Jumps and stochastic volatility are two helpful refinements into Black Scholes’s arbitrage-free pricing paradigm. Mastering them would require advanced calculus and probability theory. Still, we will give an intuitive explanation of why we need jumps without any formula. In this blog, we would only focus on jumps and talk about stochastic volatility in the next blog.
1. There are jumps in the real world, such as the 20% SPX drop on Oct 29, 1987. The pricing Q measure should be consistent with the historical measure P. If jumps are possible in the historical measure, it should exist in the pricing measure.
2. Jump is a flexible modeling tool. A process with jump and diffusion is flexible enough to model any static stochastic process. Just as you all might already have known, any continuous bounded function can be approximated by polynomial functions in Taylor expansion. Similarly, for any probability distribution to describe one-day stock return, as long as the distribution stays the same every day, we can use jump and diffusion to decompose it. The theory is called the Levy process theory.
3. Jump helps to model near-expiry out the money option. There are situations where the option is about to expire and deep out the money while the price is still not zero. This situation indicates traders are betting whether there is some shocking news happens later. In a simplified world, we use three methods to model three market behaviors:
(1) the determined trend to model the diffusion of public information;
(2) diffusion process (Brownian motion) to model the stochastic supply-demand imbalance;
(3) jumps to model unexpected, shocking news.
Mathematically, the stock move by diffusion is proportional to the square root of time. In contrast, the possibility that stock move by Poisson jump is proportional to time. If time is close to zero, the jump term has greater order of magnitude and dominates the dynamics to generate skew (t vs. square root of t).
In short, for traders who heavily depend on near-expiry OTM options, or who are trading exotic options related to realized volatility/tail risk, adding jumps in the option pricing models would be helpful.