Jointly Estimating Jump-Diffusion Frequency and Jump Sizes
Summary
The document explains why jump intensity cannot be interpreted independently of the assumed jump-size distribution. To estimate a model’s jump frequency, it recommends fitting intensity and the other model parameters jointly, using maximum likelihood or generalized method of moments. The example is a log-price jump-diffusion with Brownian variation, Poisson arrivals, and independent jump sizes; the return density can be evaluated numerically before estimation.
It contrasts a double-exponential jump distribution whose tails begin at zero with a displaced version whose tails start away from zero. In the described simulation and estimation comparison, both specifications can produce similar tail behavior, yet the model allowing small jumps attributes more return noise to jumps and estimates a higher intensity with lower diffusion volatility. This illustrates that inferred frequency depends on model structure. The comparison is illustrative, and its parameter relationship is not presented as universal across datasets or specifications.
Key ideas
- Jump intensity depends on the assumed distribution of jump sizes.
- Estimate intensity jointly with the remaining jump-diffusion parameters using MLE or GMM.
- The example uses a Poisson jump process with double-exponential jump sizes and Brownian diffusion.
- Allowing jumps near zero can lead to a higher estimated intensity and lower estimated diffusion volatility.
- Different specifications may reproduce similar tail behavior while assigning variation differently between jumps and diffusion.
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# How to estimate lambda for Jump-Diffusion Process from Empirical data?
# 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?
## Answer by LocalVolatility (score 1, accepted)
https://quant.stackexchange.com/a/38645
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 $Y_i$:
$$ f_Y(x) = p \eta_+ e^{-\eta_+ \left( x - \kappa_+ \right)} \mathrm{1} \left\{ x \geq \kappa_+ \right\} + (1 - p) \eta_- e^{\eta_- \left( x - \kappa_- \right)} \mathrm{1} \left\{ x \leq \kappa_- \right\}. $$
- 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.
- 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:
- The diffusion coefficient $\sigma$ is larger and
- 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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.