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The Merton Jump-Diffusion Model: Pricing with Discontinuities

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The Formal Theorem

Let St S_t be the asset price following the jump-diffusion process defined by the SDE:
dStSt−=(μ−λk)dt+σdWt+d(∑i=1Nt(Yi−1)) \frac{dS_t}{S_{t-}} = (\mu - \lambda k) dt + \sigma dW_t + d(\sum_{i=1}^{N_t} (Y_i - 1))
where Wt W_t is a standard Brownian motion, Nt N_t is a Poisson process with intensity λ \lambda , and Yi Y_i are i.i.d. jump sizes such that ln⁡(Yi)∼N(μJ,σJ2) \ln(Y_i) \sim N(\mu_J, \sigma_J^2) . The constant k=E[Yi−1]=exp⁡(μJ+12σJ2)−1 k = E[Y_i - 1] = \exp(\mu_J + \frac{1}{2}\sigma_J^2) - 1 . The price of a European call option with strike K K and maturity T T is given by the weighted sum of Black-Scholes prices:
C(S,K,T)=∑n=0∞e−λ′T(λ′T)nn!CBS(S,K,T,σn,rn) C(S, K, T) = \sum_{n=0}^{\infty} \frac{e^{-\lambda' T}(\lambda' T)^n}{n!} C_{BS}(S, K, T, \sigma_n, r_n)

Analytical Intuition.

Imagine the financial market as a calm sea, where the Black-Scholes model captures the gentle, predictable rise and fall of tides through Wt W_t . However, the real ocean is prone to rogue waves—sudden, violent market shocks triggered by earnings surprises or geopolitical crises. Robert Merton introduced the 'Jump-Diffusion' model to account for these discontinuities. The price dynamics are no longer a single smooth path but a hybrid: a continuous 'diffusion' component blended with a discrete 'Poisson' process. When a jump occurs (at rate λ \lambda ), the asset price instantly gaps to a new level defined by the jump size Yi Y_i . To price options, we decompose the total variance into these two distinct forces. We observe that if exactly n n jumps occur before maturity, the price is simply the Black-Scholes formula adjusted for the mean and variance of those n n jumps. By summing all possible jump scenarios (weighted by their Poisson probability), we arrive at a robust framework that captures the 'fat tails' and 'volatility smiles' that elude standard, smooth-path models.
CAUTION

Institutional Warning.

Students often struggle with the drift adjustment term λk \lambda k . It is vital to recognize this as the 'martingale correction'; without subtracting the expected value of the jumps from the drift, the asset price process would not be a martingale under the risk-neutral measure, leading to arbitrage opportunities.

Academic Inquiries.

01

Why is the jump size modeled as a log-normal distribution?

The log-normal assumption for Yi Y_i ensures that the asset price St S_t remains strictly positive, as the exponential of a normal distribution is always positive, preventing the impossibility of negative stock prices.

02

Does the Merton model resolve the volatility smile?

Yes, unlike Black-Scholes, the Merton model incorporates 'kurtosis' or fat tails due to the jump component, which generates the observed volatility smile for short-dated options.

Standardized References.

  • Definitive Institutional SourceMerton, R. C., 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). The Merton Jump-Diffusion Model: Pricing with Discontinuities: Visual Proof & Intuition. Retrieved from https://nicefa.org/library/advanced-stochastic-processes/the-merton-jump-diffusion-model--pricing-with-discontinuities

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