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Let $(X_t)_{t\geq 0}$ denote a Geometric Brownian Motion $$\frac{dX_t}{X_t} = \mu_X dt + \sigma_X dW^X_t,\ \ \ X(0) = X_0$$ such that $X_t$ is lognormally distributed $\forall t > 0$ $$X_t = X_0 e^{(\mu_X - \frac{1}{2}\sigma_X ^2)t + \sigma_X W_t^X}$$ Let $(Y_t)_{t\geq 0}$ denote an Arithmetic Brownian Motion $$dY_t = \mu_Y dt + \sigma_Y dW_t^Y,\ \ \ ... 5 Thanks to @Phun and @oliversm I solved the problem. So I'm posting here the solution in case someone will need it. Under Black-Scholes assets dynamics are determined by a Geometric Brownian Motion, and we can define the price of a security at time t+\Delta t as:$$S_{t+\Delta t}=S_{t}\exp\left(\left(r-\frac{1}{2}\sigma^{2}\right)\Delta t+\sigma\sqrt{\...
Starting from the Black-Scholes model that $$\dfrac{dS}{S} = \mu \:dt + \sigma\:dW_t$$ where $W_t$ is a standard Brownian motion, and $\sigma$ and $\mu$ are constant where $\sigma > 0$. Here $W_t$ is a Brownian motion under the physical measure $\mathbb{P}$. We can then use Girsanov's theorem to change the measure to risk neutral measure $\mathbb{Q}$ ...