Distribution of Black Scholes call option price at time 0<t <T

Question

Does anyone know how to find the probability law (distribution) under P* of a Black Scholes Call Option price $C_t$ for $0 < t < T $?

(Under P*, $ dC_t = \frac{\partial c}{\partial s}\sigma S_t dW_t^{*} + rcdt $, where $C_t = c(s,t)$, $t \in [0,T]$ )

I'm expecting it will not be geometric Brownian motion but I'm not sure how to prove it.

Thanks!

Answer 1

You're correct. The distribution is not even close to log normal.

Think about an at-the-money call option.

It has roughly $50$% probability (i.e, $N(d_2)\approx. 0.5$ for an ATM call) of expiring out of the money.

In such cases, the value is zero.

It also has roughly a $50$% chance of expiring in the money, with a density function that decreases to zero as the expiry value increases.

So, the overall density function has a "Dirac Delta-like" behavior at $0$ with an area of roughly $50$% followed by a decreasing, convex behavior for values greater than zero.