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At time t, the value of the asset is St and at time t + At, for At > 0, the asset price is either increased to uSt or decreased to dSt, 0 0 and the asset pays a continuous dividend yield D > 0. Assume that the log-returns log(Sesat) under the risk-neutral measure have the following distribution log(St+at) ~ N [(r-D- St where o is the asset price volatility. D = 202)At,o-At] (i) Show by using the first two moments of St+At given St and the properties of log-normal distribution that pu + (1 - Pd=er-D)At u? + (1 - p)d? = e(2(r-D)+02)^t where and (1-7) are the risk-neutral probabilities of upward and downward movement of the asset price, respectively. (ii) By setting u = - show that elr-D)At-d u = A + VA2 1 d= A - VA2 1 and = ? u-d where A = [e(r=D)At + elsD+o?)at By expanding using Taylor series formula on u and d up to O(At) show that (iii) u = covAt and d = e-ovat and to ensure P, (1-P) (0,1) deduce that 0 0, the asset price is either increased to uSt or decreased to dSt, 0 0 and the asset pays a continuous dividend yield D > 0. Assume that the log-returns log(Sesat) under the risk-neutral measure have the following distribution log(St+at) ~ N [(r-D- St where o is the asset price volatility. D = 202)At,o-At] (i) Show by using the first two moments of St+At given St and the properties of log-normal distribution that pu + (1 - Pd=er-D)At u? + (1 - p)d? = e(2(r-D)+02)^t where and (1-7) are the risk-neutral probabilities of upward and downward movement of the asset price, respectively. (ii) By setting u = - show that elr-D)At-d u = A + VA2 1 d= A - VA2 1 and = ? u-d where A = [e(r=D)At + elsD+o?)at By expanding using Taylor series formula on u and d up to O(At) show that (iii) u = covAt and d = e-ovat and to ensure P, (1-P) (0,1) deduce that 0

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