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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 $V(t,S_t$)?

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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 $dS_t\,.$

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  • $\begingroup$ 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)? $\endgroup$
    – UNOwen
    Nov 17, 2021 at 11:47
  • $\begingroup$ 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 $\omega\,.$ $\endgroup$
    – Kurt G.
    Nov 17, 2021 at 12:25
  • $\begingroup$ 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. $\endgroup$
    – Kurt G.
    Nov 17, 2021 at 12:30
  • $\begingroup$ 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 $Q$)? $\endgroup$
    – UNOwen
    Nov 17, 2021 at 12:38
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    $\begingroup$ 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. $\endgroup$
    – Kurt G.
    Nov 17, 2021 at 15:21

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