Mathematics 547 Spring 2012 Financial Mathematics Professor Peter - - PowerPoint PPT Presentation

Mathematics 547 Spring 2012 Financial Mathematics

Professor Peter Bloomfield email: Peter Bloomfield@ncsu.edu http://www.stat.ncsu.edu/people/bloomfield/courses/ma547/

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Single Period Models Financial market instruments

• Underlying assets; e.g. shares, bonds, commodities, curren-

cies;

• Derivatives: contracts with obligations that are contingent
• n behavior of the market in an underlying.
• Derivatives are used to manage and transfer the risk in the

underlying markets.

• Key issue: what is a derivative worth?

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Examples of Derivatives

• A forward contract is an agreement between two parties, A

and B. Party A agrees to buy a specified asset (the underly- ing) on a specified future date T for a specified price K, and party B agrees to deliver the asset. – The specified time T is the exercise date or maturity of the deal. The specified price is the strike price. – Both parties have obligations to meet on the exercise date. – A has the long position in the contract, and B has the short position.

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• No money changes hand up front, when A and B make the

agreement.

• Pricing issue: the choice of the strike price K.
• Counterparty risk:

A is B’s counterparty, and vice versa. Each party faces the risk that its counterparty may not be able to meet its obligation on the exercise date.

• A futures contract is a forward contract that trades on an

exchange instead of being negotiated by two parties: – The contract is standardized, not fully negotiatable. – The exchange holds reserves (margin) to eliminate risk.

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• A European call option is a contract that gives A the right,

but not the obligation, to buy a specified asset on a specified future date T for a specified price K, and B agrees to deliver the asset if called upon to do so.

• If, at maturity, the asset is trading above K, A should exercise

the option, but not if it is trading below K.

• A has no obligation, only the opportunity to profit, and B

therefore demands an up-front payment or premium.

• Pricing issue: the choice of the premium for a given strike

price K.

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• A has counterparty risk: B may fail to deliver the underlying

when called upon.

• Options are often sold by exchanges, to eliminate the risk.
• B has only execution risk: the buyer A may negotiate the

contract, then fail to pay the premium.

• A European put option is the corresponding contract that

gives A the right to sell the asset at time T for the strike price K, and B agrees to pay the price and accept the asset if called upon to do so.

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• The payoff of a European option is the value to the holder,

as a function of the underlying price at maturity, ST: – The payoff of a European call is (ST − K)+ = max (ST − K, 0) ; – The payoff of a European put is (K − ST)+ .

• Note:

If an option is physically settled, the underlying is actually delivered to one party; the payoff is calculated as if it is bought or sold at a price of ST. Many options are cash settled.

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• Options are often bought in packages, to achieve other pay-
• ffs.
• E.g.: a straddle is a call and a put, with the same strike and

maturity; the payoff is (ST − K)+ + (K − ST)+ = |ST − K| .

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Pricing a Derivative

• E.g.: a forward contract; payoff (to party A) is ST − K.
• Expectation pricing versus no arbitrage pricing.
• Suppose that at time 0, we model ST as a log-normal random

variable. – Specifically, log

• ST

S0

• ∼ N
• ν, σ2

.

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• Then

E [ST|S0] = S0 exp

• ν + σ2/2
• .
• That appears to be a candidate for the choice of the strike

price K.

• But that choice may allow an arbitrage: a risk-free profit for
• ne party.

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• Suppose that the risk-free interest rate is r:

– That is, a party can borrow unlimited sums at that interest rate, or deposit unlimited sums and receive that interest rate; – The risk-free rate is usually measured by short-term U.S. Treasury bills.

• The arbitrage: suppose that K > S0erT:

– At time 0, the seller (party B) borrows S0 and buys one share; – At time T, party B delivers the share to party A and re- ceives K, then pays off the loan for S0erT, leaving a profit

• f K − S0erT > 0.

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• Conversely, if K < S0erT:

– At time 0, party A sells short one share for S0 and invests the cash at the risk-free rate; – At time T, A has S0erT in cash, uses K to buy one share from B, under the contract; – Party A then uses the share to close out the short sale, and is left with profit S0erT − K > 0.

• We assume that such arbitrage opportunities do not exist,

so the strike price must be K = S0erT.

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• A forward contract is a very simple derivative instrument, but

the pricing strategy of looking for arbitrage opportunities is very general.

• Note that the expectation price would be the same as the

no-arbitrage price if ν + σ2 2 = rT. Finding a model that makes the expectation equal to the no-arbitrage price is also a very general strategy.

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CNN data on the March S&P 500 futures contract.

• “Level” is the current market strike for the contract.
• “Fair value” is the no-arbitrage strike, based on the previous

day’s closing value of the index.

• “Difference” indicates that traders expect the market to open

lower.

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