Chapter 13 Binomial trees.doc - Binomial trees ONE STEP BINOMIAL TREES European call No-arbitrage valuation Assume that the stock pays no dividends Read

Chapter 13 Binomial trees.doc - Binomial trees ONE STEP...

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Binomial trees ONE STEP BINOMIAL TREES European call No-arbitrage valuation Assume that the stock pays no dividends Read problem 13.1. Stock values The length of the time period between T-1 and T is 1 month. u = 42/40 = 1.05 u = S u /S d = 38/40 = 0.95 d = S d /S The risk free rate r is 8% with continuous compounding. Call option values 1 month European call option with a strike price = X = $39
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Assumption - No arbitrage opportunity for any investor. Portfolio - Buy  shares of stock and simultaneously write 1 call option on the stock. What is the value of which makes the portfolio riskless? If the stock price moves up to 42, the value of the portfolio = 42 - 3. If the stock price moves down to 38, the value of the portfolio = 38 - 0. If the portfolio is riskless, then the two values should be equal. 42 - 3 = 38 - 0 = (3 - 0)/(42 - 38) = 0.75 Therefore, the riskless portfolio consists of: Buy 0.75 shares of stock and write 1 call option on the stock. The riskless portfolio must, in the absence of arbitrage profits, earn the risk free rate of return. Therefore, value of the portfolio today = Present value of future cash flows discounted at the risk free rate 0.75 x 40 - c = [3/4 x 42 - 3]e -0.08 x 1/12 c = 0.75 x 40 - [3/4 x 42 - 3]e -0.08 x 1/12 = $1.69 Therefore, the European call option must be worth $1.69. Generalization Stock values The stock price can move either up to Su or down to Sd in 1 period. The proportional increase in the stock price when there is an up movement in the stock = u-1. The proportional decrease in the stock price when there is a down movement in the stock is d-1. Payoff from the derivative 2
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The payoff from the derivative can either be f u or f d in 1 period. Note the tree diagram which follows. Riskless portfolio 3
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Buy shares of the stock and write 1 derivative on the stock. What is the value of that makes this portfolio riskless? S u - f u = S d - f d = [f u - f d ]/[ S u - S d ] The value of the risk free portfolio is given by: S - f = [ S u - f u ]e -rT Substituting for : f = [p f u + (1-p) f d ] e -rT Equation 13.2 where p = [e rT – d]/[u - d] Equation 13.3 In the numerical example: p = [e 0.08 x 1/12 – 0.95]/[1.05 – 0.95] = 0.5669 1 - p = 1 - 0.5669 = 0.4331 Note that when we create a risk free portfolio of the stock and the call option we buy the stock and write the call option.
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