MATHEMATICS OF FINANCE
MATHEMATICS OF FINANCE MATH 628
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This 5 page Class Notes was uploaded by Evert Christiansen on Wednesday October 21, 2015. The Class Notes belongs to MATH 628 at Texas A&M University taught by Daniel Lewis in Fall. Since its upload, it has received 21 views. For similar materials see /class/226025/math-628-texas-a-m-university in Mathematics (M) at Texas A&M University.
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Date Created: 10/21/15
MATH 628 HANDOUT ONE FORWARDS AND OPTIONS These notes deal with the arbitrage free pricing of nancial derivatives in idealized capital markets For instance we assume there are no taxes or commissions information is available universally and instantaneously investors may buy or sell assets in any necessary amounts there are no defaults or unful lled contracts and so on A derivative is a security whose price is determined by the price of another asset called the underlying asset or simply the underlying Common examples are forwards futures and options A forward contract between two parties obliges one designated party to purchase the underlying asset from the other at a predetermined future time and at a predetermined price The predetermined price is called the delive price The party buying the underlying at the contract s eXpiration holds the g position and the seller the short position Futures contracts are discussed later Once a forward contract is agreed to it becomes a tradable security in its own right with a market price possibly different from the contract s delivery price and the market price of the underlying To nd a formula for the price of a forward contract at times before eXpiration requires two special assumptions the eXistence of a risk free interest rate and the absence of arbitrage opportunities Both assumptions will remain in force throughout these notes INTEREST RATE ASSUMPTION There is a single interest rate called the risk free rate at which all investors may borrow or lend any desired amount for any time period We ll use continuous compounding in our interest calculations With that convention the future value of P dollars invested for t years at 100 r per year is F P exp rt dollars Viewed the other way around P F exp rt dollars is the present value or discounted value of F dollars received t years in the future A zero coupon bond or zero is a security that pays the holder 1 at maturity and whose price at each time before maturity is the discounted value of 1 If a zero matures T years from present its price tyears from present is Bt exp r T t dollars 0 S t S T An investor can lend K dollars by buying K zeros and can borrow K dollars by selling K zeros A collection of assets forms an arbitrage portfolio iff the portfolio 1 can be set up at no net cost 2 has no possibility ofa loss but 3 has a positive probability of a gain Condition 3 is super uous in case the portfolio has a negative initial cost ie shows an immediate pro t Loosely speaking the owner of an arbitrage portfolio has a risk free chance to make a pro t Our second basic assumption denies that can happen NO ARBITRAGE ASSUMPTION aka NO FREE LUNCH ASSUMPTION There are no arbitrage portfolios Prices consistent with this principle are called arbitrage free prices One technique that investors use to raise cash for purchases is through a short sale In a short sale an investor borrows an asset sells it and later replaces the asset The short seller may replace the borrowed asset at any time unless the asset has a limited lifetime eg a zero THEOREM 11 Let Ut and Vt 0 g t g T be the prices in t years of two assets or asset portfolios Assume each can be bought or sold short at any time and in any amount 1 If UT lt VT is certain to occur then Ut lt Vt for all times 0 S t S T 2 If UT S VT is certain to occur then Ut S Vt for all times 0 S t S T 3 If UT VT is certain to occur then Ut Vt for all times 0 S t S T Note that the prices Ut and Vt depend on more than just the future time They are random variables depending on both time and the future states of the economy nancial markets etc The meaning of statements such as UT lt VT is certain to occur is that UT lt VT will hold in each future state One portfolio replicates another up to time T iff the portfolios have equal prices at all times up to and including time T To check replication it is enough to see that the portfolios have the same prices at time T Replication provides a powerful method for determining derivative prices We ll now use replication to price forward contracts Consider a forward contract that expires T years in the future and requires a payment of K for the underlying Let t 0 denote the present and for each future time t 0 S t S T let St and Vt respectively be the prices of the underlying and a long position on the forward contract THEOREM 12 From the viewpoint of the long party the arbitrage free price of the forward contractt years from present is Vt St K exp rT t In particular the nal and initial contract prices are VT ST K and V0 S0 Kexp rT COROLLARY 13 A forward contract with delivery price K can be replicated by a portfolio that is long the underlying asset and short K zeros maturing on the delivery date In the notation of the theorem 3 S0 exprT dollars defines the underlying asset s forward price for deliveg in T years sometimes called the fair price for deliveg in T years A contract with that delivery price has initial or current value 0 to both parties A European call option respectively European put option gives its holder the right but not the obligation to purchase respectively to sell the underlying asset at a predetermined future time and at a predetermined price The predetermined time is called the expiration time and the option ends on the expiration date The predetermined price is the exercise price or strike price A European option cannot be exercised before its expiration date Other types of options American call options and American put options can be exercised at any time up to and including the expiration date Unlike a forward contract the owner of an option of either type need not exercise the option An option is worthless after its exercise date but an option can never be worthless before the expiration date Why not Many widely traded options have shares of stock as the underlying The value and price behavior of these options may depend on whether or not the underlying stock pays dividends Unless speci cally stated otherwise assume the shares underlying a stock option pay no dividends until after the option expires In writing results about options K denotes the strike price t 0 the present and T the time in years to expiration The arbitrage free prices of a European call European put and the underlying asset at time t are denoted by Ct Pt and St respectively Forwards and European options have several similarities Each has a xed exercise time and at expiration the price is VT FST for a certain function F of the underlying asset39s price F is called the payout function or payoff function Three basic examples are Fs s K for the long position on a forward contract Fs maxs K0 s K for a European call and Fs maxK s0 K s for a European put THEOREM 14 PUTCALL PARITY Let r be the risk free rate The prices of a European call and European put with the same underlying expiration and strike price must satisfy Ct Pt St Kexp rT t atalltimes 0 St S T SKETCH PROOF Consider the portfolio consisting of a put and the long position on a forward contract both with the given underlying expiration and strikedelivery price At expiration the portfolio has the same value as the European call and thus the same price at all prior times DD COROLLARY 15 A European call can be replicated by a portfolio that holds a put and the long position on a forward contract with the three derivatives having the same underlying asset expiration date and exercise price COROLLARY 16 Ct 2 St K at all times t with strict inequality at times before expiration COROLLARY 17 An American call and a European call have equal prices provided they have the same expiration strike price and underlying SKETCH PROOF Let At be the price at time t of the American call The rst step is to prove At S Ct at every time To get a contradiction assume At gt Ct at some time In that event an arbitrageur could write ie create and sell an American call use part of the At proceeds to buy a European call for C t and make an immediate pro t of At Ct gt 0 dollars If the American call were exercised at some time 9 Z t then the arbitrageur would owe either zero or Se K dollars By Corollary 16 Ce 2 Se K so the arbitrageur could cover the liability by selling the European call for its market price C e The existence of this arbitrage portfolio contradicts the No Arbitrage Principle A similar argument establishes At 2 Ct and is left as a problem DE A BASIC PROBLEM Find a model for the arbitrage free price of a European style derivative that expires at a future time t T and has payout VT FST There are several approaches to this problem If a derivative can be replicated by a portfolio of assets with known prices then the derivative price is the price of the replicating portfolio This method was successful in pricing forward contracts Another approach to the basic problem is through riskneutral probabilities Again let r denote the riskfree rate For a given probability measure P write E 1X or more brie y E X for expectation with respect to P P is a riskneutral probability or riskneutral measure for an asset price Vt iff E Vt exp rt V0 at each time in the asset s life Many European style derivatives can be priced using risk neutral probabilities even if the de ning equalities hold only at times t 0 and T THEOREM If P is a riskneutral probability for the underlying then an arbitrage free price for a European style derivative worth VT FST at expiration is V0 exp r T E FST Later sections contain a proof of this result and the construction of riskneutral probabilities for various derivatives For now it is enough to note three simple cases in which risk neutral pricing works THEOREM 18 If P is a riskneutral probability for an asset then the formula V0 exp r T E 1FS1 correctly prices the asset zeros and forwards with the asset as underlying PROBLEM 11 Suppose a portfolio holds both a certain asset and the short position on a forward contract to deliver the asset in T years for K Prove that the portfolio replicates a portfolio of K zeros that mature T years from present PROBLEM 12 Assume the future prices of a given asset are known with certainty and are a function of time alone Let Wt denote the asset price t years in the future and let r be the risk free rate Prove the following a Wt g W0exp r t at all time t 2 0 b If Wt is differentiable at t 0 then W390 g W0r c If the asset can be sold short then Wt W0 exp r t Problems 13 17 are about European s le options PROBLEM 13 Fix a future time T and strike prices K lt L A bull spread is a portfolio that is a long one call with strike price K that expires at time T an b short one call with strike price L that expires at time T Find and make a sketch of the portfolio s payoff function What expectations about the price of the underlying might cause an investor to buy a bull spread What are the advantages and disadvantages of the bull spread compared to the call in a PROBLEM 14 Fix a future time T and strike prices K lt L A bear spread is a portfolio that is a long one put with strike price L that expires at time T and b short one put with strike price K that expires at time T Find and make a sketch of the portfolio s payoff function What expectations about the price of the underlying might cause an investor to buy a bear spread What are the advantages and disadvantages of the bear spread compared to the put in a PROBLEM 15 Fix a future time T three strike prices L lt M lt R and let x be the number in 01 satisfying M 1L 1 or R The portfolio that is a long or calls having strike price L and expiring at time T b long 1 0L calls having strike price R and expiring at time T and c short one call having strike price M and expiring at time T is called a butter y spread Find and make a sketch of the portfolio s payoff function Prove that the prices of the three calls satisfy M S LLt l at RT PROBLEM 16 Fix a future time T and two strike prices K lt L Let r be the risk free rate and write CtX for the price of a call expiring at time T with strike price X a Checkthat OSCtK C1LS L K whentT b Prove OSCtK CtLSL K exp rT t for alltimet 0 t T c Write the analogous inequalities for puts PROBLEM l7 Fix a strike price K and two future times T lt Tquot Write U tand Vt for the prices of European calls that both have strike price K but expires at times T and T respectively Prove Ut S Vt at all time up to and including T PROBLEM 18 LetAt and Ct respectively denote the arbitrage free prices at time t of American and European calls that expire at time T have the same strike price K and the same underlying asset Complete the proof of Corollary 17 by proving At S Ct
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