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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.

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This is easy to answer with the meta theorem given in the same chapter. Here you have two sources of randomness (W and N), and one risky asset.

Q1: Arbitrage generally happens when you have more assets than the number of random sources, but here it is the other way around, so the answer is yes.

Q2: You have one risky asset so you can delta hedge one source of randomness, say Brownian, but not both sources of randomness so the market is not complete. This is also your guide to Q4- i.e.,can you use the other call option with the longer maturity to manage the second random source? In theory yes?

Q3: Based on Q1 and Q2, can we say there are many such prices? So not unique arbitrage free price?

Q4: Pls see Q2 above

Hope this helps!

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