Pricing variance derivatives using hybrid models with stochastic interest rates

In this thesis, the research focuses on the development and implementation of two hybrid models for pricing variance swaps and variance options. Some variance derivatives (i.e., variance swap) are priced using portfolios of put and call options. However, longer-term options price not only stock variance, but also interest rate variance. By ignoring stochastic interest rates, variance derivatives utilizing this approach are overpriced. In recent months, the Federal Reserve lowered the funds rate as the equity markets fell. This created correlation between equities and interest rates. Furthermore, interest rate volatility increased. Thus, it is presently crucial to understand how stochastic interest rates and correlation impact the pricing of variance derivatives. The first model (SR-LV) is driven by two processes: the stock return follows a diffusion and the stochastic interest rate is driven by the Hull-White short rate dynamics. Local volatility is constructed with the help of Gyongy's result on recovering a Markov process from a general n-dimensional Ito process with the same marginals. In the solution for the local volatility, the joint forward density of the stock price and interest rate is derived by solving an appropriate partial differential equation. Realized variance can then be computed by Monte Carlo simulation under the forward measure where local variances are collected over each realized path and averaged. Results are presented for different levels of assumed correlation between the stock price and interest rates. Prices obtained are lower than those produced with an options portfolio and this price difference strongly depends on the volatility of the short rate. The second model (SR-SLV) adds one more dimension to the first model. In practice, volatility of a stock may change without the stock price moving. This effect is not captured in SR-LV model, but stochastic local volatility exhibits this trait. In this setting, a leverage function must be calibrated utilizing the joint density of the stock price, interest rate, and a stochastic term governed by a mean reverting lognormal model. By design, the price of variance swaps is the same as under SRLV dynamics. However, variance option prices differ from SR-LV model and are presented for different levels parameters of the new stochastic component. Although this work focuses on pricing variance derivatives, the developed methodology is extended to pricing volatility swaps and options

The valuation of variance swaps under stochastic volatility, stochastic interest rate and full correlation structure

This paper considers the case of pricing discretely-sampled variance swaps under the class of equity-interest rate hybridization. Our modeling framework consists of the equity which follows the dynamics of the Heston stochastic volatility model, and the stochastic interest rate is driven by the Cox-Ingersoll-Ross (CIR) process with full correlation structure imposed among the state variables. This full correlation structure possesses the limitation to have fully analytical pricing formula for hybrid models of variance swaps, due to the non-affinity property embedded in the model itself. We address this issue by obtaining an efficient semi-closed form pricing formula of variance swaps for an approximation of the hybrid model via the derivation of characteristic functions. Subsequently, we implement numerical experiments to evaluate the accuracy of our pricing formula. Our findings confirm that the impact of the correlation between the underlying and the interest rate is significant for pricing discretely-sampled variance swaps

Exact Pricing and Hedging Formulas of Long Dated Variance Swaps under a 3/23/2 Volatility Model

This paper investigates the pricing and hedging of variance swaps under a $3/2$ volatility model. Explicit pricing and hedging formulas of variance swaps are obtained under the benchmark approach, which only requires the existence of the num\'{e}raire portfolio. The growth optimal portfolio is the num\'{e}raire portfolio and used as num\'{e}raire together with the real world probability measure as pricing measure. This pricing concept provides minimal prices for variance swaps even when an equivalent risk neutral probability measure does not exist.Comment: 23 pages, 5 figure

Spectral methods for volatility derivatives

In the first quarter of 2006 Chicago Board Options Exchange (CBOE) introduced, as one of the listed products, options on its implied volatility index (VIX). This created the challenge of developing a pricing framework that can simultaneously handle European options, forward-starts, options on the realized variance and options on the VIX. In this paper we propose a new approach to this problem using spectral methods. We use a regime switching model with jumps and local volatility defined in \cite{FXrev} and calibrate it to the European options on the S&P 500 for a broad range of strikes and maturities. The main idea of this paper is to "lift" (i.e. extend) the generator of the underlying process to keep track of the relevant path information, namely the realized variance. The lifted generator is too large a matrix to be diagonalized numerically. We overcome this difficulty by applying a new semi-analytic algorithm for block-diagonalization. This method enables us to evaluate numerically the joint distribution between the underlying stock price and the realized variance, which in turn gives us a way of pricing consistently European options, general accrued variance payoffs and forward-starting and VIX options.Comment: to appear in Quantitative Financ

Moment Methods for Exotic Volatility Derivatives

The latest generation of volatility derivatives goes beyond variance and volatility swaps and probes our ability to price realized variance and sojourn times along bridges for the underlying stock price process. In this paper, we give an operator algebraic treatment of this problem based on Dyson expansions and moment methods and discuss applications to exotic volatility derivatives. The methods are quite flexible and allow for a specification of the underlying process which is semi-parametric or even non-parametric, including state-dependent local volatility, jumps, stochastic volatility and regime switching. We find that volatility derivatives are particularly well suited to be treated with moment methods, whereby one extrapolates the distribution of the relevant path functionals on the basis of a few moments. We consider a number of exotics such as variance knockouts, conditional corridor variance swaps, gamma swaps and variance swaptions and give valuation formulas in detail