Risk-Neutral Variance Gamma Parameters and Stock Moments

Summary

The document presents the risk-neutral Variance Gamma stock model, including its drift adjustment and the time-changed Brownian representation of the log return. It describes sigma as controlling volatility, while nu and theta jointly determine asymmetry and kurtosis; when theta is zero, the process is symmetric and nu controls excess kurtosis.

The author asks how those parameter interpretations carry over from the Variance Gamma process to the risk-neutral stock price, and how the parameters determine the first four stock-price moments. The document supplies no derivation, numerical example, or answer to those questions. Its value is therefore as a focused statement of modeling questions rather than a complete method. Any interpretation of stock-price moments would need to account for exponentiating the process and the risk-neutral drift adjustment.

Key ideas

• The risk-neutral stock model adjusts the drift using a parameter chosen to enforce risk neutrality.
• Sigma controls the Variance Gamma process's volatility.
• Nu and theta jointly shape asymmetry and kurtosis, with a symmetric case when theta is zero.
• The document asks how process parameters affect stock-price moments but does not derive the answer.

Tags

# Risk Neutral Variance Gamma


# Risk Neutral Variance Gamma












In the risk neutral version of the Variance Gamma model the stock dynamics are

$$S_T=S_0 e^{ (r-q+\omega)t + X(t;\sigma,\nu,\theta)}$$

with

$$\omega=\frac{1}{\nu}\ln\left(1-\theta \nu - \frac{\sigma^2 \nu }{2}\right)$$

and where

$$X(t;\sigma,\nu,\theta) = \theta G(t; \nu)+\sigma G(t;\nu)W_t$$

is a Variance Gamma process where $G(t;\nu)$ is a Gamma distribution with mean $t$ and variance $\nu t$, and $W$ is $N(0,1)$.

For the VG process $X(t;\sigma,\nu,\theta)$ $\sigma$ controls the volatility, while $\nu$ and $\theta$ jointly control the asymmetry and kurtosis. Specifically, if $\theta = 0$, the VG process is symmetric and $\nu$ determines the excess kurtosis, but in other cases both $\nu$ and $\theta$ are joinly needed to obtain the higher moments.

However, I wonder how these interpretations translate to the risk neutral process for $S_T$. In particular:

- Is there any simple interpretation of how $\sigma$, $\nu$ and $\theta$ affect the risk neutral process?

- How does $\sigma$, $\nu$ and $\theta$ relate to the four first moments of $S_T$ (or $\frac{ S_T} { S_0}$)? This thesis (pages 31/32) obtains these four moments for $X(t;\sigma,\nu,\theta)$, but I have not seen a similar result for $S_T$ (or $\frac{ S_T} { S_0}$).

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.