Option Pricing under Jump Diffusion Models

Question

I was wondering what the overall approach/intuition behind how to price options under Jump Diffusion Models. My understanding is under Diffusion models such as Geometric Brownian Motion (Black Sholes), allows for concept of complete markets in particular that the perfect hedging strategies thus one can replicate a call option thus leading to a pricing of the options by a no arbitrage argument. But since in Jump Diffusion models markets are incomplete, how would one approach this problem?

There may be some fundamental misunderstanding that I have with problem of pricing a derivative. So as an additional question, when someone prices a derivative what is one usual thought process in deciding a price?

Answer 1

The first consideration is to set prices which do not generate arbitrage opportunities. The existence of a risk-neutral probability measure ensures that the model is arbitrage-free. In the Black-Scholes setting, as you mentioned, the market is complete and there as a unique martingale measure, hence only one possible price for each derivative.

In a jump-diffusion setting, due to incompleteness, you have many possible risk-neutral measures, hence many ways to price a derivative. You will get an interval of possible prices which cannot be aribtraged by the underlying assets of that economy.

So that is where the problem starts, since you will have to make a choice in some specific risk-neutral measure. There is a myriad of ways to do that, each of which will be based on your own preferences and assumptions. One way is to have a utility function on the payoff and try to maximise that. The original idea of Merton, when he introduced the jump-diffusion, was to use Girsanov and only change the drift of the diffusion part and leave the jumps untouched. This has the interpretation that the price of the jump-risk is 0. His reasoning was that the jumps could be diversified away in a portfolio of many stocks (this is obviously not true since most assets are correlated).

Financial modelling with Jump Processes by by Rama Cont & Peter Tankov treats many of the alternative ways to price in incomplete markets.