3. Consider the set-up of Section 5.5.2 in Goodman &Stampfli. Derive, by directly evaluating the expectation, the Black-Scholes formula for the time-t price P of the European put option, t e [0,T. 5.5.2 The Expected Value In the case of our European call option, the final payoff is (ST X)so equation (5.14) becomes We are using the model given by equation (5.16): Now rewrite this expression. Recall that BT is a normal random variable with mean 0 and variance T. We may substitute TZ for BT, where Z is the standard normal variable (mean 0, variance 1). Then Hence and so, by the basic rule for computing an expected value. 3. Consider the set-up of Section 5.5.2 in Goodman &Stampfli. Derive, by directly evaluating the expectation, the Black-Scholes formula for the time-t price P of the European put option, t e [0,T. 5.5.2 The Expected Value In the case of our European call option, the final payoff is (ST X)so equation (5.14) becomes We are using the model given by equation (5.16): Now rewrite this expression. Recall that BT is a normal random variable with mean 0 and variance T. We may substitute TZ for BT, where Z is the standard normal variable (mean 0, variance 1). Then Hence and so, by the basic rule for computing an expected value