Chapter 7 Jump-Diffusion Models
7.1 Why Jumps Matter
Brownian motion assumes continuous price paths. But real markets exhibit:
- Earnings announcements: Stock gaps 10%+ overnight
- Central bank decisions: FX rates jump instantly
- Black swan events: Market crashes, flash crashes
- Fat tails: Extreme returns occur more often than log-normal predicts
Purely diffusive models systematically underprice out-of-the-money options.
7.2 The Poisson Process
Before jumps, we need to count random events.
Definition: \(N(t)\) is a Poisson process with intensity \(\lambda\) if:
- \(N(0) = 0\)
- Independent increments
- \(N(t) - N(s) \sim \text{Poisson}(\lambda(t-s))\) for \(t > s\)
- Paths are step functions (jumps of size 1)
Properties: - \(E[N(t)] = \lambda t\) (average \(\lambda\) events per unit time) - \(\text{Var}(N(t)) = \lambda t\) - \(P(N(t) = k) = \frac{(\lambda t)^k e^{-\lambda t}}{k!}\) - Waiting times between events: exponential with rate \(\lambda\)
Differential notation: \[dN(t) = \begin{cases} 1 & \text{with probability } \lambda \, dt \\ 0 & \text{with probability } 1 - \lambda \, dt \end{cases}\]
Key property: \((dN)^2 = dN\) (since \(dN \in \{0,1\}\) and \(1^2 = 1\)).
7.3 Compound Poisson Process
Add random jump sizes:
\[J(t) = \sum_{i=1}^{N(t)} Y_i\]
where \(Y_i\) are i.i.d. random jumps (often \(Y_i \sim N(\mu_J, \sigma_J^2)\)).
In differential form: \[dJ(t) = Y \, dN(t)\]
where \(Y\) is drawn when a jump occurs.
7.4 Merton’s Jump-Diffusion Model
Combine Brownian motion with compound Poisson jumps:
\[\boxed{\frac{dS}{S} = \mu \, dt + \sigma \, dB + (e^Y - 1) \, dN}\]
Components: - \(\mu \, dt\): Deterministic drift - \(\sigma \, dB\): Continuous diffusion (Brownian) - \((e^Y - 1) \, dN\): Discontinuous jumps
Jump structure: - Jumps arrive with intensity \(\lambda\) - When jump occurs: \(S \to S \cdot e^Y\) (multiplicative) - Typically \(Y \sim N(\mu_J, \sigma_J^2)\) (log-normal jumps)
7.5 Itô’s Lemma with Jumps
For a function \(f(S)\) where \(S\) follows jump-diffusion, Itô’s lemma becomes:
\[df = \frac{\partial f}{\partial S}dS + \frac{1}{2}\frac{\partial^2 f}{\partial S^2}(dS)^2 + [f(Se^Y) - f(S) - \frac{\partial f}{\partial S}S(e^Y - 1)]\,dN\]
The last term is the jump correction: actual jump minus first-order Taylor approximation.
Multiplication rules: - \(dB \cdot dN = 0\) (jumps and diffusion independent) - \(dt \cdot dN = 0\) - \(dN \cdot dN = dN\) (only one jump at a time)
7.5.1 Example: Log Price
Apply to \(f(S) = \log S\):
\[d(\log S) = \left(\mu - \frac{\sigma^2}{2}\right)dt + \sigma \, dB + Y \, dN\]
Integrating:
\[\log S(t) = \log S(0) + \left(\mu - \frac{\sigma^2}{2}\right)t + \sigma B(t) + \sum_{i=1}^{N(t)} Y_i\]
Log price = drift + Brownian + sum of jumps. Clean decomposition!
7.6 Risk-Neutral Pricing with Jumps
Under \(\mathbb{Q}\), discounted price must be martingale:
\[\frac{dS}{S} = (r - \lambda \kappa) \, dt + \sigma \, d\tilde{B} + (e^Y - 1) \, d\tilde{N}\]
where \(\kappa = E^{\mathbb{Q}}[e^Y - 1]\) is the expected jump size.
The drift adjustment \(-\lambda \kappa\) compensates for jumps to maintain martingale property.
If jumps are log-normal: \(Y \sim N(\mu_J, \sigma_J^2)\), then:
\[\kappa = e^{\mu_J + \sigma_J^2/2} - 1\]
7.7 Merton’s Option Pricing Formula
For a European call:
\[\boxed{C(S, t) = \sum_{n=0}^{\infty} \frac{e^{-\lambda'(T-t)}[\lambda'(T-t)]^n}{n!} \text{BS}(S, K, r_n, \sigma_n, T-t)}\]
where: - \(\lambda' = \lambda(1 + \kappa)\) is adjusted intensity - \(\text{BS}(\cdot)\) is Black-Scholes formula with modified parameters: - \(r_n = r - \lambda\kappa + \frac{n\log(1+\kappa)}{T-t}\) - \(\sigma_n^2 = \sigma^2 + \frac{n\sigma_J^2}{T-t}\)
Interpretation: Weighted average of Black-Scholes prices, each term corresponding to exactly \(n\) jumps before maturity.
7.8 The PIDE
For general derivatives, the pricing equation becomes a partial integro-differential equation:
\[\frac{\partial V}{\partial t} + (r - \lambda\kappa)S\frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2 S^2\frac{\partial^2 V}{\partial S^2}\] \[+ \lambda\int_{-\infty}^{\infty}[V(t, Se^y) - V(t,S)]f(y)\,dy = rV\]
The integral accounts for all possible jump outcomes, weighted by their probability density \(f(y)\).
Much harder to solve than Black-Scholes PDE! Typically requires numerical methods: - Monte Carlo simulation - Fourier methods (FFT) - Finite difference on jump-diffusion grid
7.9 Market Implications
7.9.1 Volatility Smile
Jump models naturally produce volatility smiles: - Out-of-the-money puts protect against downward jumps → higher implied vol - Near-the-money options less affected → lower implied vol - Creates the characteristic “smirk” in equity markets
7.10 Extensions
7.10.1 Double exponential jumps
Use asymmetric exponential distributions for up/down jumps:
\[f(y) = \begin{cases} p \eta_u e^{-\eta_u y} & y \geq 0 \\ (1-p) \eta_d e^{\eta_d y} & y < 0 \end{cases}\]
Allows different behavior for positive and negative jumps (market asymmetry).
7.11 Summary
Jump-diffusion models capture: - Discrete shocks in addition to continuous uncertainty - Fat tails and skewness in return distributions - Incomplete markets and non-hedgeable risk - More realistic market dynamics
They’re the bridge between Black-Scholes and more sophisticated Lévy process models.