Financial Mathematics, MST 30030

Lecturer: Masha Vlasenko

I Introduction: financial markets, derivatives and traders

I Interest rates

I Properties of stock options

I Some probability theory

I Binomial model for options pricing

I The Black-Scholes Model

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Literature:

I “Options, Futures, and other Derivatives” by John C. Hull

I “The Concept and Practice of Mathematical Finance” byMark Joshi.

I “The Mathematics of Financial Derivatives: A StudentIntroduction” by Paul Wilmott, Sam Howison, and J.Dewynne.

I J. Cox, S. Ross and M. Rubenstein, “Pricing: A SimplifiedApproach”, Journal of Financial Economics, 7 (1979), 229 -263.

I F. Black and M. Scholes, “The Pricing of Options andCorporate Liabilities”, Journal of Political Economy, 81(May/June 1971), 637 - 659.

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Lecture 1. Financial Derivatives

Reading: John C. Hull, “Options, Futures and Other Derivatives”,Chapter 1

A derivative is a financial instrument whose value derives from thevalues of other, more basic underlying variables.

We will consider the following types of derivatives

I forward contracts

I futures contracts

I call options

I put options

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Forward contracts

A forward contract is an agreement between two parties to buy orsell an asset for a certain price K at a certain future time T .

K is called the delivery price or the strike priceT is called the delivery time or the maturity time

An agreement to buy or sell an asset for a certain price K today iscalled a spot contract. In this case K is called the spot price ormarket price.

It might be that one party has to pay a certain price c to the other partyfor the contract.

Forward contracts are usually between two financial institutions, or

between the institution and its clients. The contract can be modified

according to special needs of the parties.

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One party of the forward contract buys the underlying asset, attime T for a price K , and we say that they take a long position.

The party agreeing to sell the underlying asset is taking a shortposition.

Let ST be the price of the asset at the maturity time T :

The payoff from a long position is defined as

ST − K ,

and the payoff from a short position

K − ST .

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Example ( forward contract for hedging foreign currency risk)

A US corporation A has to pay to a German company in six monthsone million euro for a modern machine, and wants to hedgeagainst exchange rate moves (their business is based on dollars).

A is able to make a forward contract with a bank B: according tothe contract A buys the asset e 1 million for the strike price$1, 335, 900 after maturity time. This is a binding contract.

Advantage for A: it can calculate the price for the machine withoutany currency risk.The bank has taken over the currency risk and it will require somemoney for the contract. The risk for the bank can be compensatedif the bank is able to make “complementary” forward contractswith other partners (buying instead of selling Euros) for the samestrike price.

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Consider payoffs for the above forward contract.

K = $1.3359 (the delivery price of e 1)T = six months

Recall that ST is the price of the asset at maturity.

Suppose that for example ST = 1.4000, that is ST > K . Thecorporation gets e 1,000,000 for $1, 335, 900 instead of$1, 400, 000. They pay $64, 100 less than the market value.

Suppose that for example ST = 1.3000, that is ST < K . Then thecorporation pays $35, 900 more then the market price for the euro.

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Example

Spot and forward quotes on the USD/GBP exchange rate thatmight be made by a large international bank in June 2003 (quoteis number of USD per GBP).

Bid OfferSpot 1.6281 1.62851-month forward 1.6248 1.62533-month forward 1.6187 1.61926-month forward 1.6094 1.6100

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Futures contracts

A futures contract is a standardized contract between twoparties to buy or sell a specified asset of standardized quantity andquality for a price K agreed upon today with delivery and paymentoccurring at a specified future date T . Futures contracts arenegotiated at an exchange-traded market, which acts as anintermediary between the two parties.

K is called the futures price or the strike price.T is called the delivery time or the maturity time.The party agreeing to buy the underlying asset in the future, thebuyer of the contract, is said to be long.The party agreeing to sell the asset in the future, the seller of thecontract, is said to be short.

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Exchange-traded markets

A derivatives exchange market is a market where individuals tradestandardized contracts that have been defined by the exchange.

The Chicago Board of Trade was established in 1848 to bringfarmers and merchants together. Initially its main task was tostandardize the quantities and qualities of the grains that weretraded.

Within a few years the first future-type contract was developed. Itwas known as to-arrive contract.

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Chicago Board of Trade, 1970, USDA History Collection, SpecialCollections, National Agricultural Library.

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Speculators soon became interested in the contract and foundtrading the contract to be an attractive alternative to trading thegrain itself. A rival futures exchange, the Chicago MercantileExchange (CME, www.cme.com), was established in 1919. Nowfuture exchanges exist all over the world.

Traditionally derivatives traders have used what is known as theopen outcry system. This involves traders physically meeting onthe floor of the exchange, shouting, and using a complicated set ofhand signals to indicate the trades they would like to carry out.

Exchanges are increasingly replacing the open outcry system byelectronic trading.

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Exchange-traded market vs over-the-counter market

While futures contracts are traded on exchange-traded markets,forward contracts belong to so called over-the-counter market. It isa telephone- and computer-linked nework of dealers who do notphysically meet.

The size of the market is measured in terms of the estimated totalvalue of the assets underlying the contracts. Over-the-countermarket is much larger than the exchange-traded market. By June2004, the over-to-the counter market had grown to $ 220.1 trillion(approximately five times the world gross domestic product) andthe exchange-traded market has grown to $ 49.0 trillion’.

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Call and put options

A (European) call option gives the holder the right to buy theunderlying asset for a price K at maturity time T .

A (European) put option gives the holder the right to sell theunderlying asset for a price K at maturity time T .

The price K in the contract is the exercise price or strike price.

The time T in the contract is the expiration time or maturity.

Again, there are two sides to every option contract

I long position, buying the option,

I short position, selling (or writing the option).

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Options are traded both on exchanges and over-the-countermarket.

The Chicago Board Options Exchange (CBOE, www.cboe.com)started trading call option contracts in 1973. Options had tradedprior to 1973, but the CBOE succeeded in creating an orderlymarket with well-defined contracts. Put option contracts startedtrading on the exchange in 1977.

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American options vs European

American options can be exercised at any time up to theexpiration date.

European options can be exercised only at maturity.

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Example ( call option)

John issues a call option, a contract offering the right to buy fromJohn one share of Microsoft for price K at time T . Betty sees theoffer and she agrees to the contract. She has to pay immediatelythe price c EUR for the call option to John.

Thus John obtains c EUR from Betty at time t = 0. Betty is theholder of the contract and has the right to buy a share from Johnat time T for the price K .

Question: what if John can not fulfill the obligations of thecontract?

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Responsibility

Example

John issues one million call options at time t = 0 to a sensationalprice of 10 EUR. He obtains 10 million EUR. John decides toemigrate and to disappear with the money. Who is responsible forthe damage?

I If John and his contract partners made the contract privately,say without use of an exchange market, then the partnershave lost all the money.

I If they traded at an exchange market for options then themanagement of the market will fulfill the obligations of thecontracts. If John is a trader of a bank which is registered tothe exchange market, the exchange market can pursue thebank to reimburse the damage. Details depend on theregulations of the market.

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Payoff

Example (long position in a call option)

An investor buys a European call option with a strike price ofK = $60 to purchase 100 Microsoft shares. The current stockprice S0 = $58, and the option is for T = 4 months. The price ofthe option to purchase 1 share is $5.

Scenario 1: ST < K . The options won’t be exercised since theinvestor can buy directly on the market for a lower price ST . Theinvestor loses his initial investment of $500.

Scenario 2: ST > K . For example, suppose that ST = $75. Theoptions will be exercised and the investor makes a profit of100 × $15 = $1500. The net profit to investor is $1000.

The payoff is max(ST − K , 0) (ignoring the initial cost of theoption).

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Example (long position in a put option)

An investor buys a European put option with a strike price ofK = $90 to sell 100 shares in IBM. Suppose S0 = $85, expirationdate is T = 3 months. The price of option to sell 1 share is $7.

Scenario 1: ST > K . The options won’t be exercised since theinvestor can sell the shares on the market for a higher price. Theinvestor loses his initial investment of $700.

Scenario 2: ST < K . For example, suppose that ST = $75. Theinvestor buys 100 shares at $75 each, then sells them for $90 each,making a profit of $1, 500 (ignoring the transaction costs). Thenet profit to investor is $800.

The payoff is max(K − ST , 0) (ignoring the initial cost of theoption).

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Payoff (summary)

Let us ignore as usual the initial cost of the derivative. Then thepayoff is given by

I long position in a forward contract: ST − K

I short position in a forward contract: K − STI long position in a call option: max(ST − K , 0)

I short position in a call option: min(K − ST , 0)

I long position in a put option: max(K − ST , 0)

I short position in a put option: min(ST − K , 0)

(We’re supposing we have European options.)

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