Applying Itô’s Lemma to an Exponential Function of Geometric Brownian Motion

Summary

The document asks how to specify a geometric Brownian motion for a stock price and derive the dynamics of a function of that price. It states the usual drift-and-diffusion form for the stock and considers the function exp(t times the square of the stock price). The answer corrects the proposed Itô expression by including the squared stock level in the second-order drift term and the function’s price derivative in the diffusion term.

It then lists the time derivative, first price derivative, and second price derivative, and substitutes them to give the resulting drift and stochastic components in proportional form. This illustrates how Itô’s lemma accounts for curvature through the second derivative, beyond ordinary chain-rule differentiation. The example is narrowly mathematical and assumes the stated constant drift and volatility model; it does not discuss estimation, risk-neutral pricing, or empirical fit. As with any formula transcription, the differential notation and parentheses should be checked carefully when applying it.

Key ideas

• A geometric Brownian stock model has drift proportional to the stock level and diffusion proportional to the stock level.
• Itô’s lemma includes a second-derivative adjustment in the drift of a nonlinear function of the stock.
• The diffusion term for the transformed process includes the first derivative of the function.
• For the exponential function of time and squared stock price, derivatives can be substituted directly into Itô’s formula.

Tags

# How to compute the dynamic of stock using Geometric Brownian Motion?


# How to compute the dynamic of stock using Geometric Brownian Motion?












I have been given the following question:

> Given that $S_t$ follows Geometric Brownian Motion, write down the dynamic of $S_t$ and then compute the dynamic of $f(t,S_t) = e^{tS^{2}}$

For the first part of the question, I have got this answer: $$dS_t = \mu S_tdt + \sigma S_t dWt$$

Is it correct?

And for the second part, I know that the price $f(t,S_t)$ follows the process $$df = (\frac{\partial f}{\partial t}+\mu S_t \frac{\partial f}{\partial S_t}+\frac{1}{2} \sigma ^2S_t\frac{\partial^2f}{\partial S_t^2})dt +\sigma S_t dWt$$

I am having trouble finding the answer using this process and given the information.

Any help is appreciated.

## Answer by ZRH (score 4)

https://quant.stackexchange.com/a/44223

The above equation should correctly read as follows:

$df=\big(\frac{\partial f}{\partial t}+\mu S_t \frac{\partial f}{\partial S_t}+\frac{1}{2}\sigma^2 S_t^2\frac{\partial^2 f}{\partial S_t^2}\big)+\sigma S_t \frac{\partial f}{\partial S_t}dW$

Using:

(a) $\frac{\partial f}{\partial t}=S_t^2f$

(b) $\frac{\partial f}{\partial S_t}=2S_ttf$

(c) $\frac{\partial^2 f}{\partial S_t^2}=2tf+4S_t^2t^2f$

The Stochastic Differential Equation (SDF) governing the dynamics of $f$ becomes:

$\frac{df}{f}=dt \big(S_t^2+2 \mu S_t^2t+\sigma^2S_t^2t+2\sigma^2S_t^4t^2 \big)+2S_t^2t\sigma dW$

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.