Kou Double Exponential Jump-Diffusion Lab

Simulate Steven G. Kou's (2002) double exponential jump-diffusion model. Price European options via characteristic function Fourier inversion, analyze asymmetric upward/downward jump distributions, inspect volatility smiles, and evaluate Greeks.

Model Parameters
Contract & Asset
100.0
100.0
4.00%
1.50%
1.00 Y
Jump-Diffusion Dynamics
16.0%
3.0 /yr
30% (Up)
25.0
10.0
Kou Call Price
$0.00
Put: $0.00
Kou Implied Vol (IV)
0.00%
BS Flat: $0.00
Kou - BS Spread
$0.00
Total Vol: 0.00%
Compensator ($\zeta$)
0.00%
Mean Jump: 0.00%
Delta ($\Delta$) 0.0000
Gamma ($\Gamma$) 0.0000
Vega ($\mathcal{V}$) $0.00
Theta ($\Theta$) $0.000
Interactive Dynamic Visualizer
Dual 5x5 Analytical Sensitivity Matrices
Matrix 1: Up-Jump Prob ($p$) vs. Downward Decay ($\eta_2$) on 90% OTM Put Implied Vol (%)

Evaluates how jump directionality $p$ and crash severity $\eta_2$ drive downside crash put premiums.

Matrix 2: Strike ($K$) vs. Expiry ($T$) on Kou European Call Option Price ($)

Maps option pricing across moneyness and maturity tenors under double exponential jump-diffusion.

Quantitative Foundations: The Kou (2002) Jump-Diffusion Model

1. Asset Price Dynamics & Compound Poisson Jumps

Classical Black-Scholes-Merton (1973) theory assumes that asset prices follow continuous geometric Brownian motion. However, financial markets frequently experience abrupt, discontinuous price shocks caused by unexpected macroeconomic announcements, earnings surprises, or geopolitical crises. Robert C. Merton (1976) introduced jump-diffusion by augmenting Brownian motion with compound Poisson jumps having Gaussian jump magnitudes.

Steven G. Kou (2002) proposed a major breakthrough: replacing Gaussian jumps with an asymmetric double exponential jump distribution. Under the risk-neutral measure $\mathbb{Q}$, the asset price evolves according to:

$$\frac{dS_t}{S_{t-}} = (r - q - \lambda \zeta) dt + \sigma dW_t + d\left(\sum_{i=1}^{N_t} (V_i - 1)\right)$$

where:

  • $W_t$: Standard Brownian motion modeling normal continuous price fluctuations with diffusion volatility $\sigma$.
  • $N_t$: Poisson process with constant intensity $\lambda$ (expected number of jumps per year).
  • $V_i$: Jump magnitude such that $Y_i = \ln(V_i)$ represents the percentage jump in log-asset price.
  • $\zeta = \mathbb{E}[e^Y - 1]$: Risk-neutral jump compensator ensuring that $\mathbb{E}^{\mathbb{Q}}[S_T] = S_0 e^{(r-q)T}$.
2. The Double Exponential Distribution

In Kou's formulation, the log jump size $Y$ has an asymmetric double exponential density:

$$f_Y(y) = p \eta_1 e^{-\eta_1 y} \mathbf{1}_{\{y \ge 0\}} + q \eta_2 e^{\eta_2 y} \mathbf{1}_{\{y < 0\}}$$

where $p \in [0, 1]$ is the probability of an upward jump, $q = 1 - p$ is the probability of a downward jump, $\eta_1 > 1$ governs the exponential decay of upward jumps (with mean $1/\eta_1$), and $\eta_2 > 0$ governs the exponential decay of downward jumps (with mean $1/\eta_2$).

3. The Memoryless Property & Analytical Tractability

The decisive advantage of Kou's double exponential distribution over Merton's normal distribution is the memoryless property of the exponential distribution:

$$\mathbb{P}(Y > s + t \mid Y > s) = \mathbb{P}(Y > t)$$

In continuous-time boundary crossing problems (such as barrier options, lookback options, and American early exercise), when the asset price jumps over a barrier $H$, the conditional distribution of the overshoot across $H$ remains exponentially distributed with the exact same parameter! This eliminates the complex overshoot integrals required in Gaussian jump models, allowing exact, closed-form solutions for exotic derivatives and American options.

4. Jump Compensator & Characteristic Function

The risk-neutral expectation of the jump magnitude is:

$$\zeta = \mathbb{E}[e^Y - 1] = \frac{p \eta_1}{\eta_1 - 1} + \frac{q \eta_2}{\eta_2 + 1} - 1 \quad (\text{requiring } \eta_1 > 1)$$

The characteristic function of the jump magnitude $Y$ is:

$$\phi_Y(u) = \mathbb{E}[e^{i u Y}] = \frac{p \eta_1}{\eta_1 - i u} + \frac{q \eta_2}{\eta_2 + i u}$$

And the characteristic function of $\ln(S_T)$ is:

$$\phi(u) = \exp\left( i u [\ln S_0 + (r - q - \lambda \zeta - \frac{1}{2}\sigma^2)T] - \frac{1}{2}\sigma^2 u^2 T + \lambda T (\phi_Y(u) - 1) \right)$$

European option prices are evaluated via Gil-Pelaez Fourier inversion:

$$C(S_0, K, T) = S_0 e^{-q T} P_1 - K e^{-r T} P_2$$
Kou Jump-Diffusion Mastery Quiz

1. How does the Kou (2002) model mathematically specify the distribution of asset price jumps?

2. What key mathematical property makes Kou's model analytically superior for pricing barrier and American options?

3. How can Kou's parameters be calibrated to reflect the typical downside volatility skew in equity index options?

4. Why is the condition $\eta_1 > 1$ strictly required in Kou's model?

Frequently Asked Questions

The Kou (2002) model, developed by Steven G. Kou in "A Jump-Diffusion Model for Option Pricing" (Management Science), is a continuous-time financial asset pricing model where log-asset returns follow Brownian motion superimposed with a compound Poisson process whose jump sizes have an asymmetric double exponential distribution: $f_Y(y) = p \eta_1 e^{-\eta_1 y}$ for $y \ge 0$ and $q \eta_2 e^{\eta_2 y}$ for $y < 0$. This formulation captures both continuous market volatility and discontinuous market shocks with heavy tails and skewness.

While Robert C. Merton's (1976) model assumes jump sizes are normally distributed, Kou assumes jumps follow an asymmetric double exponential distribution. This distinction has profound mathematical implications: because the exponential distribution possesses the memoryless property, the distribution of first-passage times and overshoot across boundaries can be computed analytically. Consequently, Kou's model yields closed-form analytical solutions not only for plain-vanilla European options, but also for path-dependent barrier options and American options, which are analytically intractable under Merton's normal jumps.

The memoryless property of the exponential distribution states that $\mathbb{P}(Y > s + t \mid Y > s) = \mathbb{P}(Y > t)$. In option pricing, when an asset price jumps across a barrier (such as a barrier option knock-out level or an American early-exercise boundary), the overshoot distribution conditioned on jumping over the boundary is again exponentially distributed with the exact same parameter. This eliminates the notorious 'overshoot problem' in continuous-time boundary analysis, enabling exact analytical solutions for exotic and American derivatives.

The parameter $p$ specifies the proportion of jumps that are upward rallies (with mean $1/\eta_1$), while $q = 1 - p$ specifies the proportion of jumps that are downward drops (with mean $1/\eta_2$). In equity index markets, setting $p < 0.50$ and $\eta_2 < \eta_1$ creates a market regime where crashes are both more frequent and larger in expected magnitude than rallies. This generates pronounced negative skewness in the risk-neutral distribution, driving up out-of-the-money put prices and producing the steep downward-sloping volatility skew observed empirically in equity markets.

Under the risk-neutral measure $\mathbb{Q}$, the discounted underlying asset price $S_t e^{-rt}$ must be a martingale. Because jumps introduce a non-zero expected price percentage change $\mathbb{E}[e^Y - 1] = \zeta = \frac{p \eta_1}{\eta_1 - 1} + \frac{q \eta_2}{\eta_2 + 1} - 1$ (requiring $\eta_1 > 1$ for finite expectation), the continuous drift of the asset must be compensated by subtracting $\lambda \zeta$: $\text{drift} = r - q - \lambda \zeta - \frac{1}{2}\sigma^2$. This guarantees that $\mathbb{E}[S_T] = S_0 e^{(r-q)T}$, ensuring that derivatives are priced free of static arbitrage.
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