Merton Jump-Diffusion Option Pricing Lab

Simulate Robert C. Merton's (1976) jump-diffusion framework. Model discontinuous compound Poisson jumps, lognormal shock distributions, analytical series expansion pricing, and short-maturity implied volatility skew surfaces.

Market Calibration Presets:

Model Parameters

$
$
Years
%
%
%

Poisson Jump Dynamics

0.75 jumps/yr
Expected number of Poisson jump events per year.
-12.0%
Average percentage shift when a jump occurs (<0 = crash risk, >0 = supply shock).
15.0%
Standard deviation of the jump size distribution (governs tail thickness).
Merton Call Option Price
$3.98
BS Parity: $3.54 (+$0.44 jump prem)
Merton Put Option Price
$3.24
BS Parity: $2.80 (+$0.44 jump prem)
Implied Black-Scholes Vol σᵢ
22.84%
Continuous σ: 15.00%
Total Annualized Volatility σᵛ
22.98%
Excess Kurtosis: +2.85

Quantitative Visualizer

Merton Implied Vol Smile Black-Scholes Benchmark Selected Strike (K)

Robert C. Merton (1976) Analytical Series Expansion Formula

C_Merton = ∑_{n=0}^{∞} [ e^{-λ′ T} (λ′ T)^n / n! ] · C_BS(S₀, K, T, σ_n, r_n, q)

Where λ′ = λ(1 + κ) with mean jump κ = e^{μᵃ + 0.5δ²} - 1, conditioned variance σ_n² = σ² + nδ²/T, and conditioned rate r_n = r - λκ + n ln(1 + κ)/T.

Sensitivity Grid 1: Jump Intensity (λ) vs. Mean Jump (μᵃ)

Evaluates 90% OTM Put Implied Volatility σᵢ(0.90 S₀), quantifying downside crash risk and tail hedging cost.

Sensitivity Grid 2: Jump Volatility (δ) vs. Expiry (T)

Evaluates ATM Implied Volatility across maturity horizons, showing the decay of jump excess kurtosis.

Institutional Mechanics of the Merton Jump-Diffusion Model

1. Discontinuous Jumps vs. Continuous Diffusion

Continuous diffusion models (like Black-Scholes) assume that over any small time increment Δt, the maximum price change is bounded by O(√Δt). Consequently, the probability of a 10% move occurring over a single day approaches zero in standard models. Merton's introduction of compound Poisson jumps dq_t provides the necessary mathematical apparatus to represent discrete informational shocks:

  • Continuous Drift & Diffusion: (r - q - λκ) dt + σ dW_t captures ordinary market trading, microstructure noise, and gradual drift.
  • Compound Poisson Jumps: (Y - 1) dq_t captures sudden earnings announcements, geopolitical crises, macro rate surprises, and flash crashes.
  • Compensator λκ dt: Guarantees that the discounted stock price S_t e^{-(r-q)t} remains a true martingale under the risk-neutral pricing measure.

2. The Term Structure of Implied Volatility Skew

A celebrated empirical feature of jump-diffusion models is how they replicate the term structure of market skew:

  • Short Maturities (T < 0.25): Jumps dominate the tails. Because a single jump can instantly breach out-of-the-money strikes, the implied volatility smile is sharp, steep, and pronounced.
  • Long Maturities (T > 2.0): By the Central Limit Theorem, the sum of many independent Poisson jumps converges toward a normal distribution. As a result, excess kurtosis decays as 1/T, and the smile flattens out toward a continuous diffusion regime.
  • Skew Asymmetry: A negative mean jump (μ_J < 0) naturally generates the persistent downward equity skew observed across global stock indices since the 1987 crash.

Merton Jump-Diffusion Mastery Quiz

Frequently Asked Questions

Introduced by Nobel laureate Robert C. Merton in 1976, the jump-diffusion model extends the standard Black-Scholes-Merton continuous diffusion framework by incorporating sudden, discrete jumps governed by a compound Poisson process. While continuous Brownian motion cannot generate significant probability for large market moves over short time horizons, Merton's model captures market crashes, earnings announcements, and macroeconomic shocks, resolving the empirical failure of Black-Scholes to price short-dated out-of-the-money options accurately.

Merton proved that if jump risk is diversifiable, the European option price is equal to an infinite series of standard Black-Scholes option prices, where each term is weighted by the Poisson probability of observing exactly n jumps over the option's life. In each term, the volatility and interest rate are conditioned on the occurrence of n jumps: sigma_n^2 = sigma^2 + n * delta^2 / T and r_n = r - lambda * kappa + n * ln(1 + kappa) / T. Because Poisson probabilities decay super-exponentially, summing 30 to 50 terms yields exact analytical pricing.

Jumps introduce excess kurtosis (fat tails) and skewness into the return distribution. When the mean jump size mu_J is negative (typical in equity indices where shocks represent market crashes), deep out-of-the-money puts gain substantial probability, producing a steep downward-sloping implied volatility skew. The jump volatility parameter delta widens both tails, creating the classic U-shaped smile wings. As time to expiry T increases, the central limit theorem causes diffusion to dominate and the jump-induced excess kurtosis fades.

The model features three jump parameters alongside the standard Black-Scholes parameters: jump intensity lambda (the average number of jump events per year), mean log-jump size mu_J (the expected percentage shift per jump), and jump size standard deviation delta (the uncertainty or dispersion around the jump magnitude). Together with continuous volatility sigma, these determine the total annualized variance sigma_total^2 = sigma^2 + lambda * (mu_J^2 + delta^2).

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