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【正文】 = * 20 ? The value of the option is therefore = 8 Options, Futures, and Other Derivatives, 4th edition 169。 2023 by John C. Hull Irrelevance of a Stock’s E(R) Proof: (continued) ? Let’s call pu the probability of an increase in the stock price and pd=1 pu the probability of a stock decrease S0D f = [pu(S0uDfu)+ pd(S0dDfd)] ekT where k is the appropriate rate for the risk involved ? However, D is chosen such that S0uDfu= S0dDfd and we know that pd=1 pu ? Substituting, S0D f = [pu(S0uDfu)+ (1 pu)(S0uDfu)] ekT = (S0uDfu)erT as since this is riskfree, k = r ? No pu’s or pd’s left, thus probability of stock increase is irrelevant ? 16 Options, Futures, and Other Derivatives, 4th edition 169。 2023 by John C. Hull A TwoStep Example 20 22 18 *?? ??? ??Ddudp TrNote the change in the formula for p with a multistep tree ? Figure ? Each time step is 3 months, r is still 12%. 24 Options, Futures, and Other Derivatives, 4th edition 169。 2023 by John C. Hull Assignment ? , , , , , , , , , Assignment Questions 32 Options, Futures, and Other Derivatives, 4th edition 169。 2023 by John C. Hull ? Figure , X = 52, u = , d = , r = 5%, and T = 2 12 What Happens When an Option is American? 72 0 48 4 32 20 60 40 12 50 A B C D F E 6282 . 0 8 . 0 2 . 1 8 . 0 e e * ? ? ? ? ? ? ? D d u d p T r ? Rule: ?The value of the option at the final nodes is the same for the European option ?At earlier nodes it is the greater of The value given by () The payoff from early exercise 28 Options, Futures, and Other Derivatives, 4th edition 169。 2023 by John C. Hull RiskNeutral Valuation ? The variable ? does NOT appear in the solution to the option value using the Binomial Method ? Thus, the solution is independent of all variables affected by risk preference ? The solution is therefore the same in a riskfree world as it is in the real world ? Hence, we can assume that the world is risk neutral ? This leads to the principle of riskneutral valuation ? 20 Options, Futures, and Other Derivatives, 4th edition 169。 2023 by John C. Hull Generalization (continued) ? Substituting for D we obtain ? = [ p ?u + (1 – p )?d ]e–rT where p e du drT? ??12 Options, Futures, and Other Der
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