We propose a class of equity models whose volatility, L\'evy measure , and killing rate all have local stochastic state-dependence. In this framework we find a closed form solution for the price of any European-style option.  Additionally, for a certain sub-class of models, we find an exact expression for the induced implied volatility smile. To illustrate our framework, we perform specific computations for hybrid CEV/L\'evy model (which we christen as the "C\'EV" model)  .
We consider a class of assets whose risk-neutral pricing dynamics are described by an exponential L\'evy-type process subject to default. The class of processes we consider features locally-dependent drift, diffusion and default-intensity as well as a locally-dependent L\'evy measure . Using techniques from regular perturbation theory and Fourier analysis, we derive a series expansion for the price of a European-style option. We also provide precise conditions under which this series expansion converges to the exact price. Additionally, for a certain subclass of assets in our modeling framework, we derive an expansion for the implied volatility induced by our option pricing formula. The implied volatility expansion is exact within its radius of convergence. As an example of our framework, we propose a class of CEV-like L\'evy-type models. Within this class, approximate option prices can be computed by a single Fourier integral and approximate implied volatilities are explicit (i.e., no integration is required). Furthermore, the class of CEV-like L\'evy-type models is shown to provide a tight fit to the implied volatility surface of S P500 index options .
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propose
consider
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equity models whose volatility,
assets whose risk-neutral pricing dynamics are described by an exponential L\'evy-type process subject to default. The class of processes we consider features locally-dependent drift, diffusion and default-intensity as well as a locally-dependent
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, and killing rate all have local stochastic state-dependence. In this framework we find a closed form solution
. Using techniques from regular perturbation theory and Fourier analysis, we derive a series expansion
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any
a
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We also provide precise conditions under which this series expansion converges to the exact price.
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sub-class of models, we find an exact expression for the induced implied volatility smile. To illustrate
subclass of assets in our modeling framework, we derive an expansion for the implied volatility induced by our option pricing formula. The implied volatility expansion is exact within its radius of convergence. As an example of
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perform specific computations for hybrid CEV/L\'evy model (which we christen as the "C\'EV" model)
propose a class of CEV-like L\'evy-type models. Within this class, approximate option prices can be computed by a single Fourier integral and approximate implied volatilities are explicit (i.e., no integration is required). Furthermore, the class of CEV-like L\'evy-type models is shown to provide a tight fit to the implied volatility surface of S
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P500 index options