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Pricing Power Options within the Heston Framework

Year 2013, Volume: 1 Issue: 1, 1 - 9, 01.04.2013

Siti N.i. Ibrahim John G. O'hara Nick Constantinou

Abstract

Numerous studies have presented evidence that certain financial assets may exhibit stochastic volatility or jumps, which cannot be captured within the Black-Scholes environment. This work investigates the valuation of power options when the variance follows the Heston model of stochastic volatility. A closed form representation of the characteristic function of the process is derived from the partial differential equation (PDE) of the replicating portfolio. The characteristic function is essential for the computation of the European power option prices via the Fast Fourier Transform (FFT) technique. Numerical results are presented. © 2012 Published by NTMSCI Selection and/or peer review under responsibility of NTMSCI Publication Society

Keywords

Power Option Partial Differential Equation Heston Model Characteristic Function Fast Fourier Transform

References

  • Bastami, A. A., Belic, M. R., & Petrovic, N. Z. (2010). Special solutions of the Riccati equation with applications to the Gross-Pitaevskii nonlinear PDE. Electronic Journal of Differential Equations, 2010 (66), 1-10. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu.
  • Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637-659.
  • Boyle, P. P. (1977). Options: A Monte Carlo approach. Journal of Financial Economics, 4 (3), 323-338.
  • Carr, P., & Madan, D. (1999). Option valuation using the fast Fourier transform. Journal of Computational Finance, 2 (4), 61-73.
  • Esser, A. (2003). General valuation principles for arbitrary payoffs and applications to power options under stochastic volatility. Financial
  • Markets and Portfolio Management, 17 (3), 351-372. Gatheral, J. (2006). The Volatility Surface: A Practitioner's Guide. John Wiley and Sons.
  • Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of
  • Financial Studies, 6 (2), 327-343. Lord, R., Koekkoek, R., & van Dijk, D. (2010). A comparison of biased simulation schemes for stochastic volatility models. Quantitative Finance, 10 (2), 177-194.
  • Pillay, E., & O'Hara, J. G. (2011). FFT based option pricing under a mean reverting process with stochastic volatility and jumps. Journal of
  • Computational Applied Mathematics, 235 (12), 3378-3384.

Pricing Power Options under the Heston Dynamics using the FFT

Year 2013, Volume: 1 Issue: 1, 1 - 9, 01.04.2013

Siti N.i. Ibrahim John G. O'hara Nick Constantinou

Abstract

References

  • Bastami, A. A., Belic, M. R., & Petrovic, N. Z. (2010). Special solutions of the Riccati equation with applications to the Gross-Pitaevskii nonlinear PDE. Electronic Journal of Differential Equations, 2010 (66), 1-10. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu.
  • Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637-659.
  • Boyle, P. P. (1977). Options: A Monte Carlo approach. Journal of Financial Economics, 4 (3), 323-338.
  • Carr, P., & Madan, D. (1999). Option valuation using the fast Fourier transform. Journal of Computational Finance, 2 (4), 61-73.
  • Esser, A. (2003). General valuation principles for arbitrary payoffs and applications to power options under stochastic volatility. Financial
  • Markets and Portfolio Management, 17 (3), 351-372. Gatheral, J. (2006). The Volatility Surface: A Practitioner's Guide. John Wiley and Sons.
  • Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of
  • Financial Studies, 6 (2), 327-343. Lord, R., Koekkoek, R., & van Dijk, D. (2010). A comparison of biased simulation schemes for stochastic volatility models. Quantitative Finance, 10 (2), 177-194.
  • Pillay, E., & O'Hara, J. G. (2011). FFT based option pricing under a mean reverting process with stochastic volatility and jumps. Journal of
  • Computational Applied Mathematics, 235 (12), 3378-3384.
There are 10 citations in total.

Details

Journal Section Articles
Authors

Siti N.i. Ibrahim This is me

John G. O'hara This is me

Nick Constantinou This is me

Publication Date April 1, 2013
Published in Issue Year 2013 Volume: 1 Issue: 1

Cite

APA Ibrahim, S. N., O’hara, J. G., & Constantinou, N. (2013). Pricing Power Options under the Heston Dynamics using the FFT. New Trends in Mathematical Sciences, 1(1), 1-9.
AMA Ibrahim SN, O’hara JG, Constantinou N. Pricing Power Options under the Heston Dynamics using the FFT. New Trends in Mathematical Sciences. April 2013;1(1):1-9.
Chicago Ibrahim, Siti N.i., John G. O’hara, and Nick Constantinou. “Pricing Power Options under the Heston Dynamics Using the FFT”. New Trends in Mathematical Sciences 1, no. 1 (April 2013): 1-9.
EndNote Ibrahim SN, O’hara JG, Constantinou N (April 1, 2013) Pricing Power Options under the Heston Dynamics using the FFT. New Trends in Mathematical Sciences 1 1 1–9.
IEEE S. N. Ibrahim, J. G. O’hara, and N. Constantinou, “Pricing Power Options under the Heston Dynamics using the FFT”, New Trends in Mathematical Sciences, vol. 1, no. 1, pp. 1–9, 2013.
ISNAD Ibrahim, Siti N.i. et al. “Pricing Power Options under the Heston Dynamics Using the FFT”. New Trends in Mathematical Sciences 1/1 (April 2013), 1-9.
JAMA Ibrahim SN, O’hara JG, Constantinou N. Pricing Power Options under the Heston Dynamics using the FFT. New Trends in Mathematical Sciences. 2013;1:1–9.
MLA Ibrahim, Siti N.i. et al. “Pricing Power Options under the Heston Dynamics Using the FFT”. New Trends in Mathematical Sciences, vol. 1, no. 1, 2013, pp. 1-9.
Vancouver Ibrahim SN, O’hara JG, Constantinou N. Pricing Power Options under the Heston Dynamics using the FFT. New Trends in Mathematical Sciences. 2013;1(1):1-9.