Visualize how sudden jumps affect asset price paths compared to standard diffusion processes.
Understanding Jump Diffusion
The jump diffusion model extends standard geometric Brownian motion by adding random jumps to account for sudden market movements:
dSt = μStdt + σStdWt + JtStdNt
Where:
- St = Asset price at time t
- μ = Drift rate
- σ = Volatility of diffusion component
- Wt = Standard Brownian motion
- Nt = Poisson process (counts jumps)
- Jt = Random jump size (log-normal in this model)
This model better captures the "fat tails" observed in real market returns compared to Black-Scholes.
Key Parameters
- Jump Intensity (λ): Average number of jumps per year
- Jump Size (μJ): Mean of the log jump size distribution
- Jump Volatility (σJ): Std dev of the log jump size distribution
In options pricing, jump risk leads to higher implied volatilities for short-dated options.