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FIN 524B Derivatives
Generalization: Binomial Trees Recall from last lecture The time zero value of an option in an n-step binomial tree (without path-dependence) is given as f0 = e −rn∆t n∑ k=0 ( n k ) qk(1− q)n−kf(n, k) (1) where each time step describes a time period ∆t, and the option has maturity n∆t = T . f(n, k) denotes the value of the option after n time steps and exactly k ≤ n up movements (final nodes). q = e r∆t−d u−d is the risk-neutral probability of an up-jump, where r is the annual risk-free rate, u is the factor by which the underlying stock jumps up and d is the factor by which the stock jumps down( n k ) = n!k!(n−k)! Professor Linda M. Schilling Olin Business School FIN 524B Derivatives 2 / 12 Binomial Coefficients Binomial coefficients form the Pascal’s triangle ⇒ Trick to count paths for large trees From first to second layer: 1-step tree From first to third layer: 2-step tree From first to fourth layer: 3-step tree Professor Linda M. Schilling Olin Business School FIN 524B Derivatives 3 / 12 Application: Large Binomial Tree Exercise Consider a time horizon of one year T = 1, split into 4 time periods n = 4 each with length 3 months ∆t = 3/12. Assume the current stock price in the market equals S0 = 50. The risk-free rate equals r = 0.05. The stock increases or drops by 10% in each period, i.e. jumps up or down by factors u = 1.1, d = 0.9 at every time step. The stock pays no dividends. You are interested in a European Call option on the stock with strike K = 50 and maturity T = 1 year (a) Calculate the risk-neutral probability of an up-jump of the stock. b) Calculate the values of the underlying stock and then the values of the European Call with strike K = 50 in all final nodes. c) Price a European Call option on the stock with strike K = 50 and maturity T = 1 year in the given binomial tree. Professor Linda M. Schilling Olin Business School FIN 524B Derivatives 4 / 12 Application: Large Binomial Tree Solution (a) Calculate the risk-neutral probability of an up-jump of the stock. For each jump the stock makes, the risk-neutral probability of an up-jump equals q = e0.05×3/12 − 0.9 1.1− 0.9 = 0.5629 Professor Linda M. Schilling Olin Business School FIN 524B Derivatives 5 / 12 Application: Large Binomial Tree Solution continued b) Calculate the value of a European Call option on the stock with strike K = 50 and maturity T = 1 year in all final nodes. The stock jumps n = 4 times. Therefore Suuuu = S0u 4 = 73.205, Suuud = S0u 3d = 59.895, Suudd = S0u 2d2 = 49.005, Suddd = S0ud 3 = 40.095, and Sdddd = 32.805 The value of the Call option is given by fET = max(ST −K, 0). Denote by f(4, k) the value of the Call option after 4 jumps when exactly k up jumps have occured. [e.g. since there is no path-dependence, f(4, 1) = fuddd = fdudd = fdddu, similarly f(4, 2) = fuudd = fudud = fduud, i.e. for the final option value only the number of up jumps matters, not the sequence in which they occur.] The option values in the final nodes are therefore f(4, 4) = fuuuu = 23.205, f(4, 3) = fuuud = 9.895, f(4, 2) = fuudd = 0, f(4, 1) = fuddd = 0, and f(4, 0) = fdddd = 0. Professor Linda M. Schilling Olin Business School FIN 524B Derivatives 6 / 12 Application: Large Binomial Tree Solution continued c) Price the European Call on the stock with strike K = 50 and maturity T = 1 year in the given binomial tree.