# Modelling considerations for a jump model

The Problem:

Suppose I have a simple jump model for an asset price

$$dS = S(t-)[\mu dt + YdN(t)]$$

where $$N(t)$$ is a Poisson process and $$Y_i$$ are the jump sizes (assume independece of $$N(t)$$ and $$Y_i$$). Also for simplicity assume that $$Y_i$$ can only take two values, lets say $$a$$ and $$b$$. What constraints are there for $$\mu$$, $$a$$ and $$b$$?

I should also add that we can assume the existence of a money market account with constant short rate: $$B(t) = B_0e^{rt}$$

My initial thoughts:

In order to preserve positivity of stock prices I need both $$a$$ and $$b$$ to be greater than $$-1$$. I also need to consider when there might be arbitrage in my model. However I can't work out what additional constraints I need to impose. Can anyone help?

Thanks in advance for any responses.

Note that, \begin{align*} d\big( e^{-\mu t}S_t \big) &= -\mu e^{-\mu t}S_t dt + e^{-\mu t}S_{t-}(\mu dt + Y_t dN_t)\\ &=e^{-\mu t}S_{t-}Y_t dN_t. \end{align*} From the Doleans-Dade exponential formula, \begin{align*} e^{-\mu t}S_t = \prod_{0 That is, \begin{align*} S_t = e^{\mu t} \prod_{0 Therefore, both $$a$$ and $$b$$ are greater than $$-1$$.
Let $$\lambda$$ be the intensity of the Poisson process $$N$$. Moreover, for ease of exposition, we assume that $$E(Y_s)=L$$, for $$s\ge 0$$, is a constant. Let \begin{align*} M_t = N_t-\lambda t \end{align*} and \begin{align*} X_t = \int_0^t Y_s dN_s -\lambda L t. \end{align*} Then, $$\{M_t, t \ge 0 \}$$ is a martingale. Moreover, for $$u\ge v \ge 0$$ and the information set $$\mathscr{F}_v$$ at time $$v$$, \begin{align*} E(X_u-X_v\mid \mathscr{F}_v) &= E\bigg(\int_v^u Y_s dN_s \mid \mathscr{F}_v\bigg) - \lambda L (u-v)\\ &= E\bigg(\int_v^u Y_s dM_s \mid \mathscr{F}_v\bigg) + \lambda E\bigg(\int_v^u Y_s ds \mid \mathscr{F}_v\bigg) - \lambda L (u-v)\\ &=0. \end{align*} That is, $$\{X_t, t \ge 0\}$$ is a martingale.
To rule out any arbitrage opportunity, $$e^{-rt}S_t$$ has to be a martingale. Note that \begin{align*} d\big( e^{-r t}S_t \big) &= -r e^{-r t}S_t dt + e^{-r t}S_{t-}(\mu dt + Y_t dN_t)\\ &= e^{-r t}S_{t-}\big((\mu-r+\lambda L)dt + dX_t \big). \end{align*} Then, $$\mu = r - \lambda L$$.