# Arbitrage free in a Black-Scholes/Poisson model

I am trying to solve the following exercise from Bjork's Arbitrage Theory in Continuous Time:

Consider a model for the stock market where the short rate of interest $$r$$ is a deterministic constant. We focus on a particular stock with price process $$S$$. Under the objective probability measure $$P$$ we have the following dynamics for the price process. $$dS(t) = \alpha S(t)dt + \sigma S(t)dW(t) + \delta S(t^-)dN(t)$$ Here $$W$$ is a standard Wiener process whereas $$N$$ is a Poisson process with intensity $$\lambda$$. We assume that $$\alpha, \sigma,\delta$$ and $$\lambda$$ are known to us. The $$dN$$ term is to be interpreted in the following way:

• Between the jump times of the Poisson process $$N$$, the $$S$$-process behaves just like ordinary geometric Brownian motion.
• If $$N$$ has a jump at time $$t$$ this induces $$S$$ to have a jump at time $$t$$. The size of the $$S$$-jump is given by $$S(t) - S(t^-) = \delta\cdot S(t^-)$$

Discuss the following questions.

1. Is the model free of arbitrage?
2. Is the model complete?
3. Is there a unique arbitrage free price for, say, a European call option?
4. Suppose that you want to replicate a European call option maturing in January 1999. Is it posssible (theoretically) to replicate this asset by a portfolio consisting of bonds, the underlying stock and European call option maturing in December 2001?

### Q2

The model is complete if we can find a replicating portfolio for it. The replicating portfolio can be built using a bond, with deterministic price process: $$dB = rBdt$$ and a certain number of stock shares $$S_i$$, with stochastic price process: $$dS_i = \alpha_i S_i dt + S_i\sum_j\sigma_{ij} dW_j + \delta_i S_i dN_i,$$ where $$W_j$$ are independent standard Wiener processes, and $$N_i$$ are independent standard Poisson processes.

To build a replicating portfolio, we need some theoretical tools, like proving some form of the Ito's lemma for jump processes, but in principle, supposing that such a thing exists, we should probably manage to build a replicating portfolio.

The jump processes probably require some care when imposing the self-financing constraint in the portfolio, but basically they act as random, instantaneous injections or withdrawals of the money that can be used to rebalance the portfolio.

### Q1, Q3 and Q4

I am not sure about my answer to Q2, and have no idea on how to approach Q1, Q3, and Q4. Any help would be appreciated.