(8.9) Proposition 8.9 Let u, d, 36 and the risk-neutral SDE for S be as in Proposition 8.5. Then if we set ad 2 1, (8.9) u=eam and d=e_am as Alb)0. Comment: The condition u d = 1 is for convenience. To determine u, d and 13 in terms of a and At requires three conditions. Equating risk-neutral means and risk-neutral variances gives only two. Setting 16 = % is an alternative convenient condition. The values of u and d which use 16 = % are derived below in Proposition 8.17. (8.17) Proposition 8.17 Let u, d, p and the risk-neutral SDE for S be as in Proposition 8.5. Then if we set p = ? u = er At (1+ Veo2 At - 1 (8.17) and d = er At (1 - Veo2 At - 1 as At -> 0. 38 Use e2rAt = 1+ 2rAt+ O(At2) and e("to ) At = 1 + r+ 2 At+ O(At?) for At - 0, for example.(8.5) Proposition 8.5 Let u > d > 0 be such that either S(t+At) = S(t) u or S(t + At) = S(t) d, and let the risk-neutral SDE for S be dS = r Sdt +o SdW, where r is the riskless rate. Define p as the risk-neutral probability that S(t + At) = Su. Then er At (8.5) - d p = U - er At and 1- p = u - d u - d4. Assuming ud : 1 or 33 : 1/2, Proposition 8.9 and Proposition 8.17 in the notes (page 87 and page 89) give formulas for the 'up' and 'down' factors u and d in the riskneutral formulation of the binomial option pricing model. Consider a call option with strike price E : 9.90 and time to maturity T t : At : 1/ 12. Assume 7' : 0.05, 0 : 0.30, and 3(t) : 10.00. (a) Compute the values of u, d, and the riskneutral probability 33 for each u, d pair from the two propo sitions. (The general formula for f) is given in Proposition 8.5. The 13 in Proposition 8.17 needs no calculation 7 Why?) (b) Compute the two possible maturity call prices Cu and Cd, for assumption ud : 1 and another two possible maturity prices for the assumption g3 : 1 / 2. (It will rst be necessary to compute the Sn and Sal using the two different values of u and d from part (a).) (c) Use 0 : 6\"" At (13 Cu + (1 13) 04) to compute the price C of the call when we assume ud : 1 and again when we assume f) : 1 / 2. (d) The riskneutral model says that the two call prices in part (c) should be the same. Suggest a reason for the difference we obtained in part (c) by repeating parts (a), (b), and (c) using T t 2 At = 1/52 instead