# How to estimate lambda for Jump-Diffusion Process from Empirical data?

So, I have really no idea how to go about this, but how would I go about choosing sensible parameter values for a basic jump-diffusion simulation, namely $\lambda$ ? For example, getting the average frequency of such huge changes occurring in a short period, and maybe their average size.

The only idea I have is to apply maximum likelihood estimation to an entire specified jump-diffusion model based on some data, but that seems like a whole lot of work to just get an idea for a reasonable jump process.

What would be a simple, yet theoretically respectable method of doing this?

## 1 Answer

TLDR:

The jump frequency depends on how you specify the jump size distribution. If you want the $\lambda$ to actually represent the jump frequency under a certain jump-diffusion model, then you should jointly estimate all model parameters, e.g. using maximum likelihood estimation (MLE) or generalized method of moments (GMM).

Example:

Consider a general jump-diffusion model for the logarithmic asset price process $X_t = \ln \left( S_t / S_0 \right)$

$$X_t = \gamma t + \sigma W_t + \sum_{i = 1}^n Y_i$$

where $Y_i$ are the i.i.d. jump sizes and the Poisson process $N$ has a intensity $\lambda$. You can relatively easily compute the log-return density via e.g. the Fang and Oosterlee (2008) COS method and then run a MLE jointly for all model parameters. Consider e.g. the following specification of the density of $Y_i$:

$$f_Y(x) = p \eta_+ e^{-\eta_+ \left( x - \kappa_+ \right)} \mathrm{1} \left\{ x \geq \kappa_+ \right\} + (1 - p) \eta_- e^{\eta_- \left( x - \kappa_- \right)} \mathrm{1} \left\{ x \leq \kappa_- \right\}.$$

1. original model: For $\kappa_+ = \kappa_- = 0$, this is the Kou (2002) double exponential jump diffusion model. The jump size density has two exponential tails that start at the origin.

2. displaced model: For $\kappa_+ > 0$ and $\kappa_- < 0$, the two tails are being displaced away from the origin.

If you estimate both of the above models using MLE then you will find that under the displaced model:

1. The diffusion coefficient $\sigma$ is larger and

2. the intensity $\lambda$ smaller.

The reasons for this is that in the original model, both the diffusion and the jumps generate small return noise. Thus you need more jumps overall in the original model to obtain the same overall number of large jumps. This is compensated for by a smaller diffusion coefficient.

The below plot illustrates this using the Levy measure. It was generated by (i) fixing the parameters of the displaced model, (ii) simulating a time-series of logarithmic returns and then (iii) using MLE to infer the matching original model parameters. We see that the tail behaviour is almost identical. The original model also generates jumps in $\left[ \kappa_-, \kappa_+ \right]$ and thus needs a higher frequency of them. • So it all in the end depends upon calculating the transition density somehow, as in Sahalia's work? – Coolio2654 Mar 7 '18 at 2:00
• If you want to use MLE then yes, you need the density. In case of normally distributed jumps, the log-return is still normal conditional on $N_t$ and you can express it as a sum. For more general jump size distributions, I’d recommend the COS method mentioned in my answer. You don’t need a density when using GMM however but could directly use the characteristic function evaluated at certain points as the “moments” - search for “empirical characteristic function”. – LocalVolatility Mar 7 '18 at 7:22
• Thank you for pointing me in the direction of another method than MLE that could be much more tractable, then. – Coolio2654 Mar 7 '18 at 7:29