Black–Scholes Stock Drift Under Real-World and Risk-Neutral Measures
Summary
The document clarifies two commonly seen stock-price expressions in the Black–Scholes model. For a geometric Brownian motion with arithmetic drift μ and volatility σ, applying Itô’s lemma gives a log-price exponent that includes the volatility correction, −σ²/2. Thus, the expression with μ alone in the exponent is not the solution to the stated stochastic differential equation.
It also distinguishes the probability measures used for different purposes. Under the real-world measure, the stock’s arithmetic drift is μ; under the risk-neutral measure used in option pricing, that drift is the risk-free rate r. The corresponding Brownian motion is measure-specific, and Girsanov’s theorem relates the two formulations. The explanation is conceptual and does not specify assumptions such as dividends or market completeness, so the stated risk-neutral drift applies to the simplified setup described in the document.
Key ideas
- The solution to a geometric Brownian motion includes a negative half-variance term in the exponent.
- The arithmetic drift under the real-world measure is denoted by μ.
- In the stated option-pricing setup, the risk-neutral arithmetic drift is the risk-free rate.
- The Brownian motion changes with the measure, and Girsanov’s theorem connects the formulations.
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Full text
# Which expression of $S_t$ to use under the Black-Scholes model?
# 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?
## Answer by byouness (score 1)
https://quant.stackexchange.com/a/39930
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.