Solving Geometric Brownian Motion with Time-Varying Drift and Volatility
Summary
The document derives the solution to a geometric Brownian motion whose drift and volatility vary with time. Applying Itô's lemma to the logarithm of the asset price produces a drift term adjusted by half the instantaneous variance, alongside a stochastic integral weighted by the time-varying volatility. Integrating from the initial time gives the log price, and exponentiating yields the asset price as its initial value times the exponential of those two integrals.
The solution generally cannot be simplified further without specifying the functions for drift and volatility. Under the stated deterministic time-dependent coefficients, the log price remains normally distributed, so the asset price is lognormally distributed. The discussion identifies the analytical form but does not work through particular coefficient functions or calculate moments explicitly.
Key ideas
- Applying Itô's lemma to the log price introduces a drift correction equal to half the instantaneous variance.
- The solution uses time integrals of adjusted drift and a stochastic integral of volatility.
- The asset price is obtained by exponentiating the integrated log-price expression.
- Further simplification requires specifying the time-dependent drift and volatility functions.
- With the stated coefficient structure, the asset price remains lognormally distributed.
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# Solution to geometric Brownian motion with time dependent volatility and drift?
# Solution to geometric Brownian motion with time dependent volatility and drift?
I am able to compute the general solution of a standard geometric Brownian motion, but I'm struggling to find the general solution for a GBM where volatility and mean depend on time, $$\text{d}S_t = \mu(t) S_t\text{d}t+\sigma(t) S_t\text{d}W_t.$$
The general solution for a standard geometric Brownian, $\text{d}S_t = \mu S_t\text{d}t+\sigma S_t\text{d}W_t$ can be computed by firstly separating the variables $\frac{\text{d}S_t}{S_t} = \mu \text{d}t+\sigma \text{d}W_t$, then taking integration on both sides $\int\frac{\text{d}S_t}{S_t} = \int \mu dt+\sigma dW_t$. Since $\frac{\text dS_t}{S_t}$ links to the derivative of $\ln(S_t)$, the proceeding step constitutes the Itô calculus and results in $\ln(S_t) = (\mu - \frac{1}{2} \sigma^2)t + \sigma W_t$. Then taking exponential on both sides and plugging in the initial condition $S(0)$ we obtain the analytical solution $S(t) = S(0) e^{(\mu - \frac{1}{2} \sigma^2)t+ \sigma W_t}$
However, when $\mu$ and $\sigma$ are time dependent $\text{d}S_t = \mu(t) S_t\text{d}t+\sigma(t) S_t\text{d}W_t$, the solution is totally different and I tried applying the same methods I used in a standard geometric Brownian motion but the solution is not correct. I have found some material online but it doesn't seem to make sense to me ... I am able to continue up until integrating on both sides, then after that I don't know what to do.
## Answer by Kevin (score 3, accepted)
https://quant.stackexchange.com/a/61936
Consider the generalised geometric Brownian motion $$\text{d}S_t = \mu(t)S_t \text{d}t+\sigma(t)S_t \text{d}W_t.$$
Using Itô's Lemma, you get $$\text{d}\ln(S_t) = \left(\mu(t)-\frac{1}{2}\sigma^2(t)\right)\text{d}t+\sigma(t) \text{d}W_t.$$ Thus, by definition of an SDE, $$\ln(S_t) =\ln(S_0)+\int_0^t \left(\mu(s)-\frac{1}{2}\sigma^2(s)\right)\text{d}s+\int_0^t\sigma(s) \text{d}W_s.$$ Thus, $$S_t =S_0\exp\left(\int_0^t \left(\mu(s)-\frac{1}{2}\sigma^2(s)\right)\text{d}s+\int_0^t\sigma(s) \text{d}W_s\right).$$ You cannot simplify these integrals without assuming what your drift and variance are. You can however compute moments of the stock price, etc. The process is still log-normally distributed.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.