Applying Itô’s Lemma to a Mean-Reverting Stock Option

Summary

The document asks how to apply Itô’s lemma to an option value modeled as a function of time and a stock price. The stock has a drift that pulls its level toward a parameter P at a rate set by omega, and its diffusion term is proportional to the stock price. Applying the standard one-dimensional formula produces a time component, a first-derivative term multiplied by the stock’s drift, a second-derivative term from the diffusion variance, and a stochastic term proportional to the option’s sensitivity to the stock.

The expression shown in the document is the direct Itô expansion for the stated process: the diffusion coefficient is S, so its squared value multiplies half the second derivative. This is a calculus step, not a full option-pricing result. To derive a pricing equation or price, further assumptions and conditions—such as a hedging or pricing framework, interest rate, and terminal payoff—would be needed.

Key ideas

• For a function of time and a stochastic stock price, Itô’s lemma includes time, drift, curvature, and stochastic terms.
• The stock’s drift term contributes through the option value’s first derivative with respect to the stock.
• The diffusion coefficient is squared in the second-derivative term.
• The stochastic component is the stock diffusion multiplied by the option’s first derivative.
• The Itô expansion alone does not determine an option price or a pricing equation.

Tags

# Ito's Lemma in option pricing for a stock satisfying $dS=\frac{P-S}{\omega}dt+SdW_t$


# Ito's Lemma in option pricing for a stock satisfying $dS=\frac{P-S}{\omega}dt+SdW_t$












Suppose a stock follows the stochastic differential equation

$$dS=\frac{P-S}{\omega}dt+SdW_t,$$

such that $W_t$ is a wiener process, $\omega\in\mathbb{R}^+$, and $P_t,S_t\in\mathbb{R}$. If the value of an option is $V(t,S_t)$, what is the value of $dV(t,S_t)$ given by Itô's Lemma (i.e. similar to the stochastic derivation of the Black-Scholes formula)? Any help would be much appreciated.

So far I have,

$$dV=\left(\frac{\partial V}{\partial t}+\frac{P-S}{\omega}\frac{\partial V}{\partial S}+\frac{S^2}{2}\frac{\partial^2 V}{\partial S^2}\right)dt+S\frac{\partial V}{\partial S} dW_t,$$

but I'm not sure whether this is correct or how best to simplify.

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