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.


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