A generalization of Rubinstein's “Pay now, choose later”

Abstract

This article provides quasi-analytic pricing formulae for forward-start options under stochastic volatility, double jumps, and stochastic interest rates. Our methodology is a generalization of the Rubinstein approach and can be applied to several existing option models. Properties of a forward-start option may be very different from those of a plain vanilla option because the entire uncertainty of evolution of its price is cut off by the strike price at the time of determination. For instance, in contrast to the plain vanilla option, the value of a forward-start option may not always increase as the maturity increases. It depends on the current term structure of interest rates.

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

“…We then explicitly investigate the impact and parameter sensitivities of stochastic interest rates and stochastic volatility on the prices of forward starting options. Finally, we tackle the issue of model risk and compare our framework with the Black and Scholes (1973) and Heston (1993) model, respectively, considered in Rubinstein (1991) and Guo and Hung (2008) for the valuation of forward starters.…”

“…Note that in principle it is also possible, following the lines of Rubinstein (1991), Guo and Hung (2008) and Ahlip and Ruthowski (2009), to express the forward starting option price as the expected value of a future call option price, which can be evaluated using similar techniques as the evaluation of formula (11), but results in a pricing formula containing two integrals. However, working out the equivalent expectation (11) results in a pricing formula which only contains one integral.…”

“…Lucic (2003), Hong (2004) and Kruse and Nögel (2005) relax the constant volatility assumption and consider the pricing of forward starting options under Heston (1993) stochastic volatility. The pricing of forward starting options under stochastic volatility with independent stochastic interest rates was considered by Guo and Hung (2008), Ahlip and Rutkowski (2009) and Nunes and Alcaria (2009). The framework employed in this study distinguishes itself from these models by a closed-form pricing formula and an explicit, rather than implicit, incorporation of the correlation between the underlying and the term structure of interest rates.…”

“…Stochastic interest rates are crucial for the pricing of forward starting options because securities with forward starting features often have a longdated maturity and are therefore much more interest rate sensitive, e.g. see Guo and Hung (2008) or Kijima and Muromachi (2001). The addition of interest rates as a stochastic factor has been the subject of empirical investigations most notably by Bakshi, Cao and Chen (2000).…”

A Generalization of Gray and Whaley's Option

“…For instance, Rubinstein (1991) provides a pricing formula of standard forward-start option for which the strike price is set at a future time point such that the option becomes ATM at that time point. Guo and Hung (2008) There are also several studies on the valuation of rainbow options. Stultz (1982) uses the solution of partial di¤erential equations to derive the pricing formula for rainbow option on the maximum or minimum of two assets.…”