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Quantitative Finance · Glosarium

Apa itu Variance gamma model?

Definition 13.6 Derivatives and Volatility · Bab 13 — Jumps and Lévy Models

The variance gamma model runs a Brownian motion with drift θ\theta and volatility σ\sigma on a random clock gtg_t, a gamma process with mean rate one and variance rate ν\nu (a subordinator, One Quant Book 4, chapter 6): xt=ωt+θgt+σWgtx_t=\omega t+\theta g_t+\sigma W_{g_t}, with ω\omega set so that E[ext]=1\E[e^{x_t}]=1. Its characteristic function is E[eiuxt]=eiuωt(1−iuθν+12σ2νu2)−t/ν\E[e^{iux_t}]=e^{iu\omega t}\bigl(1-iu\theta\nu+\frac12\sigma^2\nu u^2\bigr)^{-t/\nu}, and ω=1νln⁡(1−θν−12σ2ν)\omega=\frac1\nu\ln\bigl(1-\theta\nu-\frac12\sigma^2\nu\bigr).

Contoh

Example 13.7 (Moments that fade)

For the Merton model fitted above, the skewness of the log-return is −2.90-2.90 at one week, −1.39-1.39 at one month and −0.40-0.40 at one year, and the excess kurtosis 15.2, 3.5 and 0.29: exactly the T−1/2T^{-1/2} and T−1T^{-1} laws. The fitted variance gamma model is within a few percent of the same numbers, since both were fitted to the same month. That is why a Lévy model fitted at one expiry cannot fit the others. The market’s skew decays like T−0.44T^{-0.44} (chapter 12), close to the Lévy exponent −12-\frac12 at short expiries, but at long expiries it is held up by the persistence of volatility, which no Lévy process has.

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