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Risk Management and Financial Institutions, 4th Edition by John C. Hull

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APPENDIX F Valuing American Options

To value American-style options, we divide the life of the option into n time steps of length Δt. Suppose that the asset price at the beginning of a step is S. At the end of the time step it moves up to Su with probability p and down to Sd with probability 1 − p. For an investment asset that provides no income the values of u, d and p are given by

numbered Display Equation

with

numbered Display Equation

where r is the risk-free rate and σ is the volatility.

Figure F.1 shows the tree constructed for valuing a five-month American put option on a non-dividend-paying stock where the initial stock price is 50, the strike price is 50, the risk-free rate is 10%, and the volatility is 40%. In this case, there are five steps so that Δt = 0.08333, u = 1.1224, d = 0.8909, a = 1.0084, and p = 0.5073. The upper number at each node is the stock price and the lower number is the value of the option.

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FIGURE F.1 Binomial Tree from DerivaGem for American Put on Non-Dividend-Paying Stock

At the final nodes of the tree, the option price is its intrinsic value. For example, at node G, the option price is 50 − 35.36 = 14.64. At earlier nodes, we first calculate a value assuming that the option is held for ...

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