What is the Merton Jump-Diffusion Model? Modeling Discontinuous Returns

Financial markets do not move in continuous paths. Learn how the Merton Jump-Diffusion model incorporates Poisson processes to account for sudden market crashes.

I analyzed the price action of pharmaceutical stocks during clinical trial announcements to prove why the standard Brownian motion assumption fails in the presence of 'gaps.' The Merton Jump-Diffusion Model extends the Black-Scholes-Merton model framework by adding a jump component to the continuous price path, allowing for the discrete, non-marginal changes we see during earnings or geopolitical shocks.

In Merton's model, the asset price $S_t$ follows the stochastic differential equation: $dS_t = (\mu - \lambda k) S_t dt + \sigma S_t dW_t + S_{t-} (J - 1) dN_t$ where $N_t$ is a Poisson process with intensity $\lambda$, $J$ is the jump size drawn from a log-normal distribution, and $k$ is the expected jump size. This framework captures both the continuous diffusion and the discrete jumps in asset prices.

The primary insight of Merton is that jump risk is non-diversifiable. Because jumps are discontinuous, they cannot be perfectly hedged using the underlying asset in continuous time. This creates a 'market incompleteness' where the risk-neutral measure is not unique.

1. Jump Intensity ($\lambda$): The frequency of jumps per year. 2. Mean Jump Size: The expected percentage move when a jump occurs. 3. Volatility of Jumps: The variance in the magnitude of the jumps.

Simulating this requires generating a Poisson number of events and then drawing jump magnitudes from a log-normal distribution.