Assume the price, S, of a non-dividend-paying stock follows geometric Brownian motion with drift r and volatility o. Consider a perpetual call option on S. The option is exercised when S = §, and the payoff on the option is $-X, where X is the exercise price of the option. Assume all investors are risk-neutral and the instantaneous risk-free rate of interest is a constant, r. Note that the drift of the stock-the instantaneous rate of return on the stock-is equal to r. Of course, this makes sense, since the return on all traded assets in a risk-neutral world must be the risk-free rate of interest. Compute the value of the call option C(S;$) and the optimal exercise policy $ = arg max[C(S; $)].

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Step 1 A call option gives the right to the buyer to buy a stock at an exercise price X An option i... View the full answer