Euler Scheme for Jump-Diffusion models

Question

Jump-diffusion models (as Merton) have following SDE: $$dS_t=\mu S_tdt+\sigma S_t dW_t+S_tdJ_t$$ where $$J_t=\sum_{i=1}^{N_t}(\xi_i - 1)$$ $\xi_i$ - i.i.dn $N_t$ - Poisson process

Do we in Euler scheme have sth like this?

$$S_{t+\Delta t}=S_t+\mu S_t\Delta t+\sigma S_t\Delta W_t+S_t\Delta J_t$$

where $$\Delta J_t = \sum_{i=N_{t}}^{N_{t+\Delta t}}(\xi_i -1)$$ So to calculate $\Delta J_t$ we have to smulate random variable from Poisson distribution $\lambda \Delta t$ which denotes number of jumps between $t$ and $t+\Delta t$ and then simulate this number of jumps $\xi$, am I right? I know that this SDE has a solution, but I want to compare results. Which number of $N$ (time steps) is typically optimal to aproximate solution very well?

Answer 1

Commonly, we employ the Euler scheme for $\Delta\ln(S_t)$, not for $\Delta S_t$.

Let us specify the jump part as

$$ S_{t+}=S_{t}J\Rightarrow dS_t=S_t(J-1) $$ where $J$ is a strictly positive random variable. (NB: Under Merton we would have $\ln(J)\sim N(\mu_J,\sigma_J^2)$ and $\mathrm{E}(J)=e^{\mu_J+\frac{1}{2}\sigma_J^2}$)

And for the solution scheme we arrive at:

$$ \begin{align} \frac{dS_t}{S_t}&=\mu dt + \sigma dW_t+(J-1)dN_t\\ y_t&=\ln{S_t}\\ \Rightarrow dy_t&=\left( \mu-\frac{1}{2}\sigma^2 \right)dt+\sigma dW_t+\left(\ln (S_{t+})-(ln S_t)\right)dN_t\\ &=\left( \mu-\frac{1}{2}\sigma^2 \right)dt+\sigma dW_t+\ln(J) dN_t\\ \Rightarrow y_t&=\left( \mu-\frac{1}{2}\sigma^2 \right)t+\sigma W_t + \sum_{i=i}^{N_t}\ln(J_i)\\ \Rightarrow S_t&=S_0e^{\left( \mu-\frac{1}{2}\sigma^2 \right)t+\sigma W_t} \prod_{i=1}^{N_t}J_i\\ \end{align} $$

Let's assume that we have the Merton jump diffusion model here. Then the Euler discretization is:

$$ \begin{align} y_t&\leftarrow y_0\\ \epsilon_{1,t} & \sim N(0,\sigma^2)\\ \epsilon_{2,t} & \sim N(\mu_J,\sigma_J^2)\\ N_t&\sim \left\{ \begin{array}{1} 0 & p=e^{-\lambda\Delta t}\\ 1 & 1-p\end{array} \right. \\ y_{t+\Delta t}&\leftarrow y_t+\left(\mu-\frac{1}{2}\sigma^2\right)\Delta t+\sigma\sqrt{\Delta t}\epsilon_1+N_t\epsilon_{2,t} \end{align} $$ and $S_{t}=e^{y_t}$ accordingly. And if you simulate under the risk-neutral measure, then of course $\mu=r_f-\lambda\mathrm{E}^{\mathbb{Q}}(J-1)$.

HTH?