What is a jump diffusion model, and how does it differ from the standard geometric Brownian motion used in the Black-Scholes-Merton model?

A Jump Diffusion Model is a type of financial model that aims to incorporate both continuous price changes, captured by the Geometric Brownian Motion (GBM), and sudden large movements or jumps, captured by a jump process, in the price dynamics of an asset. This model was first proposed by Robert C. Merton in 1976 as a means to improve the accuracy of the Black-Scholes-Merton (BSM) option pricing model, which employs standard geometric Brownian motion.

In the standard Geometric Brownian Motion used in the BSM model, the price dynamics of an asset are defined by the following stochastic differential equation (SDE):

dS_t = μS_tdt + σS_tdW_t

Where:

- S_t is the asset price at time t

- μ is the expected return (drift) of the asset

- σ is the volatility of the asset

- dt is the incremental time

- dW_t ∼ 𝒩(0, dt) is the increment of a standard Wiener process (Brownian motion) i.e., dW_t follows a normal distribution with mean 0 and variance dt

However, Geometric Brownian motion does not capture the jump component in asset prices, which can be observed in real financial data.

To account for these jumps, the Jump Diffusion Model integrates a Poisson jump process into the standard GBM. The SDE of a Jump Diffusion model is as follows:

dS_t = (μS_t + λζ)dt + σS_tdW_t + S_tdY_t

Where:

- λ is the intensity of the jump process or the expected number of jumps per unit of time

- ζ is the average jump size or the mean of the jump distribution

- dY_t is the increment of a compound Poisson process or the jump process, representing the impact of jumps in the asset price

The key difference between the Jump Diffusion model and the standard GBM of the BSM model is the incorporation of the jump process through the additional term S_tdY_t. With this addition, Jump Diffusion models provide a better fit to empirical data which often exhibit leptokurtosis (fat tails) and skewness, features that are not captured by the standard GBM alone.

To illustrate a possible application of the Jump Diffusion model, consider its use in option pricing. Merton developed a closed-form solution for European-style options pricing using a Jump Diffusion model, known as the Merton Jump Diffusion Model:

$$C(S_t, t) = e^{-\lambda T} \sum_{n=0}^\infty \frac{(\lambda T)^n}{n!} C_{BS}(S_t, t, \bar{\sigma}_n, \bar{r}_n)$$

Where:

- C(S_t, t) is the option’s price

- C_BS(S_t, t, σ̄_n, r̄_n) is the standard Black-Scholes price of an option with modified parameters σ̄_n and r̄_n

- T is the time to maturity

- e is the base of the natural logarithm

- $\bar{\sigma}_n = \sqrt{\sigma^2 + \frac{n \eta^2}{T}}$, representing the modified volatility that incorporates the impact of jumps

- $\bar{r}_n = r + \zeta \lambda - \frac{n \gamma}{T}$, representing the modified drift rate that incorporates the impact of jumps

- η and γ are the standard deviation and skewness parameters of the jump size distribution, respectively

In summary, the Jump Diffusion model extends the standard Geometric Brownian Motion used in the BSM model to account for asset price jumps not captured by simple diffusion processes. This allows the model to better fit empirical data in finance, which often exhibit sudden large movements (jumps) and fat-tailed return distributions.