We illustrate how to compute local risk minimization (LRM) of call options for exponential Levy models. We have previously obtained a representation of LRM for call options; here we transform it into a form that allows use of the fast Fourier transfo
rm method suggested by Carr & Madan. In particular, we consider Merton jump-diffusion models and variance gamma models as concrete applications.
We obtain explicit representations of locally risk-minimizing strategies of call and put options for the Barndorff-Nielsen and Shephard models, which are Ornstein--Uhlenbeck-type stochastic volatility models. Using Malliavin calculus for Levy process
es, Arai and Suzuki (2015) obtained a formula for locally risk-minimizing strategies for Levy markets under many additional conditions. Supposing mild conditions, we make sure that the Barndorff-Nielsen and Shephard models satisfy all the conditions imposed in Arai and Suzuki (2015). Among others, we investigate the Malliavin differentiability of the density of the minimal martingale measure. Moreover, some numerical experiments for locally risk-minimizing strategies are introduced.
Let $H$ be the quaternion algebra. Let $g$ be a complex Lie algebra and let $U(g)$ be the enveloping algebra of $g$. We define a Lie algebra structure on the tensor product space of $H$ and $U(g)$, and obtain the quaternification $g^H$ of $g$. Let $S
^3g^H$ be the set of $g^H$-valued smooth mappings over $S^3$. The Lie algebra structure on $S^3g^H$ is induced naturally from that of $g^H$. On $S^3$ exists the space of Laurent polynomial spinors spanned by a complete orthogonal system of eigen spinors of the tangential Dirac operator on $S^3$. Tensoring $U(g)$ we have the space of $U(g)$-valued Laurent polynomial spinors, which is a Lie subalgebra of $S^3g^H$. We introduce a 2-cocycle on the space of $U(g)$-valued Laurent polynomial spinors by the aid of a tangential vector field on $S^3$. Then we have the corresponding central extension $hat g(a)$ of the Lie algebra of $U(g)$-valued Laurent polynomial spinors. Finally we have the a Lie algebra $hat g=hat g(a)+Cd$ which is obtained by adding to $hat g(a)$ a derivation $d$ which acts on $hat g(a)$ as the radial derivation. When $g$ is a simple Lie algebra with its Cartan subalgebra $h$, We shall investigate the weight space decomposition of $(hat g, ad(hat h))$, where $hat h=h+Ca+Cd$ . The previo
