A closed-form solution for outperformance options with stochastic correlation and stochastic volatility

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October 2015, 11(4): 1185-1209. doi: 10.3934/jimo.2015.11.1185

A closed-form solution for outperformance options with stochastic correlation and stochastic volatility

Jacinto Marabel Romo ^1,

1.	BBVA and University Institute for Economic and Social Analysis, University of Alcalá, Mailing address: c/ Sauceda, 28 Edicio ASIA, 28050, Madrid, Spain

Received January 2014 Revised September 2014 Published March 2015

Abstract
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Outperformance options allow investors to benefit from a view on the relative performance of two underlying assets without taking any directional exposure to the evolution of the market. These structures exhibit high sensitivity to the correlation between the underlying assets and are usually priced assuming constant instantaneous correlations.
This article considers a multi-asset model based on Wishart processes that accounts for stochastic volatility and for stochastic correlations between the assets returns, as well as between their volatilities. Under the assumptions of the model this article provides semi-closed form solutions for the price of outperformance options. The article shows that the price of these options depends crucially on the term structure of the correlation corresponding to the assets returns. Furthermore, the comparison of the prices obtained under this model and under other models with constant correlations commonly used by financial institutions reveals the existence of a stochastic correlation premium.

Keywords: Outperformance option, Wishart process, stochastic volatility, correlation term structure., stochastic correlation.

Mathematics Subject Classification: 42B10, 65R1.

Citation: Jacinto Marabel Romo. A closed-form solution for outperformance options with stochastic correlation and stochastic volatility. Journal of Industrial and Management Optimization, 2015, 11 (4) : 1185-1209. doi: 10.3934/jimo.2015.11.1185

References:

[1]	M. Avellaneda and Y. Zhu, An e-arch model for the term structure of implied volatility of fx options, Applied Mathematical Finance, 11 (1997), 81-100. doi: 10.2139/ssrn.15150.
[2]	G. Bakshi, C. Cao and R. Stelzer, Do call prices and the underlying stock always move in the same direction?, Review of Financial Studies, 13 (2000), 549-584. doi: 10.1093/rfs/13.3.549.
[3]	C. Ball and W. Torous, Stochastic correlation across international stock markets, Journal of Empirical Finance, 7 (2000), 373-388. doi: 10.1016/S0927-5398(00)00017-7.
[4]	O. Barndorff-Nielsen and R. Stelzer, The multivariate supou stochastic volatility model, Mathematical Finance, 23 (2013), 275-296. doi: 10.1111/j.1467-9965.2011.00494.x.
[5]	F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy, 81 (1973), 637-654. doi: 10.1086/260062.
[6]	P. Boyle, Options: A Monte Carlo approach, Journal of Financial Economics, 4 (1977), 323-338. doi: 10.1016/0304-405X(77)90005-8.
[7]	N. Branger and M. Muck, Keep on smiling? volatility surfaces and the pricing of quanto options when all covariances are stochastic, Journal of Banking and Finance, 36 (2012), 1577-1591.
[8]	M. Bru, Wishart processes, Journal of Theoretical Probability, 4 (1991), 725-751. doi: 10.1007/BF01259552.
[9]	P. Carr and L. Wu, Variance risk premiums, Review of Financial Studies, 22 (2009), 1311-1341. doi: 10.1093/rfs/hhn038.
[10]	R. Cont and J. Da Fonseca, Dynamics of implied volatility surfaces, Quantitative Finance, 2 (2002), 45-60. doi: 10.1088/1469-7688/2/1/304.
[11]	J. Da Fonseca, M. Grasselli and C. Tebaldi, Option pricing when correlations are stochastic: An analytical framework, Review of Derivatives Research, 10 (2007), 151-180. doi: 10.1007/s11147-008-9018-x.
[12]	J. Da Fonseca, M. Grasselli and C. Tebaldi, A multifactor volatility Heston model, Quantitative Finance, 8 (2008), 591-604. doi: 10.1080/14697680701668418.
[13]	T. Daglish, J. Hull and W. Suo, Volatility surfaces, theory, rules of thumb and empirical evidence, Quantitative Finance, 7 (2007), 507-524. doi: 10.1080/14697680601087883.
[14]	E. Derman, Outperformance options, in The Handbook of Exotic Options: Instruments, Analysis, and Application (ed. I. Nelken), 1996.
[15]	E. Derman, Regimes of volatility, Risk, 4 (1999), 55-59.
[16]	E. Derman and I. Kani, The volatility smile and its implied tree, Quantitative Strategies Research Notes, Goldman Sachs, New York.
[17]	E. Derman, I. Kani and J. Zou, The local volatility surface: unlocking the information in index option prices, Quantitative Strategies Research Notes, Goldman Sachs, New York, 52 (1996), 1-25. doi: 10.2469/faj.v52.n4.2008.
[18]	E. Derman and P. Wilmott, Perfect models, imperfect world, Business Week, 59.
[19]	D. Duffie, J. Pan and K. Singleton, Transform analysis and asset pricing for affine jump-diffusions, Econometrica, 68 (2000), 1343-1376. doi: 10.1111/1468-0262.00164.
[20]	B. Dupire, Pricing with a smile, Risk, 7 (1994), 18-20.
[21]	S. Fischer, Call option pricing when the exercise price is uncertain, and the valuation of index bonds, Journal of Finance, 33 (1978), 169-176. doi: 10.2307/2326357.
[22]	R. Franks and E. Schwartz, The stochastic behavior of market variance implied in the price of index options, The Economic Journal, 101 (1991), 1460-1475. doi: 10.2307/2234896.
[23]	J. Gatheral, The Volatility Surface. A Practitioner's Guide, John Wiley and Sons, Hoboken, New Jersey, 2006.
[24]	C. Gourieroux and R. Sufana, Derivative Pricing with Multivariate Stochastic Volatility: Application to Credit Risk, Les Cahiers du CREF, Working Paper, 2005. doi: 10.2139/ssrn.757312.
[25]	S. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies, 6 (1993), 327-343. doi: 10.1093/rfs/6.2.327.
[26]	J. Hull and W. Suo, A methodology for assessing model risk and its application to the implied volatility function model, Journal of Financial and Quantitative Analysis, 37 (2002), 297-318. doi: 10.2307/3595007.
[27]	J. Hull and A. White, The pricing of options on assets with stochastic volatilities, Journal of Financial and Quantitative Analysis, 42 (1987), 281-300. doi: 10.2307/2328253.
[28]	T. Hurd and Z. Zhou, A fourier transform method for spread option pricing, SIAM Journal on Financial Mathematics, 1 (2010), 142-157. doi: 10.1137/090750421.
[29]	M. Krekel, J. de Kock, R. Korn and T. Man, An analysis of pricing methods for basket options, Wilmott magazine, 82-89.
[30]	R. Lee, The moment formula for implied volatility at extreme strikes, Mathematical Finance, 14 (2004), 469-480. doi: 10.1111/j.0960-1627.2004.00200.x.
[31]	M. Leippold and F. Trojani, Asset pricing with matrix jump diffusions, Working paper, Imperial College London, University of St. Gallen, (2008), p57. doi: 10.2139/ssrn.1274482.
[32]	A. Lewis, Option Valuation under Stochastic Volatility with Mathematica Code, Finance Press, Newport Beach, California, 2000.
[33]	M. Loretan and W. English, Evaluating correlation breakdowns during periods of market volatility, Working paper, Board of Governors of the Federal Reserve System International Finance, (2000), p33. doi: 10.2139/ssrn.231857.
[34]	J. Marabel Romo, Fitting the skew with an analytic local volatility function, International Review of Applied Financial Issues and Economics, 3 (2011), 721-736.
[35]	J. Marabel Romo, Worst-of options and correlation skew under a stochastic correlation framework, International journal of Theoretical and Applied Finance, 7.
[36]	W. Margrabe, The value of an option to exchange one asset for another, Journal of Finance, 33 (1978), 177-186. doi: 10.2307/2326358.
[37]	M. Pan, Y. Liu and H. Roth, Term structure of return correlations and international diversification: evidence from european stock markets, The European Journal of Finance, 7 (2001), 144-164. doi: 10.1080/13518470151141477.
[38]	M. Rubinstein, Implied binomial trees, Journal of Finance, 49 (1994), 771-818. doi: 10.2307/2329207.
[39]	B. Solnik, C. Boucrelle and Y. Le Fur, International market correlation and volatility, Financial Analysts Journal, 52 (1996), 17-34. doi: 10.2469/faj.v52.n5.2021.

show all references

References:

[1]	M. Avellaneda and Y. Zhu, An e-arch model for the term structure of implied volatility of fx options, Applied Mathematical Finance, 11 (1997), 81-100. doi: 10.2139/ssrn.15150.
[2]	G. Bakshi, C. Cao and R. Stelzer, Do call prices and the underlying stock always move in the same direction?, Review of Financial Studies, 13 (2000), 549-584. doi: 10.1093/rfs/13.3.549.
[3]	C. Ball and W. Torous, Stochastic correlation across international stock markets, Journal of Empirical Finance, 7 (2000), 373-388. doi: 10.1016/S0927-5398(00)00017-7.
[4]	O. Barndorff-Nielsen and R. Stelzer, The multivariate supou stochastic volatility model, Mathematical Finance, 23 (2013), 275-296. doi: 10.1111/j.1467-9965.2011.00494.x.
[5]	F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy, 81 (1973), 637-654. doi: 10.1086/260062.
[6]	P. Boyle, Options: A Monte Carlo approach, Journal of Financial Economics, 4 (1977), 323-338. doi: 10.1016/0304-405X(77)90005-8.
[7]	N. Branger and M. Muck, Keep on smiling? volatility surfaces and the pricing of quanto options when all covariances are stochastic, Journal of Banking and Finance, 36 (2012), 1577-1591.
[8]	M. Bru, Wishart processes, Journal of Theoretical Probability, 4 (1991), 725-751. doi: 10.1007/BF01259552.
[9]	P. Carr and L. Wu, Variance risk premiums, Review of Financial Studies, 22 (2009), 1311-1341. doi: 10.1093/rfs/hhn038.
[10]	R. Cont and J. Da Fonseca, Dynamics of implied volatility surfaces, Quantitative Finance, 2 (2002), 45-60. doi: 10.1088/1469-7688/2/1/304.
[11]	J. Da Fonseca, M. Grasselli and C. Tebaldi, Option pricing when correlations are stochastic: An analytical framework, Review of Derivatives Research, 10 (2007), 151-180. doi: 10.1007/s11147-008-9018-x.
[12]	J. Da Fonseca, M. Grasselli and C. Tebaldi, A multifactor volatility Heston model, Quantitative Finance, 8 (2008), 591-604. doi: 10.1080/14697680701668418.
[13]	T. Daglish, J. Hull and W. Suo, Volatility surfaces, theory, rules of thumb and empirical evidence, Quantitative Finance, 7 (2007), 507-524. doi: 10.1080/14697680601087883.
[14]	E. Derman, Outperformance options, in The Handbook of Exotic Options: Instruments, Analysis, and Application (ed. I. Nelken), 1996.
[15]	E. Derman, Regimes of volatility, Risk, 4 (1999), 55-59.
[16]	E. Derman and I. Kani, The volatility smile and its implied tree, Quantitative Strategies Research Notes, Goldman Sachs, New York.
[17]	E. Derman, I. Kani and J. Zou, The local volatility surface: unlocking the information in index option prices, Quantitative Strategies Research Notes, Goldman Sachs, New York, 52 (1996), 1-25. doi: 10.2469/faj.v52.n4.2008.
[18]	E. Derman and P. Wilmott, Perfect models, imperfect world, Business Week, 59.
[19]	D. Duffie, J. Pan and K. Singleton, Transform analysis and asset pricing for affine jump-diffusions, Econometrica, 68 (2000), 1343-1376. doi: 10.1111/1468-0262.00164.
[20]	B. Dupire, Pricing with a smile, Risk, 7 (1994), 18-20.
[21]	S. Fischer, Call option pricing when the exercise price is uncertain, and the valuation of index bonds, Journal of Finance, 33 (1978), 169-176. doi: 10.2307/2326357.
[22]	R. Franks and E. Schwartz, The stochastic behavior of market variance implied in the price of index options, The Economic Journal, 101 (1991), 1460-1475. doi: 10.2307/2234896.
[23]	J. Gatheral, The Volatility Surface. A Practitioner's Guide, John Wiley and Sons, Hoboken, New Jersey, 2006.
[24]	C. Gourieroux and R. Sufana, Derivative Pricing with Multivariate Stochastic Volatility: Application to Credit Risk, Les Cahiers du CREF, Working Paper, 2005. doi: 10.2139/ssrn.757312.
[25]	S. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies, 6 (1993), 327-343. doi: 10.1093/rfs/6.2.327.
[26]	J. Hull and W. Suo, A methodology for assessing model risk and its application to the implied volatility function model, Journal of Financial and Quantitative Analysis, 37 (2002), 297-318. doi: 10.2307/3595007.
[27]	J. Hull and A. White, The pricing of options on assets with stochastic volatilities, Journal of Financial and Quantitative Analysis, 42 (1987), 281-300. doi: 10.2307/2328253.
[28]	T. Hurd and Z. Zhou, A fourier transform method for spread option pricing, SIAM Journal on Financial Mathematics, 1 (2010), 142-157. doi: 10.1137/090750421.
[29]	M. Krekel, J. de Kock, R. Korn and T. Man, An analysis of pricing methods for basket options, Wilmott magazine, 82-89.
[30]	R. Lee, The moment formula for implied volatility at extreme strikes, Mathematical Finance, 14 (2004), 469-480. doi: 10.1111/j.0960-1627.2004.00200.x.
[31]	M. Leippold and F. Trojani, Asset pricing with matrix jump diffusions, Working paper, Imperial College London, University of St. Gallen, (2008), p57. doi: 10.2139/ssrn.1274482.
[32]	A. Lewis, Option Valuation under Stochastic Volatility with Mathematica Code, Finance Press, Newport Beach, California, 2000.
[33]	M. Loretan and W. English, Evaluating correlation breakdowns during periods of market volatility, Working paper, Board of Governors of the Federal Reserve System International Finance, (2000), p33. doi: 10.2139/ssrn.231857.
[34]	J. Marabel Romo, Fitting the skew with an analytic local volatility function, International Review of Applied Financial Issues and Economics, 3 (2011), 721-736.
[35]	J. Marabel Romo, Worst-of options and correlation skew under a stochastic correlation framework, International journal of Theoretical and Applied Finance, 7.
[36]	W. Margrabe, The value of an option to exchange one asset for another, Journal of Finance, 33 (1978), 177-186. doi: 10.2307/2326358.
[37]	M. Pan, Y. Liu and H. Roth, Term structure of return correlations and international diversification: evidence from european stock markets, The European Journal of Finance, 7 (2001), 144-164. doi: 10.1080/13518470151141477.
[38]	M. Rubinstein, Implied binomial trees, Journal of Finance, 49 (1994), 771-818. doi: 10.2307/2329207.
[39]	B. Solnik, C. Boucrelle and Y. Le Fur, International market correlation and volatility, Financial Analysts Journal, 52 (1996), 17-34. doi: 10.2469/faj.v52.n5.2021.

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