Adding jumps to diffusion models of stocks
Question
What is a jump-diffusion model, and what effect do jumps have on option prices?
Short answer
It adds Poisson-driven discontinuous jumps to the usual Brownian diffusion of the asset price (as in Merton's model). Jumps fatten the tails of returns, which raises the value of out-of-the-money options and produces a volatility smile or skew, especially at short maturities. They also make the market incomplete: jump risk cannot be removed by delta hedging alone, so prices depend on assumptions about jump risk premia.
Adapted from Wilmott, Frequently Asked Questions in Quantitative Finance, FAQ 31, p.142; question and answer rewritten by SelfMadeQuant.com.
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Open the worked solutionLessons behind this question
- Defining the Poisson Process: Three Equivalent Constructions
- Option Mechanics: Calls, Puts, Moneyness, Intrinsic Value, and Exchange vs. OTC Markets
- Option Trading Strategies: Spreads, Combinations, and Payoff Engineering
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