Accounting for stochastic interest rates, stochastic volatility and a general correlation structure in the valuation of forward starting options

Abstract

A quantitative analysis on the pricing of forward starting options under stochastic volatility and stochastic interest rates is performed. The main finding is that forward starting options not only depend on future smiles, but also directly on the evolution of the interest rates as well as the dependency structures among the underlying asset, the interest rates, and the stochastic volatility: compared to vanilla options, dynamic structures such as forward starting options are much more sensitive to model specifications such as volatility, interest rate, and correlation movements. We conclude that it is of crucial importance to take all these van Haastrecht and PelsserJournal of Futures Markets DOI: 10.1002/fut factors explicitly into account for a proper valuation and risk management of these securities. The performed analysis is facilitated by deriving closed-form formulas for the valuation of forward starting options, hereby taking the stochastic volatility, stochastic interest rates as well the dependency structure between all these processes explicitly into account. The valuation framework is derived using a probabilistic approach, enabling a fast and efficient evaluation of the option price by Fourier inverting the forward starting characteristic functions.

Valuation of forward start options under affine jump-diffusion models

“…Similarly to Kruse and Nögel (2005) or van Haastrecht and Pelsser (2011), two types of forward start options will be priced: European-style forward start call options on the underlying asset price and on the rate of return of the underlying asset. For both contracts, the option premium is paid on the purchase date but the options' life will only start on a future date (previous to the options' expiry date, and known as the determination time of the strike).…”

Random Time Forward-Starting Options

“…The previous method can be adapted to cover also the case of affine stochastic interest rate market model proposed in [20], where the interest rate follows a Hull and White model and the stochastic volatility is a Ornstein-Uhlenbeck process.…”

“…The main result is an analytic formula for the call case, derived by using the probabilistic approach combined with the Fourier inversion technique, as developed in Carr and Madan (1999) (see [10]). Van Haastrecht and Pelsser (2011), ( [20]), exploiting Fourier inversion of characteristic functions, developed a quantitative analysis of the pricing of forward starting options under stochastic volatility and stochastic interest rates (with Ornstein-Uhlenbeck dynamics), confirming that these not only depend on future smiles, but are also very sensitive to model specifications such as volatility, interest rate and correlation movements, concluding that it is of crucial importance to take all these factors into account for a correct valuation and risk management of these securities. More recently, Ramponi (2012) (see [27]) extended this Fourier analysis to regime switching models for the asset price.…”

Pricing of long-dated commodity derivatives: Do stochastic interest rates matter?

“…In response, a class of hybrid pricing models emerged, predominantly with applications in equity, insurance and foreign exchange markets, for instance Ballotta and Haberman (2003), Schrager and Pelsser (2004), van Haastrecht, Lord, Pelsser, and Schrager (2009), van Haastrecht and Pelsser (2011) and Grzelak, Oosterlee, and van Weeren (2012). In particular, van Haastrecht et al (2009), van Haastrecht and Pelsser (2011) and Grzelak et al (2012) discuss numerical solutions of models combining the stochastic volatility Schöbel and Zhu (1999) model and the stochastic interest rate Hull and White (1990) model, with full correlations between the underlying processes. van Haastrecht et al (2009) apply the hybrid model to the valuation of insurance options with long-term equity or foreign exchange (FX) exposure.…”