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

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

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.
Citation: Jacinto Marabel Romo. A closed-form solution for outperformance options with stochastic correlation and stochastic volatility. Journal of Industrial & 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. doi: 10.2139/ssrn.15150. Google Scholar

[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. doi: 10.1093/rfs/13.3.549. Google Scholar

[3]

C. Ball and W. Torous, Stochastic correlation across international stock markets,, Journal of Empirical Finance, 7 (2000), 373. doi: 10.1016/S0927-5398(00)00017-7. Google Scholar

[4]

O. Barndorff-Nielsen and R. Stelzer, The multivariate supou stochastic volatility model,, Mathematical Finance, 23 (2013), 275. doi: 10.1111/j.1467-9965.2011.00494.x. Google Scholar

[5]

F. Black and M. Scholes, The pricing of options and corporate liabilities,, Journal of Political Economy, 81 (1973), 637. doi: 10.1086/260062. Google Scholar

[6]

P. Boyle, Options: A Monte Carlo approach,, Journal of Financial Economics, 4 (1977), 323. doi: 10.1016/0304-405X(77)90005-8. Google Scholar

[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. Google Scholar

[8]

M. Bru, Wishart processes,, Journal of Theoretical Probability, 4 (1991), 725. doi: 10.1007/BF01259552. Google Scholar

[9]

P. Carr and L. Wu, Variance risk premiums,, Review of Financial Studies, 22 (2009), 1311. doi: 10.1093/rfs/hhn038. Google Scholar

[10]

R. Cont and J. Da Fonseca, Dynamics of implied volatility surfaces,, Quantitative Finance, 2 (2002), 45. doi: 10.1088/1469-7688/2/1/304. Google Scholar

[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. doi: 10.1007/s11147-008-9018-x. Google Scholar

[12]

J. Da Fonseca, M. Grasselli and C. Tebaldi, A multifactor volatility Heston model,, Quantitative Finance, 8 (2008), 591. doi: 10.1080/14697680701668418. Google Scholar

[13]

T. Daglish, J. Hull and W. Suo, Volatility surfaces, theory, rules of thumb and empirical evidence,, Quantitative Finance, 7 (2007), 507. doi: 10.1080/14697680601087883. Google Scholar

[14]

E. Derman, Outperformance options,, in The Handbook of Exotic Options: Instruments, (1996). Google Scholar

[15]

E. Derman, Regimes of volatility,, Risk, 4 (1999), 55. Google Scholar

[16]

E. Derman and I. Kani, The volatility smile and its implied tree,, Quantitative Strategies Research Notes, (). Google Scholar

[17]

E. Derman, I. Kani and J. Zou, The local volatility surface: unlocking the information in index option prices,, Quantitative Strategies Research Notes, 52 (1996), 1. doi: 10.2469/faj.v52.n4.2008. Google Scholar

[18]

E. Derman and P. Wilmott, Perfect models, imperfect world,, Business Week, (). Google Scholar

[19]

D. Duffie, J. Pan and K. Singleton, Transform analysis and asset pricing for affine jump-diffusions,, Econometrica, 68 (2000), 1343. doi: 10.1111/1468-0262.00164. Google Scholar

[20]

B. Dupire, Pricing with a smile,, Risk, 7 (1994), 18. Google Scholar

[21]

S. Fischer, Call option pricing when the exercise price is uncertain, and the valuation of index bonds,, Journal of Finance, 33 (1978), 169. doi: 10.2307/2326357. Google Scholar

[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. doi: 10.2307/2234896. Google Scholar

[23]

J. Gatheral, The Volatility Surface. A Practitioner's Guide,, John Wiley and Sons, (2006). Google Scholar

[24]

C. Gourieroux and R. Sufana, Derivative Pricing with Multivariate Stochastic Volatility: Application to Credit Risk,, Les Cahiers du CREF, (2005). doi: 10.2139/ssrn.757312. Google Scholar

[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. doi: 10.1093/rfs/6.2.327. Google Scholar

[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. doi: 10.2307/3595007. Google Scholar

[27]

J. Hull and A. White, The pricing of options on assets with stochastic volatilities,, Journal of Financial and Quantitative Analysis, 42 (1987), 281. doi: 10.2307/2328253. Google Scholar

[28]

T. Hurd and Z. Zhou, A fourier transform method for spread option pricing,, SIAM Journal on Financial Mathematics, 1 (2010), 142. doi: 10.1137/090750421. Google Scholar

[29]

M. Krekel, J. de Kock, R. Korn and T. Man, An analysis of pricing methods for basket options,, Wilmott magazine, (): 82. Google Scholar

[30]

R. Lee, The moment formula for implied volatility at extreme strikes,, Mathematical Finance, 14 (2004), 469. doi: 10.1111/j.0960-1627.2004.00200.x. Google Scholar

[31]

M. Leippold and F. Trojani, Asset pricing with matrix jump diffusions,, Working paper, (2008). doi: 10.2139/ssrn.1274482. Google Scholar

[32]

A. Lewis, Option Valuation under Stochastic Volatility with Mathematica Code,, Finance Press, (2000). Google Scholar

[33]

M. Loretan and W. English, Evaluating correlation breakdowns during periods of market volatility,, Working paper, (2000). doi: 10.2139/ssrn.231857. Google Scholar

[34]

J. Marabel Romo, Fitting the skew with an analytic local volatility function,, International Review of Applied Financial Issues and Economics, 3 (2011), 721. Google Scholar

[35]

J. Marabel Romo, Worst-of options and correlation skew under a stochastic correlation framework,, International journal of Theoretical and Applied Finance, 7 (). Google Scholar

[36]

W. Margrabe, The value of an option to exchange one asset for another,, Journal of Finance, 33 (1978), 177. doi: 10.2307/2326358. Google Scholar

[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. doi: 10.1080/13518470151141477. Google Scholar

[38]

M. Rubinstein, Implied binomial trees,, Journal of Finance, 49 (1994), 771. doi: 10.2307/2329207. Google Scholar

[39]

B. Solnik, C. Boucrelle and Y. Le Fur, International market correlation and volatility,, Financial Analysts Journal, 52 (1996), 17. doi: 10.2469/faj.v52.n5.2021. Google Scholar

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. doi: 10.2139/ssrn.15150. Google Scholar

[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. doi: 10.1093/rfs/13.3.549. Google Scholar

[3]

C. Ball and W. Torous, Stochastic correlation across international stock markets,, Journal of Empirical Finance, 7 (2000), 373. doi: 10.1016/S0927-5398(00)00017-7. Google Scholar

[4]

O. Barndorff-Nielsen and R. Stelzer, The multivariate supou stochastic volatility model,, Mathematical Finance, 23 (2013), 275. doi: 10.1111/j.1467-9965.2011.00494.x. Google Scholar

[5]

F. Black and M. Scholes, The pricing of options and corporate liabilities,, Journal of Political Economy, 81 (1973), 637. doi: 10.1086/260062. Google Scholar

[6]

P. Boyle, Options: A Monte Carlo approach,, Journal of Financial Economics, 4 (1977), 323. doi: 10.1016/0304-405X(77)90005-8. Google Scholar

[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. Google Scholar

[8]

M. Bru, Wishart processes,, Journal of Theoretical Probability, 4 (1991), 725. doi: 10.1007/BF01259552. Google Scholar

[9]

P. Carr and L. Wu, Variance risk premiums,, Review of Financial Studies, 22 (2009), 1311. doi: 10.1093/rfs/hhn038. Google Scholar

[10]

R. Cont and J. Da Fonseca, Dynamics of implied volatility surfaces,, Quantitative Finance, 2 (2002), 45. doi: 10.1088/1469-7688/2/1/304. Google Scholar

[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. doi: 10.1007/s11147-008-9018-x. Google Scholar

[12]

J. Da Fonseca, M. Grasselli and C. Tebaldi, A multifactor volatility Heston model,, Quantitative Finance, 8 (2008), 591. doi: 10.1080/14697680701668418. Google Scholar

[13]

T. Daglish, J. Hull and W. Suo, Volatility surfaces, theory, rules of thumb and empirical evidence,, Quantitative Finance, 7 (2007), 507. doi: 10.1080/14697680601087883. Google Scholar

[14]

E. Derman, Outperformance options,, in The Handbook of Exotic Options: Instruments, (1996). Google Scholar

[15]

E. Derman, Regimes of volatility,, Risk, 4 (1999), 55. Google Scholar

[16]

E. Derman and I. Kani, The volatility smile and its implied tree,, Quantitative Strategies Research Notes, (). Google Scholar

[17]

E. Derman, I. Kani and J. Zou, The local volatility surface: unlocking the information in index option prices,, Quantitative Strategies Research Notes, 52 (1996), 1. doi: 10.2469/faj.v52.n4.2008. Google Scholar

[18]

E. Derman and P. Wilmott, Perfect models, imperfect world,, Business Week, (). Google Scholar

[19]

D. Duffie, J. Pan and K. Singleton, Transform analysis and asset pricing for affine jump-diffusions,, Econometrica, 68 (2000), 1343. doi: 10.1111/1468-0262.00164. Google Scholar

[20]

B. Dupire, Pricing with a smile,, Risk, 7 (1994), 18. Google Scholar

[21]

S. Fischer, Call option pricing when the exercise price is uncertain, and the valuation of index bonds,, Journal of Finance, 33 (1978), 169. doi: 10.2307/2326357. Google Scholar

[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. doi: 10.2307/2234896. Google Scholar

[23]

J. Gatheral, The Volatility Surface. A Practitioner's Guide,, John Wiley and Sons, (2006). Google Scholar

[24]

C. Gourieroux and R. Sufana, Derivative Pricing with Multivariate Stochastic Volatility: Application to Credit Risk,, Les Cahiers du CREF, (2005). doi: 10.2139/ssrn.757312. Google Scholar

[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. doi: 10.1093/rfs/6.2.327. Google Scholar

[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. doi: 10.2307/3595007. Google Scholar

[27]

J. Hull and A. White, The pricing of options on assets with stochastic volatilities,, Journal of Financial and Quantitative Analysis, 42 (1987), 281. doi: 10.2307/2328253. Google Scholar

[28]

T. Hurd and Z. Zhou, A fourier transform method for spread option pricing,, SIAM Journal on Financial Mathematics, 1 (2010), 142. doi: 10.1137/090750421. Google Scholar

[29]

M. Krekel, J. de Kock, R. Korn and T. Man, An analysis of pricing methods for basket options,, Wilmott magazine, (): 82. Google Scholar

[30]

R. Lee, The moment formula for implied volatility at extreme strikes,, Mathematical Finance, 14 (2004), 469. doi: 10.1111/j.0960-1627.2004.00200.x. Google Scholar

[31]

M. Leippold and F. Trojani, Asset pricing with matrix jump diffusions,, Working paper, (2008). doi: 10.2139/ssrn.1274482. Google Scholar

[32]

A. Lewis, Option Valuation under Stochastic Volatility with Mathematica Code,, Finance Press, (2000). Google Scholar

[33]

M. Loretan and W. English, Evaluating correlation breakdowns during periods of market volatility,, Working paper, (2000). doi: 10.2139/ssrn.231857. Google Scholar

[34]

J. Marabel Romo, Fitting the skew with an analytic local volatility function,, International Review of Applied Financial Issues and Economics, 3 (2011), 721. Google Scholar

[35]

J. Marabel Romo, Worst-of options and correlation skew under a stochastic correlation framework,, International journal of Theoretical and Applied Finance, 7 (). Google Scholar

[36]

W. Margrabe, The value of an option to exchange one asset for another,, Journal of Finance, 33 (1978), 177. doi: 10.2307/2326358. Google Scholar

[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. doi: 10.1080/13518470151141477. Google Scholar

[38]

M. Rubinstein, Implied binomial trees,, Journal of Finance, 49 (1994), 771. doi: 10.2307/2329207. Google Scholar

[39]

B. Solnik, C. Boucrelle and Y. Le Fur, International market correlation and volatility,, Financial Analysts Journal, 52 (1996), 17. doi: 10.2469/faj.v52.n5.2021. Google Scholar

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