Pricing variance derivatives using hybrid models with stochastic interest rates

Abstract

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.