Risk-Neutral Constraints in Variance Gamma Option Pricing
Summary
The document asks whether European options can be priced in closed form when the variance gamma model’s drift, volatility, and variance rate are random. It explains that unrestricted random coefficients generally do not yield a closed-form price: risk-neutral pricing imposes a constraint on the drift and variance parameters so the discounted asset price remains a martingale. The stated inequality links the gamma variance rate to the Brownian drift and volatility.
As an alternative, the response suggests making volatility stochastic through a separate process, such as a square-root process, while retaining gamma time subordination. The characteristic-function derivation would then use the chosen stochastic-volatility model’s conditional characteristic function in place of the ordinary Brownian one. A closed-form characteristic function may be possible if the volatility model is selected carefully. The document offers a modeling direction rather than a complete derivation or option-pricing formula, and it gives no empirical tests or comparison of model performance.
Key ideas
- Risk-neutral pricing requires the discounted asset price to be a martingale, constraining variance gamma parameters.
- Randomizing drift, volatility, and variance rate generally prevents a closed-form solution under the stated setup.
- A separate stochastic-volatility process can be combined with gamma time subordination.
- A tractable volatility specification may allow a closed-form characteristic function, which can support option pricing.
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# Answer by pbr142 (score 1)
# Closed form european option prices for a variance gamma process with a randomly distributed drift, volatility, and variance rate
Does an option pricing model with a closed form European option price exist that takes into account randomly distributed drift, volatility, and variance rate?
I prefer a modification to the variance gamma model, but a modification to any other model is welcome.
## Answer by pbr142 (score 1)
https://quant.stackexchange.com/a/10855
In general, there cannot be a closed-form solution of a random coefficients VG model. The reason is the drift-restriction that needs to be imposed to ensure that the discounted price process is a martingale under the risk-neutral measure. Using the bank account as numeraire, the restriction is $$ \frac{1}{\beta} > \theta + \frac{\sigma^2}{2} $$ where $\beta$ is the variance rate of the gamma subordinator, and $\theta$ and $\sigma$ are the drift and diffusion coefficient of the driving Brownian motion.
What can be done is to make the volatility of the driving process stochastic using a second (independent) process. So you could have a square-root process driving the volatility while the main process is subordinated to a gamma time. The book by Barndorff-Nielsen and Shiryaev, Chapter 12, contains some hints how to do that. The procedure would be analogous to the derivation of the regular VG characteristic function, but you would get the conditional characteristic function of the stochastic volatility model chosen instead of the normal distribution characteristic function from a regular Brownian motion. If you carefully select the volatility model, you can take the expectation with respect to the gamme time and get a closed-form solution of the characteristic function.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.