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# lecture11 - Lecture 11: Risk-Neutral Valuation Steven...

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Unformatted text preview: Lecture 11: Risk-Neutral Valuation Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 117944400 http://www.cs.sunysb.edu/ skiena Risk-Neutral Probabilities We can use an arbitrage argument to set the right probability of an upward move ( ) as a function of the risk- free rate. At any point, investors can either (a) hold \$1 stock or (b) invest \$1 at the risk-free rate r . A risk-neutral investor would not care which portfolio they owned if they had the same return. Setting equal the returns from the stock ( +(1- ) / ) and the risk-free portfolio ( 1+ r ), we can solve for to determine the risk-neutral probability . But in truth, investors are not risk-neutral. In order to take the riskier investment they must be paid a premium. Single-Step Option Pricing Binomial trees price options using the idea of risk-neutral valuation . Suppose a stock price is currently at \$20, and will either be at \$22 or \$18 in three months. What is the price of a European call option for a strike price of \$21? Clearly, this reduces to determining the probability of the upward price movement. Risk Neutral Valuation The risk-neutral investor argument for setting this probability can be applied if we set up two portfolios which are of provably of equal risk and value. We will construct two riskless portfolios, one involving the stock and the other the risk-free rate. Using Options to Eliminate Risk A riskless portfolio can be created by buying shares of stock and selling a short position in 1 call option, such that the value of the portfolio is the same whether the stock moves up or down. If the stock moves to \$22, our portfolio will be worth \$22- \$1 1 , since we must pay the return of the option we sold....
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## lecture11 - Lecture 11: Risk-Neutral Valuation Steven...

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