Interpreting Return as Percentage Price Change in Black–Scholes
Summary
The document interprets the stock return term in the Black–Scholes setting as a proportional, rather than dollar, rate of price change. Dividing the instantaneous price change by the current price expresses movement as a percentage change. The discrete-time analogue given is the change in log price per unit of time, whose continuous-time limit is the time derivative of log price.
A second answer connects the return parameter in a geometric Brownian motion model to the annualized drift: if the random component is removed, the proportional price change equals that drift. The discussion uses simplified notation and does not fully reconcile the stochastic differential equation’s noise term with the stated return expression. In a stochastic model, the realized price change includes random movement as well as drift, so the drift should not be read as the return realized over every instant.
Key ideas
- Dividing a price change by the current price expresses it as a proportional change.
- The continuous-time log-price rate corresponds to the instantaneous proportional price change.
- In the Black–Scholes model, the drift parameter represents the modeled annual rate of return.
- Realized changes in a stochastic price process include both drift and random movement.
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Full text
# Rate of return in Black-Scholes model
# Rate of return in Black-Scholes model
The rate of return of a stock is denoted $\frac{dS}{S dt}$ where $S$ is the solution to the SDE modeling the price of a stock. Can someone give an explanation of the rate of return and what it is supposed to represent in this context.
## Answer by Alex C (score 1, accepted)
https://quant.stackexchange.com/a/48862
In discrete time the (annualized) logarithmic return is defined as $\frac{\Delta \ln(S)}{\Delta t}=\frac{\ln(S_{t+\Delta t})-\ln(S_t)}{\Delta t}$
In continuous time this becomes $\frac{d \ln(S)}{d t}=\frac{1}{S}\frac{d S}{d t}$
Note that $\frac{d S}{dt}$ is the instantaneous rate of change in price, dividing it by $S$ turns it into a percentage change in price. And that is how we usually assess stock price changes, in percentage terms.
## Answer by user18399 (score 2)
https://quant.stackexchange.com/a/48859
A fundamental premise of the BS model is that equity prices move according to a Weiner process. Moreover, when there are a series of many small random movements in the share price the track that it is tracing can be assumed to be geometric Brownian motion.
This process is then symbolically defined as (see Ito's lemma)
> $$dS = \mu S dt + \sigma dz$$
Where
- $\mu$ is constant and represents the return on the share reported as an annual rate
- $\sigma$ is constant and represents the share's volatility also reported as an annual rate
- $dt$ is an infinitesmal passage of time
- $dz$ represents a term which generates randomness into the movement of the Share price, $S$.
In this process, if there is no randomness, $dz=0$, thus \begin{align} dS &= \mu S dt \\ \implies \frac{dS}{S dt} &= \mu \end{align}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.