Martingale Correction for Exponential Lévy Price Processes

# Discounted price process - martingale


# Discounted price process - martingale












I have a process $S_{t}=S_{0}e^{\left(r-q\right)t+mt+X_{t}}$, where $X_t$ is a Levy process and I want to check for which $m$ the process $e^{-(r-q)t}S_t$ is a martingale. The third condition of a martingale states that for $s\leq t$ $$E(e^{-(r-q)t}S_t|F_s)=e^{-(r-q)s}S_s,$$ where $F_s$ is the filtration generated by the process $S_t$.

Many authors write that this process is a martingale when $E(e^{-(r-q)t}S_t)=S_0$ i.e. when $m=-\frac{1}{t}\ln\left(\phi_{X_{t}}\left(-i\right)\right)$, where $\phi_{X_t}$ is the characteristic function of $X_t$.

Why don't they condition on $F_s$ when they verify that the process is a martingale?

## Answer by Kevin (score 4, accepted)

https://quant.stackexchange.com/a/59365

An exponential Lévy process is typically modelled via $$ S_t = S_0\exp\left(\left(r-q+\omega\right)t+X_t\right),$$ where $X_t$ is a Lévy process with $X_0=0$. A Lévy process includes three model features: a linear drift, diffusive shocks and jumps (which may be large and rare or small and frequent). The number $\omega$ is called martingale correction or Jensen's correction and ensures the martingale property.

For our standard finance theory to work, the reinvested and discounted stock price, $S_te^{-(r-q)t}$, needs to be a martingale under $\mathbb{Q}$ (assuming constant interest rates and dividend yields). Let $(\mathcal{F}_t)$ denote the natural filtration of $X_t$. Then, for any $s\leq t$, \begin{align*} \mathbb{E}^\mathbb{Q}[S_t|\mathcal{F}_s] &= \mathbb{E}^\mathbb{Q}[S_0e^{(r-q+\omega)t+X_s+(X_t-X_s)}|\mathcal{F}_s] \\ &= S_0e^{(r-q+\omega)t} e^{X_s} \mathbb{E}^\mathbb{Q}[e^{X_t-X_s}] \\ &= S_s e^{(r-q+\omega)(t-s)} \mathbb{E}^\mathbb{Q}[e^{X_{t-s}}], \end{align*} where we used that $X_s$ is $\mathcal{F}_s$-measurable, and $X_t-X_s\overset{d}{=} X_{t-s}$ is independent of $\mathcal{F}_s$, see here.

Let $\varphi_{X_t}(u)=\mathbb{E}[e^{iuX_t}]$ be the characteristic function of the Lévy process $X_t$. The Lévy-Khintchine formula states that $\varphi_{X_t}(u)=e^{t\Psi(u)}$ which follows from the infinite divisibility of a Lévy process. The function $\Psi$ is called the characteristic exponent and captures the drift, diffusion and jump components of $X_t$.

Then, \begin{align*} \mathbb{E}^\mathbb{Q}[S_t|\mathcal{F}_s] &= S_s e^{(r-q+\omega)(t-s)} \varphi_{X_{t-s}}(-i) \\ &= S_s e^{(r-q+\omega)(t-s)} e^{(t-s)\Psi(-i)}. \end{align*} Hence, setting $\omega=-\Psi(-i)$ yields \begin{align*} \mathbb{E}^\mathbb{Q}[S_t|\mathcal{F}_s] &= S_s e^{(r-q)(t-s)}, \end{align*} which in turn implies that the discounted reinvested stock price is indeed a $\mathbb{Q}$-martingale.

Note that $$\omega=-\Psi(-i)=-\frac{1}{t}\ln\left(\varphi_{X_t}(-i)\right)$$ is independent of time. Thus, for an exponential Lévy process, the martingale property is ensured to hold if you verify that $\mathbb{E}^\mathbb{Q}[S_t]=S_0e^{(r-q)t}$.