chapter2 - Chapter 2: Binomial Trees Chapter 2. Binomial...

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Chapter 2: Binomial Trees Chapter 2. Binomial Trees Section 1. A Simple Example Example 1.1. A stock price is currently $20, and it is known that at the end of 3 months it will be either $22 or $18. Assume the risk-free rate is 12% per annum . We want to value a European call option to buy the stock for $21 in 3 months. The option will have one of two values at the end of 3 months. If the stock price turns out to be $22, the value of the option will be $1; otherwise, the value of the option will be $0. Time zero Time one Stock price=$22 Option price = $1 Stock price: $20 Option price:$c Stock price=$18 Option price=$0 Consider a portfolio consisting of a long position in Δ shares of the stock and a short position in call option position. We calculate Δ that makes the portfolio riskless. Namely Or Δ =0.25 Regardless of whether the stock price moves up or down, the value of the portfolio is always $4.5 at the end of the option. Riskless portfolio must earn the risk-free rate of interest by “no-arbitrage principle”. Since the risk-free rate is 12% per annum, the value of the portfolio today must be the present value of $4.5. It is 4.5*exp(-0.12/4) = 4.367.
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Chapter 2: Binomial Trees The value of the stock price today is known to be $20. Suppose the option price is denoted by c. Then 20*0.25 – c = 5 – c = 4.367 Then c = $0.633 This shows that in the absence of arbitrage opportunities, the current value of the option must be $0.633. If the call option were not priced at $0.633, a sure profit would be possible. Say the option c = $1. An arbitrageur can implement the following strategy: Borrowed $4 at 12%, to be paid back at the end of three months Short a call option, receive $1 Buy 0.25 shares of stock At the end of three months, the arbitrageur has $4.5 for each possible level of the stock, and $4.5 is more than enough to pay $4 * exp(0.12/4) = 4.1218. If the option were priced at a price which is less than $0.633, the arbitrageur can follow the strategy in opposite position! 1.2. A One-Step Binomial Model Consider a stock price whose price is 0 S and the option on the stock whose current price is f . Suppose that the option lasts for time T and that during the life of the option the stock price can either move up from 0 S to a new level 0 S u, where u>1, or down from 0 S to a new level 0 S d, where d < 1. If the stock price moves up to 0 S u, we suppose that the payoff from option is u f ; if the stock price moves down to 0 S d, we suppose that the payoff from the option is . d f Consider a portfolio consisting of a long position in Δ shares and a short position in one option. Solve Δ by d u f d S f u S - Δ = - Δ 0 0
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Chapter 2: Binomial Trees Or d S u S f f d u 0 0 - - = Δ In this case, the portfolio is riskless and must earn the risk-fee interest rate. Let r denote the risk-free interest rate. The present value of the portfolio is
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chapter2 - Chapter 2: Binomial Trees Chapter 2. Binomial...

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