EXERCISE IV. A.2. Why do we have the square root of At multiplying the volatility? What will go wrong if we use some other power? Consider the following model, = exp (mAt+ 0At")+1) , (IV.A.4) Sn where Y,, is a random variable with zero mean and variance 1, and the power a of At is some fixed real number. Attempt to calculate the limiting distribution of S(7) = Sw in the limit N -> co and At =T/N with a fixed 7'. We have done it in class for o = 1/2. Consider two cases: (a) a 1/2. Hint: We want to calculate the distribution of In(Sw/So) which we can write as SN SN SN-1 SN SN-1 In = In = In + In + . .. + In So SN-1SN-2 So SN-1 SN-2. Use model (IV.A.4) with n = N-1, N-2, .. .,0 to get an expression similar to equation (IV.1.17) and then attempt to use the CLT.Chaining this recursion from time 0 to time T is steps of size at (N - T/ at steps altogether), we get N Sr - SoII (1 + pat + OVALY.). Then the log-return over the period from 0 to T is N In (Sr/So) - In (1 + pat + ovaLY.) Let us expand the logarithm into Taylor series In(1 + x) = x - ja' + $23-... resulting in In (1 + pat + ovalY. ) - (ust toVARY.) -" (uAt + OVALY.) +... and drop the terms of order At3/2 and higher This results in the approximation In (Sr/So) = (mat toVAIN -" (ovair.)"). (IV.1.17) Remembering that at - T/N, we have N In (Sr/So) AT 2 In the middle term, we recognize the form of the Law of Large Numbers, 1 N 2 CY? = E(Y?) - 1. The last term fits the Central Limit Theorem, VN SYN(0, 1). As a result, we obtain an already familiar form, In (ST/So) - ( 1 -, ) T + ovTZ, Z ~ N(0, 1 ). (IV.1.18)