Merton Jump-Diffusion Pricing for European Call Options
Summary
The article explains how adding instantaneous random jumps to geometric Brownian motion changes the assumptions behind Black–Scholes option pricing. Jump arrivals are modeled with a Poisson process, while jump sizes are treated as random and lognormally distributed. Because jumps prevent perfect continuous delta hedging, the market becomes incomplete and option prices are no longer uniquely fixed by the same replication argument.
For European calls, the article presents Merton’s semi-closed-form valuation as a weighted sum of Black–Scholes prices with adjusted rates and volatilities, approximating the infinite series with a finite number of terms. A C++ implementation illustrates the calculation and reports a sample call value above the Black–Scholes benchmark under its chosen parameters, consistent with the additional modeled jump risk. This example is parameter-specific; it does not establish that jump-diffusion prices are always higher, and the finite sum and modeling assumptions limit the calculation. The approach addresses European vanilla options rather than the exotic contracts mentioned as a later topic.
Key ideas
- A Poisson process models jump arrivals with an intensity parameter and independent arrivals over time.
- Lognormally distributed jump sizes extend geometric Brownian motion with discontinuous price moves.
- Jumps make perfect continuous delta hedging impossible and leave option prices bounded rather than uniquely fixed.
- Merton’s European call valuation combines weighted Black–Scholes prices in a finite-series approximation.
- The sample comparison reflects one set of parameters and does not imply jump-diffusion prices are always higher.
Tags
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.