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Home / Financial Tools / Merton Jump-Diffusion Option Pricing & Fat-Tail Smile Engine

Graduate-Level Modeling · Derivatives Pricing & Jump Risk

Merton Jump-Diffusion Option Pricing & Fat-Tail Smile Engine

Black-Scholes assumes prices move continuously. Real markets gap, earnings surprises, defaults, macro shocks. Merton (1976) added a Poisson jump process on top of diffusion and found a closed form anyway: a probability-weighted sum of ordinary Black-Scholes prices.

How To Use This Model

Reading This Tool

Set the diffusive volatility as before, then layer in jump risk: how often jumps arrive (λ), their average size and direction (μJ), and their dispersion (σJ). The price is an infinite sum over "exactly n jumps occurred," each term a plain Black-Scholes price at an adjusted rate and volatility, weighted by its Poisson probability. This tool sums 40 terms, more than enough for any realistic jump intensity.

Then look at the smile tab: price across strikes, back out the Black-Scholes implied volatility that would reproduce each price, and watch a smile appear purely from jump risk, with zero stochastic volatility anywhere in the model. This is the cleanest possible demonstration that smiles do not require Heston-style stochastic volatility; discrete jump risk alone is sufficient.

Diffusion & Jump Parameters

Spot and strike range are normalized to $100 so results read directly as moneyness. Negative mean jump size, the default, reflects the well-documented empirical asymmetry: equity markets jump down on shocks far more violently than they jump up.

Model Price & Smile Diagnostics

E[jump]: –

ATM Call (K = $100)

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ATM Implied Volatility

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90–110 Skew

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Pure-Diffusion BS Price

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Implied Volatility Smile Generated By Jump Risk Alone

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Where The Price Actually Comes From

P(0 Jumps By T)

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P(1 Jump By T)

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P(2+ Jumps By T)

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Effective Blended Volatility

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Poisson-Weighted Contribution To The ATM Price By Jump Count

Why Jumps Alone Build A Smile

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Reading The Jump Distribution

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Merton vs. Heston: Two Roads To The Same Smile

Both models produce implied volatility smiles from a Black-Scholes baseline with zero smile, but through entirely different mechanics. Heston smooths the return distribution continuously through randomly evolving variance. Merton adds discrete, rare, large moves on top of ordinary diffusion. Empirically, short-dated smiles look more jump-like (steep, then flattening fast with maturity) while longer-dated smiles look more stochastic-volatility-like, which is exactly why practitioner models increasingly combine both, jumps for the short end, stochastic volatility for the term structure.

Derivatives Pricing & Jump Risk

The Model & Closed-Form Solution

dSt/St = (r−λk)dt + σdWt + dJt,   Jt = Σi=1Nt(Yi−1),  ln Yi ~ N(μJ, σJ2)
C = Σn=0∞ [e−λ′T(λ′T)n/n!] · BS(S,K,T,rn,σn)
λ′ = λ(1+k),  k = eμJ+σJ2/2−1,  rn = r−λk+n·ln(1+k)/T,  σn2 = σ2 + nσJ2/T

Each term of the sum is exactly a Black-Scholes price conditional on exactly n jumps having occurred by T, weighted by the Poisson probability of that count under the risk-neutral compensated intensity λ′. This implementation sums 40 terms (verified against an independent scipy computation to eight decimal places, and confirmed to collapse exactly onto plain Black-Scholes as λ → 0) and recovers implied volatilities by bisection against a double-precision Black-Scholes.

When To Actually Use This Model

  • Teaching why fat tails and volatility smiles can arise from discrete event risk, independent of any stochastic volatility mechanism.
  • Pricing intuition around earnings, binary regulatory, or macro-announcement dates, single-name options ahead of known catalysts behave far more like this model than like pure diffusion.
  • Explaining short-dated index put skew, where crash risk (large, negative, infrequent jumps) is priced far more aggressively than a diffusion-only model would ever generate.

Key Assumptions & Limitations

  • Jump sizes are lognormally distributed and independent of the diffusion, real crash dynamics often show jump clustering and jump-size dependence on the current volatility regime.
  • The model is not perfectly hedgeable, jump risk cannot be fully replicated with the underlying alone, so the risk-neutral pricing here relies on jump risk being diversifiable or priced via a specific market price of jump risk, an assumption real markets only partially support.
  • Parameters are constant; practitioners recalibrate λ, μJ and σJ to the current smile rather than estimating them from historical jump frequency alone.

Foundational References

Merton, R. C. (1976). Option Pricing When Underlying Stock Returns Are Discontinuous. Journal of Financial Economics, 3(1-2), 125-144.

Kou, S. G. (2002). A Jump-Diffusion Model for Option Pricing. Management Science, 48(8), 1086-1101.

Bates, D. S. (1996). Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options. The Review of Financial Studies, 9(1), 69-107.

Want the smile from stochastic volatility instead of jumps?

Open The Heston Stochastic Volatility Engine →