Chapter 1 Assets and Financial Derivatives An … 1 Assets and Financial Derivatives An Introduction I-Liang Chern September 7, 2016 I-Liang Chern Chapter 1 Assets and Financial DerivativesAn

Chapter 1Assets and Financial Derivatives

An Introduction

I-Liang Chern

September 7, 2016

I-Liang Chern Chapter 1 Assets and Financial DerivativesAn IntroductionSeptember 7, 2016 1 / 55

Outline

Assets: bonds, stocks, derivatives

Financial derivatives: forward contracts, futures, options

Hedge risk and speculation

Goal of this course: how to price a financial derivatives?


References

I-Liang Chern, Financial Mathematics, 2000

Y-L Zhu, X Wu and I-L Chern, Derivative Securities andDifference Methods, 2004.

Raymond Chan’s Lecture Note on Financial Mathematics

John Hull, Options, Futures, and Other Derivative Securities

P. Wilmott, S. Howison and J. Dewynne, The mathematicsof Financial Derivatives, A Student Introduction.

Wiki

Investopedia


AssetsAssets represent value of ownership that can be converted intocash.

Tangible assets:I Commodity (oil, electricity, wheat, gold )I Foreign currencyI House, building, equipments

Intangible assertsI Copyrights, trademarks, patterns, computer programsI Financial asserts

F Bank depositesF Debt instrument: bonds, mortgages, loansF Equity: stocksF Derivatives: futures, options


Asset price, markets

Asset prices are settled by trading

Many assets are traded in markets

There are commodity markets, currency markets, stockmarkets, debt markets, ...


Commodity Markets

Commodity markets, where physical assets such as oil,gold, copper, wheat or electricity are traded.

The prices of these products are unpredictable but oftenshow seasonal effects. Scarcity of the product results inhigher price.

Commodities are usually traded by people who have noneed of the raw material. For example, they may just bespeculating on the direction of gold.

Most trading is done on the futures market, making deals tobuy or sell the commodity at some time in the future. Thedeal is then closed out before the commodity is due to bedelivered.


Currency Markets

Currency markets or foreign exchange markets, wherecurrencies are bought and sold. What matters in such adealing is the exchange rate.

The fluctuation in exchange rates is unpredictable, but thereis a link between exchange rates and the interest rates inthe two countries.

For example, if the interest rate on dollars is higher than theinterest rate on pounds sterling, we would expect to seesterling depreciating against the dollars. Central banks canuse interest rates as a tool for manipulating exchange rates,but only to a degree.







Debt instrumentDebt instruments are issued by anyone who borrows money— firms, governments, and households.They include

I Corporate bondsI Government bondsI Residential and commercial mortgagesI Consumer loans

Bond is a kind of the fixed-income security. It reflects a“promise” by a borrower, the seller of the bond, to repay theamount borrowed at a specific time, plus interest at anagreed-upon rate.These debt instruments are also called fixed-incomeinstruments.


Bonds

US treasury bonds (T-bonds)I Discount bonds: pay the owner only the face value of the

bond at the time of maturity.I Coupon bonds: pay the face value at maturity, and they also

generate fixed, periodic payments known as coupons overthe lifetime of the bond. Eg. every 0.5 or 1 year. This is thecoupon and would often be a fixed rate of interest. At theend of your fixed term you get a final coupon and the returnof the principal, the amount on which the interest wascalculated. http://www.savingsbonds.gov/instit/auctfund/work/work.htm


http://www.savingsbonds.gov/instit/auctfund/work/work.htm

http://www.savingsbonds.gov/instit/auctfund/work/work.htm

Example of T-bondOn Jan 5, 2007, the 6-month T-bill with par value US$1,000 maturing at July 5, 2007 wasquoted on Bloomberg.com as US$975.5. The half-yearly interest rate on that day wastherefore 24.5/975.5 ≈ 2.51%.

The 30-year T-bond issued on Jan 4, 2006 at par value US$1,000 has coupons of US$45each year for 30 years. That means the buyer of this bond will receive US$45 on Jan 4every year for 30 years and on Jan 4, 2036, he will receive US$1,045. The 30-yearfixed-rate interest rate is therefore equal to 4.5% per annum.

Note that since interest rate fluctuates, bond prices also fluctuate accordingly. The30-year T-bond issued on Jan 4, 2006 was quoted as US$962.19 on Jan 5, 2007.

What did it say about the interest rate on that day? It means that the 29-year fixed-rateinterest rate should be like 45/962.19 ≈ 4.68% per annum. Seewww.bloomberg.com/markets/rates/index.html.


www.bloomberg.com/markets/rates/index.html

Bond Markets

The bond market is the channel through whichgovernments and corporations that need to borrow moneyare matched with investors who have funds to lend. Such amarket provides monetary liquidity that is vital to aneconomy’s health.

Bond dealers at security firms act as intermediaries, buyingfrom issuers and selling to buyers. This activity forms theprimary market for bonds.

Bond dealers also maintain an active secondary market inbonds, bidding for bonds, already issued, that investorswish to sell.


Dealers also offer bonds, already issued, from theirinventory to investors who wish to buy. It is possible to sellbonds short (to be explained later), just as in the case ofstock.

Bonds are bought by institutions such as insurancecompanies, mutual funds, pension funds, and financialinstitutions as well as by individual investors. They are alsoactively traded, and the average holding period for the30-year Treasury bonds (bonds issued by the USgovernment) is two weeks.


Market value of bondsOnce a bond is issued, its secondary market price goes up or down, partly in response tothe general level of interest rates.

Interest rate changes in response to a number of factors: changes in demand and supplyof credit, fiscal policy, Federal Reserve policy, exchange rates, economic conditions, and,most important for the bond market, changes in perceptions regarding inflation.

By inflation, we mean a persistent rise in prices of basic goods. This price pattern lowersthe value of bonds as it reduces the future buying power of the fixed payments from thebond.

The bond market often reacts negatively to positive economic news because of the fear ofrising inflation. Conversely, negative news, such as higher unemployment, reducesinflationary concerns and tends to raise bond prices.

When interest rates go up, the price of a bond goes down because its coupon ratebecomes less desirable than the rates of newly issued bonds of similar quality. If interestrates fall, the bond’s coupon rate becomes more attractive to investors, which drives upthe price.







Equity, Share and Stock

If you have a bright idea for a new product or service, thenyou could raise capital to realize this idea. The investors inthe company give you some cash, and in return you givethem a contract stating how much of the company they own.Usually these investors are called shareholders.

Equity is the claim of the ownership of a firm. Equitysecurities issued by corporations are called common stocksor shares.

When someone buys a share, he is buying ownership ofpart of a company and becomes a shareholder in thatcompany.


Stock Market

Primary market: When a stock is issued initially in public(initial public offerings)

Secondary market: subsequent trading after IPO.

Market organization

Stock exchanges: Euronext, New York Exchange, Nasdaq,Hong Kong Exchange

Over-the-Counter market: where buy/sell contracts aremade between the two counterparties.


Types of StocksOn the basis of ownership

Common stocks: own part of the company, has voting right,no dividend, lower priority when the company is distributingsurplus money.

Preferred stocks: dividend, no voting right, enjoy greaterpriority when the company is distributing surplus money.


Stock split

It is customary for shares in the US to have prices between$10 and $100. A company whose share price rises abovethis range will usually issue new shares to bring it back.This is called a stock split. That is, the existing shares are“split” into more shares.

For example, in a 3-for-1 stock split, three new shares areissued to replace each existing share.


Stock dividend

Stock dividend: a company issuing more shares to itsexisting shareholders. For example, a 20% stock dividendmeans that investors receive one new share for each fivealready owned. The 20% stock dividend referred to isessentially the same as a 6-for-5 stock split. Thus, a stockdividend has no effect on either the assets or the earningpower of a company.


Cash dividend

Cash dividend: These are payments to shareholders out ofthe profits made by the company concerned. Therefore, itreflects in today’s share price. Suppose that our asset paysa dividend of amount D at time t = td. Suppose the dividendyield, which is given by d = D/S, is a constant. Then theshare price after paying dividend is

S(td+) = (1− d)S(td−).







Derivatives-I

Derivatives (derivative securities or contingent claims) arefinancial instruments that derive their values from the pricesof one or more other assets such as equity securities,fixed-income securities, foreign currencies, or commodities.

This includesI Forwards contractsI FuturesI OptionsI Swaps, credit derivatives, ...

Their main function is to serve as tools for managingexposures to the risks associated with the underlyingassets.


Derivatives-II

The person or firm who formulates this contract and offers itfor sale is termed the writer, or in short (sell) position.

The person or firm who purchases the contract is termedthe holder, or in long position.

The stock that the contract is based on is termed theunderlying equity.


Derivatives Markets – I (wiki)

Exchanges: Futures, OptionsI Examples: Euronext, Chicago Mercantile Exchange, ...I For each contract, there are two parties, long one (who buy)

and short one (who sell).I The sum of all the long positions must be equal to the sum of

all the short positions. In other words, risk is transferred fromone party to another.

I The notional positions was $81 trillion by the end of March2008.


Derivatives Markets - II (wiki)

Over-the-counter: forward contracts, swaps, creditderivatives, ...

I These consist of investment banks with traders who makemarkets in these derivatives, and clients such as hedgefunds, commercial banks, government-sponsoredenterprises, etc.

I The total notional amount of all the outstanding positionswas $615 trillion in 2009.


Why financial derivatives?One is said to hedge a risk when reducing one’s exposure to aloss entails giving up the possibility of a gain. For example,farmers who sell their future crops at a fixed price in order toeliminate the risk of a low price at harvest time give up thepossibility of profiting from higher prices at harvest time.


Who are the participants in the derivativesmarkets

Hedgers:

Speculators:

Arbitrageurs


Forward ContractsA forward contract is an agreement which allows the holder ofthe contract to buy or sell a certain asset at or by a certain dayat a certain price. Here,

the certain day—maturity or expiration date,

the certain price—delivery price

the person who write the contract (has the asset) is calledin short position

the person who holds the contract is called in long position.

It is traded OTC.


Examples of Forward Contract-I

Example 1. Suppose today’s price of corn is US$3 perbushel. You want to buy 5,000 bushels of corn for deliverytwo months later.

You can sign an agreement with someone who is willing tosell you this amount at price, say US$3.05, at such futuretime. Then you make your uncertain risk more certain.


Examples of Forward Contract-IIExample 2. Suppose it is May 30 now and you need to payyour tuition to Oxford University £1,000 pound by Sept. 1.You can earn HK$15,600 in the summer. The exchangerate now is £1.00 = HK$15.40. What should you do?1. Try your luck: If the exchange rate is less than HK$15.60by August 31, you earn some extra cash besides theregistration fee. If the exchange rate is higher thanHK$15.60, you ... ?2. Buy a forward on British pounds: Look for a forwardcontract on British pounds that entitles you to useHK$15,600 to buy £1,000 on August 31. Then you haveeliminated, or hedged, your risk.Forward contracts eliminate your risk.


Examples of Forward Contract-IIIExample 3. Suppose it is May 31, and the Chief FinancialOfficer (CFO) of a corporation knows that the corporationwill receive £1 million in three months (on August 31) andwants to hedge against exchange rate moves. The CFOcan contact a bank, find out that the exchange rate for athree-month forward contract on pounds sterling is, say15.600, and agree to sell £1 million. The corporation thenhas a short forward contract on sterling, i.e. it has agreedthat on August 31, it will sell £1 million to the bank forHK$15.6 million. The bank has a long forward contract onsterling, i.e. it has agreed that on August 31 it will buy £1million for HK$15.6 million. Both sides have made a bindingcommitment.


FuturesA futures contract is very similar to a forward contract. Futurescontracts are usually traded through an exchange, orclearinghouse, which standardizes the terms of the contracts.The exchange helps to eliminate the risk of default of eitherparty through a margin account. Each party has to pay an initialmargin as deposit at the inception of the contract. The profit orloss from the futures position is calculated every day and thechange in this value is paid from one party to the other. Themaintenance margin is the minimum level below which theinvestor is required to deposit additional margin.


Example of Future ContractMr. Chan takes a long (buy) position of one contract in corn (5,000 bushels) for Marchdelivery at a price of US$3.682 per bushel at the Chicago Board of Trade (CBOT). Seequotation at http://www.cbot.com/.CBOT

It requires maintenance margin of US$700 with an initial margin markup of 135%, i.e. theinitial margin is US$945 which Mr. Chan and the seller each has to deposit into thebroker’s account on the first day they enter the contract.

The next day the price of this contract drops to US$3.652. This represents a loss ofUS$0.03 × 5,000 = US$150. The broker will take this amount from Mr. Chan’s marginaccount and deposit it to the seller’s margin account. It leaves Mr. Chan with a balance ofUS$795.

The following day the price drops again to US$3.552. This represents an additional lossof US$500, which is again deducted from the margin account. As this point the marginaccount is US$295, which is below the maintenance level. The broker calls Mr. Chan andtells him that he must deposit at least US$405 in his margin account, or his position will beclosed out, i.e. both sides agree to settle the contract at this point. Once the position isclosed, Mr. Chan will not be able to earn back any money in the future even if the pricerises above US$3.682.


http://www.cbot.com/.CBOT

Forward Contract and Future Contract

Forward Contract Future Contractnature customized contract standarized contractTrading over the counter through exchangesliquidity less liquid highly liquidcounter party risk high negligibleSettlement delivery (at the end) closed out prior to maturityMargin no margin compulsorily needs to be paid by the parties


Payoff of forward contract or futures

Long position (who holds): Λ = ST − K

Short position (who sells): Λ = K − ST

where

K = delivery price

ST = assert price at maturity


K

ST

Λ

future (long)

(a) left

K

ST

Λ

future (short)

(b) right

Figure: Payoff of a future, long position (left) and short position (right)


Options

There are two kinds of options — call and put.

A call (put) option is a contract between two parties, inwhich the holder has the right to buy (sell) and the writerhas the obligation to sell (buy) an asset at certain time inthe future at a certain price.

The holder is called in long position, while the writer iscalled in short position.

The underlying assets of an option can be commodity,stocks, stock indices, foreign currencies, or future contracts.


Exercise feature of Options

European options : Options can only be exercised at thematurity date.

American options : Options can be exercised any time up tothe maturity date.


Example of Options-INotation

t current time

T maturity date

S current asset price

ST asset price at time T

E strike price

c premium, the price of call option

r bank interest rate

σ volativity


Example of Options-IIAn investor buys 100 European call options on XYZ stock with strike price:

E = 140, St = 138

T = 2 months, c = 5 (the price of one call option).

If at time T, ST > E, then he should exercise this option. The payoff is100× (ST − E) = 100× (146− 140) = 600. The premium is 5× 100 = 500. Hence, heearns $100.

If ST ≤ T, then he should not exercise his call contracts. The payoff is 0.

The payoff function for a call option is Λ = max{ST − E, 0}. One needs to pay premium(ct) to buy the options. Thus the net profit from buying this call is

Λ− cter(T−t).


Example of Options-IIISuppose a call option is

today is t = 03/09/2016,

expiration is T = 31/12/2016 ,

the strike price E = 250

for some stock. If ST = 270 at expiration, which is smaller than the strike price, we shouldexercise this call option, then buy the share for 250, and sell it in the market immediately for 270.The payoff Λ = 270− 250 = 20. If ST = 230, we should give up our option, and the payoff is 0.Suppose the share take 230 or 270 with equal probability. Then the expected profit is

12× 0 +

12× 20 = 10.

Ignoring the interest of bank, then a reasonable price for this call option should be 10. IfST = 270, then the net profit= 20− 10 = 10. This means that the profits is 100% (He paid 10 forthe option). If ST = 230 the loss is 10 for the premium. The loss is also 100%. On the other hand,if the investor had instead purchased the share for 250 at t, then the corresponding profit or lossat T is ±20. Which is only ±8% of the original investment. Thus, option is of high risk and withhigh return.


Payoff of call options

Long call: Λ = max{ST − E, 0}I If ST > E, then the holder exercise the call at price E then sell

at price ST

I If ST ≤ E, then he gives up the call option.

Short call: Λ = min{E − ST , 0}I If ST > E, he has the obligation to sell the asset at price E,

thus he needs to buy the asset at price ST then sell it at priceE. He losses ST − E.

I If ST ≤ E, the holder will give up the call option. So the writerlosses nothing.


K

ST

Λ

call option (long):max{ST−K,0}

(a) left

K

ST

Λ

call option (short):−max{ST−K,0}=min{K−S

T,0}

(b) right

Figure: Payoff of a call, long position (left) and short position (right)


Payoff of put options

Long put: Λ = max{E − ST , 0}I If ST < E, the holder has the right to sell the asset at E, then

buy it back at ST . Thus the payoff is E − ST .I If ST ≥ E, the holder just gives up the option.

Short put: Λ = min{ST − E, 0}I If ST < E, he has the obligation to buy the asset at E. He

then sell it at ST . Thus he losses E − ST .I If ST ≥ E, the holder will give up the put option, So the writer

losses nothing.


K

ST

Λ

put option (long):max{K−ST,0}

(a) left

K

ST

Λ

put option (short):−max{K−ST,0}=min{S

T−K,0}

(b) right

Figure: Payoff of a put, long position (left) and short position (right)


Premium of an option-IThe person who hold an option has the right and the counterparty has the obligation to fulfill the contract. Thus, a premiumshould be paid by the holder to the writer. Below is a portion of acall option copied from the Financial Times.

the current time t = Feb 3

the expiration T = end of Feb,

T − t ≈ 10 days

St = 2872

E 2650 2700 2750 2800 2850 2900 2950 3000c 233 183 135 89 50 24 9 3


Premium of an option-II

2600 2650 2700 2750 2800 2850 2900 2950 3000 3050−50

0

50

100

150

200

250

K

c

Figure: The FT-SE index call option values versus exercise price.


Premium of an option-IIIThe premium of a call option depends on

St (current asset price)

E (strike price)

T (future time)

r (interest rate)

D (dividend)

σ (volativity)

increase in call option price ??intuitive reasonsS(t) ↑ potential payoff (S(T)− E)+ ↑E ↓ potential payoff (S(T)− E)+ ↓T ↑ more “time value”r ↑ less money upfront to pay for E, i.e. E ↓D ↓ S(t) ↓σ ↑ risk is higher








Chapter 1 Assets and Financial Derivatives An … 1 Assets and Financial Derivatives An Introduction I-Liang Chern September 7, 2016 I-Liang Chern Chapter 1 Assets and Financial DerivativesAn

Documents