American Institute of Mathematical Sciences

Advanced
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

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 & 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

[1]

J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the Sakawa-Shindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391-406. doi: 10.3934/mcrf.2016008
[2]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437
[3]

Seung-Yeal Ha, Dongnam Ko, Chanho Min, Xiongtao Zhang. Emergent collective behaviors of stochastic kuramoto oscillators. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1059-1081. doi: 10.3934/dcdsb.2019208
[4]

María J. Garrido-Atienza, Bohdan Maslowski, Jana  Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088
[5]

Ardeshir Ahmadi, Hamed Davari-Ardakani. A multistage stochastic programming framework for cardinality constrained portfolio optimization. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 359-377. doi: 10.3934/naco.2017023
[6]

Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175
[7]

Longxiang Fang, Narayanaswamy Balakrishnan, Wenyu Huang. Stochastic comparisons of parallel systems with scale proportional hazards components equipped with starting devices. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021004
[8]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213
[9]

Shihu Li, Wei Liu, Yingchao Xie. Large deviations for stochastic 3D Leray-$ \alpha $ model with fractional dissipation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2491-2509. doi: 10.3934/cpaa.2019113
[10]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521
[11]

Xingchun Wang, Yongjin Wang. Variance-optimal hedging for target volatility options. Journal of Industrial & Management Optimization, 2014, 10 (1) : 207-218. doi: 10.3934/jimo.2014.10.207
[12]

Emma D'Aniello, Saber Elaydi. The structure of $ \omega $-limit sets of asymptotically non-autonomous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 903-915. doi: 10.3934/dcdsb.2019195
[13]

Chih-Chiang Fang. Bayesian decision making in determining optimal leased term and preventive maintenance scheme for leased facilities. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020127
[14]

Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027
[15]

Min Li. A three term Polak-Ribière-Polyak conjugate gradient method close to the memoryless BFGS quasi-Newton method. Journal of Industrial & Management Optimization, 2020, 16 (1) : 245-260. doi: 10.3934/jimo.2018149

2019 Impact Factor: 1.366

Tools

Metrics

  • PDF downloads (64)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences