Abstracts
Thursday June 5, 14:00-14:30 | session 8.7 | Stochastic Volatility | room I
In this work we derive new closed-form pricing formulas for common volatility derivatives such as realized variance and VIX options. Our approach is based on the classical methodology of expanding a density function in a finite sum of polynomials weighted by a kernel. Approximations based on the Gaussian distribution such as Edgeworth or Gram-Charlier expansions have been successfully employed by a number of authors in the context of equity options.
However these expansions are not quite suitable for volatility/variance densities as they inherently assign positive mass to the negative real line. Here we develop option-prices approximations via expansions which instead are based on kernels defined on the positive real line. Specifically, we consider a rather flexible family of distributions which generalizes the gamma kernel associated with the classical Laguerre expansions.
The method can be employed whenever the moments of the underlying variance distribution are known. It provides fast and accurate price-computations and therefore it represents a valid and possibly more robust alternative to pricing techniques based on Laplace transform inversions.
We illustrate the accuracy of the developed approximations for realized variance options and for VIX options in the Heston (1993) model as well as in the jump-diffusion SVJJ model proposed by Duffie et al. (2000).
Thursday June 5, 14:30-15:00 | session 8.7 | Stochastic Volatility | room I
In this work we suggest a suitable framework for the SABR model of so-called weighted spaces, on which Feller-like properties hold, with a view on applicability to approximation schemes for unbounded payoffs. We first describe a recently introduced flexible functional analytic framework extending the Feller property towards unbounded functions of controlled growth. Motivated by an analysis of the sub-eigenspaces of the SABR-infinitesimal generator, we accordingly construct spaces on which semigroups of positive bounded operators are in fact strongly continuous. The result is crucial to prove convergence of splitting schemes for payoffs with controlled growth.
Thursday June 5, 15:00-15:30 | session 8.7 | Stochastic Volatility | room I
We consider a tractable affine stochastic volatility model that generalizes the seminal Heston (1993) by augmenting it with jumps in the instantaneous variance. This model class can be seen as a particular instance of the double-jump model of Duffie, Pan and Singleton (2000) allowing for jumps only at the variance level. Embedded in this modeling framework is also the case of variance processes described by the pure-jump Ornstein-Uhlenbeck type processes introduced by Barndorff-Nielsen and Shepard (2001).
In this framework, we consider options written on the realized variance and we examine how the particular choice of the distribution of jumps impacts the associated implied volatility smile. In particular, we show that by selecting alternative jump distributions, one obtains fundamentally different shapes of the implied volatility of variance smile - some clearly at odds with the upward-sloping volatility skew observed in variance-derivatives markets. For example, we find that the Gamma distribution leads to a downward-sloping volatility skew, the Inverse Gamma distribution predicts an upward-sloping skew, while the Inverse Gaussian distribution leads to a frown of the implied volatility surface.
Our analysis is based on the classical Tauberian theorems and the more recent results of Lee (2004) and Gulisashvili (2012) on implied volatility asymptotics. Specifically, we derive easy-to-check sufficient conditions for the asymptotic behavior of volatility of variance for small as well as large strikes, given the particular distribution of variance jumps.
Thursday June 5, 15:30-16:00 | session 8.7 | Stochastic Volatility | room I
We introduce downward volatility jumps into a general framework of modeling the term structure of variance. With variance swap data alone, we find that downward volatility jumps are associated with a resolution of policy uncertainty, in particular through statements from Federal Open Market Committee meetings and speeches of Federal Reserve chairmen, and that such jumps are priced with positive risk premia, which reflect the premia for the ``put protection'' offered by the Federal Reserve. On the modeling side, we explore the structural differences and relative goodness-of-fits of factor specifications, and find that a log-volatility model with two Ornstein-Uhlenbeck factors and two-sided jumps is superior in capturing the volatility dynamics.