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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?

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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:

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

Second, the use of one drift or the other depends on which measure you are considering:

  • 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 = $\mu$: $$dS_t = S_t\left(\mu dt + \sigma dW^{\mathbb{P}}_t \right)$$

Girsanov's theorem enables you to move from one to the other.

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