Jump-Diffusion Model

Definition

A jump-diffusion model augments the standard geometric Brownian motion for asset prices with a Poisson jump component. Asset returns are driven by two sources of randomness: a continuous diffusion (capturing normal day-to-day fluctuations) and a discrete jump process (capturing sudden large moves such as crash risk or earnings surprises). The jump arrival rate is governed by a Poisson process with intensity $\lambda$ , and jump sizes are drawn from a specified distribution (typically lognormal).

Key Ideas

Two-component returns: Total return = drift + diffusion + jump. The diffusion component generates the volatility clustering captured by stochastic volatility (SV)/generalized autoregressive conditional heteroskedasticity (GARCH) models; the jump component generates the fat left tail and extreme skewness that diffusion models cannot match.
Characteristic function tractability: European option prices under jump-diffusion (combined with stochastic volatility) are computed via characteristic function inversion. The characteristic function of the log price is available in closed form under affine dynamics, enabling efficient numerical integration.
Pricing vs. hedging asymmetry: Jump risk is difficult to hedge — a jump that arrives once every 1–2 years is rarely realized at daily/weekly rebalancing intervals. This creates the key Bakshi-Cao-Chen (BCC, 1997) finding: the SVJ (stochastic volatility with jumps) model prices better than pure SV but does not hedge better, because at 1–5 day horizons the realized jump frequency is effectively zero.
Misspecification persistence: Even the best empirical models require implausible implied parameters. BCC (1997) find that implied $\rho$ is $2$ – $3\times$ the time-series estimate and implied $\sigma_v$ is $\approx 4\times$ its maximum likelihood (ML) value. Structural misspecification and empirical usefulness are orthogonal.

How It Works

Merton (1976): The Original Jump-Diffusion

The foundational model adds a compound Poisson jump to geometric Brownian motion:

$\frac{dS}{S} = (\mu - \lambda \bar{k})\, dt + \sigma\, dz + (e^{J} - 1)\, dq$

where $dq$ is a Poisson increment with intensity $\lambda$ (arrivals per year), $J \sim \mathcal{N}(\mu_J - \frac{1}{2}\sigma_J^2, \sigma_J^2)$ is the log jump size, and $\bar{k} = e^{\mu_J} - 1$ is the mean jump. The drift is adjusted by $\lambda\bar{k}$ to maintain the risk-neutral pricing relationship.

Under risk neutrality, the European call price is a weighted sum of Black-Scholes prices:

$C = \sum_{n=0}^{\infty} \frac{e^{-\lambda' T}(\lambda' T)^n}{n!} \cdot \mathrm{BS}(\sigma_n, r_n)$

where $\lambda' = \lambda(1+\bar{k})$ is the risk-adjusted jump intensity, $r_n = r - \lambda\bar{k} + n\ln(1+\bar{k})/T$ , and $\sigma_n^2 = \sigma^2 + n\sigma_J^2/T$ . This series converges rapidly in practice.

Limitations. Jumps and volatility are independent; volatility is constant between jumps; cannot capture the volatility smile's term-structure flattening at longer maturities.

Equilibrium extension (Amin-Ng 1993): When jump risk is systematic — correlated with aggregate consumption — risk neutrality no longer applies to the jump component. Amin and Ng (1993) derive the equilibrium counterpart of Merton's series (eq. 27) under constant proportional risk aversion (CPRA) preferences: the infinite-sum structure is preserved, but jump sizes are priced through the stochastic discount factor rather than under the risk-neutral measure.

BCC (1997): SVJ and the SVSI-J Nested Framework

Bakshi, Cao, and Chen (1997) nest four models within a single general specification:

Stock price (eq. 1): $\frac{dS}{S} = [R(t) - \lambda\mu_J]\, dt + \sqrt{V(t)}\, d\omega_S(t) + J(t)\, dq(t)$

Volatility — Cox-Ingersoll-Ross (CIR) mean-reverting square root (eq. 2): $dV(t) = [\theta_v - \kappa_v V(t)]\, dt + \sigma_v \sqrt{V(t)}\, d\omega_v(t), \qquad \mathrm{Cov}[d\omega_S, d\omega_v] = \rho\, dt$

Interest rate — CIR (eq. 5): $dR(t) = [\theta_R - \kappa_R R(t)]\, dt + \sigma_R \sqrt{R(t)}\, d\omega_R(t)$

Jump distribution (eq. 3): $\ln[1+J(t)] \sim \mathcal{N}(\ln[1+\mu_J] - \frac{1}{2}\sigma_J^2,\; \sigma_J^2)$ , Poisson intensity $\lambda$ , mean jump $\mu_J$ , log-std $\sigma_J$ .

Total instantaneous variance (eq. 4): $V_{\text{total}}(t) = V(t) + \lambda\bigl[\mu_J^2 + (e^{\sigma_J^2}-1)(1+\mu_J)^2\bigr]$

The second term is the jump contribution to variance ( $V_J = \lambda[\mu_J^2 + \sigma_J^2 (1+\mu_J)^2]$ approximately).

Special cases (nested by setting parameters to zero):

Model	$\lambda$	$\sigma_R$	$\sigma_v$
BS (Black-Scholes)	0	0	0
SV (stochastic volatility)	0	0	free
SVSI (SV + stochastic interest rates)	0	free	free
SVJ (SV + jumps)	free	0	free
SVSI-J (all)	free	free	free

European call price (eq. 8): $C(t,\tau) = S(t)\,\Pi_1 - K\,B(t,\tau)\,\Pi_2$

where $\Pi_1$ and $\Pi_2$ are risk-adjusted and risk-neutral exercise probabilities, each recovered by characteristic function inversion (eq. 9). The characteristic function of $\ln S(T)$ under the SVSI-J dynamics factors into contributions from $V(t)$ , $R(t)$ , and the jump process, all of which are affine in the state variables.

Implied parameter estimation (eq. 17): On each trading day t, implied structural parameters $\hat\theta_t$ are the values minimizing the total squared pricing error across all available option contracts: $\hat\theta_t = \arg\min_\theta \sum_{i} [C_i^{\text{mkt}} - C_i^{\text{model}}(\theta)]^2$

Empirical Results: Three-Yardstick Horse Race (BCC 1997)

Data: 38,749 S&P 500 call options, June 1988–May 1991 (Berkeley Option Database). Six moneyness bins × three maturity bins = 18 categories.

Implied parameters (All Options, Table III):

Model	$\kappa_v$	$\sigma_v$	$\rho$	$\lambda$	$\mu_J$	SSE (Sum of Squared Errors)
BS	—	—	—	—	—	69.60
SV	1.15	0.39	$-0.64$	—	—	10.63
SVSI	0.98	0.42	$-0.76$	—	—	10.68
SVJ	2.03	0.38	$-0.57$	0.59/yr	$-5\%$	6.46

Performance rankings:

Dimension	Ranking
Internal consistency	$\text{SVJ} \succ \text{SV} \approx \text{SVSI} \succ \text{BS}$
Out-of-sample pricing	$\text{SVJ} \succ \text{SVSI} \succ \text{SV} \succ \text{BS}$
Hedging (single-instrument)	$\text{SV} \succ \text{SVJ} \succ \text{SVSI} \succ \text{BS}$
Hedging (delta-neutral)	$\text{SV} \approx \text{SVSI} \approx \text{SVJ} \gg \text{BS}$

Key mechanism: With $\lambda \approx 0.59$ per year, the expected inter-jump interval is $\approx 1.7$ years. At 1–5 day hedging frequencies, jumps essentially never occur, so jump risk cannot be exploited in dynamic hedging. The SVJ model's pricing advantage over SV is therefore not reflected in hedging performance.

BSDV control: A Black-Scholes delta-plus-vega-neutral strategy (BSDV) — using a second option to hedge both delta and vega — nearly closes the hedging gap between BS and the SV-class models for non-in-the-money (ITM) calls. This shows that most of the SV models' hedging advantage is mechanical (using a second option instrument) rather than structural.

Why It Matters

Jump-diffusion models capture the implied volatility smile's skew and term-structure behavior better than pure SV or pure diffusion models, particularly for short-maturity out-of-the-money (OTM) options where the smile is steepest.
The BCC (1997) three-yardstick methodology — distinguishing static pricing accuracy from dynamic hedging performance — is a standard template for empirical option pricing studies.
The pricing-hedging divergence (SVJ prices best but doesn't hedge better than SV) has practical implications: adding model complexity to improve option pricing may not improve risk management at typical rebalancing frequencies.
Jump risk contributes to crash-state pricing: the implied mean jump $\mu_J \approx -5\%$ captures the negative skewness associated with crash events priced into OTM puts.

Open Questions

Whether the CIR square-root volatility process is the correct specification, or whether affine but non-square-root processes (Stochastic Alpha Beta Rho (SABR), Heston with different elasticity) improve performance.
Whether jump intensity $\lambda$ should itself be stochastic (self-exciting Hawkes processes, regime-switching $\lambda$ ) to capture clustering of large moves.
How to jointly model jumps in both returns and volatility — Eraker-Johannes-Polson (2002) find evidence for contemporaneous jumps in both processes.