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Applying Itô’s Lemma to a Mean-Reverting Diffusion with Stable Drift

Article Quant Q&A · Author: UNOwen
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Summary

The document considers an asset whose price follows a mean-reverting diffusion with a time-varying stable-process drift, then asks whether option values require a modified Itô formula. The answer says the stated price process remains continuous and has quadratic variation determined by its Brownian diffusion term. Therefore, the standard Itô formula applies to a sufficiently smooth option value as a function of time and price; substituting the given price dynamics supplies its drift and diffusion contributions.

The key modeling distinction is between a stable random variable appearing in the drift and a process with discontinuous jumps. In the setup presented, the stable component enters the time-integrated drift, while the Brownian term supplies the quadratic variation. The response does not derive a pricing equation or discuss risk-neutral assumptions, boundary conditions, or numerical stability, so it addresses the calculus question rather than providing a complete option-pricing model.

Key ideas

  • A continuous price process with Brownian diffusion uses the standard Itô formula.
  • The quadratic variation in the stated model comes from the Brownian price component.
  • The stable random input appears in the drift and does not by itself require a jump-process Itô formula.
  • Applying the formula requires substituting the full price dynamics into the differential of the option value.
  • The response does not establish a risk-neutral pricing model or derive an option valuation equation.

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# Ito's lemma for option pricing with Levy-alpha stable drift


# Ito's lemma for option pricing with Levy-alpha stable drift












Consider

$$dS=\omega\left(\Lambda-S\right)dt+\sigma_S S dW_t,$$

such that such that $W_t$ is a Wiener process, $\sigma_S$ is constant, $\omega: t\rightarrow\mathbb{R}$ represents anticipated drift and is a stable process, and $\Lambda:t\rightarrow\mathbb{R}^+$ is deterministic. Numerically, the discrete form is

$$S_{t+\Delta t}=\Delta t\left(\sigma_S S_t\xi_t+(\zeta_t\Delta t+\omega_t)(\Lambda-S_t)\right)+S_t$$

where $\zeta\sim\mathcal{S}(\alpha,0,c,0)$ is Lévy alpha-stable distributed. Note that the characteristic function of $\zeta_t$ is

$$\varphi=e^{-\left|ct\right|^{\alpha}}.$$

So, when $\alpha=1$, we have the Cauchy distribution. Does a suitably modified version of Ito's lemma exist for the value $dV(t,S_t)$ of an option $V(t,S_t$)?

## Answer by Kurt G. (score 1, accepted)

https://quant.stackexchange.com/a/68808

By definition, your $S$ is a continuous process and $$ d\langle S\rangle_t=\sigma_S^2\,S_t^2\,dt\,. $$ The applicable Ito formula is the traditional one $$ dV(t,S_t)=\partial_tV(t,S_t)\,dt+\partial_SV(t,S_t)\,dS_t+\frac{1}{2}\partial^2_SV(t,S_t)\,d\langle S\rangle_t\,. $$ Now plug in your expressionfor $dS_t\,.$

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.