Jumps in Asset Prices

In Part I of this book, asset prices and factor processes were represented by diffusion processes, driven by correlated Brownian motions. In Part II we extend the theory — using as far as possible the same general approach — to jump-diffusion processes, where the driving Brownian motions are augmented by a class of Poisson random measures, which we introduce in Section 7.1 below. There are at least three good reasons for doing so:1. The diffusion framework cannot accommodate credit-related assets such as CDS (credit default swaps). A CDS is a swap consisting of regular premium payments one one side and, on the other, a contingent payment should some reference entity trigger a default event. Obviously, the main modelling question is how to represent the default time and the size of the contingent payment. There are many ways to do this, but even if default isrepresented by, say, a barrier hitting time by some diffusion process, the fact remains that the CDS investor's portfolio will take a hit at the moment of default, so the portfolio value cannot be represented as a continuous process.2. The asset price distributions implied by diffusion models can be unrealistic. Specifically, all the models presented so far have ‘thin tails’, i.e. the asset return distributions are similar to the Gaussian distribution in the tails. It is however well known that even quite standard financial data series such as stock indices have fatter tails than that: thin tails are the exception rather than the rule. From an asset allocation point of view, this may or may not be a significant factor. It may be that inaccurate representation of the tails is swamped by other deficiencies of the model such as — notably — the inability to estimate growth rates over long time horizons. On the other hand, from a risk-management perspective tail behaviour is the only thing that matters when one is computing VaR or CVaR, so one could hardly be satisfied if this were grossly misrepresented.3. The absence of explicit liquidity modelling in the diffusion model means that some risks certainly present in practice are ignored. To see the point, consider the simple Merton model of Chapter 1. The decision variable h(t) is the fraction of wealth invested in the risky asset, and this can be adjusted at will. This implies that it is always possible to avoid running into bankruptcy in this model. For example, suppose h(t) ≡ h*, a constant as in the Merton strategies; then the portfolio value is log-normal and the probability of hitting zero in finite time is zero. To see what is happening in detail, suppose the initial stock price is $100 and our investor is 10 times leveraged with initial capital $1000, so he starts with $10,000 in stock, financed by borrowing $9,000 from the bank. If there is a crash and the stock price drops by 15% to $85 then the investor is wiped out: his stock is now worth $8,500 while he still owes $9,000 to the bank. If, however, he can trade sufficiently fast that he can rebalance the portfolio to maintain the 10-to-1 leverage ratio every time the stock price falls by $1 then he loses a lot of money but remains solvent. The evolution of his portfolio is shown in Figure 7.1, and its final net value is $181.77…