Matthew Lorig

A Fast Mean-Reverting Correction to Heston's Stochastic Volatility Model

We propose a multiscale stochastic volatility model in which a fast mean-reverting factor of volatility is built on top of the Heston stochastic volatility model. A singular perturbative expansion is then used to obtain an approximation for European option prices. The resulting pricing formulas are semianalytic, in the sense that they can be expressed as integrals. Difficulties associated with the numerical evaluation of these integrals are discussed, and techniques for avoiding these difficulties are provided. Overall, it is shown that computational complexity for our model is comparable to the case of a pure Heston model, but our correction brings significant flexibility in terms of fitting to the implied volatility surface. This is illustrated numerically and with option data.

Explicit Implied Volatilities for Multifactor Local-Stochastic Volatility Models

We consider an asset whose risk-neutral dynamics are described by a general class of local-stochastic volatility models and derive a family of asymptotic expansions for European-style option prices and implied volatilities. Our implied volatility expansions are explicit; they do not require any special functions nor do they require numerical integration. To illustrate the accuracy and versatility of our method, we implement it under five different model dynamics: CEV local volatility, quadratic local volatility, Heston stochastic volatility,

A Family of Density Expansions for Lévy-Type Processes

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Second Order Multiscale Stochastic Volatility Asymptotics: Stochastic Terminal Layer Analysis & Calibration

stochastic volatility, and SABR local-stochastic volatility.

The Smile of Certain Lévy-Type Models

We consider a defaultable asset whose risk-neutral pricing dynamics are described by an exponential Levy-type martingale subject to default. This class of models allows for local volatility, local default intensity, and a locally dependent Levy measure. Generalizing and extending the novel adjoint expansion technique of Pagliarani, Pascucci, and Riga (2013), we derive a family of asymptotic expansions for the transition density of the underlying as well as for European-style option prices and defaultable bond prices. For the density expansion, we also provide error bounds for the truncated asymptotic series. Our method is numerically efficient; approximate transition densities and European option prices are computed via Fourier transforms; approximate bond prices are computed as finite series. Additionally, as in Pagliarani et al. (2013), for models with Gaussian-type jumps, approximate option prices can be computed in closed form. Sample Mathematica code is provided.

Spectral Decomposition of Option Prices in Fast Mean-Reverting Stochastic Volatility Models

Multiscale stochastic volatility models have been developed as an efficient way to capture the principle effects on derivative pricing and portfolio optimization of randomly varying volatility. The recent book Fouque, Papanicolaou, Sircar and S{\o}lna (2011, CUP) analyzes models in which the volatility of the underlying is driven by two diffusions -- one fast mean-reverting and one slow-varying, and provides a first order approximation for European option prices and for the implied volatility surface, which is calibrated to market data. Here, we present the full second order asymptotics, which are considerably more complicated due to a terminal layer near the option expiration time. We find that, to second order, the implied volatility approximation depends quadratically on log-moneyness, capturing the convexity of the implied volatility curve seen in data. We introduce a new probabilistic approach to the terminal layer analysis needed for the derivation of the second order singular perturbation term, and calibrate to S&P 500 options data.

The exact smile of certain local volatility models

We consider a class of assets whose risk-neutral pricing dynamics are described by an exponential Levy-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 Levy 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 Levy-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)....

A Taylor Series Approach to Pricing and Implied Vol for LSV Models

Using spectral decomposition techniques and singular perturbation theory, we develop a systematic method to approximate the prices of a variety of European and path-dependent options in a fast mean-reverting stochastic volatility setting. Our method is shown to be equivalent to those developed in [J.-P. Fouque, G. Papanicolaou, and R. Sircar, Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press, Cambridge, UK, 2000] but has the advantage of being able to price options for which the methods of [J.-P. Fouque, G. Papanicolaou, and R. Sircar, Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press, Cambridge, UK, 2000] are unsuitable. In particular, we are able to price double-barrier options. To our knowledge, this is the first time that double-barrier options have been priced in a stochastic volatility setting in which the Brownian motions driving the stock and volatility are correlated.

Small-time asymptotics for a general local-stochastic volatility model with a jump-to-default: curvature and the heat kernel expansion

We introduce a new class of local volatility models. Within this framework, we obtain expressions for both (i) the price of any European option and (ii) the induced implied volatility smile. As an illustration of our framework, we perform specific pricing and implied volatility computations for a CEV-like example. Numerical examples are provided.

Variance Swaps on Defaultable Assets and Market Implied Time-Changes

Using classical Taylor series techniques, we develop a unified approach to pricing and implied volatility for European-style options in a general local-stochastic volatility setting. Our price approximations require only a normal CDF and our implied volatility approximations are fully explicit (ie, they require no special functions, no infinite series and no numerical integration). As such, approximate prices can be computed as efficiently as Black-Scholes prices, and approximate implied volatilities can be computed nearly instantaneously.