## Glossary

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# NIG model

The Normal Inverse Gaussian (NIG) model \begin{eqnarray*} S_t&=&S_0\exp(rt+X(t)), \end{eqnarray*} where $X(t)$ is NIG Lévy process. \begin{eqnarray*} E[e^{iy\ln(S(T))}|S(t)]&=&\exp(iy\ln(S(t))+iyr(T-t)+A(t,T,iy))\\ A(t,T,iy)&=&(T-t)\delta\left(\sqrt{\alpha^2-\beta^2}-\sqrt{\alpha^2-(\beta+iy)^2}\right)\\ &&-iy(T-t)\delta\left(\sqrt{\alpha^2-\beta^2}-\sqrt{\alpha^2-(\beta+1)^2}\right), \end{eqnarray*} where $\alpha>|\beta+1|,~\delta>0$.

### Origin of NIG

The original NIG distribution depend on four parameters $(\alpha,\beta,\delta,\mu)$ and it is related to two independent Brownian motions $W_1$ and $W_2$. Let $W_1$ be a Brownian motion starting at $\mu$ with drift $\beta$ and let $W_2$ be a Brownian motion starting at 0 with drift $\sqrt{\alpha^2-\beta^2}$. Let $\tau_\delta=\inf\{s>0:W_2(s)>\delta\}$. Now $X=W_1(\tau_\delta)$ has a NIG distribution with parameters $(\alpha,\beta,\delta,\mu)$ and $E[e^{iyX}]=\exp(iy\mu+\delta(\sqrt{\alpha^2-\beta^2}-\sqrt{\alpha^2-(\beta+iy)^2}))$ In order to get the right model for stocks we should choose $\mu=-\delta(\sqrt{\alpha^2-\beta^2}-\sqrt{\alpha^2-(\beta+1)^2})$.

### References

Barndorff-Nielsen, O. E., 1997. Processes of Normal Inverse Gaussian Type. Finance and Stochastics 2, 41-68.