# Which expression of $S_t$ to use under the Black-Scholes model?

I am currently looking at example exam questions relating to the evolution of a stock price under the Black-Scholes model. However, I am confused by seemingly inconsistent expressions used for the evolution of the stock price $S_t$.

For a stock $S$ within a BMS stochastic market with

• Drift $\mu$
• Volatility $\sigma$
• Interest rate $r$

both of the following expressions are used to model the price of $S$ at time $t$:

$$S_t = S_0 \exp(\mu t + \sigma W_t)$$

and

$$S_t = S_0 \exp \left( \left( r - \frac{\sigma^2}{2} \right) t + \sigma W_t \right)$$

Assuming that both of these expressions are correct, when should one be used rather than the other? The difference seems to lie entirely in the fact that sometimes $\mu$ is used as the drift coefficient and sometimes $\left( r - \frac{\sigma^2}{2} \right)$ is used as the drift coefficient.

Can someone please explain the use of each of these?

First, if you assume that the diffusion equation is: $dS_t = S_t (\mu dt + \sigma dW_t)$, then $S_t = S_0 \exp(\mu t + \sigma W_t)$ is not correct! It should instead be:
$$S_t = S_0 \exp\left(\left(\mu - \frac{\sigma ^2}{2} \right) t + \sigma W_t\right)$$
• Under the risk-neutral measure $\mathbb{Q}$ (used to price options on $S$), the drift = $r$: $$dS_t = S_t\left(rdt + \sigma dW^{\mathbb{Q}}_t \right)$$
• Under the real-world measure $\mathbb{P}$ the drift = $\mu$: $$dS_t = S_t\left(\mu dt + \sigma dW^{\mathbb{P}}_t \right)$$