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Derivatives

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QFI QF Exam 2021 PAK Study Manual Intro-Maths-Fin-1

1

Intro-Maths-Fin-1 Financial Derivatives (A Brief Introduction)

Background This chapter deals with the two basic building blocks of financial derivatives:

1. Options and

2. Forwards and futures.

We briefly introduce the third class of derivative: swap. We see how a complex swap can be decomposed into a number of forwards and options.

Definitions Derivatives securities are financial contracts that ‘derive’ their value from cash market instruments such as stocks, bonds, currencies and commodities. At the time of the maturity of the derivative contract, denoted by T, the price F(T) of the derivative asset is completely determined by the market price of the underlying asset (𝑆𝑇). For instance, the value at maturity (T) of a call option of strike (K) written on an asset (𝑆𝑇) is:

𝑴𝒂𝒙[𝑺𝑻 − 𝑲; 𝟎] “In the reading of PDE (later readings on Partial Differential Reading), the maturity payoff will also be described as

a boundary condition of the PDE”

Types of derivatives We group derivatives into three general headings:

1. Futures, Forwards, Repos, Reverse Repos and Flexible Repos (Basic building blocks)

2. Options and

3. Swaps

We let (𝑆𝑡) represent the price of the relevant cash instrument, which we call the underlying asset. We list five main groups of underlying assets:

1. Stocks (These are claims on “real” returns)

2. Currencies

3. Interest rates: Interest rate in not an asset, so we are referring to the direction of interest rates. The

assets are Treasuries, bonds.

4. Indexes (S&P 500) and

QFI QF Exam 2021 PAK Study Manual Intro-Maths-Fin-1

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5. Commodities: they are not financial assets either, they are goods in kind.

There is another method for classifying the underlying asset:

1. The cash and carry markets and

2. The price discovery markets

So:

This new classification is important to us.

In the cash and carry market, one can borrow at risk-free rates, buy and store the product, and insure it

until the expiration date of any derivative contract.

Pure cash and carry market have one property: Information about demand and supplies of the

underlying instrument should not influence the spread between cash and futures (forward) prices.

In the Pure cash and carry market, any relevant information concerning future supplies and demands of

the underlying instrument is expected to make the cash price and the future price change by the same

amount (This is not so, in the price discovery market).

In the price discovery market, it is physically impossible to buy the underlying instrument for cash and

store it until some future expiration date. That strategy (of borrowing, buying and storing) is no longer

applicable.

In the price discovery market, any information about the future supply and demand of the underlying commodity cannot influence the corresponding cash price.

Expiration Date At the expiration of the forward/futures contract, we expect:

𝐸𝑥𝑝𝑖𝑟𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑢𝑡𝑢𝑟𝑒𝑠 𝑜𝑟 𝑓𝑜𝑟𝑤𝑎𝑟𝑑 𝑐𝑜𝑛𝑡𝑟𝑎𝑐𝑡 = 𝐹(𝑇) = 𝑆𝑇

“As you read about these instruments, it will help to see how they can be used to manage risks in your insurance company. The study manual has a problem that helps along that line.”

Forward and Futures Forwards and Futures are linear instruments (while options are nonlinear instruments).

“Options are non-linear instruments because the derivative of the payoff function changes sign around the strike price. Later on, in the Wilmot readings on Jensen’s inequality, we will see that the payoff is a convex function of the

underlying asset” A long forward contract is an obligation to buy an underlying asset at a specified forward price (the strike price K) on a known date (the maturity T).

QFI QF Exam 2021 PAK Study Manual Intro-Maths-Fin-1

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The payoff diagram for a simplified long position in a forward contract of strike K is shown below:

The upward slopping line indicates the profit (when 𝑆𝑇>K) or the loss (when 𝑆𝑇<K) of the long

position.

As the maturity value of the underlying asset (the cash price) exceeds the pre-agreed strike or the

forward price (K), the holder of the forward realizes a profit (𝑆𝑇 − 𝐾).

The slope of the line is 1 (The payoff is a linear function of the underlying asset).

Futures and forwards are similar instruments. The major differences between them can be stated briefly as follows:

1. Futures are traded in organized exchanges and forwards are custom-made and traded over the counter.

2. Futures exchanges are cleared through exchanges clearing houses and there is a mechanism designed to

reduce default risk. Forwards are not cleared and there is default risk.

3. Futures contracts are market to market. Every day, the contract is settled and a new contract is entered.

Repos, Reverse Repos, and Flexible Repos1

1 You will get to read this in the Handbook of Fixed Income Securities and in the Management of Investment

Portfolio. So, here is a preliminary exposure to the concept.

QFI QF Exam 2021 PAK Study Manual Intro-Maths-Fin-1

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In a Repo (Repurchase agreement), one party sells securities to another party in return for cash, with an agreement to repurchase the securities (or equivalent) at a pre-agreed price (the repurchase price) and pre-agreed time (the maturity date). The long position in a repo (the buyer) acts as a lender of cash, and the short position (the seller) as a borrower. For the party selling the security and agreeing to repurchase it in the future, the transaction is a repo. For the party on the other end, the transaction is a reverse repo. The securities are used as a collateral in this transaction. Let:

𝑆0 = 𝑇ℎ𝑒 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑠𝑎𝑙𝑒 𝑝𝑟𝑖𝑐𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑒𝑐𝑢𝑟𝑖𝑡𝑦

𝑇 = 𝑇ℎ𝑒 𝑚𝑎𝑡𝑢𝑟𝑖𝑡𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑜𝑛𝑡𝑟𝑎𝑐𝑡

𝑋𝑇 = 𝑇ℎ𝑒 𝑎𝑔𝑟𝑒𝑒𝑑 𝑢𝑝𝑜𝑛 𝑟𝑒𝑝𝑢𝑟𝑐ℎ𝑎𝑠𝑒 𝑝𝑟𝑖𝑐𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑜𝑙𝑙𝑎𝑡𝑒𝑟𝑎𝑙

𝐹𝑇 = 𝑇ℎ𝑒 𝑓𝑜𝑟𝑤𝑎𝑟𝑑 𝑝𝑟𝑖𝑐𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑜𝑙𝑙𝑎𝑡𝑒𝑟𝑎𝑙 We have:

𝑇ℎ𝑒 𝑅𝑒𝑝𝑜 𝑅𝑎𝑡𝑒 = 𝑇ℎ𝑒 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 𝑜𝑛 𝑡ℎ𝑒 𝑟𝑒𝑝𝑜 𝑡𝑟𝑎𝑛𝑠𝑎𝑐𝑡𝑖𝑜𝑛 = 𝑋𝑇 − 𝑆0 And

𝑇ℎ𝑒 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑜𝑛 𝑡ℎ𝑒 𝑙𝑜𝑎𝑛 = 𝐹𝑇 − 𝑆0 Repos are used to raise short term capital and are classified as money market instruments. There are three broad categories of repos:

1) Overnight repos: A one day maturity repo transaction. 2) Term repos: This is a repo with a specified maturity. 3) Open repo: This repo has no end date.

A flexible repo is a repo with a flexible withdrawal schedule. Therefore, the party holding the collateral can sell it in parts before or at the maturity of the repo. There are two types of flexible repos:

1) Secured: The municipality/customer receives collateral in the form of Treasury bonds, GNMA bonds, agency MBS/CMO.

2) Unsecured: The customer does not receive collateral. The deal commands a higher spread. Differences between flexible repos and traditional repos:

1) Convexity due to cash withdrawals, 2) Formal written auction like trade, 3) Enhanced documentation, 4) Counterparties are usually municipal bond issuers.

Options

QFI QF Exam 2021 PAK Study Manual Intro-Maths-Fin-1

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Forwards and futures obligate the holder to deliver or accept the delivery of the underlying instrument at expiration. Options, on the other hand give the owner the right (not the obligation) to purchase or sell an asset. A European style call option on a security S is the right to buy the security at a preset strike price K. That right may be exercised at the expiration date of the option T.

“Be able to define a European put option as well, and the American call and put” Reasons why traders may want to calculate the arbitrage-free price of a call option

1. New contract: before the option is first issued, a trader may want to know the price that the option will trade at.

2. Mispricing: A trader may want to calculate the arbitrage free value of an option to determine if the option is mispriced in the market.

Some Notation The expiration of the option is time T. The value of the option at any time t (t<T) is:

𝐶𝑡 If there are no commissions and/or fees, and there is no bid-ask spreads on the underlying asset and the derivative, then at maturity of the option, the option value can only take two possible values:

1. The option expires in the money, in which case:

𝑆𝑇 > 𝐾, 𝑎𝑛𝑑 𝑡ℎ𝑢𝑠 𝐶𝑇 = 𝑆𝑇 − 𝐾 = 𝑇ℎ𝑒 𝑚𝑎𝑡𝑢𝑟𝑖𝑡𝑦 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑜𝑝𝑡𝑖𝑜𝑛 = 𝑇ℎ𝑒 𝑜𝑝𝑡𝑖𝑜𝑛′𝑠 𝑝𝑎𝑦𝑜𝑓𝑓 Or:

2. The option expires out the money, in which case:

𝑆𝑇 < 𝐾, 𝑎𝑛𝑑 𝑡ℎ𝑢𝑠 𝐶𝑇 = 0 = 𝑇ℎ𝑒 𝑚𝑎𝑡𝑢𝑟𝑖𝑡𝑦 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑜𝑝𝑡𝑖𝑜𝑛 = 𝑇ℎ𝑒 𝑜𝑝𝑡𝑖𝑜𝑛′𝑠 𝑝𝑎𝑦𝑜𝑓𝑓 With this, we can simply say that the maturity payoff of the long position in a call option is given by:

𝑪𝑻 = 𝑴𝒂𝒙[𝑺𝑻 − 𝑲; 𝟎]

QFI QF Exam 2021 PAK Study Manual Intro-Maths-Fin-1

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The profit from a call option is the payoff (𝐶𝑇) minus the premium (𝐶0)

Options are nonlinear instruments (see the convexity/concavity of the payoff)

Swaps A swap is an agreement between two counterparties for selling and purchasing cash flows involving various currencies, interest rates and a number of other financial assets. The counterparties borrow in sectors where they have an advantage and then exchange the interest payments. In a simple IRS, at the end, both counterparties secure a lower rates and the swap dealer will earn a fee. One method for pricing swaps and swaptions is to decompose them into forwards and options. The forwards can then be priced separately, and the corresponding value of the swap can be determined from these numbers. Two example of swaps

1) The simple IRS (Interest Rate Swap): Each counterparty borrows in sector where it has an advantage and they both exchange the payments.

2) The Cancelable Swaps: In this swap, each party has the option to cancel the transaction before maturity.

They come in two flavors: Callable swaps and Puttable swaps. Some properties of Cancelable swaps

Popular among institutions with an obligation in which they are to repay principal before maturity. The embedded option on the swap can be exercised to honor such liabilities.

QFI QF Exam 2021 PAK Study Manual Intro-Maths-Fin-1

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They can be used as hedge.

They allow institutions to mitigate maturity mismatch between assets and liabilities due to prepayments or early surrenders.

Past exams SOA Spring 2015 QFIC Q11 on Repo (Must Read) SOA Spring 2013 APM Q3 (Must Read)

Practice questions Question 1 List the differences between forwards and futures:

1. Futures are traded in organized exchanges and forwards are custom-made and traded over the counter.

2. Futures exchanges are cleared through exchanges clearing houses and there is a mechanism designed to

reduce default risk. Forwards are not cleared and there is default risk.

3. Futures contracts are market to market. Every day, the contract is settled and a new contract is entered.

Question 2 Assume that the S&P 500 index is at 100. For a single premium of $100, a life insurance company had sold the following type of products: Contract 1: promising to pay 100 at maturity of the contract in 5 years plus any excess of the S&P 500 over its initial value of 100. Contract 2: The Company promises to pay the excess of two quantities:

o 90% of the initial premium accumulated at 2% per annum and o The proceeds from the investment of the initial premium into a fund that performs exactly like the S&P

500 at the maturity of the contract in 10 years. For each contract, determine the following:

1. The maturity of the contract

2. The type of derivative embedded in the liability and the underlying asset.

3. Identify the payoff of the liability, and the strike amount

Question 3 Describe the bull call spread option

QFI QF Exam 2021 PAK Study Manual Intro-Maths-Fin-1

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Solution 1. Contract 1 has a maturity of 5 years and contract 2 has a maturity of 10 years. 2 and 3. Embedded derivative in Contract 1: A call option on the S&P 500 of maturity 5 and strike 100. Embedded derivative in contract 2: Payoff is:

𝑀𝑎𝑥(0.9 × 100 × (1.02)10; 𝐹10) Where:

𝐹10 = 𝑡ℎ𝑒 𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑢𝑛𝑑 𝑡ℎ𝑎𝑡 𝑚𝑖𝑚𝑖𝑐𝑠 𝑡ℎ𝑒 𝑆&𝑃 500 The payoff is equal to:

𝑀𝑎𝑥(109.7; 𝐹10) = 𝐹10 + 𝑀𝑎𝑥(0; 109.7 − 𝐹10) = 𝐹10 + 𝑃𝑎𝑦𝑜𝑓𝑓 𝑜𝑓 𝑎 𝑝𝑢𝑡 𝑜𝑝𝑡𝑖𝑜𝑛 Thus the embedded derivative is a put option on the underlying fund of strike 109.7 and maturity 10 years. Bull Call Spread Option (Neftci practice problem 6 on page 11)

A bull spread call is a strategy that involves purchasing call options at a specific strike price while also selling the same number of calls of the same asset and expiration date but at a higher strike.

A bull call spread is used when a moderate rise in the price of the underlying asset is expected.

The maximum profit in this strategy is the difference between the strike prices of the long and short

options, less the net cost of options.