Bates model क्या है?

Example 13.5 (Three models on one expiry, one on all)

Fitted to the one-month smile of chapter 9’s surface (nine strikes within two standard moves), Merton returns $\sigma=9.8\%$, $\lambda=3.16$, $\mu_J=-5.8\%$, $\delta=4.6\%$ with a root-mean-square error of 0.18 volatility point. Kou returns $\sigma=8.4\%$, $\lambda=11.7$, $p=0.24$, $\eta_1=85$, $\eta_2=30.4$, with an error of 0.06. Variance gamma (next section) has an error of 0.24 (Figure 13.2). Carried to other expiries, all three fail. At one year Merton’s skew is $-0.07$ against the market’s $-0.25$. At one week variance gamma’s is $-3.66$ against $-1.44$ (Figure 13.1). Bates fitted to all eight expiries from one week to two years has an error of 0.34 point, with $v_0=0.018$, $\kappa=2.06$, $\bar v=0.062$, $\eta=0.85$, $\rho=-0.70$ and jumps $\lambda=2.34$, $\mu_J=-3.6\%$, $\delta=3.0\%$. Its skew is $-1.36$ at one week and $-0.23$ at one year.