# Why Jumps in Option Pricing models?

The Bates model adds a Jump process to the Underlying. I understand this may represent observed time series more realistically, but why would one care about this in option pricing?

The option price is just an expected value, which does not have any jump. I would think that adding jumps is just the same as increasing volatility.

• "The Volatility Surface" by Jim Gatheral.Chapter 5 – user16891 Aug 17 '15 at 11:22

The problem is that what some mean when they say "volatility" is BS implied vol from an option price. What some others mean when they say "volatility" is some diffusion parameter from a drift diffusion model (with or without jumps). These are the same value in the log normal model of stock prices but different for many other models including those with jumps. Therefore, a model with jumps would have a higher option implied volatility as mention by @Farahvartish in the comments, but it may have its own diffusion parameter which could be called volatility that would be lower than the option implied volatility.

To answer your question as to why anyone would care about better modelling, it comes down to arbitrage and hedging portfolios. If I can better price and hedge an option position than you can, I can make money from inconsistent pricing in the market.

Diffusion brings about a standard deviation which increases with the square root of time (just like in Brownian motion), while jumps add variability proportional to time (since the jump times are a Poisson process). So they are quite different.

Experience shows that sharp stock market moves do occur (in connection with big news events for example), so modeling both a diffusion component and a jump component (as Merton 1976 did) can be worthwhile.

• The question is why jumps would be important for option pricing, not whether it reflects the underlying. As you say we can have the same variability by adjusting the jump and diffusion parameters. – emcor Aug 18 '15 at 6:29
• At a fixed maturity it is indeed the case that a jump model and a diffusion model can give the same results. But the models will give different results at different maturities. In one vol varies $O(T)$ in the other $O(\sqrt T)$. – noob2 Aug 18 '15 at 14:20

Jumps are totally different from volatility. Imagine a stock whose price has jumps but has no volatility. The asset pricing implications for options on that stock are totally different than from a stock with volatility. Below I simulated 3 stock paths: (i) Jumps and volatility, (2) Only Jumps and (3) No jumps but higher volatility.

As you can imagine the asset pricing implications are very different for each case - and increasing volatility is not the same as jumps.

• In some cases, jumps explain the volatility surface comes from.So jumps are not totally different from volatility. – user16891 Aug 17 '15 at 11:39
• – user16891 Aug 17 '15 at 12:08
• If we have a jump process and a diffusion process with same volatility, I would expect the option prices to be the same as well? – emcor Aug 17 '15 at 13:19
• @Farahvartish In practice jumps are very different from volatility. Think about how jumps would affect optimal hedging. – meh Aug 17 '15 at 14:08
• emcor, no! Here's a code to value options under the Merton (1976) jump-diffusion. Just play around with parameters and check that the options have different values! volopta.com/ComputerCode/Matlab/… – phdstudent Aug 17 '15 at 14:20