Mathematics 547 Spring 2012 Financial Mathematics Professor Peter Bloomfield email: Peter Bloomfield@ncsu.edu http://www.stat.ncsu.edu/people/bloomfield/courses/ma547/ 1
Single Period Models Financial market instruments • Underlying assets; e.g. shares, bonds, commodities, curren- cies; • Derivatives: contracts with obligations that are contingent on 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? 2
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. 3
• 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. 4
• 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 . 5
• 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. 6
• The payoff of a European option is the value to the holder, as a function of the underlying price at maturity, S T : – The payoff of a European call is ( S T − K ) + = max ( S T − K, 0) ; – The payoff of a European put is ( K − S T ) + . • 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 S T . Many options are cash settled . 7
• Options are often bought in packages, to achieve other pay- offs. • E.g.: a straddle is a call and a put, with the same strike and maturity; the payoff is ( S T − K ) + + ( K − S T ) + = | S T − K | . 8
Pricing a Derivative • E.g.: a forward contract; payoff (to party A) is S T − K . • Expectation pricing versus no arbitrage pricing. • Suppose that at time 0, we model S T as a log-normal random variable. – Specifically, � � S T � ν, σ 2 � log ∼ N . S 0 9
• Then ν + σ 2 / 2 � � E [ S T | S 0 ] = S 0 exp . • 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 one party. 10
• 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 > S 0 e rT : – At time 0, the seller (party B) borrows S 0 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 S 0 e rT , leaving a profit of K − S 0 e rT > 0. 11
• Conversely, if K < S 0 e rT : – At time 0, party A sells short one share for S 0 and invests the cash at the risk-free rate; – At time T, A has S 0 e rT 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 S 0 e rT − K > 0. • We assume that such arbitrage opportunities do not exist, so the strike price must be K = S 0 e rT . 12
• 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 + σ 2 2 = rT. Finding a model that makes the expectation equal to the no-arbitrage price is also a very general strategy. 13
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. 14
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