Home > An Introduction to the Merton's Jump Diffusion Model

Merton’s Jump Diffusion Model

1. The theory

Merton has proposed a model where the underlying asset price process is driven by a Brownian motion and a compound Poisson process. The jump size is assumed to follow a Gaussian distribution (log scale). Let St be the asset price process and process Xt satiesfies:

St = ert + Xt. 

Under Merton’s assumption, Xt follows the stochastic differential equation:

d Xt = γ d t + σ d Wt + 
Nt
∑
i=1
 Yi     (1)

where Wt is a standard Wiener process, Nt is a Possion random variable with intensity λ t, and Yi, i=1,2,… are i.i.d. random variables with law:

  Yi ∼ N(m, δ2)

γ in equation (1) is chosen in a way such that eXt is a martingale (the risk-neutral assumption). Under the Merton’s model, we have:

  γ = −
σ2
2
 − λ⎛
⎝		em+δ2/2−1⎞
⎠

Thus the Merton’s jump diffusion model is parameterized with five parameters:

The Levy density ν(·) for such a process is simply:

  ν(y) = λ f(y)

where f(·) is the density function for a Gaussian distribution with mean m and standard deviation δ.

2. Vanilla Option Pricing using Merton’s Jump Diffusion Model

For European style vanilla options, Merton’s Jump Diffusion model has an analytic solution. Consider a call option with time-to-maturity T and strike K. The payoff of this option is:

H(S) = max[0, (S−K)]

The risk-neutral price of this call option at time 0 is then:

c(S) = e−rT E[H(SerT+XT)]

The analytic form of the right hand side of the above equation is given as:

c(S) = 
 
∑
n≥0
e−lambda T (λ T)n
n!
CHBS(T,Sn,σn),     (2)

where σn2 = σ2 + nδ2/T,

Sn=S·exp⎡
⎣		nm+
nδ2
2
−λ T exp(m+δ2/2) + λ T⎤
⎦

and CHBS(T, S, σ) is the price of an European option with time-to-maturity T and payoff H(S) in a Black-Scholes model with volatility σ. The proof can be found in various sources [e.g. [1], they actually have a typo in the equation].

The analytic solution (2) is highly recommended for most real world applications because it is generally much faster and achieves better accuracy compared to other methods, for instance the Fourier transform methods.

We complete this session by giving the characteristic function for XT, which is used in the Fast Fourier solver [2]:

Φ(u) = E(eiuXt)  = exp⎛
⎝		iuγ T −u2Tσ2/2 + λ T eium−u2δ2/2− λ T⎞
⎠		     (3)

3. Calibration Result

Figure (1) shows the calibration results for Heston’s model and Merton’s Jump Diffusion model using the SPX market price data.

Figure 1: Merton’s Jump Diffusion Model vs Heston Model, Calibrated using SPX data. (April 25, 2011)

For calibration of the heston model, I borrowed code from Letian Wang [SITE].

References

[1]
Rama Cont and Peter Tankov, Financial Modelling With Jump Processes, CHAPMANHALL/CRC.
[2]
Peter Carr, Option Valuation Using the Fast Fourier Transform, 1999
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