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Delta Hedging in Discrete Time under Stochastic Interest Rate

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  • Flavio ANGELINI
  • Stefano HERZEL

Abstract

We examine the e?ect of stochastic interest rate on the Delta hedging strategy in discrete time when hedging a contingent claim written on a risky asset. The performance of the hedging is mainly measured by the variance of the error. We consider a simple two-dimensional model of the type Black-Scholes combined with the Vasicek model, allowing for correlation between the stock and the interest rate. Within this model, we perform the Delta hedging ?rst by implementing the strategy by taking into account the stochasticity of interest rate and then by using a plain Black-Scholes Delta with deterministic rate. The di?erences between the two performances can be relevant, mainly depending on the correlation and on the relation between the standard deviation of the risky asset and that of the interest rate. We also consider Delta hedging for an interest rate option in the Cox-Ingersoll and Ross model. The analysis is done by applying a general result for the e?cient computation of expected value and variance of the hedging error of a certain class of strategies, which include the Delta strategy.

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Paper provided by Università di Perugia, Dipartimento Economia, Finanza e Statistica in its series Quaderni del Dipartimento di Economia, Finanza e Statistica with number 110/2012.

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Length: 22 pages
Date of creation: 19 Oct 2012
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Handle: RePEc:pia:wpaper:110/2012

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  1. Jin-Chuan Duan & Jean-Guy Simonato, 1995. "Estimating and Testing Exponential Affine Term Structure Models by Kalman Filter," CIRANO Working Papers 95s-44, CIRANO.
  2. van Haastrecht, Alexander & Lord, Roger & Pelsser, Antoon & Schrager, David, 2009. "Pricing long-dated insurance contracts with stochastic interest rates and stochastic volatility," Insurance: Mathematics and Economics, Elsevier, vol. 45(3), pages 436-448, December.
  3. Flavio Angelini & Stefano Herzel, 2009. "Evaluating Discrete Dynamic Strategies in Affine Models," Quaderni del Dipartimento di Economia, Finanza e Statistica 71/2009, Università di Perugia, Dipartimento Economia, Finanza e Statistica.
  4. Ales Čern� & Jan Kallsen, 2008. "Mean-Variance Hedging And Optimal Investment In Heston'S Model With Correlation," Mathematical Finance, Wiley Blackwell, vol. 18(3), pages 473-492.
  5. Ales Čern�, 2007. "Optimal Continuous-Time Hedging With Leptokurtic Returns," Mathematical Finance, Wiley Blackwell, vol. 17(2), pages 175-203.
  6. Friedrich Hubalek & Jan Kallsen & Leszek Krawczyk, 2006. "Variance-optimal hedging for processes with stationary independent increments," Papers math/0607112, arXiv.org.
  7. Darrell Duffie & Rui Kan, 1996. "A Yield-Factor Model Of Interest Rates," Mathematical Finance, Wiley Blackwell, vol. 6(4), pages 379-406.
  8. Flavio Angelini & Stefano Herzel, 2007. "Measuring the error of dynamic hedging: a Laplace transform approach," Quaderni del Dipartimento di Economia, Finanza e Statistica 33/2007, Università di Perugia, Dipartimento Economia, Finanza e Statistica.
  9. Vasicek, Oldrich Alfonso, 1977. "Abstract: An Equilibrium Characterization of the Term Structure," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 12(04), pages 627-627, November.
  10. Cox, John C & Ingersoll, Jonathan E, Jr & Ross, Stephen A, 1985. "A Theory of the Term Structure of Interest Rates," Econometrica, Econometric Society, vol. 53(2), pages 385-407, March.
  11. Vasicek, Oldrich, 1977. "An equilibrium characterization of the term structure," Journal of Financial Economics, Elsevier, vol. 5(2), pages 177-188, November.
  12. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-43.
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