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.
- Is the model free of arbitrage?
- Is the model complete?
- Is there a unique arbitrage free price for, say, a European call option?
- 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.