# Can someone please help me answer this question about Black-Scholes model? (risk-neutral & true probability of the call option) [closed]

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

• This is a basic question and can be looked up in any textbook on option theory
– T123
Feb 27 at 8:01
• 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. Feb 27 at 10:31
• 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 $e^{-r(T-t)}$
– dm63
Feb 27 at 11:25

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.