Deriving the Geometric Brownian Motion Solution with Itô’s Lemma
Summary
The document explains how to motivate the closed-form solution to the stock-price stochastic differential equation used in Black–Scholes. Rather than guessing the exponential solution, it applies Itô’s formula to a transformation of the price process. Since the diffusion term in the original equation scales with the price, the proposed transformation is chosen so its derivative cancels that dependence.
This condition leads to a logarithmic transformation, whose derivative is the reciprocal of price. Applying Itô’s formula to the log process then yields dynamics that can be integrated to obtain the exponential form. The reasoning is presented as an intuitive route to selecting the transformation, not as a fully rigorous derivation. It explains the key idea behind the solution but does not discuss parameter estimation, empirical fit, or limitations of the Black–Scholes assumptions.
Key ideas
- The stock price in the Black–Scholes model follows a geometric Brownian motion.
- Itô’s formula describes how a transformed process inherits drift and diffusion terms.
- Choosing a logarithm makes the price-dependent diffusion coefficient constant.
- Integrating the transformed process gives the exponential solution for the original price.
- The transformation choice is motivated intuitively rather than proved through a complete rigorous argument.
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# How to get Geometric Brownian Motion's closed-form solution in Black-Scholes model?
# How to get Geometric Brownian Motion's closed-form solution in Black-Scholes model?
The Black Scholes model assumes the following dynamics for the underlying, well known as the Geometric Brownian Motion: $$dS_t=S_t(\mu dt+\sigma dW_t)$$
Then the solution is given: $$S_t=S_0\,e^{\left(\mu-\frac{\sigma^2}{2}\right)t+\sigma W_t}$$
It can be shown by Ito Lemma on function $f(t,W_t)=\ln S_t$ that this solution is correct as it leads to above dynamics.
But how do we solve the above SDE originally to find this solution?
Guessing the above solution to apply Ito seems unlikely to me.
## Answer by Kiwiakos (score 7, accepted)
https://quant.stackexchange.com/a/14113
If by 'solve' you mean how do we know that $\ln S_t$ is the right change of variable, then you can go by the following (not rigorous) line of thought:
- Ito's fomula suggests that given an SDE $$dX_t = \mu(X_t,t)dt+\sigma(X_t,t)dW_t$$ and a function $f(x,t)$: the SDE for the process $Y_t=f(X_t,t)$ will satisfy $$dY_t = [f_t(X_t,t) + f_x(X_t,t)\mu(X_t,t) + \frac{1}{2}f_{xx}(X_t,t)\sigma^2(X_t,t)]dt+f_x(X_t,t)\sigma(X_t,t)dW_t$$
- Now the SDE for the spot price is, as you wrote, given by $$dS_t = \mu S_tdt+\sigma S_t dW_t$$: hence a transformation $f$ applied to this SDE will have dynamics given by $$dY_t = [...]dt+f_x(S_t,t)\sigma S_t dW_t$$
- We want the transformation to kill the dependence of volatility on the spot, therefore we want to 'solve' something like '$f_x(S_t,t)\sigma S_t = const.$' which essentially means '$f_x(x,t) = \frac{1}{x}$'. This points towards the guess $$f(x,t)=\ln x$$.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.