Princeton FIN 501 - Dynamic Arbitrage (23 pages)

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Dynamic Arbitrage



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Dynamic Arbitrage

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Pages:
23
School:
Princeton University
Course:
Fin 501 - Asset Pricing I:Pricing Models

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INSTITUTIONAL FINANCE Lecture 08 Dynamic Arbitrage to Replicate Non Linear Payoffs Originally prepared by Ufuk Ince Ekaterina Emm supplemental material courtesy of Wei Xiong Princeton University 1 BINOMIAL OPTION PRICING Consider a European call option maturing at time T with strike K CT max ST K 0 K 0 no cash flows in between Is there a way to statically replicate this payoff Not using just the stock and risk free bond required stock position changes for each period until maturity as we will see Need to dynamically hedge compare with static hedge such as hedging a forward forward or hedge using put put call call parity Replication strategy depends on specified random process of stock price need to know how price evolves over time Binomial Cox Rubinstein Ross model is canonical ASSUMPTIONS Assumptions Stock St k which hi h pays no dividend di id d Over each period of time stock price moves from S to either uS or dS i i d over time so that final distribution of ST is binomial uS S dS Suppose length of period is h and risk free rate is given by R erh No arbitrage u R d Note simplistic model but as we will see with enough periods begins to look more realistic A ONE PERIOD BINOMIAL TREE Example of a single period model S 50 u 2 d 0 5 R 1 25 100 50 25 What is value of a European call option with K 50 Option payoff max ST K 0 50 C 0 Use replication p to p price SINGLE PERIOD SINGLE PERIOD REPLICATION Consider a long position of in the stock and B dollars in d ll i bond b d Payoff from portfolio uS RB 100 1 25B S B 50 B dS RB 25 1 25B Define Cu as option payoff in up state and Cd as option payoff in down state Cu 50 Cd 0 here Replicating strategy must match payoffs Cu uS RB Cd dS RB SINGLE PERIOD REPLICATION Solving these equations yields Cu Cd S u d uC C dCu B d R u d In previous example example 2 3 and B 13 33 B 13 33 so the option value is C S B 20 Interpretation of sensitivity of call price to a change in the stock price Equivalently tells you how to hedge risk of



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