Jump-Diffusion Models & Merton's Model

Mathematical foundation

Merton's jump-diffusion model describes asset price dynamics using the following stochastic differential equation:

$\frac{dS}{S} = (\mu - \lambda k)dt + \sigma dW + (J - 1)dN$

Where:

• $S$ is the asset price
• $\mu$ is the drift rate
• $\sigma$ is the volatility
• $dW$ is a Wiener process
• $\lambda$ is the jump intensity (average number of jumps per year)
• $k = E[J-1]$ is the average jump size
• $dN$ is a Poisson process with intensity $\lambda$
• $J$ is the jump magnitude (typically log-normally distributed)

Components of jump-diffusion models

Continuous component

The continuous part follows geometric Brownian motion, similar to the Black-Scholes Model for Option Pricing:

$(\mu - \lambda k)dt + \sigma dW$

Jump component

The jump component introduces sudden price changes:

$(J - 1)dN$

This captures market shocks, announcements, and other discrete events that cause immediate price movements.

Option pricing under Merton's model

The option price is a weighted average of Black-Scholes prices, each assuming a different number of jumps:

$C(S,t) = \sum_{n=0}^{\infty} \frac{e^{-\lambda T}(\lambda T)^n}{n!} C_{BS}(S,t,\sigma_n)$

Where:

• $C_{BS}$ is the Black-Scholes option price
• $\sigma_n$ is the adjusted volatility incorporating n jumps
• $T$ is time to expiration

Risk management applications

Portfolio hedging

Jump-diffusion models improve delta hedging strategies by accounting for:

1. Continuous price changes
2. Sudden market movements
3. Tail risk events

Risk metrics

The model enhances calculation of:

• Value at Risk (VaR)
• Expected shortfall
• Option-adjusted spreads

Market calibration

Parameter estimation

Key parameters requiring calibration:

1. Jump intensity ($\lambda$)
2. Jump size distribution
3. Continuous volatility ($\sigma$)

Market data sources

Calibration typically uses:

• Option prices across strikes and maturities
• Historical price data
• Implied volatility surfaces

Limitations and extensions

Model constraints

1. Assumes constant parameters
2. May overfit to historical data
3. Computational complexity in implementation

Modern extensions

1. Stochastic volatility with jumps
2. Time-varying jump intensities
3. State-dependent jump sizes

Applications in modern markets

High-frequency trading

Jump-diffusion models help in:

• Risk assessment for algorithmic trading
• Market making strategies
• Statistical arbitrage

Derivatives pricing

Enhanced accuracy for:

• Exotic options
• Structured products
• Over-the-counter (OTC) derivatives