How Jump Risk Differs from Diffusion Volatility in Option Pricing
Summary
The document explains why adding jumps to an option pricing model represents a different risk from simply increasing diffusion volatility. In a diffusion model, variability grows with the square root of time, while a Poisson jump component contributes variability in proportion to time. Sudden price moves tied to news can therefore produce return distributions and option values that a smooth diffusion with higher volatility may not reproduce.
It also distinguishes a model’s diffusion parameter from Black–Scholes implied volatility, which is the volatility input that reproduces a market option price under that formula. With jumps, those values need not match. More realistic jump modeling can matter for pricing and hedging portfolios, and better estimates may expose inconsistent market prices. The discussion is conceptual and gives no calibration procedure, measured performance, or model comparison results; the usefulness of a jump model depends on how well it captures the risks relevant to the options being traded.
Key ideas
- A jump component captures abrupt price moves that a continuous diffusion may not represent.
- Diffusion variability and jump variability scale differently over time.
- Black–Scholes implied volatility is not necessarily the same as a jump model’s diffusion parameter.
- Jump modeling can affect option valuation and hedging, potentially revealing inconsistent prices.
Tags
Full text
# Why Jumps in Option Pricing models? # 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. ## Answer by user25064 (score 2, accepted) https://quant.stackexchange.com/a/19361 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. ## Answer by nbbo2 (score 2) https://quant.stackexchange.com/a/19365 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. ## Answer by phdstudent (score 1) https://quant.stackexchange.com/a/19354 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.
Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.