Now, let's not assume that the dividend rate (q) is 0%. Our second approach for pricing...

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Now, let's not assume that the dividend rate (q) is 0%. Our second approach for pricing derivatives was the "martingale method." We proceeded in the following way: Under certain assumptions about the market there's a way to "shift the probabilities" for "outcomes" for the stock-price path. Select a traded asset, called the numeraire (N). We're given the stochastic process for the numeraire. Now, there's a way to adjust the evolution of S so that, under this new process, the price of any traded asset divided by the numeraire's price, V(t)/N(t), is a martingale. That means that the SDE for V(t)/N(t) has no "dt" term. We call this new probability rule the "martingale measure" associated to N, and we denote it QN. [a] Suppose we are in the Black-Scholes framework (as in Problem 1), and we choose as our numeraire the "bank account": dB = r B dt, B(0) = 1. As given, in the "real world" (P-measure) the stock propagates as: dS = a S dt+s SdW (P-world) In the "Q = QB world" S(t)/B(t) must be a martingale. It can be shown that in the Q-world the process for the stock is: ds = (r - q) S dt + s S dw (Q-world) Why is this result very plausible? [b] Show how this result, and the fact that, for any derivative V. V(t)/B(t) must be a martingale (under Q), leads immediately to our famous pricing formula. V(S,t) = exp(-t(T-t))E[Payoff formula (function of S(T))] or, equivalently, V(S.t) = exp(-t(T-t)) EQ[Payoff formula (function of S(T))] Now, let's not assume that the dividend rate (q) is 0%. Our second approach for pricing derivatives was the "martingale method." We proceeded in the following way: Under certain assumptions about the market there's a way to "shift the probabilities" for "outcomes" for the stock-price path. Select a traded asset, called the numeraire (N). We're given the stochastic process for the numeraire. Now, there's a way to adjust the evolution of S so that, under this new process, the price of any traded asset divided by the numeraire's price, V(t)/N(t), is a martingale. That means that the SDE for V(t)/N(t) has no "dt" term. We call this new probability rule the "martingale measure" associated to N, and we denote it QN. [a] Suppose we are in the Black-Scholes framework (as in Problem 1), and we choose as our numeraire the "bank account": dB = r B dt, B(0) = 1. As given, in the "real world" (P-measure) the stock propagates as: dS = a S dt+s SdW (P-world) In the "Q = QB world" S(t)/B(t) must be a martingale. It can be shown that in the Q-world the process for the stock is: ds = (r - q) S dt + s S dw (Q-world) Why is this result very plausible? [b] Show how this result, and the fact that, for any derivative V. V(t)/B(t) must be a martingale (under Q), leads immediately to our famous pricing formula. V(S,t) = exp(-t(T-t))E[Payoff formula (function of S(T))] or, equivalently, V(S.t) = exp(-t(T-t)) EQ[Payoff formula (function of S(T))]

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a ANSWER This result is very plausible because it makes sense that the rate of return of the stock ... View the full answer