Generic pricing of FX, inflation and stock options under stochastic interest rates and stochastic volatility

A. van Haastrecht, A. Pelsser

Research output: Contribution to JournalArticleAcademicpeer-review

Abstract

We consider the pricing of FX, inflation and stock options under stochastic interest rates and stochastic volatility, for which we use a generic multi-currency framework. We allow for a general correlation structure between the drivers of the volatility, the inflation index, the domestic (nominal) and the foreign (real) rates. Having the flexibility to correlate the underlying FX/inflation/stock index with both stochastic volatility and stochastic interest rates yields a realistic model that is of practical importance for the pricing and hedging of options with a long-term exposure. We derive explicit valuation formulas for various securities, such as vanilla call/put options, forward starting options, inflation-indexed swaps and inflation caps/floors. These vanilla derivatives can be valued in closed form under Schö bel and Zhu [Eur. Finance Rev., 1999, 4, 23-46] stochastic volatility, whereas we devise an (Monte Carlo) approximation in the form of a very effective control variate for the general Heston [Rev. Financial Stud., 1993, 6, 327-343] model. Finally, we investigate the quality of this approximation numerically and consider a calibration example to FX and inflation market data. © 2011 Taylor & Francis.
Original languageEnglish
Pages (from-to)665-691
JournalQuantitative Finance
Volume11
Issue number4
DOIs
Publication statusPublished - 2011

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Pricing
Inflation
Stochastic volatility
Stochastic interest rates
Stock options
Interest rate volatility
Approximation
Correlates
Hedging
Derivatives
Market data
Finance
Calibration
Swaps
Currency
Put option
Heston
Correlation structure
Stock index
Control variate

Cite this

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title = "Generic pricing of FX, inflation and stock options under stochastic interest rates and stochastic volatility",
abstract = "We consider the pricing of FX, inflation and stock options under stochastic interest rates and stochastic volatility, for which we use a generic multi-currency framework. We allow for a general correlation structure between the drivers of the volatility, the inflation index, the domestic (nominal) and the foreign (real) rates. Having the flexibility to correlate the underlying FX/inflation/stock index with both stochastic volatility and stochastic interest rates yields a realistic model that is of practical importance for the pricing and hedging of options with a long-term exposure. We derive explicit valuation formulas for various securities, such as vanilla call/put options, forward starting options, inflation-indexed swaps and inflation caps/floors. These vanilla derivatives can be valued in closed form under Sch{\"o} bel and Zhu [Eur. Finance Rev., 1999, 4, 23-46] stochastic volatility, whereas we devise an (Monte Carlo) approximation in the form of a very effective control variate for the general Heston [Rev. Financial Stud., 1993, 6, 327-343] model. Finally, we investigate the quality of this approximation numerically and consider a calibration example to FX and inflation market data. {\circledC} 2011 Taylor & Francis.",
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Generic pricing of FX, inflation and stock options under stochastic interest rates and stochastic volatility. / van Haastrecht, A.; Pelsser, A.

In: Quantitative Finance, Vol. 11, No. 4, 2011, p. 665-691.

Research output: Contribution to JournalArticleAcademicpeer-review

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AU - Pelsser, A.

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AB - We consider the pricing of FX, inflation and stock options under stochastic interest rates and stochastic volatility, for which we use a generic multi-currency framework. We allow for a general correlation structure between the drivers of the volatility, the inflation index, the domestic (nominal) and the foreign (real) rates. Having the flexibility to correlate the underlying FX/inflation/stock index with both stochastic volatility and stochastic interest rates yields a realistic model that is of practical importance for the pricing and hedging of options with a long-term exposure. We derive explicit valuation formulas for various securities, such as vanilla call/put options, forward starting options, inflation-indexed swaps and inflation caps/floors. These vanilla derivatives can be valued in closed form under Schö bel and Zhu [Eur. Finance Rev., 1999, 4, 23-46] stochastic volatility, whereas we devise an (Monte Carlo) approximation in the form of a very effective control variate for the general Heston [Rev. Financial Stud., 1993, 6, 327-343] model. Finally, we investigate the quality of this approximation numerically and consider a calibration example to FX and inflation market data. © 2011 Taylor & Francis.

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