# 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 $$V(t,S_t$$)?

## 1 Answer

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 $$dS_t\,.$$

• So, would Ito's formula give $dV=\left(\frac{\partial V}{\partial t}+\omega(\Lambda-S)\frac{\partial V}{\partial S}+\frac{\sigma^2 S^2}{2}\frac{\partial^2 V}{\partial S^2}\right)dt+\sigma S\frac{\partial V}{\partial S} dW_t$? How is $\omega$ dealt with, since it is a stochastic process (i.e. with risk-neutral pricing for constant $\omega$, the $\omega(\Lambda-S)$ term drops out)? Nov 17, 2021 at 11:47
• Is $S$ a stock that does not pay dividends ? If so your equation containing the drift $\omega(\Lambda-S)\,dt$ is not under the risk-neutral measure $Q$. Under $Q$ the drift is (as we know) $r_t\,S_t\,dt\,.$ In other words, if you are only interested in risk-neutral pricing you don't even have to worry about $\omega\,.$ Nov 17, 2021 at 12:25
• Or let me be more precise: under $Q$ your $\Lambda$ must be zero and your $\omega_t$ can be viewed as the negative of riskless interest rate $r_t$. Without further background of what you want to achieve it is hard to say more. Nov 17, 2021 at 12:30
• So, when performing Monte Carlo simulation for the non-dividend paying call option (i.e. the discounted value of $\mathbb{E}(S_t-K)$ where $S_t$ can be defined by $dS$), what framework is this (the expected value of the call is dependent on $\Lambda$ and $\omega$, so this isn't under $Q$)? Nov 17, 2021 at 12:38
• It isn't under $Q\,.$ If it is under the real-world measure the authors of your model should know. Under $Q$ the non dividend paying stock is a martingale when divided by the savings account. I.e. $\exp(-\int_0^tr_s\,ds)\,S_t$ is a $Q$-martingale. This implies that the drift must be $r_tS_t\,dt\,.$ No other drift is possible. Nov 17, 2021 at 15:21