Risk neutral measure for jump processes

Question

Assume we model the dynamics of a tradable asset as follows $$ S_t = S_0 \exp\left[\sigma W_t +(\alpha-\beta\lambda-\frac{1}{2}\sigma^2)t+J_t \right] $$ where $W_t$ is a standard Brownian motion independent from $J_t = \sum_{i=1}^{N_t} Y_i$ a compound Poisson process.

What conditions should $\alpha$ and $\beta$ verify for this dynamics to be a valid risk-neutral dynamics?

Answer 1

Assume a constant risk-free rate $r$ and no dividends. Generalisation is straightforward.

To preclude arbitrage opportunities, under the risk-neutral measure $\Bbb{Q}$, the discounted asset price process should be a $\Bbb{Q}$-martingale i.e. $$ S_0 = \Bbb{E}^\Bbb{Q}_0 \left[ e^{-rt} S_t \right] \iff \Bbb{E}^\Bbb{Q}_0 \left[ S_t \right] = S_0 \exp(rt) \tag{1} $$

Now, rewriting your equation as \begin{align} S_t &= S_0 \exp(\alpha t) \mathcal{E}(\sigma W_t) \exp(-\beta \lambda t + J_t) \end{align} where $ J_t = \sum_{i=1}^{N_t} Y_i $ denotes a compound Poisson process with $\{Y_i\}_{i=1}^\infty$ i.i.d. random variables and $N_t$ a Poisson process of intensity $\lambda$, and taking the expectation under $\Bbb{Q}$ bearing in mind that the Wiener process is independent from the compound Poisson process yields \begin{align} \Bbb{E}_0^\Bbb{Q} [S_t] &= S_0 \exp(\alpha t) \exp(-\beta \lambda t ) \Bbb{E}_0^\Bbb Q[\exp(J_t)] \tag{2} \end{align}

Comparing the above expressions, we see that $(1)$ is consistent with $(2)$ if and only if $ \alpha = r $ and $ \beta $ is such that $ \Bbb{E}_0^\Bbb{Q}[ \exp(J_t) ] = \exp(\beta \lambda t ) $

Evaluating $\Bbb{E}_0^\Bbb{Q}[ \exp(J_t) ]$ gives \begin{align} \Bbb{E}_0^\Bbb{Q}[ \exp(J_t) ] &= \Bbb{E}_0^\Bbb{Q} \left[ \exp \left(\sum_{i=1}^{N_t} Y_i\right) \right] \\ &= \Bbb{E}_0^\Bbb{Q} \left[ \Bbb{E}_t^\Bbb{Q} \left[ \exp \left(\sum_{i=1}^{N_t} Y_i\right) \right] \right] \\ &= \sum_{n=0}^\infty \Bbb{E} \left[ \exp\left(\sum_{i=1}^n Y_i\right) \right] \Bbb{Q}(N_t = n) \\ &= \sum_{n=0}^\infty \prod_{i=1}^n \Bbb{E} \left[\exp(Y_i) \right] \Bbb{Q}(N_t = n) \\ &= \sum_{n=0}^\infty \left( \Bbb{E} \left[ \exp(Y_1) \right] \right)^n \Bbb{Q}(N_t = n) \\ &= e^{-\lambda t} \sum_{n=0}^\infty \left( \Bbb{E} \left[ \exp(Y_1) \right] \right)^n \frac{(\lambda t)^n}{n!} \\ &= \exp \left(\Bbb{E} \left[ \exp(Y_1) \right] - 1)\lambda t \right) \end{align} thereby showing that, under $\Bbb{Q}$, it is enough that $$ \alpha = r,\ \ \beta = \Bbb{E} \left[ \exp(Y_1) \right] - 1 $$ to preclude arbitrage opportunities.