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Merton's Jump-Diffusion Model: Incorporating Discontinuities in Asset Prices

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The Formal Theorem

Let St S_t be the price of an asset following a jump-diffusion process on a filtered probability space. Under the risk-neutral measure, the dynamics are given by the stochastic differential equation:
dSt=(r−λκ)Stdt+σStdWt+St−d(∑i=1Nt(Yi−1)) dS_t = (r - \lambda \kappa) S_t dt + \sigma S_t dW_t + S_{t^-} d(\sum_{i=1}^{N_t} (Y_i - 1))
where r r is the risk-free rate, σ \sigma is the diffusion volatility, Wt W_t is a standard Brownian motion, Nt N_t is a Poisson process with intensity λ \lambda , Yi Y_i represents the random jump size such that Yi>0 Y_i > 0 , and κ=E[Yi−1] \kappa = E[Y_i - 1] is the expected relative jump size.

Analytical Intuition.

Standard Black-Scholes assumes asset paths are continuous, like a smooth, flowing river. But markets are often chaotic, characterized by 'gaps' or 'shocks'—sudden news events, geopolitical crises, or earnings surprises that cause prices to teleport rather than glide. Merton’s Jump-Diffusion model introduces these discontinuities by layering a Poisson process over the existing Brownian motion. Think of it as a hybrid beast: the diffusion component σdWt \sigma dW_t handles the constant, day-to-day 'noise' of the market, while the jump component Nt N_t acts as a series of intermittent 'quantum leaps.' When Nt N_t triggers, the price St S_t jumps by a random factor Yi Y_i . This allows us to capture 'fat tails' and the volatility smile observed in option markets, acknowledging that reality isn't just a random walk—it's a walk interrupted by bursts of intense, discontinuous re-pricing.
CAUTION

Institutional Warning.

Students often struggle to distinguish between the jump intensity λ \lambda and the jump magnitude Y Y . Remember: λ \lambda dictates the frequency of 'shocks,' while the distribution of Y Y determines how violent those shocks are. They are independent parameters.

Academic Inquiries.

01

Why is the term −λκ -\lambda \kappa included in the drift?

It acts as a compensator. Since the Poisson process has a non-zero mean jump size, we must subtract its expected contribution to the drift to ensure the discounted asset price remains a martingale under the risk-neutral measure.

02

Does this model solve the volatility smile problem?

Partially. By allowing for jumps, the model introduces excess kurtosis (fat tails) in the return distribution, which effectively flattens the implied volatility curve compared to the standard Black-Scholes model.

Standardized References.

  • Definitive Institutional SourceMerton, R. C. (1976). Option pricing when underlying stock returns are discontinuous.

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Institutional Citation

Reference this proof in your academic research or publications.

NICEFA Visual Mathematics. (2026). Merton's Jump-Diffusion Model: Incorporating Discontinuities in Asset Prices: Visual Proof & Intuition. Retrieved from https://nicefa.org/library/advanced-stochastic-processes/merton-s-jump-diffusion-model--incorporating-discontinuities-in-asset-prices

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