Valuing qualitative options with stochastic volatility

Abstract

We find a closed-form formula for valuing a time-switch option where its underlying asset is affected by a stochastically changing market environment, and apply it to the valuation of other qualitative options such as corridor options and options in foreign exchange markets. The stochastic market environment is modeled as a Markov regime-switching process. This analytic formula provides us with a rapid and accurate scheme for valuing qualitative options with stochastic volatility.

Keywords:

• Stochastic volatility
• Regime-switching environment
• Alternative investment
• Qualitative option
• Corridor option

Notes

†Sometimes, researchers and option dealers call a corridor option a range (or range-accrual) option. Its payoff function can be constructed from the difference in payoffs of two time-switch options.

‡We cannot find any statistics for the trading volume, but studies (e.g. Haug (1997), Linetsky (1999), Pechtl (1999), and Fusai (2000)) inform us that these types of options are now very popular in the global derivatives market. Also, as far as we know, a large number of the structured and hybrid products traded in the over-the-counter markets of Hong Kong and Korea contain these types of options written on underlying assets or variables such as stocks, currencies, and interest rates.

†In order to apply Girsanov's theorem, one should check the integral condition described in Theorem 5.2.3 of Shreve (2004). Since μ(t) and σ(t) are positive and constant across each regime, we can easily check that the condition holds under our setup.