
In 1993 oil prices fell from $20 to $15, leading to billion $ margin calls German parent company exchanges the US subsidiary management and closed the
positions at a loss Creditor stepped in with $2.4 billion rescue package Stock price dropped form 64 to 25 DM
Introduction to Risk Management
Misuse of derivatives
Entity Date Instrument Loss ($million)
Barings, UK Feb. 1995 Stock index futures 1 330
233 year old bank 28-year old trader Nicholas Leeson
Large exposure to the Japanese stock market (via futures) Baring’s position in Nikkei 225 added up to $7 billion Jan&Feb 1995 market fell by 15%, which lead to large losses Yet the exposure was increased 23 Feb Nicholas Lesson walked out of his job
The bank went bankrupt Nicholas went to jail (43 months), then worked as an accountant for Galway United
Football Club
Introduction to Risk Management
Misuse of derivatives
Lessons for Derivative users
Define the risk limits
Do not assume you can outguess the market
Do not underestimate the benefits of diversification
Carry out scenario analysis
Monitor traders carefully
Do not ignore liquidity risk
Short-term funding might create liquidity problems
Make sure you understand the trades you are doing
Make sure a hedger doesn't become a speculator
Introduction to Risk Management
Risk and exposure
Why manage risk?
Types of risk
Risk management process
Risk management instruments
Misuse of the derivatives
Introduction to Risk Management
2. Bonds Interest
Rates and Swaps
Zero-coupon bonds and coupon paying bonds
The yield curve and the term structure of interest
rates
Duration, Convexity
Interest Rate Risk and Immunisation of Bonds
Portfolios
Bonds
Comparative Advantage
Swap Design
Valuation of Swaps
Various Interest Rates, Risk Free Rate Proxy
Measuring Interest Rates, Zero Rates
Swaps
Interest
Rates
Bonds, Interest Rates and Swaps
An interest rate quantifies the amount of money borrower pays the lender.
The interest rate depends on: (a) the credit risk of the borrower, the higher the risk the higher the interest rate.
Interest rates change in time.
Bonds, Interest Rates and Swaps Interest Rates
(b) the time to maturity
Various Interest Rates:
Treasury Rates interest rates paid on Treasury bills and bonds
usually assumed that the risk of default is zero
used a proxy of risk-free rate
LIBOR London Interbank Offered Rate
quoted by British Bankers’ Association
for maturities up to 12 months in all major currencies
at what rates banks make large wholesale deposits with each other
(i.e. at what rate banks loan money to other banks)
LIBID at what rates banks accept large wholesale deposits from other banks
at any time LIBOR rate > LIBID rate
Ask rate: how much I want to get from you if I deposit money with you
Bid rate: how much I want to pay to you if you deposit money with me
Repo Rates Repo or repurchase agreement
Enter a contract where securities are sold and later repurchased at a higher price,
with the difference in price creating the interest rate (called repo rate)
Overnight
Indexed Swap
Rate
Overnight indexed swap is a swap where a fixed rate for a period is exchanged for
the geometric average of the overnight rates during the period
The fixed rate is called the overnight indexed swap rate.
Defaults:Russia 1991 Mexico 1982 Argentina 2005
USA 1862 UK 1932Sweden 1812 Germany 1948Denmark 1813…and many many more
Geometric average of a, b, c : 3𝑎 ∗ 𝑏 ∗ 𝑐
Bonds, Interest Rates and Swaps Interest Rates: Various Interest Rates
Risk-free Interest Rate
The rate is used in valuation of assets (eg. CAMP, or valuation of derivatives).
time
Treasure RatesOvernight Index Swaps
(OIS)LIBOR
Believed to be at
artificially low level
because of tax and
regulatory issues
Use Eurodollar futures
and interest rate swaps
to extend the risk-free
LIBOR curve beyond
12 months
Financial crisis:
difficult to borrow
money at interbank
market
what next?
Rates used as a proxy of risk free rate in pricing derivatives
Bonds, Interest Rates and Swaps Interest Rates: Risk Free Rate Proxies
Negative Interest Rate
Recently some central Banks adopted negative interest rates for cash ‘parked’ with them
European Central Bank Bank of JapanSwiss National Bank
Danish Central Bank
Swedish Central Bank Rate on excess
reserves cut to
minus 0.1%
Bonds, Interest Rates and Swaps Interest Rates: Risk Free Rate Proxies
June 2014time
29.01.2016Dec 2014
To put more money into the market
Measuring Interest Rates
If Rc is a rate of interest with continuous compounding and Rm is the equivalent rate with compounding
m times per annum and A is the amount invested for n years, then:
𝐴𝑒𝑥𝑝 𝑅𝑐𝑛 = 𝐴 1 +𝑅𝑚𝑚
𝑚𝑛
thus: 𝑅𝑐 = 𝑚 ln 1 +𝑅𝑚𝑚
and
𝑅𝑚 = 𝑚 𝑒𝑥𝑝𝑅𝑐𝑚
− 1
Example: If semi-annually compounded rate is 10% what is the equivalent continuously compounded rate?
m = 2 Rm = 0.1 Rc = ?
A = 2 ln 1 +0.1
2A = 0.09758
Bonds, Interest Rates and Swaps Interest Rates: Measuring Interest Rates
Zero Rates
N-year zero-coupon rate is the rate earned on investment that starts today and last for n years,
with no intermediate payments and all the interest and principal realized at the end of n years.
Example: If today you invest £100 and in 5 years time you receive back £128.40 what is the continuously compounded
zero coupon rate (R)?
A 100 𝑒𝑅∗5 = 128.4
R = 0.049996
A 5R = ln128.4
100= 0.24998
Bonds, Interest Rates and Swaps Interest Rates: Zero Rates
Definitions
A bond is a contract that commits the issuer to make a definite sequence of payments until a
specified terminal date.
Bonds are an example of fixed-income securities (like savings account) – you deposit money (`pay the price`) in
order to receive a certain stream of income (interest) at some fixed/certain dates.
The payment made each period is known as the coupon.
The amount paid at the terminal date is the maturity value (par value, face value or bond’s principal).
Notation:
Date today, t
Maturity date, T
Maturity Value, m
Time to maturity: τ = T - t
Coupon, c
Price of bond today: p
The (annual) yield from holding the bond: y
Yield is a single discount rate that applied to all cash flows of the bond gives the price of the bond equal to its market price.
Bonds, Interest Rates and Swaps Bonds: Definitions
Zero-coupon bonds
A zero-coupon bond is one that pays m at the maturity date, and nothing else (i.e. c = 0).
Since it costs p to buy the bond today, and m is received after τ time periods, it must be the case that:
Let yτ be the yield to maturity of the zero-coupon bond.
yτ is the constant annual rate of return that would be received if the bond was held until maturity.
𝑝 1 + 𝑦ττ = 𝑚
Rearranging:
𝑝 =𝑚
1 + 𝑦ττ
Negative relationship between yield to
maturity and the bond price.
Moreover:
𝑦τ =𝑚
𝑝
1τ
− 1
Restrictive monetary policy which increases
rf must increase y1, which in turn brings
about a fall in bond prices.
In this section assume annual compounding
Bonds, Interest Rates and Swaps Bonds: Zero-coupon Bonds
Note that the curve that shows p as a function of yτ is:
Negatively sloped (the higher the yield, the lower the price)
Convex from below (for successive increases in the yield, the smaller are the reductions in price).
Bonds, Interest Rates and Swaps Bonds: Zero-coupon Bonds
4𝑒−0.10469∗0.5 + 4𝑒−0.10536∗1 + 104𝑒−𝑅∗1.5 = 96
The yield curve and the term structure of interest rates
Consider the yield to maturity yτ on zero-coupon bonds with different times to maturity τ.
Bootstrapping zero rates
A curve showing the relationship between yτ and τ is known as the yield curve.
The shape of the yield curve represents the term structure of interest rates.
Zero rates can be determined from Treasury bills and coupon-bearing bonds.
Example: What is the 1.5 year zero rate (R) if 0.5 year zero rate is 10.469%; 1 year rate is 10.536% and bond that
pays coupon of 4 every 6 months and lasts for 1.5 year with par of 100, sells for 96?
4𝑒−0.10469∗0.5 + 4𝑒−0.10536∗1 + 104𝑒−𝑅∗1.5 = 96
𝑒−1.5𝑅 = 0.85196
𝑅 = −ln(0.85196)
1.5= 0.10681
Bonds, Interest Rates and Swaps Bonds: Yield Curve
Example of yield curve:
Assumptions:
Linear between bootstrapping points
Horizontal before the 1st and after the
last bootstrapping point.
Bonds, Interest Rates and Swaps Bonds: Yield Curve
The most important determinant of the shape of the yield curve is expectations of future movements of interest rates.
This dependence is summed up by the expectations hypothesis.
Bonds, Interest Rates and Swaps Bonds: Yield Curve
Example of yield curve:
Expectations hypothesis:
Assume that the market consist of only two (zero-coupon) bonds: one year to maturity (short-term) and two years to
maturity (long-term). Suppose that initially, yields on the two bonds are equal.
If the one-year yield is expected to rise next year, investors will have a preference for the one-year bonds, since they
will mature in one year, and the proceeds can be invested at one-year bonds commencing next year, at a higher rate.
This preference would cause investors to sell two-year bonds and buy one-year bonds, bringing about a fall in the price
of two-year bonds, and a rise in the price of one-year bonds.
In turn this will cause the yield on two-year bonds to rise above that of one-year bonds. The yield curve will have a
positive slope.
Therefore, theory predicts that if investors expect interest rates to rise, the yield curve will be positively sloped
Conversely, if investors expect interest rates to fall, the yield curve will be
Alternative explanations for the sign of the slope of yield curve:
(i) liquidity preference theory and
(ii) market segmentation theory.
Expectations of future
movements in the
interest rate can
therefore be deduced
from the slope of the
yield curve.
negatively sloped.
Bonds, Interest Rates and Swaps Bonds: Yield Curve
Coupon-paying Bonds
Consider a bond that promises to pay a coupon of c per year for τ years, plus the maturity value m when the bond
terminates at maturity.
This is equal to the present value of the future stream of payments arising if the bond is held to maturity:
Again price of the bond today is given by p.
𝑝 =𝑐
1 + 𝑦+
𝑐
1 + 𝑦 2 +𝑐
1 + 𝑦 3 +⋯+𝑐 +𝑚
1 + 𝑦 τ
The yield to maturity of this coupon-paying bond is defined as the value of y that solves the equation above.
Bonds, Interest Rates and Swaps Bonds: Coupon-paying Bonds
Par Yield
Par yield (cp) for a certain maturity bond is the coupon rate that causes the bond price (p) equal
to its par value (m).
𝑚 =𝑐𝑝
1 + 𝑦1+
𝑐𝑝1 + 𝑦2
2 +𝑐𝑝
1 + 𝑦33 +⋯+
𝑐𝑝 +𝑚
1 + 𝑦ττ
Bonds, Interest Rates and Swaps Bonds: Coupon-paying Bonds
Macaulay Duration
As for zero-coupon bonds, the price of a bond is a negative and convex function of its yield to maturity. For coupon-
paying bonds, the nature of this relationship is of considerable interest.
However, it is an unsatisfactory measure, because it depends on the units in which the bond is being measured.
The above represents the responsiveness of p to y.
𝑑𝑝
𝑑𝑦
We therefore use instead:
𝐷 = −1 + 𝑦
𝑝
𝑑𝑝
𝑑𝑦=1
𝑝
1 × 𝑐
(1 + 𝑦)+
2 × 𝑐
1 + 𝑦 2 +3 × 𝑐
1 + 𝑦 3 +⋯+τ × 𝑐 + 𝑚
1 + 𝑦 τ
MACAULAY DURATION An elasticity of price with respect to changes in yield.
We have also changed the sign so that the measure is positive.
= −𝑐
1 + 𝑦 2 −2𝑐
1 + 𝑦 3 −3𝑐
1 + 𝑦 4 −⋯−τ 𝑐 + 𝑚
1 + 𝑦 τ+1
Y = a + bX
If X increases by 1 unit, Y
increases by b units
Bonds, Interest Rates and Swaps Bonds: Coupon-paying Bonds: Duration
𝐷 = 𝑖=1𝑛 𝑐𝑖𝜏𝑖𝑒
−𝑦𝜏𝑖
𝑝
𝑝 =𝑐
1 + 𝑦+
𝑐
1 + 𝑦 2+
𝑐
1 + 𝑦 3+⋯+
𝑐 +𝑚
1 + 𝑦 τ
Example: Suppose that a bond with maturity value m = £100 pays a coupon of c = £5 for two years (τ = 2). If y = 4%,
then the price of the bond is:
and the Macauley Duration is:
𝑝 =5
1 + 0.04+
5 + 100
1 + 0.04 2= 101.886
𝐷 =1
101.886
1 × 5
(1 + 0.04)+2 × (5 + 100)
1 + 0.04 2 =
Duration is so-called is because it is interpreted in the time dimension:
1.953
In the example, the time to maturity is τ = 2 years, and the Macauley Duration is somewhat less than 2 (i.e. ‘the average
time to payment’ is less than 2 years).
For coupon-paying bonds:
𝐷 < τ
For zero-coupon bonds:
𝐷 = τ
The higher the coupon, ceteris
paribus, the lower the value of D.Entire payment is made after
τ periods.
Duration measures how long on average the
holder of the bond needs to wait before receiving
cash payments.
D is a weighted average of the times at which
payments are received, with weights being equal
to the proportion of the bond’s total present value
provided by the cash flow at time t (=1,.., τ).
Bonds, Interest Rates and Swaps Bonds: Coupon-paying Bonds: Duration
Interest Rate Risk
Buying and selling bonds is not a risk-free activity.
Macauley Duration can be used as a measure of interest rate risk.
Interest rate risk reflects the impact of Central Bank monetary policy. If the Central Bank raises the cost of borrowing,
all bond yields are likely to rise, and therefore all bond prices will fall, so holders of bonds will suffer a loss.
One type of risk is interest rate risk.
There is negative relationship between bond price p and bond yield y, that for small changes in y can be described by:
if y is expressed with compounding m times a year, or by:
if y is expressed with continuous compounding.
∆𝑝
𝑝= −
𝐷∆𝑦
1 +𝑦𝑚
∆𝑝
𝑝= −𝐷∆𝑦
Bonds, Interest Rates and Swaps Bonds: Coupon-paying Bonds: Duration
Example: Bond that pays semi-annually coupon of £5 has 3 years to maturity and face value of £100. The yield to maturity is
0.12. The bond price is 94.213. What happens to the bond price if yield to maturity increases by 10 basis points (i.e. by 0.001)
and the duration is 2.653?
So the new bond price is:
Interest Rate Risk
∆𝑝 = −𝑝𝐷∆𝑦
= −94.213 ∗ 2.653 ∗ 0.001
= −0.24995
94.213 − 0.24995 = 93.9631
=A2/EXP(B2*$C$1)
=SUM(C2:C7)
=(A2*B2)/EXP($C$1*B2)
=E8/C8
Bonds, Interest Rates and Swaps Bonds: Coupon-paying Bonds: Duration
𝐷 = 𝑖=1𝑛 𝑐𝑖𝜏𝑖𝑒
−𝑦𝜏𝑖
𝑝
Larger changeSmall change
Bonds, Interest Rates and Swaps Bonds: Coupon-paying Bonds: Duration
Example: Bond that pays semi-annually coupon of £5 has 3 years to maturity and face value of £100. The yield to maturity is
0.12. The bond price is 94.213. What happens to the bond price if yield to maturity increases by 10 basis points (i.e. by 0.001)
and the duration is 2.653?
Interest Rate Risk
Immunisation of Bond Portfolios
Immunisation strategies (a.k.a. neutral hedge strategies) can be used to eliminate interest rate risk.
They are used by organisations that have predictable liabilities, e.g. knowing that they will be paying a client £1m in 5 years
time. The principle of immunisation is:
Example: The contract to pay the client £1m in 5 years time clearly has D = 5 years.
Choose a bond portfolio that has the same overall Macauley Duration (D) as that of the liabilities.
Duration of the portfolio is a weighted average of duration on the bonds included in the portfolio, with weights being the
proportion of portfolio being allocated to particular bond.
The company could immunise by purchasing a zero-coupon bond with time to maturity 5 years and maturity value £1m
(which has the same D of 5 years). However, this assumes that such a bond is available to be purchased.
What the firm therefore needs to do is purchase a portfolio of bonds with overall Macauley Duration of 5 years. If two
bonds are available, bond 1 with D = 3 and bond 2 with D = 6, then the firm could immunise by purchasing a portfolio
consisting of bonds 1 and 2 in the proportion 1:2, since the overall D of this portfolio would be:
1 × 3 + 2 × 6
3= 5
Bonds, Interest Rates and Swaps Bonds: Immunisation
Convexity
The duration relationship applies only to small changes in yields (it is a
linear measure).
in case of continuous compounding:
Convexity measure the curvature of how the price change as the yield change. Convexity can be measured as:
Convexity (C) helps to improve to model the relationship for larger
changes in yields.
∆𝑝
𝑝= −𝐷∆𝑦 +
1
2𝐶 ∆𝑦 2
𝐶 =1
𝑝
𝑑2𝑝
𝑑𝑝2
Bonds, Interest Rates and Swaps Bonds: Convexity
Swap is an agreement to exchange cash flows in the future.
Most popular swaps are plain vanilla interest rate swap (where fixed rate on a given principal is exchanged for a
floating rate on the same principal) and fixed-for-fixed currency swaps.
In interest rate swap the principle is not being exchanged (thus it’s called notional principle) and at every payment
date one party remits the difference between the two payments to the other side.
Currency swap usually involves exchanging principle (both at the beginning and at the end of the swap) and
interest payments in one currency for principle and interest payments in the other currency.
Most swaps are over the counter agreements.
Bonds, Interest Rates and Swaps Swaps
Comparative advantage
Comparative advantage comes from the lack of constant spread on quotes offered on two products to two parties. One
party will have comparative advantage in one product, the other in the other.
Portugal produces both products at a lower cost.
The difference is cost between 2 countries for cloth is
10 (hours) and for wine is 40 (hours).
Portugal has comparative advantage in Wine.
England has comparative advantage in Cloth.
England produces 1 Cloth, brings it to Portugal and exchanges it for 1.125(= 1+ 10/80) Wine.
The Wine is brought back to England…it is worth 1.35 (=(1.125*120)/100) Cloth.
We are 0.35 Cloth better off.
Portugal produces 1 Wine, brings it to England and exchanges it for 1.2 (= 1+20/100) Cloth.
The Cloth is brought back to Portugal…where it is worth 1.35(=(1.2*90)/80) Wine.
We are 0.35 Wine better off.
Cloth Wine
Portugal 90 80
England 100 120
Minimum Labour Hours Required
for Production
Commodity
Bonds, Interest Rates and Swaps Swaps
Who has a comparative advantage in fixed rate loan?
The difference in the spreads can lead to potential profit that could be exploited by a swap contract.
Company ALIBOR + 1%
LIBOR + 2.25%
5%
5%
Company B
+ LIBOR + 2.25%
- LIBOR + 1%
- 5 %
Total: (-) 3.75%
- LIBOR + 2.25%
+ 5%
- 5 %
Total: (-) LIBOR + 2.25%
each company 0.25% better off
For example, each party is 0.25% better off.
Total profit from swap: (2.5% - 1%) – (5% - 4%) = 1.5% - 1% = 0.5%
How much of the 0.5% each party gets? Depends on its bargaining power.
Bonds, Interest Rates and Swaps Swaps
Comparative advantage
Who has a comparative advantage in fixed rate loan?
The difference in the spreads can lead to potential profit that could be exploited by a swap contract.
Company ALIBOR + 1%
LIBOR + 2.25%
5%
5%
Company B
+ LIBOR + 1.25%
- LIBOR + 1%
- 4 %
Total: (-) 3.75%
- LIBOR + 1.25%
+ 4%
- 5 %
Total: (-) LIBOR + 2.25%
each company 0.25% better off
Total profit from swap: (2.5% - 1%) – (5% - 4%) = 1.5% - 1% = 0.5%
How much of the 0.5% each party gets? Depends on its bargaining power.
Bonds, Interest Rates and Swaps Swaps
Comparative advantage
alternative design
4%
LIBOR+1.25%
Company ALIBOR + 1%
LIBOR + 2.30%
5%
5%
+ LIBOR + 2.20%
- LIBOR + 1%
- 5 %
Total: 3.8%
- LIBOR + 2.30%
+ 5%
- 5 %
Total: LIBOR + 2.3%
Total: 0.1%
Financial intermediary netting out 0.1% and each company gets only 0.2%.
Company B
5%
LIBOR + 2.20%
INTERMEDIARY
Bonds, Interest Rates and Swaps Swaps
Comparative advantage
Valuation of Swaps
When swap is first initiated it is worth zero. The value of swap changes with time.
Plain vanilla interest rate swap can be perceived as a difference between two bonds.
There are two valuation approaches of swaps:
(1) in terms of bond price or
(2) in terms of Forward Rate Agreements.
Therefore to the floating-rate payer, a swap can be seen as having a long position in a fixed-rate bond and a short
position in a floating rate bond. The value of the swap is determined by:
Currency swap can be valued as a difference between two bonds (in two different currencies D-domestic
currency, F-foreign currency, S0 spot exchange rate) that were converted to common currency:
The above mentioned swaps can be also valued as the sum of the Forward Rate Agreements, where each FRA comes
from the exchange of cash-flows during the life of the swap and its maturity.
𝑉𝑠𝑤𝑎𝑝 = 𝐵𝑓𝑖𝑥 − 𝐵𝑓𝑙𝑒𝑥
𝑉𝑠𝑤𝑎𝑝 = 𝐵𝐷 − 𝑆0𝐵𝐹
Bonds, Interest Rates and Swaps Swaps
Zero-coupon bonds and coupon paying bonds
The yield curve and the term structure of interest
rates
Duration, Convexity
Interest Rate Risk and Immunisation of Bonds
Portfolios
Bonds
Comparative Advantage
Swap Design
Valuation of Swaps
Various Interest Rates, Risk Free Rate Proxy
Measuring Interest Rates, Zero Rates
Swaps
Interest
Rates
Bonds, Interest Rates and Swaps
EXERCISE
3. Futures and
Forwards
4. Introduction
to Options
Lecture 2:
3. Futures and
Forwards
Definitions
Payoffs from Forwards Contract
Forward Prices
Valuation of Forward Contracts
Forward Rates
Hedging
Forwards and Futures
Forward and future contracts are agreements to buy or sell an asset at a certain future time (the
maturity date) for a certain price (the delivery price).
They can be contrasted with a spot contract, which is an agreement to buy or sell an asset today.
One of the parties to the forward/future contract assumes a long
position and agrees to buy the asset at the future date at the
agreed price.
The other party assumes a short position and agrees to sell the
asset on the same future date at the same agreed price.
Underlying (asset): bond, stock, index, currency, commodity (gold, oil, wheat)
Forwards and Futures Definitions
Forwards and Futures Definitions
FORWARD FUTURES
Private contract between two parties
Non standardized
Usually one specified delivery date
Settled at the end of contract
Delivery or final cash settlement usually
takes place
Some credit risk
Traded on exchange
Standardized contracts
Range of delivery dates
Settled daily
Contract is usually closed out prior to
maturity
Virtually no credit risk
Margin Account
Initial Margin
Maintenance Margin
Margin Call
Notice of intention to deliver presented by
seller to exchange
Forwards and Futures Definitions
Payoffs from Forward Contract
The payoff from holding a long position in a forward contract on one unit of an asset is:
where K is the delivery price and ST is the spot price of the asset at maturity of the contract.
We are buying for K something worth ST
- K
STK
Payoff
𝑆𝑇 − 𝐾
Forwards and Futures Payoffs
The payoff from a short position in a forward contract on one unit of an asset is:
We are selling for K something worth ST
K
Payoffs from Forward Contract
𝐾 − 𝑆𝑇
Forwards and Futures Payoffs
K ST
Payoff
Short Selling
Short selling = selling an asset that is not owned.
It is done by borrowing the asset from someone who does own it, selling it, and then buying it back at a later date, and
finally returning it to the party from whom it was borrowed.
Such a trade is profitable if the price of the asset has fallen over the period between the sale and the repurchase.
BORROW & SELLBUY BACK &
RETURNPROFIT ?
Forwards and Futures Forward prices: Short Selling
Forward Prices
The easiest type of forward contract to value is one written on an asset that provides the holder with no income, such as
non-dividend paying stock or zero-coupon bond.
Consider a long forward contract to purchase a non-dividend paying stock in 3 months.
Assume that the current stock price is £40 and the 3-month risk-free interest rate is 5% per annum.
time
Suppose that the forward price is relatively high at £43.
borrow £40 at rf of 5% pa buy 1 share for £40, short forward contract to
sell 1 share in 3 mths
now in 3 mths
deliver the share and receive £43
pay back loan: 40e0.05*(3/12) = 40.50
Profit:
£43 - £40.50 = £2.50Arbitrager can lock in risk free
profit of £2.50
Assumptions: no transaction costs borrow/ lend at rf
Forwards and Futures Forward prices
time
Suppose that the forward price is relatively low at £39.
short 1 share (& receive £40) invest £40 at rf
take long position in 3 month forward contract
nowin 3 mths
£40 investment grows to:40e0.05*(3/12) = 40.50
pay £39 and take delivery of 1 share use the share to close out the short
position
Profit:
£40.50 - £39 = £1.50
Arbitrager can lock in risk free profit of £1.50
Under what circumstances do arbitrage opportunities not exist?
Arbitrage opportunities arise whenever the forward price is above £40.50or below £40.50.
Thus for there to be no arbitrage, the forward price must be exactly £40.50.
Forwards and Futures Forward prices: No income
General Formula (no income case)
Consider a forward contract on an asset that provides no income.
The current (spot) price of the asset is St, τ is the time to maturity, r is the risk-free rate, Ft is the forward price.
The relationship between the spot price and the forward price is:
If Ft > Sterτ , arbitrageurs can buy the asset and short forward contracts on the asset.
If Ft < Sterτ , they can short the asset and enter into long forward contracts on it.
𝐹𝑡 = 𝑆𝑡𝑒𝑟𝜏
Forwards and Futures Forward prices: No income
Forward market
Spot market
Buy at the market where asset is cheaper…sell at the market where it’s more expensive
EXMPLE: Consider a long forward contract to purchase a stock in 3 months that pays £1 in 1 month time.
Assume that the current stock price is £40 and the 3-month risk-free interest rate is 5% per annum.
time
Suppose that the forward price is relatively high at £42.
borrow £39.00416 (=40 – 0.996) at rf of 5% pa for 3 months
borrow £0.996 at rf for 1 month buy 1 share for £40, short forward contract to sell 1
share in 3 mths
now in 3 mths
deliver the share and receive £42
pay back loan: 39.00416 e0.05*(3/12) = 39.49477
Profit:
£42 - £39.49477 = £2.50523
in 1 mth
receive £1 dividend
pay back short term loan: £1
PV(£1) = 1e-0.05*(1/12) = £0.996
Known income I
Forwards and Futures Forward prices: known income
Consider forwards contract on an asset that provides an income with a present value of I during the life of a forward
contract.
The relationship between the spot price and the forward price is:
If Ft > (St-I)erτ , arbitrageurs can buy the asset and short forward contracts on the asset.
If Ft < (St-I)erτ , they can short the asset and enter into long forward contracts on it.
In case Ft> (St-I)erτ, we need to borrow PV of (St-I) for
period of τ, and I for period until we receive the I.
General Formula (known income I)
𝐹𝑡 = 𝑆𝑡 − 𝐼 𝑒𝑟𝜏
Forwards and Futures Forward prices: known income
Consider a long forward contract to purchase a stock in 3 months. Stock pays a dividend yield of 2%.
Assume that the current stock price is £40 and the 3-month risk-free interest rate is 5% per annum.
time
Suppose that the forward price is relatively high at £43.
borrow £40 at rf of 5% pa buy 1 share for £40, short forward contract to sell
1 share in 3 mths
now in 3 mths
deliver the share and receive £43 receive divided:
40e0.02*(3/12) – 40 = 0.2005 pay back loan:
40e0.05*(3/12) = 40.50
Profit:
£43 - £40.50 + £0.2005 =
£2.7005
Forwards and Futures Forward prices: known yield
Known yield q
Consider forwards contract on an asset that provides a known yield q (i.e. income as percentage of the asset’s price at
the time the income is paid is known).
The relationship between the spot price and the forward price is:
If Ft > Ste(r-q)τ , arbitrageurs can buy the asset and short forward contracts on the asset.
If Ft < Ste(r-q)τ , they can short the asset and enter into long forward contracts on it.
General Formula (known yield q)
𝐹𝑡 = 𝑆𝑡𝑒𝑟−𝑞 𝜏
Forwards and Futures Forward prices: known yield
Valuing Forward Contract
The value of forward contract at the time it was first entered is zero.
At the later date the value can be positive of negative.
If K is the delivery price, τ is the time to maturity, r is the risk-free rate, Ft the forward price that would apply if
contract was negotiated today, the value of the (long) forward today (f) on no income paying asset can be defined as:
The value of forward on asset that pays I income is:
𝑓 = (𝐹𝑡 − 𝐾)𝑒−𝑟τ
The value of forward on asset that pays q yield is:
𝑓 = 𝑆𝑡 − 𝐼 − 𝐾𝑒−𝑟τ
𝑓 = 𝑆𝑡𝑒−𝑞τ − 𝐾𝑒−𝑟τ
= 𝑆𝑡 −𝐾𝑒−𝑟𝜏
Forwards and Futures Valuation of Forward Contracts
Forward interest rates are interest rates implied by current zero rates for periods of time in the future.
If R1 and R2 are the zero rates for maturities T1 and T2, RF is the forward interest rate for the period between T1 and T2, then:
which shows that if the zero curve is upward sloping between T1 and T2, (i.e. R2>R1), then RF>R2.
𝑅𝐹 =𝑅2𝑇2 − 𝑅1𝑇1𝑇2 − 𝑇1
= 𝑅2 + 𝑅2 − 𝑅1𝑇1
𝑇2 − 𝑇1
=𝑅2𝑇2 − 𝑅2𝑇1 + 𝑅2𝑇1 − 𝑅1𝑇1
𝑇2 − 𝑇1
=𝑅2 𝑇2 − 𝑇1 + 𝑅2𝑇1 − 𝑅1𝑇1
𝑇2 − 𝑇1
R1
R2
RF
𝑒𝑅1𝑇1+𝑅𝐹 𝑇2−𝑇1 = 𝑒𝑅2𝑇2
𝑅1𝑇1 + 𝑅𝐹 𝑇2 − 𝑇1 = 𝑅2𝑇2
Forward Rates
Forwards and Futures Forward Rates
Example:
Calculate year-2 forward rate (i.e. a rate of interest for year 2 that combined with 1-year zero interest
provides the same overall interest as the 2-year zero rate), knowing that R1 = 0.03 and R2 = 0.04.
𝑅𝐹 =𝑅2𝑇2 − 𝑅1𝑇1𝑇2 − 𝑇1
=0.04 × 2 − 0.03 × 1
2 − 1
= 0.05
Forwards and Futures Forward Rates
Hedging with Futures
Basis risk arise when (i) the asset underlying futures contract is different than the asset whose price is to be hedged.
(ii) hedge require contract to be closed out before its delivery.
The basis (b) is defined as:
𝐵𝑎𝑠𝑖𝑠 = 𝑆𝑝𝑜𝑡 𝑝𝑟𝑖𝑐𝑒 𝑜𝑓 𝑎𝑠𝑠𝑒𝑡 𝑡𝑜 𝑏𝑒 ℎ𝑒𝑑𝑔𝑒𝑑 − 𝐹𝑢𝑡𝑢𝑟𝑒𝑠 𝑝𝑟𝑖𝑐𝑒 𝑜𝑓 𝑐𝑜𝑛𝑡𝑟𝑎𝑐𝑡 𝑢𝑠𝑒𝑑
If the asset to be hedged and the asset underlying the futures is the same, the base is zero at the maturity of futures contract.
Forwards and Futures Hedging: Hedging with Futures
Cross Hedging
When asset underlying the futures contract is different to the asset whose price is being hedged the hedge is referred to
as cross hedging.
Hedge ratio is the ratio of the size of the position taken in futures contracts to the size of the exposure.
Define ΔS as the change in the spot price S during the period of time equal to the life of the hedge and ΔF change in
futures price F during a period of time equal to the life of the hedge.
ℎ∗ = 𝜌𝜎𝑆𝜎𝐹
When both assets are the same, it is natural to use hedge ratio of 1.
When assets vary it might be optimal to use different ratio.
The hedger should choose the value of the hedge ratio that minimizes the variance of the value of the hedged position.
Minimum variance hedge ratio (h*) can be defined as:
where σS is the standard deviation of ΔS, σF is the standard deviation of ΔF, and ρ is the correlation coefficient between
ΔS and ΔF.
If ρ = 1 and σF = σS, h* = 1.
Forwards and Futures Hedging: Cross Hedging
To calculate the optimal number of contracts to be entered (N*) one should multiply the size of the position being
hedged (QA) by minimum variance hedge ratio, and divide the product by the size of one futures contract (QF):
𝑁∗ =ℎ∗𝑄𝐴𝑄𝐹
In practise: Choose contract with closes delivery date to the exposure…but later delivery than exposure Choose contract on asset whose price is highly correlated with price of exposed asset
Forwards and Futures Hedging: Cross Hedging
Definitions
Payoffs from Forwards Contract
Forward Prices
Valuation of Forward Contracts
Forward Rates
Hedging
Forwards and Futures
EXERCISE
4. Introduction
to Options
European Options
Pay-off Diagrams,
Bounds of Option Prices
Introduction to Options
Introduction to Options
Introduction to Options
Top 10
Introduction to Options
During 2012 477 Million equity contracts were traded on the CBOE, representing options on 47.7 billion shares of underlying stock
S&P500 :
(1) total market capitalization (USD) as of 31 Jan 2014: 16 872 585 650 000
(2) average daily volume in 2013 around 3 000 000 000 (shares)
Introduction to Options
CALL OPTION PUT OPTION
A Call Option gives the holder the right to buy
the underlying asset on (or maybe before) a
certain date (the expiry date) for a certain price
(the strike price).
A Put Option gives the holder the right to sell
the underlying asset on (or maybe before) a
certain date (the expiry date) for a certain price
(the strike price).
Like forwards and futures, options can be written on a
stock, foreign exchange, market index,....
In forwards/ futures the contract creates an obligation to both parties.
The holder of the options has the right to exercise the option, whereas the person writing the option has an obligation to comply with holder decision.
The owner, or holder, of an option – who is said to adopt a long position – acquires the option by paying the option price
(premium) to the writer – who is said to adopt a short position.
If the holder of a call option chooses to exercise the option, he pays the strike price to the writer in exchange for the asset.
If the holder of a put option chooses to exercise the option, he delivers the asset to the writer, who simultaneously pays the
strike price to the holder.
European OptionsIntroduction to Options
EUROPEAN CALLOPTION
EUROPEAN PUTOPTION
A security that gives its owner the right, but not
the obligation, to purchase a specified asset for a
specified price, known as the strike price, at some
date in the future (the expiry date).
A security that gives its owner the right, but
not the obligation, to sell a specified asset for a
specified price, known as the strike price, at
some date in the future (expiry).
Notations:
t is the current date T is the expiry date τ = T – t is the time to expiry.
St is the current (underlying) stock price. ST is the stock price at expiry.
K is the strike price. r is the risk-free rate of interest.
ct is the current price (or the current value) of a Call Option
pt is the current price (or the current value) of a Put Option
European Options can only be exercised on the expiry date.
American options, in contrast, can be exercised at any time up to the expiry date.
For the time being, we restrict attention to European Options because they are more straightforward to analyse.
Options that expire unexercised are said to die, and are worthless
European OptionsIntroduction to Options
If you are the holder of a call option, you want the stock price at expiry to exceed the strike price.
K
STK
Payoff
Then, you exercise the option to buy at the strike price, and immediately sell at a profit ST - K.
If the stock price at expiry is less than the strike price, you let the option die.
Payoff diagram for a Call Option
A call option for which the
current stock price St is
above the strike price K is
said to be in the money.
A call option for which the
current stock price St is
below the strike price K is
said to be out of the money.
A call option for which the
current stock price St is
equals the strike price K is
said to be at the money.
out of the money
at the money
Long position in a Call Option
𝑚𝑎𝑥 0, 𝑆𝑇 − 𝐾
European Options: Pay-off DiagramsIntroduction to Options
-K
STK
Payoff
Short position in a Call Option
Issuer of the option faces pay off or loss, thus he needs to be compensated to be willing to write the option (option premium).
European Options: Pay-off Diagrams
Payoff diagram for a Call Option
Introduction to Options
K
STK
Payoff
Long position in a Put Option
out of the money
at the money
𝑚𝑎𝑥 0, 𝐾 − 𝑆𝑇
European Options: Pay-off Diagrams
Payoff diagram for a Put Option
Introduction to Options
- K
STK
Payoff
Short position in a Put Option
European Options: Pay-off Diagrams
Payoff diagram for a Call Option
Introduction to Options
A call gives the right to buy the underlying.
Thus the call payoff is always less that the value of underlying at time T, ST.
For the put the maximum value obtained at expiry is K, thus current value must be:
𝑐𝑡 ≤ 𝑆𝑡
Therefore the value of the call at time t must be less or equal to the value of the underlying:
𝑝𝑡 ≤ 𝐾𝑒𝑥𝑝(−𝑟𝜏)
European Options: Bounds of Option Prices
Bounds of Option Prices
Introduction to Options
Consider portfolio consisting of call option and a sum of money equal to Kexp(-rτ).
At T in case ST > K, this portfolio is worth the same as the underlying.
Similarly, consider portfolio consisting of put and underlying.
At T in case K > ST put is exercised and portfolio is worth K, otherwise put is not exercised and portfolio is worth ST.
Therefore at time T a portfolio of put and underlying when compared to cash currently worth Kexp(-rτ), will be worth at least
the same as the cash:
𝑐𝑡 + 𝐾𝑒𝑥𝑝(−𝑟𝜏) ≥ 𝑆𝑡
Since the cash and option produce a payoff equal to or greater than that of the underlying, thus the value of portfolio must then
equal to or greater than that of the underlying:
𝑝𝑡 + 𝑆𝑡 ≥ 𝐾𝑒𝑥𝑝(−𝑟𝜏)
Otherwise it is worth more than the underlying (i.e. K).
which implies:
𝑐𝑡 ≥ 𝑆𝑡 − 𝐾𝑒𝑥𝑝(−𝑟𝜏)
thus:
𝑝𝑡 ≥ 𝐾𝑒𝑥𝑝 −𝑟𝜏 − 𝑆𝑡
European Options: Bounds of Option Prices
Bounds of Option Prices
Introduction to Options
European Options
Pay-off Diagrams,
Bounds of Option Prices
Introduction to Options
5.Option Valuation
Lecture 3:
Option Valuation
Binomial Model
Black-Scholes Model
Put-Call Parity
Option Valuation
100
110
90
100
120
80
t=0 t=0.5 t=1
0*0.5+10*0.5=54.878
19.512
11.897
10*0.5+30*0.5=20
4.878*0.5+19.512*0.5=12.195
Consider an European put option with time to expiry of 1 year, and a strike price of 110.
The current price of the underlying is 100. Divide the time to expiry into two 6-month intervals.
Suppose that in each interval, the price can either rise by 10 or fall by 10, with equal probabilities.
The risk-free rate is 5% per annum, simply compounded.
The price movements can
be represented by a diagram
called a binomial tree.
An underlying assumption
is that the underlying price
follows a binomial process .
The value calculation proceeds backwards from T to t. Each step involves:
finding the terminal value of the option;
calculating its expected value of the option; and finally
discounting it by the risk-free rate (make sure that you use the right rate).
Risk-neutral valuation on
objective probabilities.
0
10
30
What is the value of the option?
European Options: Valuation: Binomial Model
Binomial Trees
Option Valuation
11.897
4.878*0.5+19.512*0.5=12.195
0*0.5+10*0.5=5
10*0.5+30*0.5=20
19.512
4.878
100
110
90
10010
1200
8030
t=0 t=0.5 t=1
0*0.5+10*0.5=5>4.878
>19.51
>11.897
10*0.5+30*0.5=20
4.878*0.5+19.512*0.5=12.195
0.4 0.6
0.4 0.6
0.4 0.6
Suppose that the probabilities of rise& fall were 40/60 instead of 50/50.
Without doing any further calculation, can you determine how the option price would change?
European Options: Valuation: Binomial Model
Binomial Trees
Option Valuation
100
110
90
100
120
80
t=0 t=0.5 t=1
10*0.5+0*0.5=54.878
0
2.3795
0*0.5+0*0.5=0
4.878*0.5+0*0.5=2.439
Now, let’s redo the question above, but assuming an European call option instead.
Suppose that the probabilities of rise & fall were 60/40 instead of 50/50.
Without doing any further calculation, can you determine how the option price would change?
0
0
10
What if we don’t know the probabilities? 1. No-Arbitrage Argument Valuation
2. Risk Neutral Valuation with Risk Neutral Probabilities
5exp(-0.05*0.5)=4.878
European Options: Valuation: Binomial Model
Binomial Trees
Option Valuation
S0
f
S0UfU
S0DfD
No Arbitrage Argument
Consider a stock whose price is S0 and option on the stock whose current price is f.
Option lasts for time T, and in that time the stock price moves to either S0U (where U > 1) or to S0D (where D < 1).
fU is option payoff if stock moved to S0U and fD option payoff is stock moved to S0D.
Consider a portfolio consisting of a long position in Δ shares and a short position in one option.
Calculate Δ that makes the portfolio riskless (i.e. portfolio has the same payoff regardless if the stock price increased or
decreased):
𝑆0𝑈∆ − 𝑓𝑈 = 𝑆0𝐷∆ − 𝑓𝐷 ∆ =𝑓𝑈 − 𝑓𝐷
𝑆0𝑈 − 𝑆0𝐷
S0UΔ - fU
S0DΔ - fD
S0Δ - f
European Options: Valuation: Binomial Model: No Arbitrage ArgumentOption Valuation
For arbitrage opportunities not to exist the riskless portfolio must earn risk-free interest rate.
If r is the risk-free interest rate, then the present value of the portfolio is:
(𝑆0𝑈∆ − 𝑓𝑈)exp(−𝑟𝑇) = (𝑆0𝐷∆ − 𝑓𝐷)exp(−𝑟𝑇)
whereas the cost of creating this portfolio today is:
𝑆0∆ − 𝑓
Therefore:
𝑆0∆ − 𝑓 = (𝑆0𝑈∆ − 𝑓𝑈)exp(−𝑟𝑇)
𝑓 = 𝑆0∆(1 − 𝑈𝑒𝑥𝑝 −𝑟𝑇 ) + 𝑓𝑈exp(−𝑟𝑇)
Let’s substitute 𝑓𝑈−𝑓𝐷
𝑆0𝑈−𝑆0𝐷for Δ:
𝑓 = 𝑆0𝑓𝑈 − 𝑓𝐷
𝑆0𝑈 − 𝑆0𝐷(1 − 𝑈𝑒𝑥𝑝 −𝑟𝑇 ) + 𝑓𝑈exp(−𝑟𝑇)
No Arbitrage Argument
European Options: Valuation: Binomial Model: No Arbitrage ArgumentOption Valuation
=𝑓𝑈 1 − 𝐷𝑒𝑥𝑝(−𝑟𝑇) + 𝑓𝐷 𝑈𝑒𝑥𝑝(−𝑟𝑇) − 1
𝑈 − 𝐷
𝑓 = 𝑆0𝑓𝑈 − 𝑓𝐷
𝑆0𝑈 − 𝑆0𝐷(1 − 𝑈𝑒𝑥𝑝 −𝑟𝑇 ) + 𝑓𝑈exp(−𝑟𝑇)
= exp(−𝑟𝑇) 𝑝𝑓𝑈 + (1 − 𝑝)𝑓𝐷 where: 𝑝 =exp(𝑟𝑇) − 𝐷
𝑈 − 𝐷
The model allows to price an option when stock price movements are given by a one-step binominal tree, under the
assumption there are no arbitrage opportunities in the market.
=𝑓𝑈 − 𝑓𝐷 − 𝑈𝑒𝑥𝑝 −𝑟𝑇 𝑓𝑈 + 𝑈𝑒𝑥𝑝 −𝑟𝑇 𝑓𝐷 + 𝑓𝑈 exp −𝑟𝑇 𝑈 − 𝑓𝑈 exp −𝑟𝑇 𝐷
𝑈 − 𝐷
= exp(−𝑟𝑇)𝑓𝑈 exp(𝑟𝑇) − 𝐷 + 𝑓𝐷 𝑈 − exp(𝑟𝑇)
𝑈 − 𝐷
European Options: Valuation: Binomial Model: No Arbitrage Argument
No Arbitrage Argument
Option Valuation
20f
22fU =1
18fD = 0
Example: Stock price today is equal to 20, and in 3 months it will be either 22 or 18.
What is a value of 3 month European call option with a strike price of 21.
The risk free rate is 12% (continuous compounding).
22∆ − 1 = 18∆ − 0
4∆ = 1
∆ = 0.25
18∆ − 0 = 18 × 0.25 = 4.5 4.5 exp −rT = 20∆ − 𝑓 4.5 exp −0.12 ×3
12= 5 − 𝑓
4.367005 = 5 − 𝑓 𝑓 = 0.632995
Step 1: Calculate Δ
Step 2: Calculate portfolio
value at horizonStep 3: Calculate portfolio value today, and thus calculate f
European Options: Valuation: Binomial Model: No Arbitrage Argument
No Arbitrage Argument
Option Valuation
Risk Neutral Valuation
Utility:
The usual assumptions are that u’(.) > 0 and u”(.) < 0 .
Utility Function
In risk-neutral world, risk-neutral investors do not increase the expected return they require from an investment to compensate for
increased risk.
in economics it is the fundamental measure of value.
Utility function u(x): tells us the unit of “satisfaction” that x gives us.
in finance, x usually represents the amount of money or profit.
two assumptions are normally required regarding the function u(.) :
1) slope of the function;
2) curvature, i.e. how the function “bends”.
This implies positive but decreasing marginal utility.
When x is random, then u(x) becomes a random variable.
The assumption about the curvature becomes critical as it implies the view towards risks.
In this aspect, we may classify utility functions according to their risk preferences.
European Options: Valuation: Binomial Model: Risk Neutral ValuationOption Valuation
E(W) X
U(X)
CE
U(CE)=E(U(W))
U(E(W))
Suppose that an individual holds a lottery that yields $0 or $100 with equal probabilities.
This lottery gives the expected return of $50 = 0.5*$0 + 0.5*$100
Risk-averse individuals prefer receiving the sure sum of $50 to being given a lottery whose expected return is $50.
This may be described as:𝑢 50 > 0.5 × 𝑢 0 + 0.5 × 𝑢(100)
A general condition for risk
aversion is that u”(.) < 0
(concave).
𝑢 40 = 0.5 × 𝑢 0 + 0.5 × 𝑢(100)
40 50
RP risk premium
CE certainty equivalent
E(W) expected value of uncertain
payment
E(U(W)) expected utility of the
uncertain payment
U(E(W)) utility of the expected
value of the uncertain payment
RP
Risk Preferences
European Options: Valuation: Binomial Model
Risk Neutral Valuation
Option Valuation
CE X
U(X)
E(W)
U(CE)=E(U(W))
U(E(W))
Risk-loving individuals prefer the lottery to the sure sum of $50.
This is the opposite case of risk aversion, i.e.
𝑢 50 < 0.5 × 𝑢 0 + 0.5 × 𝑢(100)
A general condition for risk
seeking is that u”(.) > 0 (convex).
50 60
RP
RP risk premium
CE certainty equivalent
E(W) expected value of uncertain
payment
E(U(W)) expected utility of the
uncertain payment
U(E(W)) utility of the expected
value of the uncertain payment
European Options: Valuation: Binomial Model
Risk Neutral Valuation
Risk Preferences
Option Valuation
E(W)=CEX
U(X)
Risk neutral Individuals who are indifferent between the lottery and the sure sum of $50
𝑢 50 = 0.5 × 𝑢 0 + 0.5 × 𝑢(100)
A general condition for risk
neutrality is that u”(.) = 0 (linear).
Real-life examples of risk preferences:
Risk-averse: Individual investors, pension
funds;
Risk-loving: Hedge funds;
Risk-neutral: Institutional investors, large
companies – Management being risk-
loving while owners being risk-averse.
European Options: Valuation: Binomial Model
Risk Neutral Valuation
Risk Preferences
Option Valuation
Risk neutrality proves very interesting since it implies that investors only care about expected returns, and not risks
associated with the investment.
Suppose there are only two assets in the economy: one risky (‘stock’) and the other riskless (‘bond’).
Risk-neutral investors will hold the stock alone – no matter how risky it is – provided that such
a stock gives a higher expected return than the bond.
If we’re willing to assume that everybody in the world is risk-neutral, then it must be the case that the returns on both
assets must be equal.
A risk-neutral world has two features that facilitate pricing derivatives:
(1) Expected return on stock (or any other instrument) is risk-free
(2) The discount rate used for the expected payoff on an option (or any other instrument) is risk-free rate.
Let 𝑝 =𝑒𝑟𝜏−𝐷
𝑈−𝐷be interpreted as the probability of an up movement in a risk-neutral world..
Thus the expected future payoff from an option in risk neutral world is:
𝑝𝑓𝑈 + (1 − 𝑝)𝑓𝐷
European Options: Valuation: Binomial Model
Risk Neutral Valuation
Option Valuation
Proof: Consider p as the probability of an up movement, the expected stock price E(ST) at time T:
𝐸 𝑆𝑇 = 𝑝𝑆0𝑈 + (1 − 𝑝)𝑆0𝐷
= 𝑝𝑆0𝑈 − 𝑝𝑆0𝐷 + 𝑆0𝐷
= 𝑝𝑆0(𝑈 − 𝐷) + 𝑆0𝐷
=𝑒𝑟𝜏 − 𝐷
𝑈 − 𝐷𝑆0(𝑈 − 𝐷) + 𝑆0𝐷
= 𝑆0𝑒𝑟𝜏 −𝑆0 𝐷 + 𝑆0𝐷
= 𝑆0𝑒𝑟𝜏
Thus stock price grows at risk free rate if p is the probability of an up movement.
European Options: Valuation: Binomial Model
Risk Neutral Valuation 𝑝 =𝑒𝑟𝜏−𝐷
𝑈−𝐷
Option Valuation
Stock price today is equal to 20, and in 3 months it will be either 22 or 18.
What is a value of 3 month European call option with a strike price of 21.
Example
The risk free rate is 12% (continuous compounding).
p could be calculated as:
Thus non-arbitrage arguments and
risk-neutral valuation give the
same results.
22𝑝 + 18 1 − 𝑝 = 20𝑒0.12×312
4𝑝 = 20𝑒0.12×312 − 18 𝑝 = 0.6523
or as:
𝑝 =𝑒𝑟𝜏 − 𝐷
𝑈 − 𝐷=𝑒0.12×
312 − 0.9
1.1 − 0.9= 0.6523
thus: 𝑓 = 0.6523 × 1 + (1 − 0.6523) × 0 𝑒−0.12×312
= 0.6523𝑒−0.12×312
= 0.633
20f
22fU
18fD = 0
= 1
European Options: Valuation: Binomial Model
Risk Neutral Valuation
Option Valuation
50C
60A
40B
484
720
3220
t=0 t=1 t=2
𝑝 =𝑒0.05∗1 − 0.8
1.2 − 0.8= 0.6282
A:
0.6282*0+0.3718*4=1.4872
1.4872*exp(-0.05)=1.41668
B:
0.6282*4+0.3718*20=9.9488
9.9488*exp(-0.05)=9.463591
C: 0.6282*1.41668+0.3718*9.463591 = 4.40725
4.40725*exp(-0.05) = 4.192306
Two-Step Binominal Trees
In order to calculate the option price at the initial node of the tree, one needs to start
with calculating option price at the final nodes and then working out option price at
the earlier nodes.
Example: Consider 2-year European put option with a strike price of 52, whose stock is currently trading at 50.
There are two 1-year steps. In each step stock price can increase by 20% or decrease by 20%. The risk-free interest rate is 5%.
C
A
B
0
4
20
Is p constant in the whole tree?
European Options: Valuation: Binomial ModelOption Valuation
Binomial Model → Black-Scholes Formula
n → ∞
n steps
European Options: Valuation: Binomial ModelOption Valuation
The more terms the more similar BM and
BS valuation
European Options: Valuation: Binomial ModelOption Valuation
put
stock
call
bond
STK
STK
0
ST
K-ST
ST
ST-K
K0
K
= =
Put-Call Parity
The put-call parity defines a relationship between the price of a call and a put – both with identical K and t.
It allows us to calculate c from p, and vice versa. The underlying assumption is that there is no arbitrage opportunities.
The parity is given by: 𝑝𝑡 + 𝑆𝑡 = 𝑐𝑡 + 𝐾𝑒−𝑟𝜏
We can prove this by considering two portfolios which always give the same payoffs at maturity:
(1) A put & a stock
(2) A call & a zero-coupon bond (or cash)
It can be shown that both portfolios give the same payoffs regardless of the terminal stock price.
ST > K ST < K
Therefore, their current values must be identical.
European Options: Put-Call ParityOption Valuation
𝑝𝑡 + 𝑆𝑡 = 𝑐𝑡 + 𝐾𝑒−𝑟𝜏
Portfolio (1) is overpriced in relation to portfolio (2).
Buy securities in portfolio (2) and (short) sell those in portfolio (1):
Buy the call and short sell both the put and the stock.
This which will generate upfront positive cash flow that should be invested at risk free rate:
If the stock price at the expiration of the option is greater than K, the call will be exercised; if the price is less than K, the put
will be exercised. In either cases the arbitrageur will end up buying one share for K. This share can be used to close the short
position, thus the net profit is equal to:
(1) A put & a stock
(2) A call & a zero-coupon bond (or cash)
𝑝𝑡 + 𝑆𝑡 − 𝑐𝑡 𝑒𝑟𝜏 − 𝐾
𝑝𝑡 + 𝑆𝑡 − 𝑐𝑡 𝑒𝑟𝜏
> 0
𝑝𝑡 + 𝑆𝑡 > 𝑐𝑡 + 𝐾𝑒−𝑟𝜏
European Options: Put-Call Parity
Put-Call Parity
Option Valuation
𝑝𝑡 + 𝑆𝑡 = 𝑐𝑡 + 𝐾𝑒−𝑟𝜏
Portfolio (2) is overpriced in relation to portfolio (1).
Buy securities in portfolio (1) and (short) sell those in portfolio (2):
Buy the put and stock and sell the call.
To do this upfront positive cash flow will be needed, that should be borrowed at risk free rate:
If the stock price at the expiration of the option is greater than K, the call will be exercised; if the price is less than K, the put
will be exercised. In either cases the arbitrageur will end up selling one share for K. This money will be used to pay back the
loan, thus the net profit is equal to:
(1) A put & a stock
(2) A call & a zero-coupon bond (or cash)
− 𝑝𝑡 − 𝑆𝑡 + 𝑐𝑡 𝑒𝑟𝜏 + 𝐾
− 𝑝𝑡− 𝑆𝑡 + 𝑐𝑡 𝑒𝑟𝜏
> 0
𝑝𝑡 + 𝑆𝑡 < 𝑐𝑡 + 𝐾𝑒−𝑟𝜏
European Options: Put-Call Parity
Put-Call Parity
Option Valuation
The formula for the value of a Call Option is:
And the formula for the value of a Put Option is:
𝑝𝑡 = exp −𝑟𝜏 𝐾Φ −𝑑2 − 𝑆𝑡Φ −𝑑1
𝑐𝑡 = 𝑆𝑡Φ 𝑑1 − exp(−𝑟𝜏)𝐾Φ 𝑑2
𝑑1 =𝑙𝑛
𝑆𝑡𝐾
+ 𝑟 +𝜎2
2𝜏
𝜎 𝜏
𝑑2 =𝑙𝑛
𝑆𝑡𝐾
+ 𝑟 −𝜎2
2𝜏
𝜎 𝜏= 𝑑1 − 𝜎 𝜏
where:
t is the current date
T is the expiry date
is the time to expiry.
St is the current (underlying) stock price.
K is the strike price.
r is the risk-free rate of interest.
σ is the volatility (σ2 is the variance of the
stock return, per unit time*)
European Options: Valuation: Black Scholes Model
Black-Scholes Formula
Option Valuation
Robert C. Merton
died in 1995
Long-Term Capital Management (LTCM)
The Black-Scholes formula calculates the price of European put and call options.
The model was proposed in 1973 by Fischer Black and Myron Scholes, who was later awarded the 1997 Nobel Prize in Economics
Science (joint with Robert C. Merton).
Myron ScholesFischer Black
European Options: Valuation: Black Scholes Model
Black-Scholes Formula
Option Valuation
Long-Term Capital Management (LTCM)
European Options: Valuation: Black Scholes ModelOption Valuation
Long-Term Capital Management (LTCM)
At the beginning of 1998:
equity of $4.72 billion
borrowed over $124.5 billion
=>assets of around $129 billion.
debt to equity ratio of over 25 to 1
The total losses by end of 1998 were found to be $4.6 billion
$1.6 bn in swaps
$1.3 bn in equity volatility
$430 mn in Russia and other emerging markets
$371 mn in directional trades in developed countries
$286 mn in Dual-listed company pairs (such as VW, Shell)
$215 mn in yield curve arbitrage
$203 mn in S&P 500 stocks
$100 mn in junk bond arbitrage
European Options: Valuation: Black Scholes ModelOption Valuation
The original model involves the methods of stochastic calculus (continuous-time finance) – which we explored earlier
during the module. Here, we will consider a simplified version of the model.
In the Black-Scholes framework, the assumption about the evolution of the Stock price is:
𝑆𝑡+∆𝑡 = 𝑆𝑡exp 𝜇∆𝑡 + 𝜎 ∆𝑡𝑍
where Z ~ N(0,1), i.e. Z is a standard normal random variable.
0
10
20
30
40
50
60
70
0 20 40 60 80 100 120
time
S
Time series resulting from the above assumption, with µ = 0.04, σ = 0.02.
European Options: Valuation: Black Scholes Model
Black-Scholes Formula
Option Valuation
The price at expiry is:
It follows that:
𝑆𝑇 = 𝑆𝑡+τ = 𝑆𝑡exp 𝜇𝜏 + 𝜎 𝜏𝑍
𝑆𝑇𝑆𝑡
= exp 𝜇𝜏 + 𝜎 𝜏𝑍
and therefore:
𝑆𝑇𝑆𝑡~𝐿𝑜𝑔𝑛𝑜𝑟𝑚𝑎𝑙 𝜇𝜏, 𝜎2𝜏
European Options: Valuation: Black Scholes Model
Black-Scholes Formula
Option Valuation
Z ~ N(0,1)
~ N(𝜇𝜏, 𝜎2𝜏)
From the properties of the Lognormal distribution, we know that:
𝐸𝑆𝑇
𝑆𝑡= 𝑒𝑥𝑝 𝜇𝜏 +
𝜎2𝜏
2= 𝑒𝑥𝑝 𝜇 +
𝜎2
2𝜏
This is a formula for the expected proportional increase in the stock price from the present to expiry.
Another formula for this is:
𝐸𝑆𝑇𝑆𝑡
= 𝑒𝑥𝑝 𝑟𝜏
This is because, in a risk-neutral world, the average return of all stocks equals the risk-free return.
For both of these formulae to be correct, it must be the case that:
𝜇 +𝜎2
2= 𝑟
European Options: Valuation: Black Scholes Model
Black-Scholes Formula
Option Valuation
Recall that a Call Option pays:
τ periods in the future.
Therefore the current value of a call option (assuming risk-neutrality) is:
𝑐𝑡 = 𝑒−𝑟𝜏𝐸 𝑚𝑎𝑥 0, 𝑆𝑇 − 𝐾
We need to evaluate the expectation:
𝑚𝑎𝑥 0, 𝑆𝑇 − 𝐾
𝐸 𝑚𝑎𝑥 0, 𝑆𝑇 − 𝐾 = 0 × 𝑃 𝑆𝑇 < 𝐾 + 𝐸 𝑆𝑇 − 𝐾 𝑆𝑇 > 𝐾 × 𝑃 𝑆𝑇 > 𝐾
= 𝐸 𝑆𝑇 − 𝐾 𝑆𝑇 > 𝐾 × 𝑃 𝑆𝑇 > 𝐾
= 𝐸 𝑆𝑇 𝑆𝑇 > 𝐾 × 𝑃 𝑆𝑇 > 𝐾 − 𝐾 × 𝑃 𝑆𝑇 > 𝐾
= 𝑆𝑡 × 𝐸𝑆𝑇𝑆𝑡
𝑆𝑇𝑆𝑡
>𝐾𝑆𝑡
× 𝑃𝑆𝑇𝑆𝑡
>𝐾
𝑆𝑡− 𝐾 × 𝑃
𝑆𝑇𝑆𝑡
>𝐾
𝑆𝑡
European Options: Valuation: Black Scholes Model
Black-Scholes Formula
Option Valuation
We know that𝑆𝑇
𝑆𝑡~𝐿𝑜𝑔𝑛𝑜𝑟𝑚𝑎𝑙 𝜇𝜏, 𝜎2𝜏 .
If 𝑌~𝐿𝑜𝑔𝑛𝑜𝑟𝑚𝑎𝑙 𝜇, 𝜎2 , then:
𝐸 𝑌| 𝑌 > 𝑐 =Φ
𝜇 − 𝑙𝑛𝑐𝜎 + 𝜎
Φ𝜇 − 𝑙𝑛𝑐
𝜎
𝑒𝑥𝑝 𝜇 +𝜎2
2
Applying this formula, we obtain:
𝐸𝑆𝑇𝑆𝑡
𝑆𝑇𝑆𝑡
>𝐾𝑆𝑡
=
Φ𝜇𝜏 − 𝑙𝑛
𝐾𝑆𝑡
𝜎 𝜏+ 𝜎 𝜏
Φ𝜇𝜏 − 𝑙𝑛
𝐾𝑆𝑡
𝜎 𝜏
𝑒𝑥𝑝 𝜇𝜏 +𝜎2𝜏
2
European Options: Valuation: Black Scholes Model
Black-Scholes Formula
Option Valuation
We also require:
𝑃𝑆𝑇𝑆𝑡
>𝐾
𝑆𝑡
Substituting to earlier equations, we obtain:
= 𝑃 𝑍 >𝑙𝑛
𝐾𝑆𝑡
− 𝜇𝜏
𝜎 𝜏= Φ
𝜇𝜏 − 𝑙𝑛𝐾𝑆𝑡
𝜎 𝜏
𝐸 𝑚𝑎𝑥 0, 𝑆𝑇 − 𝐾
= 𝑆𝑡Φ𝜇𝜏 − 𝑙𝑛
𝐾𝑆𝑡
𝜎 𝜏+ 𝜎 𝜏 𝑒𝑥𝑝 𝜇𝜏 +
𝜎2𝜏
2− 𝐾Φ
𝜇𝜏 − 𝑙𝑛𝐾𝑆𝑡
𝜎 𝜏
European Options: Valuation: Black Scholes Model
Black-Scholes Formula
Option Valuation
Since 𝜇 +𝜎2
2= 𝑟 , so exp 𝜇 +
𝜎2
2= exp(𝑟𝜏) :
𝐸 𝑚𝑎𝑥 0, 𝑆𝑇 − 𝐾
= 𝑆𝑡Φ𝑟 −
𝜎2
2 𝜏 − 𝑙𝑛𝐾𝑆𝑡
𝜎 𝜏+ 𝜎 𝜏 𝑒𝑥𝑝 𝑟𝜏 − 𝐾Φ
𝑟 −𝜎2
2 𝜏 − 𝑙𝑛𝐾𝑆𝑡
𝜎 𝜏
Also, since it is desirable to write the formula without the parameter µ, we substitute 𝜇 = 𝑟 −𝜎2
2. We obtain:
All that is now needed is the discount factor exp(-rτ).
This completes the formula for the present value of a call option.
= 𝑆𝑡Φ𝜇𝜏 − 𝑙𝑛
𝐾𝑆𝑡
𝜎 𝜏+ 𝜎 𝜏 𝑒𝑥𝑝 𝜇𝜏 +
𝜎2𝜏
2− 𝐾Φ
𝜇𝜏 − 𝑙𝑛𝐾𝑆𝑡
𝜎 𝜏
European Options: Valuation: Black Scholes Model
Black-Scholes Formula
Option Valuation
= exp(−𝑟𝜏)𝑆𝑡Φ𝑟 −
𝜎2
2𝜏 − 𝑙𝑛
𝐾𝑆𝑡
𝜎 𝜏+ 𝜎 𝜏 𝑒𝑥𝑝 𝑟𝜏 − exp(−𝑟𝜏)𝐾Φ
𝑟 −𝜎2
2𝜏 − 𝑙𝑛
𝐾𝑆𝑡
𝜎 𝜏
𝑐𝑡 = exp(−𝑟𝜏)𝐸 𝑚𝑎𝑥 0, 𝑆𝑇 − 𝐾
= 𝑆𝑡Φ𝑟 −
𝜎2
2𝜏 − 𝑙𝑛
𝐾𝑆𝑡
𝜎 𝜏+ 𝜎 𝜏 − exp(−𝑟𝜏)𝐾Φ
𝑟 −𝜎2
2𝜏 − 𝑙𝑛
𝐾𝑆𝑡
𝜎 𝜏
= 𝑆𝑡Φ𝑟 −
𝜎2
2𝜏 − 𝑙𝑛
𝐾𝑆𝑡
+ 𝜎2𝜏
𝜎 𝜏− exp(−𝑟𝜏)𝐾Φ
𝑟 −𝜎2
2𝜏 − 𝑙𝑛
𝐾𝑆𝑡
𝜎 𝜏
= 𝑆𝑡Φ𝑟 +
𝜎2
2𝜏 − 𝑙𝑛
𝐾𝑆𝑡
𝜎 𝜏− exp(−𝑟𝜏)𝐾Φ
𝑟 −𝜎2
2𝜏 − 𝑙𝑛
𝐾𝑆𝑡
𝜎 𝜏
= 𝑆𝑡Φ𝑙𝑛
𝑆𝑡𝐾
+ 𝑟 +𝜎2
2𝜏
𝜎 𝜏− exp(−𝑟𝜏)𝐾Φ
𝑙𝑛𝑆𝑡𝐾
+ 𝑟 −𝜎2
2𝜏
𝜎 𝜏
= 𝑆𝑡Φ 𝑑1 − exp(−𝑟𝜏)𝐾Φ 𝑑2
d2d1
𝑑2 = 𝑑1 − 𝜎 𝜏
European Options: Valuation: Black Scholes Model
Black-Scholes Formula
Option Valuation
𝑐 = 𝑆0𝑁 𝑑1 − exp(−𝑟𝜏)𝐾𝑁 𝑑2
= 𝑑1 − 𝜎 𝑇
John C. Hull (p.313, formula 14.20) presents the Black-Scholes formula as:
𝑑1 =𝑙𝑛
𝑆0𝐾
+ 𝑟 +𝜎2
2𝑇
𝜎 𝑇
𝑑2 =𝑙𝑛
𝑆0𝐾
+ 𝑟 −𝜎2
2𝑇
𝜎 𝑇
Note that he is using:
S0 for the current stock price (St in our analysis)
T for the time to expiry (τ in our analysis)
N(.) for the standard normal c.d.f. (Φ(.) in our analysis)
European Options: Valuation: Black Scholes Model
Black-Scholes Formula
Option Valuation
Finding the value of a put is very similar to finding the value of a call.
The present value of a European put option is:
𝑝𝑡 = 𝑒−𝑟𝜏𝐸 𝑚𝑎𝑥 0, 𝐾 − 𝑆𝑇
We need to evaluate the expectation:
𝐸 𝑚𝑎𝑥 0, 𝐾 − 𝑆𝑇 = 0 × 𝑃 𝑆𝑇 > 𝐾 + 𝐸 𝐾 − 𝑆𝑇 𝑆𝑇 < 𝐾 × 𝑃 𝑆𝑇 < 𝐾
= 𝐸 𝐾 − 𝑆𝑇 𝑆𝑇 < 𝐾 × 𝑃 𝑆𝑇 < 𝐾
= 𝐾 × 𝑃 𝑆𝑇 < 𝐾 − 𝐸 𝑆𝑇 𝑆𝑇 < 𝐾 × 𝑃 𝑆𝑇 < 𝐾
= 𝐾 × 𝑃𝑆𝑇𝑆𝑡
<𝐾
𝑆𝑡− 𝑆𝑡 × 𝐸
𝑆𝑇𝑆𝑡
𝑆𝑇𝑆𝑡
<𝐾𝑆𝑡
× 𝑃𝑆𝑇𝑆𝑡
<𝐾
𝑆𝑡
European Options: Valuation: Black Scholes Model
Black-Scholes Formula
Option Valuation
We know that𝑆𝑇
𝑆𝑡~𝐿𝑜𝑔𝑛𝑜𝑟𝑚𝑎𝑙 𝜇𝜏, 𝜎2𝜏 .
If 𝑌~𝐿𝑜𝑔𝑛𝑜𝑟𝑚𝑎𝑙 𝜇, 𝜎2 , then:
𝐸 𝑌| 𝑌 < 𝑐 =Φ
𝑙𝑛𝑐 − 𝜇𝜎 − 𝜎
Φ𝑙𝑛𝑐 − 𝜇
𝜎
𝑒𝑥𝑝 𝜇 +𝜎2
2
Applying this formula, we obtain:
𝐸𝑆𝑇𝑆𝑡
𝑆𝑇𝑆𝑡
<𝐾𝑆𝑡
=
Φ𝑙𝑛
𝐾𝑆𝑡
− 𝜇𝜏
𝜎 𝜏− 𝜎 𝜏
Φ𝑙𝑛
𝐾𝑆𝑡
− 𝜇𝜏
𝜎 𝜏
𝑒𝑥𝑝 𝜇𝜏 +𝜎2𝜏
2
European Options: Valuation: Black Scholes Model
Black-Scholes Formula
Option Valuation
We also require:
𝑃𝑆𝑇𝑆𝑡
<𝐾
𝑆𝑡
Substituting to earlier equations we obtain:
= 𝑃 𝑍 <𝑙𝑛
𝐾𝑆𝑡
− 𝜇𝜏
𝜎 𝜏= Φ
𝑙𝑛𝐾𝑆𝑡
− 𝜇𝜏
𝜎 𝜏
𝐸 𝑚𝑎𝑥 0, 𝐾 − 𝑆𝑇
= 𝐾Φ𝑙𝑛
𝐾𝑆𝑡
− 𝜇𝜏
𝜎 𝜏− 𝑆𝑡Φ
𝑙𝑛𝐾𝑆𝑡
− 𝜇𝜏
𝜎 𝜏− 𝜎 𝜏 𝑒𝑥𝑝 𝜇𝜏 +
𝜎2𝜏
2
European Options: Valuation: Black Scholes Model
Black-Scholes Formula
Option Valuation
Once again since 𝜇 +𝜎2
2= 𝑟 , so exp 𝜇 +
𝜎2
2= exp(𝑟𝜏) :
𝐸 𝑚𝑎𝑥 0, 𝐾 − 𝑆𝑇
All that is now needed is the discount factor exp(-rτ).
This completes the formula for the present value of a put option.
= 𝐾Φ𝑙𝑛
𝐾𝑆𝑡
− 𝜇𝜏
𝜎 𝜏− 𝑆𝑡Φ
𝑙𝑛𝐾𝑆𝑡
− 𝜇𝜏
𝜎 𝜏− 𝜎 𝜏 𝑒𝑥𝑝 𝜇𝜏 +
𝜎2𝜏
2
= 𝐾Φ𝑙𝑛
𝐾𝑆𝑡
− 𝑟 −𝜎2
2 𝜏
𝜎 𝜏− 𝑆𝑡Φ
𝑙𝑛𝐾𝑆𝑡
− 𝑟 −𝜎2
2 𝜏
𝜎 𝜏− 𝜎 𝜏 𝑒𝑥𝑝 𝑟𝜏
European Options: Valuation: Black Scholes Model
Black-Scholes Formula
Option Valuation
𝑝𝑡 = exp(−𝑟𝜏)𝐸 𝑚𝑎𝑥 0, 𝐾 − 𝑆𝑇
= exp −𝑟𝜏 𝐾Φ −𝑑2 − 𝑆𝑡Φ −𝑑1d2 d1
= exp(−𝑟𝜏)𝐾Φ𝑙𝑛
𝐾𝑆𝑡
− 𝑟 −𝜎2
2𝜏
𝜎 𝜏− exp(−𝑟𝜏)𝑆𝑡Φ
𝑙𝑛𝐾𝑆𝑡
− 𝑟 −𝜎2
2𝜏
𝜎 𝜏− 𝜎 𝜏 𝑒𝑥𝑝 𝑟𝜏
= exp(−𝑟𝜏)𝐾Φ𝑙𝑛
𝐾𝑆𝑡
− 𝑟 −𝜎2
2𝜏
𝜎 𝜏− 𝑆𝑡Φ
𝑙𝑛𝐾𝑆𝑡
− 𝑟 −𝜎2
2𝜏
𝜎 𝜏− 𝜎 𝜏
= exp(−𝑟𝜏)𝐾Φ𝑙𝑛
𝐾𝑆𝑡
− 𝑟 −𝜎2
2𝜏
𝜎 𝜏− 𝑆𝑡Φ
𝑙𝑛𝐾𝑆𝑡
− 𝑟 −𝜎2
2𝜏 − 𝜎2𝜏
𝜎 𝜏
= exp(−𝑟𝜏)𝐾Φ𝑙𝑛
𝐾𝑆𝑡
− 𝑟 −𝜎2
2𝜏
𝜎 𝜏− 𝑆𝑡Φ
𝑙𝑛𝐾𝑆𝑡
− 𝑟 +𝜎2
2𝜏
𝜎 𝜏
= exp(−𝑟𝜏)𝐾Φ−𝑙𝑛
𝑆𝑡𝐾
− 𝑟 −𝜎2
2𝜏
𝜎 𝜏− 𝑆𝑡Φ
−𝑙𝑛𝑆𝑡𝐾
− 𝑟 +𝜎2
2𝜏
𝜎 𝜏
= exp(−𝑟𝜏)𝐾Φ −𝑙𝑛
𝑆𝑡𝐾
+ 𝑟 −𝜎2
2𝜏
𝜎 𝜏− 𝑆𝑡Φ −
𝑙𝑛𝑆𝑡𝐾
+ 𝑟 +𝜎2
2𝜏
𝜎 𝜏
European Options: Valuation: Black Scholes Model
Black-Scholes Formula
Option Valuation
Consider put-call parity
Let’s verify the put-call parity in the context of the Black-Scholes.
𝑝𝑡 + 𝑆𝑡 = 𝑐𝑡 + 𝐾𝑒−𝑟𝜏
= 𝑝𝑡 +𝑆𝑡
= exp −𝑟𝜏 𝐾Φ −𝑑2 − 𝑆𝑡Φ −𝑑1 + 𝑆𝑡
= exp −𝑟𝜏 𝐾 1 − Φ 𝑑2 − 𝑆𝑡 1 − Φ 𝑑1 + 𝑆𝑡
= exp −𝑟𝜏 𝐾 − exp −𝑟𝜏 𝐾Φ 𝑑2 − 𝑆𝑡 + 𝑆𝑡Φ 𝑑1 + 𝑆𝑡
= exp −𝑟𝜏 𝐾 − exp −𝑟𝜏 𝐾Φ 𝑑2 + 𝑆𝑡Φ 𝑑1
= exp −𝑟𝜏 𝐾 + 𝑐𝑡
= 𝑅𝐻𝑆
𝐿𝐻𝑆
European Options: Valuation: Black Scholes Model
Put-Call Parity with Black-Scholes
Option Valuation
implied volatility
European Options: Valuation: Black Scholes Model
Working with Black-Scholes Formula
Option Valuation
The formula for the value of a Call Option is:
And the formula for the value of a Put Option is:
Let’s take a close look at the Call formula:
𝑝𝑡 = exp −𝑟𝜏 𝐾Φ −𝑑2 − 𝑆𝑡Φ −𝑑1
𝑐𝑡 = 𝑆𝑡Φ 𝑑1 − exp(−𝑟𝜏)𝐾Φ 𝑑2
1. It does not depend on the mean return on the underlying stock (µ); only on its current price (St) and volatility (σ).
2. As St becomes very large, both d1 and d2 become large,
𝑑1 =𝑙𝑛
𝑆𝑡𝐾
+ 𝑟 +𝜎2
2𝜏
𝜎 𝜏𝑑2 =
𝑙𝑛𝑆𝑡𝐾
+ 𝑟 −𝜎2
2𝜏
𝜎 𝜏= 𝑑1 − 𝜎 𝜏
∞
∞
large large
so the Call formula becomes:
so both Φ(d1) and Φ(d2) approach 1,
𝑐𝑡 = 𝑆𝑡 − exp(−𝑟𝜏)𝐾
In other words, when a Call Option is “deep in the money”, its current value is simply the current price of the stock, less
the discounted strike price (which will certainly be paid at expiry).
European Options: Valuation: Black Scholes Model
Working with Black-Scholes Formula
Option Valuation
The formula for the value of a Call Option is: 𝑐𝑡 = 𝑆𝑡Φ 𝑑1 − exp(−𝑟𝜏)𝐾Φ 𝑑2
3. If an option is very close to expiry (i.e. when τ is small), both d1 and d2 become large,
so again both Φ(d1) and Φ(d2) approach 1. Also, exp(-rτ) approaches 1
Thus the Call formula becomes:
𝑖𝑓 𝑆𝑡 > 𝐾
𝑖𝑓 𝑆𝑡 < 𝐾
In words, if an option is very close to expiry, then if it is “in the money” its value is simply the difference between the
current stock price and the strike price, and if it is “out of the money” its value is zero.
𝑐𝑡 = 𝑆𝑡 − 𝐾
0
European Options: Valuation: Black Scholes Model
Working with Black-Scholes Formula
Option Valuation
The formula for the value of a Call Option is: 𝑐𝑡 = 𝑆𝑡Φ 𝑑1 − exp(−𝑟𝜏)𝐾Φ 𝑑2
4. If the volatility (σ) is small , both d1 and d2 become large,
so again both Φ(d1) and Φ(d2) approach 1, and the Call formula becomes:
𝑖𝑓 𝑆𝑡 > exp(−𝑟𝜏)𝐾
In words, if the volatility is small, the price at expiry is known with certainty to be ST = Stexp(rτ).
This certain price at expiry has present value St, the current price.
Therefore the value of the Option is the current Stock price less the present value of the strike price, provided that the
former exceeds the latter, and zero otherwise.
𝑐𝑡 = 𝑆𝑡 − exp(−𝑟𝜏)𝐾
0 𝑖𝑓 𝑆𝑡 < exp(−𝑟𝜏)𝐾
European Options: Valuation: Black Scholes Model
Working with Black-Scholes Formula
Option Valuation
Option Valuation
Binomial Model
Black-Scholes Model
Put-Call Parity
Option Valuation
EXERCISE
6.The Greeks &
Hedging
7.Value at risk
8.American Options
and Dividends
Lecture 4:
6. The Greeks and
Hedging
The Greeks
Delta, Gamma, Theta, Vega, Rho
Volatility
Implied Volatility
Volatility Smiles and Smirks
Hedging
Delta Hedge
The Greeks and hedging
The Greeks
A sensitivity is the change in the option value resulting from a ceteris paribus change in one of the model parameters.
The option price depends on five such parameters: τ, St, K, r, σ.
𝑐𝑡 = 𝑆𝑡Φ 𝑑1 − exp(−𝑟𝜏)𝐾Φ 𝑑2 𝑝𝑡 = exp −𝑟𝜏 𝐾Φ −𝑑2 − 𝑆𝑡Φ −𝑑1
𝑑1 =𝑙𝑛
𝑆𝑡𝐾
+ 𝑟 +𝜎2
2𝜏
𝜎 𝜏
𝑑2 = 𝑑1 − 𝜎 𝜏
The sensitivities are also known as the “Greeks”, and are named: delta, gamma, theta, vega, and rho.
Thy are calculated as a partial derivative of the option price/ value (V) with respect to parameter whose impact the
sensitivity is capturing.
𝑆𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑖𝑡𝑦1 =𝜕𝑉𝑡
𝜕𝑃𝑎𝑟𝑎𝑚𝑒𝑡ℎ𝑒𝑟1
The GreeksThe Greeks and hedging
Sensitivities of Black-Scholes Call Formula:
Delta is the rate of change of the option price with respect to the price of the underlying asset.
Name (symbol) Formula Sign
Delta (Δ) 𝜕𝑐𝑡𝜕𝑆𝑡
= Φ 𝑑1+
Example: Δ = 0.6: if the stock price changes by small amount the price of the option changes by 60% of that amount
Delta is the slope of curve that
relates the option price to the
price of underlying asset
The Greeks: DeltaThe Greeks and hedging
The Greeks: DeltaThe Greeks and hedging
The Greeks: DeltaThe Greeks and hedging
Gamma is the rate of change of the option delta with respect to the price of the underlying asset.
Example: Γ = 0.6: when stock price changes by ΔS, the delta changes by 0.6* ΔS.
Name (symbol) Formula Sign
Gamma (Γ) 𝜕2𝑐𝑡𝜕𝑆𝑡
2=ϕ 𝑑1
𝜎𝑆𝑡 𝜏
+
Gamma is the second partial
derivative of the portfolio with
respect to the asset price.
It measures the curvature of the
relationship between the option
price and the stock price.
The Greeks: Gamma
Sensitivities of Black-Scholes Call Formula:
The Greeks and hedging
The Greeks: GammaThe Greeks and hedging
Theta is the rate of change of the value of the option with respect to the passage of time.
To obtain the change per calendar day’ theta needs to be divided by 365.
Name (symbol) Formula Sign
Theta (Θ) 𝜕𝑐𝑡𝜕𝜏
= −𝑆𝑡𝜎ϕ 𝑑1
2 𝜏− 𝐾𝑟𝑒𝑥𝑝(−𝑟𝜏)Φ 𝑑2
-
Theta is usually negative because as
time passes the option tends to
become less valuable.
Sensitivities of Black-Scholes Call Formula:
The Greeks: ThetaThe Greeks and hedging
Vega is the rate of change of the value of the option with respect to the volatility of the underlying asset.
Name (symbol) Formula Sign
Vega (ν) 𝜕𝑐𝑡𝜕𝜎
= 𝑆𝑡 𝜏ϕ 𝑑1+
Example: ν = 12: 1% (0.01) increase in volatility (from 20% to 21%) increases the value of the option by
approximately 0.01 * 12= 0.12
Sensitivities of Black-Scholes Call Formula:
The Greeks: VegaThe Greeks and hedging
Rho is the rate of change of the value of the option with respect to the interest rate.
Name (symbol) Formula Sign
Rho (ρ) 𝜕𝑐𝑡𝜕𝑟
= 𝜏𝐾𝑒𝑥𝑝(−𝑟𝜏) Φ 𝑑2+
Example: ρ = 5: 1% (0.01) increase in the risk free rate (from 5% to 6%) increases the value of the option by
approximately 0.01* 5 = 0.05
Sensitivities of Black-Scholes Call Formula:
The Greeks: RhoThe Greeks and hedging
Sensitivities of Black-Scholes Call Formula:
Let us verify the first two of these.
Name (symbol) Formula Sign
Delta (Δ) 𝜕𝑐𝑡𝜕𝑆𝑡
= Φ 𝑑1+
Gamma (Γ) 𝜕2𝑐𝑡𝜕𝑆𝑡
2=ϕ 𝑑1
𝜎𝑆𝑡 𝜏
+
Theta (Θ) 𝜕𝑐𝑡𝜕𝜏
= −𝑆𝑡𝜎ϕ 𝑑1
2 𝜏− 𝐾𝑟𝑒𝑥𝑝(−𝑟𝜏)Φ 𝑑2
-
Vega (ν) 𝜕𝑐𝑡𝜕𝜎
= 𝑆𝑡 𝜏ϕ 𝑑1+
Rho (ρ) 𝜕𝑐𝑡𝜕𝑟
= 𝜏𝐾𝑒𝑥𝑝(−𝑟𝜏) Φ 𝑑2+
long position
The GreeksThe Greeks and hedging
Delta (Δ)
The first thing we need to understand is what happens when we differentiate Φ(w) with respect to w. Recall that:
When we differentiate an integral with respect to the upper limit of integration, we obtain the integrand evaluated at that limit.
So:
Φ(𝑤) =
−∞
𝑤
ϕ 𝑤 𝑑𝑤
𝜕Φ(𝑤)
𝜕𝑤= ϕ 𝑤
The function we are differentiating is the Call formula:𝑐𝑡 = 𝑆𝑡Φ 𝑑1 − exp(−𝑟𝜏)𝐾Φ 𝑑2
But we have to remember that d1 and d2 both involve St,
𝑑1 =𝑙𝑛
𝑆𝑡𝐾
+ 𝑟 +𝜎2
2𝜏
𝜎 𝜏𝑑2 = 𝑑1 − 𝜎 𝜏
The Greeks: DeltaThe Greeks and hedging
So, differentiating 𝑐𝑡 = 𝑆𝑡Φ 𝑑1 − exp −𝑟𝜏 𝐾Φ 𝑑2 , we obtain:
To complete this we need to differentiate d1 and d2.
We note that d1 and d2 are both of the form:
So that:
Δ =𝜕𝑐𝑡𝜕𝑆𝑡
= Φ 𝑑1 + 𝑆𝑡ϕ 𝑑1𝜕𝑑1𝜕𝑆𝑡
− exp −𝑟𝜏 𝐾ϕ 𝑑2𝜕𝑑2𝜕𝑆𝑡
=𝑙𝑛 𝑆𝑡 + 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝜎 𝜏
𝜕𝑑1𝜕𝑆𝑡
=𝜕𝑑2𝜕𝑆𝑡
=1
𝜎 𝜏𝑆𝑡
𝑓𝑔 ′ 𝑥 = 𝑓′ 𝑥 𝑔 𝑥 + 𝑓 𝑥 𝑔′(𝑥)
𝑙𝑜𝑔𝑎
𝑏= 𝑙𝑜𝑔𝑎 − 𝑙𝑜𝑔𝑏
=𝑙𝑛 𝑆𝑡 − ln 𝐾 + 𝑐𝑜𝑛𝑠𝑡
𝜎 𝜏
𝑙𝑛𝑆𝑡𝐾
+ 𝑐𝑜𝑛𝑠𝑡
𝜎 𝜏
𝑓 ′ 𝑔(𝑥) = 𝑓′ 𝑔 𝑥 × 𝑔′(𝑥)
The Greeks: DeltaThe Greeks and hedging
We obtain:
The next important point is that the quantity in square brackets is zero.
We will not verify this analytically; we will instead verify it at seminar session.
Δ =𝜕𝑐𝑡𝜕𝑆𝑡
= Φ 𝑑1 + 𝑆𝑡ϕ 𝑑1𝜕𝑑1𝜕𝑆𝑡
− exp −𝑟𝜏 𝐾ϕ 𝑑1𝜕𝑑1𝜕𝑆𝑡
= Φ 𝑑1 + 𝑆𝑡ϕ 𝑑1ϕ(𝑑1)
𝜎 𝜏𝑆𝑡− exp −𝑟𝜏 𝐾
ϕ(𝑑2)
𝜎 𝜏𝑆𝑡
= Φ 𝑑1 +1
𝜎 𝜏ϕ 𝑑1 −
exp −𝑟𝜏 𝐾ϕ(𝑑2)
𝑆𝑡
This is highly convenient, because it means that the formula for delta is very simple:
Δ =𝜕𝑐𝑡𝜕𝑆𝑡
= Φ 𝑑1
The Greeks: DeltaThe Greeks and hedging
Note that delta is always positive (for a Call).
The value of a Call Option always rises when the current stock price rises.
However, note that delta cannot be greater than one.
This means that the value of the option never rise by more than the rise in
the price of the underlying stock.
For a deep in-the-money call, since d1 is high, delta will be close to 1, and therefore the call price will
move penny for penny with the underlying stock.
deep in the money
The Greeks: DeltaThe Greeks and hedging
Delta is sometimes called the hedge ratio.
Delta changes over time for two reasons:
1. The stock price (St) changes over time
2. The time to expiry (τ) falls over time (and since d1 involves τ, delta depends on τ).
Delta shows how many units of the underlying stock need to be short-sold for each call option
purchased for the position to be perfectly hedged over a short interval of time.
The position is perfectly hedged if losses made on the stock are offset by gains made on the option, or vice versa.
If you want your position to remain perfectly hedged, you will need to alter continuously the number of stocks held.
This is dynamic hedging. It can be costly.
Gamma tells us how much delta changes when the underlying price changes.
An option with a high gamma is little use for hedging, because the hedge would need to be readjusted constantly.
The Greeks: DeltaThe Greeks and hedging
Gamma is the second derivative of the Option value with respect to current price. It therefore represents how sensitive
delta is to the current price.
This one is easier. We have already found delta:
Differentiating again with respect to St:
Hence we obtain the second formula as presented earlier in the table.
Δ =𝜕𝑐𝑡𝜕𝑆𝑡
= Φ 𝑑1
Note that gamma is also always positive. This means that the Call value is a convex function of current stock price.
Γ =𝜕2𝑐𝑡𝜕𝑆𝑡
2
=ϕ 𝑑1
𝜎𝑆𝑡 𝜏
= ϕ 𝑑1𝜕𝑑1𝜕𝑆𝑡
= ϕ 𝑑11
𝜎 𝜏𝑆𝑡
Gamma (Γ)
The Greeks: GammaThe Greeks and hedging
Sensitivities of Black-Scholes Put Formula:
Name (symbol) Formula
Delta (Δ) 𝜕𝑝𝑡𝜕𝑆𝑡
= −Φ −𝑑1
Gamma (Γ) 𝜕2𝑝𝑡
𝜕𝑆𝑡2=ϕ 𝑑1
𝜎𝑆𝑡 𝜏
Theta (Θ) 𝜕𝑝𝑡𝜕𝜏
= −𝑆𝑡𝜎ϕ 𝑑1
2 𝜏+ 𝐾𝑟𝑒𝑥𝑝(−𝑟𝜏)Φ 𝑑2
Vega (ν) 𝜕𝑝𝑡𝜕𝜎
= 𝑆𝑡 𝜏ϕ 𝑑1
Rho (ρ) 𝜕𝑝𝑡𝜕𝑟
= −𝜏𝐾𝑒𝑥𝑝(−𝑟𝜏)Φ −𝑑2
long position
The GreeksThe Greeks and hedging
Volatility
The option price depends on five parameters: τ , St , K, r, σ. Note that all of these are known, except σ.
σ can be estimated using the historical volatility, that is, the (annualised) standard deviation of the last 30 (perhaps) daily returns.
The implicit assumption is that σ will be the same in the future as it has been in the recent past.
If you increase the number of days, you attain more accuracy in your estimate, but you are less likely to pick up recent
changes in volatility.
A better approach is to use ARCH/GARCH.
These models recognise that volatility changes over time.
They are estimated using past data and can be used to forecast future volatility
𝑐𝑡 = 𝑆𝑡Φ 𝑑1 − exp(−𝑟𝜏)𝐾Φ 𝑑2 𝑑1 =𝑙𝑛
𝑆𝑡𝐾
+ 𝑟 +𝜎2
2𝜏
𝜎 𝜏
Implied Volatility
Since the market price of an option is known, and all the parameters except σ are known, we can find the value of σ that gives
rise to an option value equal to the market price. This value of σ is the implied volatility of the underlying stock price.
Typically, many different options are available on the same underlying stock.
In theory, we expect the implied volatility to be the same for all of them, since they are measuring the same thing.
In practice, we see that implied volatility varies with the strike price.
Volatility: Implied VolatilityThe Greeks and hedging
Strike price
Implied volatility
at the money
Volatility smiles and smirks
Before 1987, implied volatility was a U-shaped function (known as a “volatility smile”) of the strike price, with minimum
around the current price.
This implies that both in-the-money and
out-of-the money options were over-priced
relative to at-the-money options. Implication:
Log-normal distribution understates the probability of
extreme movements in price of underling asset.
S
Volatility: Smiles and SmirksThe Greeks and hedging
Strike price
Implied volatility
at the moneyin the
money Call
out the
money Call
Since 1987, implied volatility has more commonly been a monotonically decreasing function of strike price (hence
“volatility smirk” or “volatility skew”).
This implies (e.g.) that in-the-money Calls
are over-priced, but out-of-the-money calls
are under-priced.
Implied distribution has heavier left tail than lognormal
S
Volatility: Smiles and Smirks
Volatility smiles and smirks
The Greeks and hedging
Question
Suppose that the stock price at time zero is S0 = £90.The continuously compounded risk free
rate is 5%, and European call option written on S with strike price £100 and time to expiry τ
= 1 year has delta of 0.352 and trades for £2.5. Find the implied volatility of the stock to
the nearest 1%.
Φ 𝑑1 = 0.352 ⟹ 𝑑1 = −0.38 𝑐𝑡 = 𝑆𝑡Φ 𝑑1 − exp(−𝑟𝜏)𝐾Φ 𝑑2
2.5 = 90 ∗ 0.352 − exp −1 ∗ 0.05 ∗ 100 ∗ Φ −0.38 − 𝑠𝑖𝑔𝑚𝑎
2.5 = 31.68 − 95.123 ∗ Φ −0.38 − 𝑠𝑖𝑔𝑚𝑎
Φ −0.38 − 𝑠𝑖𝑔𝑚𝑎 = 0.306761
−0.38 − 𝑠𝑖𝑔𝑚𝑎 = −0.51
⟹ 𝑠𝑖𝑔𝑚𝑎 = 0.13 = 13%
𝑑2 = 𝑑1 − 𝜎 𝜏
1 − 0.352 = 0.648
VolatilityThe Greeks and hedging
Hedging
The financial world is divided into speculators and hedgers.
Speculators:
take a view on the direction of some quantity such as the asset price, or volatility in the asset price, and
implement a strategy (involving purchase or sale of options) in order to profit from this view.
tend to lose if their view turns out to be incorrect.
Hedgers :
buy options that they believe to be under-priced, and simultaneously purchase something else that has the
effect of eliminating all of the risk associated with the option trade.
More traditional classification includes: speculator, arbitrageurs and hedgers:
Speculators use derivatives to bet on the future direction of price movements.
Arbitrageurs take offsetting positions in two or more instruments to lock in profit.
Hedgers use derivatives to reduce risk that they face from potential future movements in a market variable.
HedgingThe Greeks and hedging
100
120c=20
90c=0
Example 1: Perfect Hedge: Binomial pricing model with one time-period.
c=20
c=0
The current price is 100. Between now and expiry, the price either rises to 120, or falls to 90.
The strike price of the call option is 100. If you purchase the option, you either gain 20 at expiry, or you gain 0 at expiry
How do you make sure that the amount you receive at expiry is the same, regardless of the price of the underlying?
In addition to purchasing the option, you sell w units of the underlying.
What if you do not have any units to sell? You short-sell.
The value of your portfolio at expiry is then:
Either 20 – 120w
You want these values to be the same, so:
20 – 120w = -90w
or 0 – 90w
→ w = 0.67
So, for each option that you purchase,
you need to short-sell 0.67 units of the
underlying.
This amounts to a “perfect hedge” since it
results in an outcome that is invariant to
the price of the underlying.
HedgingThe Greeks and hedging
Example 2: Speculation: Straddles.
A straddle is a portfolio consisting of a call and a put with the same strike and same expiry date.
The payoff diagram is shown below:
Such a contract might be bought when the current price is in the
vicinity of the strike price (both options are At The Money), but the
investor has reason to believe there will soon be a significant change in
the price, up or down.
E.g. there will shortly be an Earnings Announcement, and it is hard to
predict whether the news will be good or bad.
Note that in order to buy a Straddle, someone must be willing to sell it.
It would be sold by someone who has the opposite view: someone who
expects the underlying price to remain stable.
These contracts are traded by those who have a view on the future direction of volatility (think of ARCH/GARCH modelling).
Investors forecasting a burst of volatility are likely to purchase Straddles.
Those forecasting a period of calm are likely to sell them.
Note that their view on the direction of the underlying price is irrelevant. For this reason, Straddles are known as volatility trades.
Buying a Straddle is a good way of reducing risk at times of high volatility.
Also, it is a very straightforward strategy which requires “no maintenance”.
However, it is not a “perfect hedge”. You can lose. E.g. if you buy a Straddle and the price doesn’t move.
Value
STK
callput
HedgingThe Greeks and hedging
Delta Hedge
You use an Econometric Model (probably ARCH/GARCH) to forecast the
volatility of the underlying stock price.
If this volatility is higher than the implied volatility of the call option, you
conclude that the call option is under-priced, and you purchase it.
You paid an amount C (0.9704) for the call option. But, by your calculations, it is
worth V (1.244902), where V > C.
If the underlying stock price falls tomorrow, value of the call option falls.
Sell w units of the underlying stock.
Value of your portfolio:
How do we choose w?
In order to construct a perfect hedge, we need to ensure that the value of the portfolio is always the same whatever happens to
the price of the underlying (S), thus:
where k is some constant that does not depend on S.
Let’s differentiate both sides with respect to S.
How should you hedge against this loss?
What if you don’t have any to sell? Short-sell.
V – w*S
V – w*S = k
Hedging: Delta HedgeThe Greeks and hedging
So, the number of units that we need to short-sell in order to create a perfect hedge is given by the option’s “delta”.
Recall that the delta of a vanilla call option is:
Δ =𝜕𝑉
𝜕𝑆= Φ 𝑑1
𝑑𝑉
𝑑𝑆− 𝑤 = 0 ⇒ 𝑤 =
𝑑𝑉
𝑑𝑆≡ ∆
where: 𝑑1 =𝑙𝑛
𝑆𝑡𝐾
+ 𝑟 +𝜎2
2𝜏
𝜎 𝜏This is delta-hedging.
A problem is that delta changes when S changes.
This means that every time the price of the underlying changes, the portfolio needs to be “re-hedged” in order to
maintain the fixed value of the portfolio.
Such re-hedging involves either further short-selling of the underlying (if S has risen) or buying back units of the
underlying (if S has fallen).
How important it is to re-hedge depends on the responsiveness of delta to changes in S. This responsiveness is given
by “gamma”:
An option with a high gamma is little use for hedging, because the hedge would need to be readjusted constantly.
Γ =𝜕2𝑐𝑡𝜕𝑆𝑡
2=ϕ 𝑑1
𝜎𝑆𝑡 𝜏
Hedging: Delta HedgeThe Greeks and hedging
Delta-hedged portfolio:
You are protected against small movements in the prices of underlying asset
As the price of underlying asset changes the delta changes as well
=> you need to re-hedge
=> dynamic hedging
Re-hedging usually done once a day
Gamma-hedged portfolio:
Between rebalancing at the trading times, Delta will drift away from zero as the underlying asset prices move.
If the portfolio is Gamma-hedged at the discrete trading times then the amount of such drift will be small
(comparable to the square of the change in underlying price).
You are protected against larger movements in the prices of underlying asset
Since underlying asset has Gamma = 0 ,
=> position in some other instrument that is non linearly depended on underlying asset needs to be taken;
=> this will affect delta, thus position in the underlying needs to be adjusted accordingly
Goal: Portfolio Delta = 0
Goal: Portfolio Gamma = 0
BTW: what is delta of a stock/
underlying asset?
1
HedgingThe Greeks and hedging
Vega-hedged portfolio:
The underlying volatilities used in hedging calculations are all estimates.
If these are incorrect then delta hedging may be incorrect, consequently it is appropriate to attempt to immunise a
portfolio against (small) errors in volatility estimates.
Just as in delta hedging, achieving a portfolio Vega of zero achieves this.
You are protected against miss-specification of the volatility
Since underlying asset has Vega = 0,
=> position in some other instrument that is non linearly depended on underlying asset needs to be taken;
In order for portfolio to be both Vega and Gamma neutral position in 2 different instruments needs to be taken
Goal: Portfolio Vega = 0
HedgingThe Greeks and hedging
Example:
Consider a delta neutral portfolio, with Gamma of -5000 and Vega of -8000.
Option 1 has Delta = 0.6, Gamma = 0.5 and Vega = 2;
Option 2 has Delta = 0.5, Gamma = 0.8 and Vega = 1.2.
To make portfolio Gamma and Vega neutral both Option 1 and 2 should be used:
and
where w1 is quantity of Option 1;
w2 is quantity of Option 2.
The delta of the portfolio after addition of Option 1 and 2 changes to:
Therefore 3 240 units of the underlying need to be sold to maintain delta neutrality.
−5000 +0.5𝑤1 + 0.8𝑤2 = 0
−8000 +2𝑤1 + 1.2𝑤2 = 0
𝑤1 = 400 𝑎𝑛𝑑 𝑤2 = 6000
= 3240400 × 0.6 + 6000 × 0.5
HedgingThe Greeks and hedging
Options, when first sold, are usually close to at the money…
so have relatively high Gammas and Vegas….
with time the price of underlying usually changes enough to make option deep in the money or out of money…
thus both Gammas and Vegas are very small…
consequently focus on delta while hedging
HedgingThe Greeks and hedging
Hedging, delta, Value of the Option…and Black-Scholes
What can we do?
1. Do nothing: have naked position
Consequences:
If ST < K option will not be exercised, we made £300 000
If ST > K option will be exercised, our cost is 100 000 (ST – K)
e.g. ST = £60, the cost is 10 * 100 000 = 1000 000, which is much greater than £300 000
2. Buy 100 000 shares as soon as we sold the option: covered position
Consequences:
If ST < K option will not be exercised, we have lost (S0 – ST)* 100 000 on position in stock
if ST = £40, we lost £900 000 on stock
If ST > K option will be exercised, we gain (K-S0)*100 000
3. Stop- loss strategy:
Buy stock if price raise above K, sell if it falls below K
Can be costly
4. Perfect Hedge would make the cost of option be equal to Black Scholes price…
Dynamic Delta Hedge
Only with dynamic delta hedge we have profit of £60 000
Example: We sold for £300 000 European call option on 100 000 shares of
a non-dividend paying stock. We also know that:
S0 = 49, K = 50, r = 0.05, σ = 0.2, τ = 0.3846
The Black- Scholes price of the option is £ 240 000.
Have we just made £60 0000 profit?
Not necessarily…
…there are risks
HedgingThe Greeks and hedging
Hedging, delta, Value of the Option…and Black-Scholes
Simulation of delta hedging.
Option closes in the money.
Cost of hedging £263 300.
Option is exercised
We get 50*100 000 for the
shares we have.
2557.8 + 2.5 – 308 = 2252.3
HedgingThe Greeks and hedging
Hedging, delta, Value of the Option…and Black-Scholes
Simulation of delta hedging.
Option closes out of the money.
Cost of hedging £256 600.
The difference in the cost
of hedging the position in
option and Black-Sholes
price come from the
frequency of hedge
rebalancing.
Still even weekly
rebalancing locks us in
profit…
HedgingThe Greeks and hedging
The Greeks
Delta, Gamma, Theta, Vega, Rho
Volatility
Implied Volatility
Volatility Smiles and Smirks
Hedging
Delta Hedge
The Greeks and hedging
EXERCISE
Derivatives and Risk Management: The Greeks….. Marta Wisniewska
7. Value at Risk
VaR
Calculating VaR:
Historical Simulation;
Model Building Approach
VaR of Option Portfolio
Value at Risk
p = (100 - X)% = 3%
Value at Risk (VaR)
Each of the Greeks (delta, gamma and vega) were describing different aspect of risk of a portfolio.
Thus, VaR is the loss level (V) over N days that has a probability of only (100 - X)% of being exceeded.
VaR is a function of:
(1) time horizon (N days) and
(2) confidence interval (X%).
VaR is the loss corresponding to the (100 - X)th percentile of the distribution of the gain in the value of the portfolio over the
next N days.
Example: If N = 5 and X = 97 what is the VaR?
Value at Risk (VaR) is a measure that attempts to summarize the total risk of a portfolio and evaluates ‘how bad things can get’.
VaR (V) can be best described by following statement:
‘We are X percent sure there will not be a loss of more than V in the next N days’.
gain over N days
VaR is a 3rd percentile of
the distribution of gain in
the value of the portfolio in
the next 5 days.
Value at Risk VaR: Introduction
Two portfolios with different distribution of gains can have the same VaR.
Expected shortfall is the expected loss during an N-day period conditional that an outcome occurs in the (100-X)% left
tail of the distribution.
gain over N days
gain over N days
Value at Risk VaR: Introduction
Time Horizon
In practice N is usually set to 1 and the usual assumption is that:
This formula is true if changes in the value of portfolio have iind (with mean 0), otherwise it is just an approximation.
Example 1: 10-day 99% VaR can be calculated as:
10 = 3.162 𝑡𝑖𝑚𝑒𝑠 𝑡ℎ𝑒 1 − 𝑑𝑎𝑦 99% 𝑉𝑎𝑅
Example 2:
What is the relationship between volatility per year σyear (used in option pricing) and volatility per day σday (used at VaR).
Assuming 252 trading days it is:
𝜎𝑦𝑒𝑎𝑟 = 𝜎𝑑𝑎𝑦 252
Daily volatility is about 6% of annual volatility.
(𝑁 − 𝑑𝑎𝑦 𝑉𝑎𝑅) = (1 − 𝑑𝑎𝑦𝑉𝑎𝑅) × 𝑁
iind: independent identical normal distribution.
Value at Risk VaR: Introduction
Calculating VaR: Historical Simulation
The distribution of daily loss in the value of the portfolio depends on the value of x market variables (v).
Define vi as the value of a market variable on Day i and suppose that today is Day n.
The ith scenario assumes that the value of the market variable tomorrow will be:
Example 1: Calculate 1-day VaR using 99% confidence level on 501 days of data.
First identify the factors affecting the Value of the portfolio.
Next calculate the daily changes in those factors.
You should have 500 such changes.
Create scenarios of the value of the portfolio based on those changes.
Since there are 500 scenarios, the 99th percentile of the distribution is the 5th highest loss.
𝑣𝑎𝑙𝑢𝑒 𝑢𝑛𝑑𝑒𝑟 𝑖𝑡ℎ 𝑠𝑐𝑒𝑛𝑎𝑡𝑖𝑜 = 𝑣𝑛𝑣𝑖𝑣𝑖−1
Value at Risk Calculating VaR: Historical Simulation
Example 2: Today the value of the portfolio is 10 000USD.
There are 4 assets in the portfolio, and their value is: (1) DJIA: 4 000USD, (2) FTSE 100: 3 000USD, (3)
CAC 40: 1 000USD and Nikkei 225: 2 000USD.
What is the one-day 99% VaR?
Scenario 1: Value of DJIA
11022.06 ×11173.59
11219.38= 10977.08
value today1st possible growth rate
Value at Risk Calculating VaR: Historical Simulation
Scenario 1: Value of the portfolio
Thus the portfolio has a gain of 14 USD under Scenario 1.
4000 ×10977.08
11022.06+ 3000 ×
5180.40
5197+ 1000 ×
4229.64
4226.81+ 2000 ×
12224.10
12006.53= 10014
Value at Risk Calculating VaR: Historical Simulation
The one-day 99% VaR can be estimated as the 5th worst loss.
This is 253,385.
Value at Risk Calculating VaR: Historical Simulation
Calculating VaR: Model-Building Approach
Consider portfolio worth P that consist of n assets with an amount αi being invested in each asset i (1 ≤ i ≤ n).
Define Δxi as the return on asset i in one day.
The change in the value of the investment in asset i in one day is αi Δxi and the change in the value of the portfolio in one
day is:
If Δxi are multivariate normal, then ΔP is normally distributed.
Assume that the expected value of Δxi change is zero (in reality it is different than zero, but small in comparison to the
standard deviation), the mean of is zero.
Therefore to calculate VaR one need to calculate the standard deviation of ΔP (σP ):
where σi is the daily volatility of the ith asset, and ρij is the correlation coefficient between returns on asset i and asset j.
𝜎𝑝2 =
𝑖=1
𝑛
𝑗=1
𝑛
ρ𝑖𝑗α𝑖α𝑗σ𝑖σ𝑗
∆𝑃 =
𝑖=1
𝑛
𝛼𝑖∆𝑥𝑖
Value at Risk Calculating VaR: Model Building Approach
Example:
Consider a portfolio consisting of 10 000GBP of shares A, and 5 000GBP of shares B.
The returns on those two shares have bivariate normal distribution with a correlation of 0.3.
The volatility of A is 2% a day and the volatility of B is 1%.
What is 10-day 99% VaR?
Share A over 1-day period has a standard deviation of 200 GBP (= 10 000* 0.02), whereas share B of 50 GBP.
The standard deviation of the portfolio is therefore:
The mean change is assumed to be zero, and the change in the value of the portfolio is normally distributed.
N(-2.33) = 0.01 means that there is 1% probability that normally distributed variable will decrease in value by more than
2.33 standard deviations.
Therefore 1-day VaR is:
Whereas 10- day VaR is:
220 × 2.33 = 512.6
10 × 512.6 = 1620
𝜎𝑃 = 2002 + 502 + 2 × 0.3 × 200 × 50 = 220
Value at Risk Calculating VaR: Model Building Approach
VaR of Option Portfolio
Consider portfolio of options of single stock. Delta of the portfolio is:
Define Δx as the percentage change in the stock price in 1 day:
We know that the approximate relationship between ΔP and Δx is:
If there are several underlying, then approximate relationship between ΔP and Δxi is similar:
Define 𝛼𝑖 = 𝑆𝑖𝛿𝑖 we have:
𝛿 =Δ𝑃
Δ𝑆
∆𝑃 =
𝑖=1
𝑛
𝛼𝑖∆𝑥𝑖
∆𝑃 =
𝑖=1
𝑛
𝑆𝑖𝛿𝑖∆𝑥𝑖
∆𝑥 =Δ𝑆
𝑆
∆𝑃 = 𝑆δΔ𝑥
Value at Risk VaR of Option Portfolio
Example:
Portfolio consists of options on stock X and Y.
The option on stock X has delta of 1 and on stock Y of 20.
X trades for 120 and Y for 30.
What is 5 day 95% VaR?
It is approximately true that:
The portfolio is assumed to be equivalent to an investment of 120 in X and of 600 in Y.
Assuming that the daily volatility of X is 2% and of Y is 1% (and assuming the correlation is 0.3), the standard deviation of
is:
Since N(-1.65) = Φ(-1.65) = 0.05, thus 5-day 95% VaR is:
∆𝑃 = 120 × 1 × ∆𝑥1 + 30 × 20 × ∆𝑥2
= 120∆𝑥1 + 600∆𝑥2
1.65 × 5 × 7.099 = 26.19
120 × 0.02 2 + 600 × 0.01 2 + 2 × 120 × 0.02 × 600 × 0.01 × 0.3 = 7.099
Value at Risk VaR of Option Portfolio
positive gamma
When gamma is positive probability distribution of the value of the portfolio is positively skewed.
long call
Tend to have less heavy left tail than the
normal distribution.
If the distribution is assumed to be normal,
then the VaR is overestimated
When options are included in portfolio linear model is approximation.
Gamma of the portfolio should be taken into account as well.
Value at Risk VaR of Option Portfolio
negative gamma
When gamma is positive probability distribution of the value of the portfolio is positively skewed.
When gamma is negative, it is negative skewed.
short call
Tend to have heavier left tail than the
normal distribution
If the distribution is assumed to be normal,
then the VaR is underestimated.
When options are included in portfolio linear model is approximation.
Gamma of the portfolio should be taken into account as well.
Value at Risk VaR of Option Portfolio
Thus both delta and gamma should be used to calculate the change in the value of portfolio.
In portfolio depended on single asset:
Using earlier formulas:
In portfolios with n underlying market variables, with each individual instrument depended on one market variable (7.18)
becomes:
∆𝑃 = 𝑆δΔ𝑥 +1
2𝑆2𝛾 ∆𝑥 2
∆𝑃 =
𝑖=1
𝑛
𝑆𝑖𝛿𝑖∆𝑥𝑖 +
𝑖=1
𝑛1
2𝑆𝑖2𝛾𝑖 ∆𝑥𝑖
2
∆𝑃 = δΔ𝑆 +1
2𝛾 ∆𝑆 2
Value at Risk VaR of Option Portfolio
VaR
Calculating VaR:
Historical Simulation;
Model Building Approach
VaR of Option Portfolio
Value at Risk
EXERCISE
Derivatives and Risk VaR Marta Wisniewska
Derivatives and Risk VaR Marta Wisniewska
8. American Options
& Dividends
American Options
Dividends
American Options and Dividends
Example: if we buy an American Call option with one year to expiry,
we can pay the strike price for the unit of the underlying at any time in the next year.
timeTt
today expiry
European Option
American Option
The right to early exercise is an advantage – the value of an American Option must be at least as high as a European option with
otherwise the same characteristics.
Value of American option can never fall below the current pay-off.
Example: if the strike is 100, and the current price of underlying is 70, the price of the put option must be at least 30.
Imagine that this condition is not met. Say the price of the option is 25.
You would buy the option for 25, and immediately exercise it, (short) selling the underlying for 100.
You would then buy it back for the current price of 70.
Your net (riskless) profit from your brief ownership of the option would be 5.
In symbols, this constraint is:
Note that there is also a constraint that the option value cannot be negative.
Why?
𝑉 ≥ 𝑚𝑎𝑥 𝐾 − 𝑆𝑡 , 0
when to exercise?
American Options are contracts that may be exercised before expiry (“Early Exercise”), whereas
European only on the expiry date
American Options
American Options
American Options and Dividends
There is a disadvantage:
the holder of an American Option needs to decide WHEN to Exercise;
this is not an easy decision.
0
20
40
60
80
100
120
140
160
0 100 200 300 400
The price of the underlying is 58.
If you exercise now (i.e. on day 189), your payoff is 42.
Do you exercise now, or do you wait?
0
20
40
60
80
100
120
140
160
0 100 200 300 400
It would have been better to wait until day 327, when the
price was 40, so payoff would have been 60.
But how were you to know this?
You are likely to formulate a rule:
As soon the price reaches S*, exercise the option.
S* will be called the optimal exercise point.
Example: American Put Option; Strike = 100; time to expiry 1 year. After 189 days, you are here:
American Options
American Options
American Options and Dividends
100
110
90
120
70
90
110
130
80
100
t=0.33t=0 t=0.67 t=1A= 5 * exp(-0.06*0.33)= 4.9
B = 20 * exp(-0.06*0.33)= 19.607
C= 2.45 * exp(-0.06*0.33)= 2.4019
D= 12.45 * exp(-0.06*0.33)= 12.265
E= 7.334 * exp(-0.06*0.33)= 7.225
We now assume that the option expires at time T, so the time to expiry is τ = T - t.
The easiest way to analyse this problem is in the context of the binomial model.
Example:
Let’s consider an American put option with time to expiry 1 year, and a strike of 100.
The current price of the underlying is 100.
Let us divide the time to expiry into three four-month intervals.
Assume that in any interval, the price can either rise by 10 or fall by 10 with equal probability.
The risk-free rate is 0.06 (continuously compounded).
B20
0A
30
10
0
0
00
At each node in the tree, we compare the
pay-off from exercising, with the
discounted expected pay-off from holding
on to the option.
Whenever the former exceeds the latter,
early exercise is rational.
C0
D
10
0
E
American Options with Expiry
American Options
American Options
American Options and Dividends
American Call Options
In the absence of dividends, early exercise is never rational on an American call option..
Proof:
You purchase an American call option with strike K and one year to expiry.
At some point in the next year (day t say), if the price of the underlying (St) is sufficiently far above K, you might consider
early exercise, pocketing the pay-off of St - K .
Instead of early exercise, you could do the following:
Hold on to the option, and short-sell one unit of the underlying, receiving an amount St at time t.
At expiry (T), either:
a. If ST < K : Buy the short-sold unit back at price ST.
Let the option die.
b. If ST < K : Exercise the option.
That is, pay K for a unit of the underlying.
Under (a), you receive St at t, and then lose an amount less than K at T.
Under (b), you receive St at t, and then lose an amount K at T.
Either way, this is better than exercising at t, with the pay-off St - K .
Hence, the value of an American call option is the same as the value of a European call option with the same strike and expiry date.
No such reasoning can be applied to put options. American puts are ceteris paribus more valuable than European puts.
American Options with Expiry
American Options
American Options
American Options and Dividends
£ 96
£ 100
Up until now, we have been assuming that no dividends are paid on the underlying stock.
Let’s relax this assumption.
A dividend is paid to the holder of a stock on a particular date – let us call this the dividend date.
[In fact, for tax reasons, the amount by which the share value falls is slightly less than the amount of the dividend, but let us
ignore this complication.]
Example: if a share price is £100 immediately before the dividend is paid, and the dividend is £4,
we will assume that the share price will be
Immediately after the dividend date, ceteris paribus, the value of the stock will fall by an amount equal to the dividend payment
£96 on the day after the dividend date.
time
Didivend: £4
dividend date
Dividends
Dividends
American Options and Dividends
100
110
90
115
75
95
t = 0 t = 0.5 t=1 A= 15 * exp(-0.06*0.5)= 14.70592
B= 2.5 * exp(-0.06*0.5)= 2.450987
C= 8.57 * exp(-0.06*0.5)= 8.3167
Example: Binomial Model with 2 periods.
Consider a European call option with time to expiry one year, and strike price 90.
The current price of the underlying is 100.
Divide the time to expiry into two six-month intervals, and assume that in each interval, the price can rise by 10 or fall
by 10 with equal probability.
Further assume that a dividend of 5 is paid at dividend date five months into the life of the option.
Find the value of the option.
C
A
B
0
5
25
85
105
↓
↓The higher the dividend, the lower
the value of the call option.
Dividends
Dividends
American Options and Dividends
100
110
90
115
75
95
t = 0 t = 0.5 t=1
Example: Binomial Model with 2 periods.
Find the value of the put option, ceteris paribus.
B
0
A
15
0
0
85
105
↓
↓
The higher the dividend, the
higher the value of the put option.
A= 7.5 * exp(-0.06*0.5)= 7.35296
B=3.676* exp(-0.06*0.5)= 3.604323
Dividends
Dividends
American Options and Dividends
Case 1: Given dividends and dividend dates
If the amounts of the dividends and the dividend dates are given, a simple adjustment needs to be made:
Compute the present value of the dividend payments, discounted using the risk-free rate.
Then simply subtract this from the current stock price St.
Then apply the Black-Scholes formula in the usual way with this downward-adjusted stock price in place of St.
Example: if the current stock price is £100,
dividends of £2 will be paid after 3 months and 9 months,
the option expires in 12 months, and the risk-free rate is 0.08,
then the present value of the two dividend payments is:
We then subtract this amount from the current stock price:
We then use the Black-Scholes formula with 96.16 as the current price in place of 100.
2.0 exp −0.08 × 0.25 + 2.0 exp −0.08 × 0.75 = 3.84
100 − 3.84 = 96.16
Dividends in the Black-Scholes formula
Dividends
Dividends
American Options and Dividends
Case 2: Dividend given as a dividend rate
Sometimes, the dividend is given as an annual dividend rate.
The stock used in the example above paid dividends of £2 every six months, and therefore £4 each year.
Since the current stock price is £100, the dividend rate in this example is 4% (or 0.04).
Let the dividend rate be δ.
The parameter δ enters the Black-Scholes formula in the following way.
As usual, we need to define the two quantities:
The formula for the value of a Call Option is:
And the formula for the value of a Put Option is:
𝑐𝑡 = 𝑆𝑡exp(−𝛿𝜏)Φ 𝑑1 − exp(−𝑟𝜏)𝐾Φ 𝑑2
𝑝𝑡 = exp −𝑟𝜏 𝐾Φ −𝑑2 − 𝑆𝑡exp(−𝛿𝜏)Φ −𝑑1
𝑑1 =𝑙𝑛
𝑆𝑡𝐾
+ 𝑟 − 𝛿 +𝜎2
2𝜏
𝜎 𝜏
𝑑2 = 𝑑1 − 𝜎 𝜏
Dividends in the Black-Scholes formula
Dividends
Dividends
American Options and Dividends
American Options
Dividends
American Options and Dividends
EXERCISE
Derivatives and Risk Management American Options… Marta Wisniewska
Derivatives and Risk Management American Options… Marta Wisniewska
9. Real Options
Test
Lecture 5:
9. Real Options
Real Options: Introduction
Market Price of Risk
Types of Real Options
Valuation of Real Options: Examples
Option Pricing in Equity Valuation
Real Options
FINANCIAL Options
OPTIONS
REAL Options
Real Options Introduction
The right, but not the obligation to undertake certain business activities, such as deferring, abandoning, expanding capital investment.
Represents decision or choices to be made during life of investment project
Deals with capital budgeting or resource allocation decision
Underlying can be illiquid, hard to trade or traded on inefficient market
The right, but not the obligation, to buy or sell underlying asset.
Underlying traded on liquid market
Real Options Introduction
Real Options Introduction
Real Options
NPV
Risk-free rate
Risk-neutral probabilities
Requires market price of risk (λ) for stochastic variables
Real Options
Valuation of potential capital investment project
NPV of a project = PV of expected future incremental cash flows
CF: Real life probabilities Risk-adjusted discount rate (from e.g. CAMP)
NPV > 0: undertake the project It creates value to the shareholders
Projects usually have options build within them Different risk than the project Different discount rate needed
𝑁𝑃𝑉 = −𝐶𝐹0 +𝐶𝐹11 + 𝑟
+⋯+𝐶𝐹𝑛
1 + 𝑟 𝑛
λ =𝜇 − 𝑟𝑓
𝜎
𝑝 =𝑟𝑓 − 𝐷
𝑈 − 𝐷
Underestimates the value of the
project with embedded option
Premium should be
paid on project
with embedded option (vs
NPV)
Introduction
A bad investment…
today
+100
-120
100*0.5 + (-120)*0.5
= -10
Real Options Introduction
A bad investment… becomes a good one
today
+100
-120
-20
+20
+80
-70
30
15
Learn at the 1st
stage
The keys to real option value come from
learning and adaptive behaviour.
NPV: expected cashflows from today’s point without considering
other pathways given what happens in the first year, etc.
Real Options Introduction
NPV
Real Options
Can the valuation be the same?
If you modified decision tree analysis to:
Estimate risk-neutral probabilities to estimate an expected value, Adjust the expected value for the market risk in the investment and Use the riskfree rate to discount cashflows in each branch
… it could yield the same values as option pricing models
Real Options Introduction
Suppose that real asset depends on several variables θi (i=1,2,..), where:
λi is the market price of risk of θi
Risk-neutral valuation: any asset dependent on θi can be valued by:
Reducing the expected growth rate of each θi from mi to mi - λisi.
Discounting cash-flows at the risk free rate
𝑑𝜃𝑖𝜃𝑖
= 𝑚𝑖𝑑𝑡 + 𝑠𝑖𝑑𝑊
𝑑1 =𝑙𝑛
𝐸(𝑉)𝐾
+𝜔2
2𝜏
𝜔 𝜏
𝐸 𝑚𝑎𝑥 𝑉 − 𝐾, 0 = 𝐸 𝑉 𝑁 𝑑1 − 𝐾𝑁 𝑑2
Assume V has lognormal distribution
𝑑2 =𝑙𝑛
𝐸(𝑉)𝐾
−𝜔2
2𝜏
𝜔 𝜏
where ω is the volatility of V
Real Options Market Price of Risk
Example: Current cost of renting 1m2 is £30. Cost is quoted as amount per 1m2 per year in 5-year
rental agreement. The expected growth rate in the cost is 12% pa, volatility 20% pa, market price of
risk 0.3. What is the value of an opportunity to pay £1m now for option to rent 100 000m2 at £35
for 5 years in 2 years time? Assume 5% risk-free rate pa.
Let V be cost per 1m2 in 2 years time.
The pay off from the option is: 100 000 × 𝐴 ×𝑚𝑎𝑥(𝑉 − 35, 0) call
The expected pay off in risk neutral world:
A: annuity factor4.5355
Expectations in risk neutral world
100 000 × 4.5355 × 𝐸 𝑚𝑎𝑥(𝑉 − 35, 0)
= 453 550 × 𝐸 𝑚𝑎𝑥(𝑉 − 35, 0)
= 453 550 × 𝐸 𝑉 𝑁 𝑑1 − 35𝑁 𝑑2
𝑑1 =
𝑙𝑛 𝐸(𝑉)35
+0.22
2 2
0.2 2
𝐸 𝑉 = 30 exp((𝑚𝑖 − 𝜆𝑖𝑠𝑖) ∗ 2)
= 30 exp((0.12 − (0.3 ∗ 0.2)) ∗ 2)
= 30 exp(0.06 ∗ 2)
= 33.83
= 1 501 500
Value of the option:
1 501 500 ∗ exp(−0.05 ∗ 2) = 1 358 600
The opportunity is worth: 1 358 600 − 1 000 000 = 358 600
Real Options Market Price of Risk
When historical data are available market price of risk can be estimated using:
λ: market price of risk
ρ: instantaneous correlation between the percentage change in the variable and returns on stock
market index
σm: volatility of return on stock market index
μm: expected return on stock market index
rf: short term risk-free rate
λ =𝜌
𝜎𝑚𝜇𝑚 − 𝑟𝑓
Example:
Percentage changes in company’s sale have a correlation of 0.3 with returns on FTSE100 index. The
volatility of FTSE100 returns is 20%pa, the expected excess returns of FTSE100 over risk-free rate
is 5%.
The market price of risk is:
λ =0.3
0.2∗ 0.05 = 0.075
Real Options Market Price of Risk
Types of Real Option
Most investment projects involve options. Those options can add substantial value to the project.
Examples of options embedded in the project:
Option to Abandon
Option to sell or close down a project.
It is an American put option on the project’s value with the strike price being the
liquidation (or resale) value less closing- down costs.
It mitigates impact of poor investment performance.
Option to Expand
Option to make further investments if conditions are favourable.
It is an American call option on the value of additional capacity. The strike price is the cost
of creating this additional capacity discounted to the time of option exercise.
The strike price depends on initial investment.
Option to Wait/ Delay
This is an American call option on the value of the project.
R&D
Oil exploration
Patent
Project can include more than one option.
Real Options Types of Real Options
Black-Scholes
OPTION Valuation Models
Binomial Model
Can price American Option
Majority of real options are exercised before maturity (early exercised)
Often underlying assets are discontinuous
Binomial tree with outcomes at each node looks like a decision tree from capital budgeting.
For European option without dividend
Can be adjusted for dividend
What about American option?
American Call Option will never be exercised prior maturity…
Still getting the inputs to a binomial model can be difficult…
Real Options Option Valuation
Valuing product patent as an option
A patent provides the company with the right to develop and market the product.
The product will be developed and marketed only if the present value of the expected cash flows
from the product sales (V) exceed the cost of development (I).
If this does not occur, the patent will not be used and non further cost will be incurred.
The payoffs from owning a product patent can be written as:
Max ( 0, V – I )
.
VI
Pay-off
Real Options Option Valuation: Examples
Input Estimation Process
St: Value of the UnderlyingAsset
PV of Cash Flows from taking the project now
σ2: Variance in value of underlying
Variance in CF of similar asset or firm Variance in PV from capital budgeting
simulation
K: Exercise Price on Option Cost of making investment in the project
τ: Expiration of the Option Life of the patent
δ: Dividend Yield Cost of delay Each year of delay means less years of CF
𝑐𝑡 = 𝑆𝑡exp(−𝛿𝜏)Φ 𝑑1 − exp(−𝑟𝜏)𝐾Φ 𝑑2𝑑1 =
𝑙𝑛𝑆𝑡𝐾 + 𝑟 − 𝛿 +
𝜎2
2 𝜏
𝜎 𝜏
𝑑2 = 𝑑1 − 𝜎 𝜏
Real Options Option Valuation: Examples
Example:
Company X, a bio-technology company, has a patent on ABC, a drug to treat multiple sclerosis, for the
next 17 years. X plans to produce and sell the drug by itself.
The key inputs on the drug are as follows:
St: PV of Cash Flows from Introducing the Drug Now = £3.422 billion
K: PV of Cost of Developing Drug for Commercial Use = £2.875 billion
τ: Patent Life = 17 years
r: Riskless Rate = 6.7% (17-year T.Bond rate)
σ2: Variance in Expected Present Values = 0.224 (Industry average firm variance for bio-tech
companies)
δ: Expected Cost of Delay = 1/17 = 5.89%
Implementing BS model:
d1 = 1.1362 N(d1) = 0.8720
d2 = -0.8512 N(d2) = 0.2076
Call = 3,422*exp(-0.0589*17)*(0.8720) - 2,875*(exp(-0.067*17)*(0.2076)
= £907 million
𝑐𝑡 = 𝑆𝑡exp(−𝛿𝜏)Φ 𝑑1 − exp(−𝑟𝜏)𝐾Φ 𝑑2
NPV of this project:
= 3422 – 2875
= £547 million
Real Options Option Valuation: Examples
Valuing natural resources
The underlying asset is the resource and the value of the asset is based upon two variables: (1) the
quantity and (2) the price of the resource.
Usually there is a cost associated with developing the resource, and the difference between the
value of the asset extracted and the cost of the development is the profit to the owner of the
resource.
Define the cost of development as X, and the estimated value of the resource as V.
The payoffs from a natural resource option can be written as:
Max ( 0, V – X )
VX
Pay-off
Real Options Option Valuation: Examples
Input Estimation Process
St: Value of Available Reserves of the Resource
Expert estimates (Geologist) PV of cash flows from the recourse
σ2: Variance in value of underlying
Based on variability of the price of the resource and variability of available reserves
K: Cost of Developing Reserve Past costs and the specifics of the investment
τ: Time to Expiration Relinquishment Period Time to exhaust inventory
δ: Net Production Revenue
(Dividend Yield) Net production revenue every year as per cent of
market value
Development lag Calculate PV of reserve based upon the lag
𝑐𝑡 = 𝑆𝑡exp(−𝛿𝜏)Φ 𝑑1 − exp(−𝑟𝜏)𝐾Φ 𝑑2𝑑1 =
𝑙𝑛𝑆𝑡𝐾 + 𝑟 − 𝛿 +
𝜎2
2 𝜏
𝜎 𝜏
𝑑2 = 𝑑1 − 𝜎 𝜏
Uncertainty about:Price, Quantity, Costs
Real Options Option Valuation: Examples
Example:
Consider oil reserve of 50 million barrels, with PV of the development cost $12 per barrel and the
development lag 2 years. Company X has the right to exploit this reserve for the next 20 years. The
marginal value per barrel of oil is $12. Once developed, the net production revenue each year will be
5% of the value of the reserves.
The key inputs on the drug are as follows:
St: Value of developed reserve discounted back the length of development lag at the
dividend yield = $12*50/(1.05)^2= $ 544.22 million
K: PV of development Costs= $12 * 50 = $ 600 million
τ: Time to expiration of the option = 20 years
r: Riskless Rate = 8%
σ2: Variance in ln(oil prices) = 0.03
δ: Dividend yield= Net production revenue/Value of reserve = 5%
Implementing BS model:
d1 = 1.0359 N(d1) = 0.8498
d2 = 0.2613 N(d2) = 0.6030
Call = 544 .22 exp(-0.05*20) (0.8498) -600 (exp(-0.08*20) (0.6030))
= $97.08 million
𝑐𝑡 = 𝑆𝑡exp(−𝛿𝜏)Φ 𝑑1 − exp(−𝑟𝜏)𝐾Φ 𝑑2
NPV of this project:
= 544.22 – 600
= - $55.78 million
Real Options Option Valuation: Examples
Option Pricing in Equity Valuation
Equity in a troubled company
Company with high leverage, negative earnings and a significant chance of
bankruptcy
Equity can be viewed as a call option (option to liquidate the company).
Natural resource companies
The undeveloped reserves can be viewed as options on the natural resource.
Start-ups or high growth companies
Companies which derive the bulk of their value from the rights to a product or a
service (eg. a patent)
In late 90s dot.com companies were valued as options to enter e-commerce market
Huge premiums
One could invest in Nokia or GE to enter the same market (lack of exclusivity)
Real Options Option Pricing in Equity Valuation
Real Options: Introduction
Market Price of Risk
Types of Real Options
Valuation of Real Options: Examples
Option Pricing in Equity Valuation
Real Options
TEST
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