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Unformatted text preview: Lecture 9: (Ch11) Binomial Models The binomial tree is a diagram representing different possible paths that might be followed by the stock price over the life of the option. The underlying assumption is that the stock price follows a random walk. • Chapter shows how we can value options under binomial models using both no-arbitrage arguments and a principal known as risk-neutral valuation. A One-step Binomial Model and a No-Arbitrage Argument • Assume absence of arbitrage opportunity • First, establish a portfolio of stock and option such that there is no uncertainty about the value of the portfolio in the next period • Because the portfolio has no risk it must earn the r isk free rate of return . • From this we deduce the value of the portfolio today and hence the value of the option today. A stock price is currently $20 In three months it will be either $22 or $18 A 3-month call option on the stock has a strike price of 21. • Suppose we write 1 call option and buy Δ shares of the stock • Can the portfolio be riskless? • Riskless means that the values of the portfolio are equal in any state. o If the price moves from $20 to $22. The value of the shares is 22 Δ and the value of the option is one. o If the prices moves down from $20 to $18, the value of the shares is 18 Δ and the value of the option is zero. o Riskless therefore implies that the in either state, the outcome will be equal. o Thus; 22 Δ- 1 = 18 Δ • Requires that Δ = 0.25, e.g., for each written option, 0.25 share stocks are needed to form a risk-free portfolio. • Regardless of how the stock price moves, the portfolio is always worth 4.5 at maturity....
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This note was uploaded on 03/30/2011 for the course FIN 3635 taught by Professor Yip during the Three '11 term at University of New South Wales.
- Three '11