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Kou’s Double-Exponential Jump Model and Its Nonsmooth Density

Article Quant Q&A · Author: DomReyes
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Summary

The document gives the double-exponential jump model associated with Kou as an example of a process with a nonsmooth density. Unlike a Merton jump-diffusion, which models jump sizes with a normal distribution, this model uses separate exponential distributions for positive and negative jumps. Their probability weights sum to one, and their decay parameters govern the two tails.

The resulting density has a join at zero where it is not smooth. The response describes the distribution as sharply peaked with heavy tails and notes that the asymmetric jump specification can represent different effects from upward and downward moves. It also highlights tractability: the model has a simple characteristic function and supports closed-form prices for many derivatives. These are qualitative claims in the response; it provides no calibration evidence or comparison of pricing performance.

Key ideas

  • Kou’s model specifies positive and negative jump sizes with separate exponential distributions.
  • The density is nonsmooth at zero, where its two exponential pieces meet.
  • The jump distribution can be asymmetric, with distinct behavior for upward and downward jumps.
  • The model is described as tractable and as allowing closed-form prices for many derivatives.

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# Which models have non-smooth densities?


# Which models have non-smooth densities?












By smooth, I mean a density $f$ that lies in the space $C^\infty$, infinitely differentiable.

Are there, in the literature, some known models where the underlying density of the state process is non-smooth?

I have only been able to find one such example: the Variance Gamma model. Would love to hear if people know of more such examples.

## Answer by Kevin (score 2)

https://quant.stackexchange.com/a/51114

A simple model would be the double exponential model from Kou (2002). It is very similar to the jump-diffusion model from Merton (1976) but instead of modelling the jump size by a normal distribution, Kou employs an asymmetric double exponential distribution (aka Laplace distribution).

The corresponding density is $$f_X(x) = p\zeta e^{-\zeta x}\mathbb{1}_{\{x\geq 0\}}+q\eta e^{\eta x}\mathbb{1}_{\{x<0\}},$$ where $p+q=1$ and $\zeta,\eta>0$ and $p,q\geq0$.

This function is not smooth (at zero) and kind of represents the pasting of two exponential distributions. Like the exponential distribution, the double exponential distribution is memoryless. The model implies a high peaked and heavy tailed distribution. Kou argues (psychologically) that upwards/downwards jumps have different effects on investors and hence he opts for the pasting of two exponential distributions. The model is quite tractable, has an easy characteristic function and allows for closed-form solutions to prices of many derivatives.

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.