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A Closed-Form Solution for Options with Ambiguity about Stochastic Volatility

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  • Gonçalo Faria

    () (RGEA, Universidad de Vigo)

  • João Correia-da-Silva

    () (CEF.UP and Faculdade de Economia, Universidade do Porto)

Abstract

We derive a closed-form solution for the price of a European call option in the presence of ambiguity about the stochastic process that determines the variance of the underlying asset's return. The option pricing formula of Heston (1993) is a particular case of ours, corresponding to the case in which there is no ambiguity (uncertainty is exclusively risk). In the presence of ambiguity, the variance uncertainty price becomes either a convex or a concave function of the instantaneous variance, depending on whether the variance ambiguity price is negative or positive. We find that if the variance ambiguity price is positive, the option price is decreasing in the level of ambiguity (across all moneyness levels). The opposite happens if the variance ambiguity price is negative. Consistently, in the former (and more natural) scenario, ambiguity aversion decreases the option's implied volatility, which helps to explain the variance premium puzzle.

Suggested Citation

  • Gonçalo Faria & João Correia-da-Silva, 2011. "A Closed-Form Solution for Options with Ambiguity about Stochastic Volatility," FEP Working Papers 414, Universidade do Porto, Faculdade de Economia do Porto.
  • Handle: RePEc:por:fepwps:414
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    as
    1. Frank Riedel, 2009. "Optimal Stopping With Multiple Priors," Econometrica, Econometric Society, vol. 77(3), pages 857-908, May.
    2. Carr, Peter & Wu, Liuren, 2004. "Time-changed Levy processes and option pricing," Journal of Financial Economics, Elsevier, vol. 71(1), pages 113-141, January.
    3. Peter Klibanoff & Massimo Marinacci & Sujoy Mukerji, 2005. "A Smooth Model of Decision Making under Ambiguity," Econometrica, Econometric Society, vol. 73(6), pages 1849-1892, November.
    4. Larry G. Epstein & Martin Schneider, 2010. "Ambiguity and Asset Markets," Annual Review of Financial Economics, Annual Reviews, vol. 2(1), pages 315-346, December.
    5. Jun Liu, 2005. "An Equilibrium Model of Rare-Event Premia and Its Implication for Option Smirks," Review of Financial Studies, Society for Financial Studies, vol. 18(1), pages 131-164.
    6. Fabio Maccheroni & Massimo Marinacci & Aldo Rustichini, 2006. "Ambiguity Aversion, Robustness, and the Variational Representation of Preferences," Econometrica, Econometric Society, vol. 74(6), pages 1447-1498, November.
    7. Peter Carr & Liuren Wu, 2003. "The Finite Moment Log Stable Process and Option Pricing," Journal of Finance, American Finance Association, vol. 58(2), pages 753-778, April.
    8. Bjørn Eraker & Michael Johannes & Nicholas Polson, 2003. "The Impact of Jumps in Volatility and Returns," Journal of Finance, American Finance Association, vol. 58(3), pages 1269-1300, June.
    9. Aleksandar Mijatovic & Paul Schneider, 2009. "Empirical Asset Pricing with Nonlinear Risk Premia," Working Papers wp09-03, Warwick Business School, Finance Group.
    10. Larry G. Epstein, 2010. "A Paradox for the “Smooth Ambiguity” Model of Preference," Econometrica, Econometric Society, vol. 78(6), pages 2085-2099, November.
    11. Peter Klibanoff & Massimo Marinacci & Sujoy Mukerji, 2012. "On the Smooth Ambiguity Model: A Reply," Econometrica, Econometric Society, vol. 80(3), pages 1303-1321, May.
    12. Johnson, Herb & Shanno, David, 1987. "Option Pricing when the Variance Is Changing," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 22(02), pages 143-151, June.
    13. Tim Bollerslev & Viktor Todorov, 2011. "Tails, Fears, and Risk Premia," Journal of Finance, American Finance Association, vol. 66(6), pages 2165-2211, December.
    14. Epstein, Larry G. & Schneider, Martin, 2003. "Recursive multiple-priors," Journal of Economic Theory, Elsevier, vol. 113(1), pages 1-31, November.
    15. Hansen, Lars Peter & Sargent, Thomas J. & Wang, Neng E., 2002. "Robust Permanent Income And Pricing With Filtering," Macroeconomic Dynamics, Cambridge University Press, vol. 6(01), pages 40-84, February.
    16. Bollerslev, Tim & Ole Mikkelsen, Hans, 1996. "Modeling and pricing long memory in stock market volatility," Journal of Econometrics, Elsevier, vol. 73(1), pages 151-184, July.
    17. Lucas, Robert E, Jr, 1978. "Asset Prices in an Exchange Economy," Econometrica, Econometric Society, vol. 46(6), pages 1429-1445, November.
    18. Lorenzo Garlappi & Raman Uppal & Tan Wang, 2007. "Portfolio Selection with Parameter and Model Uncertainty: A Multi-Prior Approach," Review of Financial Studies, Society for Financial Studies, vol. 20(1), pages 41-81, January.
    19. Bakshi, Gurdip & Cao, Charles & Chen, Zhiwu, 1997. " Empirical Performance of Alternative Option Pricing Models," Journal of Finance, American Finance Association, vol. 52(5), pages 2003-2049, December.
    20. Fabio Trojani & Paolo Vanini, 2004. "Robustness and Ambiguity Aversion in General Equilibrium," Review of Finance, European Finance Association, vol. 8(2), pages 279-324.
    21. Stein, Elias M & Stein, Jeremy C, 1991. "Stock Price Distributions with Stochastic Volatility: An Analytic Approach," Review of Financial Studies, Society for Financial Studies, vol. 4(4), pages 727-752.
    22. Bjørn Eraker, 2004. "Do Stock Prices and Volatility Jump? Reconciling Evidence from Spot and Option Prices," Journal of Finance, American Finance Association, vol. 59(3), pages 1367-1404, June.
    23. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    24. H. Henry Cao & Tan Wang & Harold H. Zhang, 2005. "Model Uncertainty, Limited Market Participation, and Asset Prices," Review of Financial Studies, Society for Financial Studies, vol. 18(4), pages 1219-1251.
    25. John C. Cox & Jonathan E. Ingersoll Jr. & Stephen A. Ross, 2005. "A Theory Of The Term Structure Of Interest Rates," World Scientific Book Chapters,in: Theory Of Valuation, chapter 5, pages 129-164 World Scientific Publishing Co. Pte. Ltd..
    26. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters,in: Theory Of Valuation, chapter 8, pages 229-288 World Scientific Publishing Co. Pte. Ltd..
    27. Egloff, Daniel & Leippold, Markus & Wu, Liuren, 2010. "The Term Structure of Variance Swap Rates and Optimal Variance Swap Investments," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 45(05), pages 1279-1310, October.
    28. Gonçalo Faria & João Correia-da-Silva, 2012. "The price of risk and ambiguity in an intertemporal general equilibrium model of asset prices," Annals of Finance, Springer, vol. 8(4), pages 507-531, November.
    29. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    30. Darrell Duffie & Jun Pan & Kenneth Singleton, 2000. "Transform Analysis and Asset Pricing for Affine Jump-Diffusions," Econometrica, Econometric Society, vol. 68(6), pages 1343-1376, November.
    31. 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 327-343.
    32. Patrick Gagliardini & Paolo Porchia & Fabio Trojani, 2009. "Ambiguity Aversion and the Term Structure of Interest Rates," Review of Financial Studies, Society for Financial Studies, vol. 22(10), pages 4157-4188, October.
    33. Rubinstein, Mark, 1994. " Implied Binomial Trees," Journal of Finance, American Finance Association, vol. 49(3), pages 771-818, July.
    34. Jianjun Miao & Bin Wei & Hao Zhou, 2012. "Ambiguity Aversion and Variance Premium," Boston University - Department of Economics - Working Papers Series WP2012-009, Boston University - Department of Economics.
    35. Hansen, Lars-Peter & Sargent, Thomas-J, 2001. "Acknowledgement Misspecification in Macroeconomic Theory," Monetary and Economic Studies, Institute for Monetary and Economic Studies, Bank of Japan, vol. 19(S1), pages 213-227, February.
    36. Wiggins, James B., 1987. "Option values under stochastic volatility: Theory and empirical estimates," Journal of Financial Economics, Elsevier, vol. 19(2), pages 351-372, December.
    37. Bollerslev, Tim & Chou, Ray Y. & Kroner, Kenneth F., 1992. "ARCH modeling in finance : A review of the theory and empirical evidence," Journal of Econometrics, Elsevier, vol. 52(1-2), pages 5-59.
    38. Camerer, Colin & Weber, Martin, 1992. "Recent Developments in Modeling Preferences: Uncertainty and Ambiguity," Journal of Risk and Uncertainty, Springer, vol. 5(4), pages 325-370, October.
    39. Lars Peter Hansen & Thomas J. Sargent, 2001. "Acknowledging Misspecification in Macroeconomic Theory," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 4(3), pages 519-535, July.
    40. Mark Rubinstein., 1994. "Implied Binomial Trees," Research Program in Finance Working Papers RPF-232, University of California at Berkeley.
    41. Johanna Etner & Meglena Jeleva & Jean‐Marc Tallon, 2012. "Decision Theory Under Ambiguity," Journal of Economic Surveys, Wiley Blackwell, vol. 26(2), pages 234-270, April.
    42. Scott, Louis O., 1987. "Option Pricing when the Variance Changes Randomly: Theory, Estimation, and an Application," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 22(04), pages 419-438, December.
    43. Gilboa, Itzhak & Schmeidler, David, 1986. "Maxmin Expected Utility with a Non-Unique Prior," Foerder Institute for Economic Research Working Papers 275405, Tel-Aviv University > Foerder Institute for Economic Research.
    44. Florence Guillaume & Wim Schoutens, 2012. "Calibration risk: Illustrating the impact of calibration risk under the Heston model," Review of Derivatives Research, Springer, vol. 15(1), pages 57-79, April.
    45. Maenhout, Pascal J., 2006. "Robust portfolio rules and detection-error probabilities for a mean-reverting risk premium," Journal of Economic Theory, Elsevier, vol. 128(1), pages 136-163, May.
    46. Hans FÃllmer & Peter Leukert, 2000. "Efficient hedging: Cost versus shortfall risk," Finance and Stochastics, Springer, vol. 4(2), pages 117-146.
    47. Joost Driessen & Pascal J. Maenhout & Grigory Vilkov, 2009. "The Price of Correlation Risk: Evidence from Equity Options," Journal of Finance, American Finance Association, vol. 64(3), pages 1377-1406, June.
    48. Cheridito, Patrick & Filipovic, Damir & Kimmel, Robert L., 2007. "Market price of risk specifications for affine models: Theory and evidence," Journal of Financial Economics, Elsevier, vol. 83(1), pages 123-170, January.
    49. Gilboa, Itzhak & Schmeidler, David, 1989. "Maxmin expected utility with non-unique prior," Journal of Mathematical Economics, Elsevier, vol. 18(2), pages 141-153, April.
    50. Itamar Drechsler & Amir Yaron, 2011. "What's Vol Got to Do with It," Review of Financial Studies, Society for Financial Studies, vol. 24(1), pages 1-45.
    51. Merton, Robert C., 1980. "On estimating the expected return on the market : An exploratory investigation," Journal of Financial Economics, Elsevier, vol. 8(4), pages 323-361, December.
    52. Ole E. Barndorff-Nielsen & Neil Shephard, 2001. "Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(2), pages 167-241.
    53. Bates, David S, 1996. "Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options," Review of Financial Studies, Society for Financial Studies, vol. 9(1), pages 69-107.
    54. Hull, John C & White, Alan D, 1987. " The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
    55. Heston, Steven L & Nandi, Saikat, 2000. "A Closed-Form GARCH Option Valuation Model," Review of Financial Studies, Society for Financial Studies, vol. 13(3), pages 585-625.
    56. Chudjakow, Tatjana & Vorbrink, Jörg, 2011. "Exercise strategies for American exotic options under ambiguity," Center for Mathematical Economics Working Papers 421, Center for Mathematical Economics, Bielefeld University.
    57. Viktor Todorov, 2010. "Variance Risk-Premium Dynamics: The Role of Jumps," Review of Financial Studies, Society for Financial Studies, vol. 23(1), pages 345-383, January.
    58. Cox, John C & Ingersoll, Jonathan E, Jr & Ross, Stephen A, 1985. "An Intertemporal General Equilibrium Model of Asset Prices," Econometrica, Econometric Society, vol. 53(2), pages 363-384, March.
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    Cited by:

    1. Gonçalo Faria & João Correia-da-Silva, 2012. "The price of risk and ambiguity in an intertemporal general equilibrium model of asset prices," Annals of Finance, Springer, vol. 8(4), pages 507-531, November.
    2. repec:spr:annopr:v:262:y:2018:i:2:d:10.1007_s10479-015-2079-y is not listed on IDEAS
    3. Chunpeng Yang & Bin Gao & Jianlei Yang, 2016. "Option pricing model with sentiment," Review of Derivatives Research, Springer, vol. 19(2), pages 147-164, July.

    More about this item

    Keywords

    Option Pricing; Stochastic Volatility; Ambiguity; Variance Premium Puzzle;

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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