How would you modify the Black-Scholes model to incorporate jumps in the stock price?

The Jump-Diffusion Model, proposed by Robert C. Merton in 1976, is an extension of the Black-Scholes model to incorporate jumps in stock prices. In this model, we assume that the stock price follows a jump-diffusion process which can be described by a stochastic differential equation (SDE) that combines a continuous diffusion process with a discrete jump process.

Let S_t denote the stock price at time t. The dynamics of S_t under the risk-neutral measure in the Jump-Diffusion model can be described by the following SDE:

dS_t = (r − λk)S_tdt + σS_tdW_t + S_tdY_t
where:

- r: the risk-free interest rate

- λ: the intensity (average frequency) of jumps per year

- k: the expected return per jump (based on log-normal distribution)

- σ: the annualized volatility of the stock price (vol(S_t))

- W_t: a standard Brownian motion

- Y_t: a compound Poisson process with jumps that have a log-normal distribution with mean μ_J and standard deviation σ_J

The dynamics of S_t can be further decomposed into two components:

1. Continuous Process (Diffusion): The first two terms on the right-hand side of the SDE represent the classical Black-Scholes dynamics, with a drift term adjusted for the jump expectation, (r − λk)S_tdt, and a diffusion term, σS_tdW_t.

2. Jump Process (Discrete): The third term on the right-hand side of the SDE corresponds to the jump component, which is described by the product of the stock price, S_t, and the increment of the compound Poisson process, dY_t.

To price European options using the Jump-Diffusion model, we can apply the risk-neutral valuation formula:

C(S_t, t) = e^{− r(T − t)}𝔼^ℚ[(S_T−K)⁺],

where C(S_t, t) denotes the price of a European call option at time t, K is the strike price, T is the time until expiration, and the expectation is taken under the risk-neutral probability measure ℚ.

However, the classic Black-Scholes formula does not apply directly to the Jump-Diffusion model, as it assumes a log-normal distribution for stock prices. To compute the option price for a Jump-Diffusion model, we typically rely on numerical methods such as:

1. Fourier Transform methods (e.g., Carr-Madan Fast Fourier Transform (FFT) method)

2. Monte Carlo simulations

3. Finite-difference methods / Partial differential equation (PDE) methods

As an example, let’s consider the Fourier Transform method. Using jump-diffusion processes, the characteristic function ϕ(u) of the log price log S_T at time T can be derived as follows:

$$\begin{aligned} \phi(u) &= \mathbb{E}^\mathbb{Q}\left[e^{iu\log S_T}\right] \\ &= \text{exp}\left\{iu\left(\log S_t + \left[r - \lambda k - \frac{\sigma^2}{2} \right](T-t)\right) \right.\\ &\qquad\left. - \frac{u^2 \sigma^2 (T-t)}{2} + \lambda(T-t)\left(e^{iu\mu_J - \frac{u^2 \sigma_J^2}{2}} -1\right)\right\}.\end{aligned}$$

Next, we can compute the option price using the inverse Fourier Transform or a Fast Fourier Transform as in the Carr-Madan approach.

In summary, the Jump-Diffusion model provides a more realistic representation of stock price dynamics by incorporating both continuous diffusion and discrete jump processes. This model can be used to price European options using various numerical methods, with the Fourier Transform method being one possible approach.