Gabriel G. Drimus

Options on Realized Variance by Transform Methods: A Non-Affine Stochastic Volatility Model

In this paper we study the pricing and hedging of options on realized variance in the 3/2 non-affine stochastic volatility model by developing efficient transform-based pricing methods. This non-affine model gives prices of options on realized variance that allow upward-sloping implied volatility of variance smiles. Hestons model [Rev. Financial Stud., 1993, 6, 327–343], the benchmark affine stochastic volatility model, leads to downward-sloping volatility of variance smiles—in disagreement with variance markets in practice. Using control variates, we propose a robust method to express the Laplace transform of the variance call function in terms of the Laplace transform of the realized variance. The proposed method works in any model where the Laplace transform of realized variance is available in closed form. Additionally, we apply a new numerical Laplace inversion algorithm that gives fast and accurate prices for options on realized variance, simultaneously at a sequence of variance strikes. The method is also used to derive hedge ratios for options on variance with respect to variance swaps.

Local Volatility of Volatility for the VIX Market

Following a trend of sustained and accelerated growth, the VIX futures and options market has become a closely followed, active and liquid market. The standard stochastic volatility models -- which focus on the modeling of instantaneous variance -- are unable to fit the entire term structure of VIX futures as well as the entire VIX options surface. In contrast, we propose to model directly the VIX index, in a mean-reverting local volatility-of-volatility model, which will provide a global fit to the VIX market. We then show how to construct the local volatility-of-volatility surface by adapting the ideas in Carr (2008) and Andreasen, Huge (2010) to a mean-reverting process.

Closed form convexity and cross-convexity adjustments for Heston prices

We present a new and general technique for obtaining closed-form expansions for prices of options in the Heston model, in terms of Black–Scholes prices and Black–Scholes Greeks up to arbitrary order. We then apply the technique to solve, in detail, the cases for the second-order and third-order expansions. In particular, such expansions show how the convexity in volatility, measured by the Black–Scholes volga, and the sensitivity of delta with respect to volatility, measured by the Black–Scholes vanna, impact option prices in the Heston model. The general method for obtaining the expansion rests on the construction of a set of new probability measures, equivalent to the original pricing measure, and which retain the affine structure of the Heston volatility diffusion. Finally, we extend the method to the pricing of forward-starting options in the Heston model.

Valuation of Options on Discretely Sampled Variance: A General Analytic Approximation

The values of options on realized variance are significantly impacted by the discrete sampling of realized variance and may be substantially higher than the values of options on continuously sampled variance (or, quadratic variation). Under arbitrary stochastic volatility dynamics, we analyze the discretization effect and obtain a simple analytical correction term to be applied to the value of options on continuously sampled variance. Our final result is remarkably compact and allows for a straightforward implementation in many of the standard stochastic volatility models proposed in the literature.

Options on Realized Variance in Log-OU Models

Abstract We study the pricing of options on realized variance in a general class of Log-OU (Ornstein–Ühlenbeck) stochastic volatility models. The class includes several important models proposed in the literature. Having as common feature the log-normal law of instantaneous variance, the application of standard Fourier–Laplace transform methods is not feasible. We derive extensions of Asian pricing methods, to obtain bounds, in particular, a very tight lower bound for options on realized variance.

Smooth and Bid-Offer Compliant Volatility Surfaces Under General Dividend Streams

Given bid-offer quotes for a set of listed vanilla options, a fundamental need of option market makers is to interpolate and extrapolate the available quotes to a full arbitrage-free surface. We propose a methodology which directly controls the trade-off between smoothness and bid-offer compliance of the resulting volatility surface. Unlike previous literature, the method applies simultaneously to all listed maturities and aims to smooth the implied risk-neutral densities. Additionally, we consider asset dynamics which allow for general dividend streams--continuous, discrete yield and discrete cash--a modelling aspect of key importance in option markets.

Volatility-of-Volatility : A Simple Model-Free Motivation

Our goal is to provide a simple, intuitive and model-free motivation for the importance of volatility-of-volatility in pricing certain kinds of exotic and structured products.