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I don't even know where to get started with this question...can someone please help me? How do I answer it?

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    $\begingroup$ This is a basic question and can be looked up in any textbook on option theory $\endgroup$
    – T123
    Feb 27 at 8:01
  • $\begingroup$ Exercise occurs when $S_T \ge K$, so you can find the prob of exercise by integrating the prob density of $S_T$ between $K$ and $\infty$. Manipulation of the integral will give the above result. $\endgroup$
    – nbbo2
    Feb 27 at 10:31
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    $\begingroup$ Alternatively you can do $d/dK$ of the call option formula , since the probability of $S>K$ is also equal to the price of a narrow call spread at $K$ (multiplied by the PV factor $e^{-r(T-t)}$ $\endgroup$
    – dm63
    Feb 27 at 11:25

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The unique solution of the stochastic differential equation that is mentioned: $$dS_t = S_t\mu dt + S_t \sigma dW_t$$ is given by: $$S_T = Se^{(\mu - \frac{\sigma^2}{2})(T-t) + \sigma (W_T-W_t)}=Se^{(\mu - \frac{\sigma^2}{2})(T-t) + \sigma \sqrt{(T-t)} X}$$ where $X\sim\mathcal{N}(0,1)$. It gives the "real-world" dynamic of the risky asset $S$.

On the other hand, the risk-neutral dynamic is almost the same but with $r$ instead of $\mu$, therefore: $$dS_t = S_trdt + S_t \sigma dW_t^Q$$ and thus $$S_T = Se^{(r - \frac{\sigma^2}{2})(T-t) + \sigma (W_T^Q-W_t^Q)} = Se^{(r - \frac{\sigma^2}{2})(T-t) + \sigma \sqrt{(T-t)} X}$$

where $W^Q$ is the Standard Brownian Motion under measure $Q$ (you can refer to Girsanov Theorem for details) and thus, $X \overset{Q}{\sim}\mathcal{N}(0,1)$.

The option with payoff $(S_T-K)_+$ is priced using the risk-neutral probability $Q$ which is induced by the no-arbitrage assumption. The option price is given by:

$$e^{-r(T-t)}\mathbb{E}_Q\left[(S_T-K)_+ | \mathcal{F}_t \right]$$

which is basically the discounted expected payoff at time $t$ under the risk-neutral measure. The formula that is given is the result of this conditional expectation within the framework of Black-Sholes model.

In short, for a) it is enough to calculate the risk-neutral probability that $S_T>K$ which means that the option will be exercised (you can purchase the asset $S$ for $K$ instead of $S_T$). Therefore, using the aforementioned dynamic (just substitute) calculate:

$$P^Q(S_T>K)$$

For b) the reasoning is exactly the same, but with $\mu$ exchanged for $r$.

For details I would refer you to any textbook or online resources on Black-Sholes model.

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