Analogy Making and the Structure of Implied Volatility Skew

Siddiqi, Hammad (2014): Analogy Making and the Structure of Implied Volatility Skew.

Abstract

An analogy based call option pricing model is put forward. The model provides a new explanation for the implied volatility skew puzzle and is consistent with empirical findings regarding leverage adjusted option returns. It also explains puzzling superior performance of covered call writing and worse-than-expected performance of zero-beta straddles. The analogy based stochastic volatility and the analogy jump diffusion models are also developed. The analogy based stochastic volatility model generates the skew even without any correlation between the stock price and volatility processes, whereas, the analogy jump diffusion does not require asymmetric jumps to generate the skew.

• Item Type: MPRA Paper
• Language: English
• Keywords: Implied Volatility Skew, Implied Volatility Smile, Analogy Making, Stochastic Volatility, Jump Diffusion, Covered Call Writing, Zero-Beta Straddle, Leverage Adjusted Option Returns
• Subjects: G - Financial Economics > G1 - General Financial Markets > G13 - Contingent Pricing ; Futures Pricing

References

Amin, K. (1993). “Jump diffusion option valuation in discrete time”, Journal of Finance 48, 1833-1863.

Anderson, Torben, Luca Benzoni, and Jesper Lund (2002). “An empirical investigation of continuous time equity return models”, Journal of Finance 57, 1239–1284.

Babcock, L., & Loewenstein, G. (1997). “Explaining bargaining impasse: The role of self-serving biases”. Journal of Economic Perspectives, 11(1), 109–126.

Babcock, L., Wang, X., & Loewenstein, G. (1996). “Choosing the wrong pond: Social comparisons in negotiations that reflect a self-serving bias”. The Quarterly Journal of Economics, 111(1), 1–19.

Bakshi G., Cao, C., Chen, Z. (1997). “Empirical performance of alternative option pricing models”, Journal of Finance 52, 2003-2049.

Ball, C., Torous, W. (1985). “On jumps in common stock prices and their impact on call option pricing”, Journal of Finance 40, 155-173.

Bates, D. (2000), “Post-‘87 Crash fears in S&P 500 futures options”, Journal of Econometrics, 94, pp. 181-238.

Beber A., Breedan F., and Buraschi A. (2010), “Differences in beliefs and currency risk premiums”, Journal of Financial Economics, Vol. 98, Issue 3, pp. 415-438.

Belini, F., Fritelli, M. (2002), “On the existence of minimax martingale measures”, Mathematical Finance 12, 1–21.

Black, F., Scholes, M. (1973). “The pricing of options and corporate liabilities”. Journal of Political Economy 81(3): pp. 637-65

Black, F., (1976). “Studies of stock price volatility changes”, Proceedings of the 1976 Meetings of the American Statistical Association, Business and Economic Statistics Section, 177–181.

Bollen, N., and Whaley, R. (2004). “Does Net Buying Pressure Affect the Shape of Implied Volatility Functions?” Journal of Finance 59(2): 711–53

Bossaerts, P., Plott, C. (2004), “Basic Principles of Asset Pricing Theory: Evidence from Large Scale Experimental Financial Markets”. Review of Finance, 8, pp. 135-169.

Carpenter, G., Rashi G., & Nakamoto, K. (1994), “Meaningful Brands from Meaningless Differentiation: The Dependence on Irrelevant Attributes,” Journal of Marketing Research 31, pp. 339-350

Chernov, Mikhail, Ron Gallant, Eric Ghysels, and George Tauchen, 2003, Alternative models of stock price dynamics, Journal of Econometrics 116, 225–257.

Christensen B. J., and Prabhala, N. R. (1998), “The Relation between Realized and Implied Volatility”, Journal of Financial Economics Vol.50, pp. 125-150.

Christie A. A. (1982), “The stochastic behavior of common stock variances: value, leverage and interest rate effects”, Journal of Financial Economics 10, 4 (1982), pp. 407–432

Coval, J. D., and Shumway, T. (2001), “Expected Option Returns”, Journal of Finance, Vol. 56, No.3, pp. 983-1009.

Cross, J. G. (1983), A Theory of Adaptive Economic Behavior. New York: Cambridge University Press.

Davis, M.H.A. (1997), “Option pricing in incomplete markets, Mathematics of Derivative Securities”, Cambridge University Press, editted by M.A.H. Dempster and S.R. Pliska, 216–226.

DeBondt, W. and Thaler, R. (1985). “Does the Stock-Market Overreact?” Journal of Finance 40: 793-805

Derman, E. (2003), “The Problem of the Volatility Smile”, talk at the Euronext Options Conference available at http://www.ederman.com/new/docs/euronext-volatility_smile.pdf

Derman, E, (2002), “The perception of time, risk and return during periods of speculation”. Quantitative Finance Vol. 2, pp. 282-296.

Derman, E., Kani, I., & Zou, J. (1996), “The local volatility surface: unlocking the information in index option prices”, Financial Analysts Journal, 52, 4, 25-36

Derman, E., Kani, I. (1994), “Riding on the Smile.” Risk, Vol. 7, pp. 32-39

Duffie, D., C. Huang. 1985. Implementing Arrow-Debreu equilibria by continuous trading of few long-lived securities. Econometrica 53 1337–1356.

Duan, Jin-Chuan, and Wei Jason (2009), “Systematic Risk and the Price Structure of Individual Equity Options”, The Review of Financial studies, Vol. 22, No.5, pp. 1981-2006.

Duan, J.-C., 1995. The GARCH option pricing model. Mathematical Finance 5, 13-32.

Dumas, B., Fleming, J., Whaley, R., 1998. Implied volatility functions: empirical tests. Journal of Finance 53, 2059-2106.

Dupire, B. (1994), “Pricing with a Smile”, Risk, Vol. 7, pp. 18-20

Edelman, G. (1992), Bright Air, Brilliant Fire: On the Matter of the Mind, New York, NY: BasicBooks.

Emanuel, D. C., and MacBeth, J. D. (1982), “Further Results on the Constant Elasticity of Variance Option Pricing Model”, Journal of Financial and Quantitative Analysis, Vol. 17, Issue 4, pp. 533-554.

Fama, E.F., and French, K.R. (1988). “Permanent and Temporary Components of Stock Prices”. Journal of Political Economy 96: 247-273

Fleming J., Ostdiek B. and Whaley R. E. (1995), “Predicting stock market volatility: a new measure”. Journal of Futures Markets 15 (1995), pp. 265–302.

Fouque, Papanicolaou, Sircar, and Solna (2004), “Maturity Cycles in Implied Volatility” Finance and Stochastics, Vol. 8, Issue 4, pp 451-477

Follmer, H., Schweizer, M. (1991) “Hedging of contingent claims under incomplete information, Applied Stochastic Analysis (M. H. A. Davis and R. J. Elliott, eds.), Gordon and Breach, New York, 389–414.

Frittelli M. (2002), “The minimal entropy martingale measure and the valuation problem in incomplete markets, Mathematical Finance 10, 39–52.

Gilboa, I., Schmeidler, D. (2001), “A Theory of Case Based Decisions”. Publisher: Cambridge University Press.

Goll, T., Ruschendorf, L. (2001) “Minimax and minimal distance martingale measures and their relationship to portfolio optimization”, Finance and Stochastics 5, 557–581

Goltz, F., and Lai, W. N. (2009), “Empirical properties of straddle returns”, The Journal of Derivatives, Vol. 17, No. 1, pp. 38-48.

Greiner, S. P. (2013), “Investment risk and uncertainty: Advanced risk awareness techniques for the intelligent investor”. Published by Wiley Finance.

Han, B. (2008), “Investor Sentiment and Option Prices”, The Review of Financial Studies, 21(1), pp. 387-414.

Heston S., Nandi, S., 2000. A closed-form GARCH option valuation model. Review of Financial Studies 13, 585-625.

Henderson, V. (2002), “Valuation of claims on nontraded assets using utility maximization”, Mathematical Finance 12, 351–373

Henderson, P. W., and Peterson, R. A. (1992), “Mental Accounting and Categorization”, Organizational Behavior and Human Decision Processes, 51, pp. 92-117.

Heston S., 1993. “A closed form solution for options with stochastic volatility with application to bond and currency options. Review of Financial Studies 6, 327-343.

Hodges, S. D., Neuberger, A. (1989), “Optimal replication of contingent claims under transaction costs”, The Review of Futures Markets 8, 222–239

Hofstadter, D., and Sander, E. (2013), “Surfaces and Essences: Analogy as the fuel and fire of thinking”, Published by Basic Books, April.

Hogarth, R. M., & Einhorn, H. J. (1992). “Order effects in belief updating: The belief-adjustment model”. Cognitive Psychology, 24.

Hull, J., White, A., 1987. The pricing of options on assets with stochastic volatilities. Journal of Finance 42, 281-300.

Hume, David (1748), “An Enquiry Concerning Human Understanding”. E-version available at http://creativecommons.org/licenses/by-nc-sa/3.0/au/

Jackwerth, J. C., (2000), “Recovering Risk Aversion from Option Prices and Realized Returns”, The Review of Financial Studies, Vol. 13, No, 2, pp. 433-451.

Jehiel, P. (2005), “Analogy based Expectations Equilibrium”, Journal of Economic Theory, Vol. 123, Issue 2, pp. 81-104.

Kahneman, D., & Frederick, S. (2002). “Representativeness revisited: Attribute substitution in intuitive judgment”. In T. Gilovich, D. Griffin, & D. Kahneman (Eds.), Heuristics and biases (pp. 49–81). New York: Cambridge University Press.

Kahneman, D., & Tversky, A. (1982), Judgment under Uncertainty: Heuristics and Biases, New York, NY: Cambridge University Press.

Keynes, John Maynard (1921), “A Treatise on Probability”, Publisher: Macmillan And Co.

Kluger, B., & Wyatt, S. (2004). “Are judgment errors reflected in market prices and allocations? Experimental evidence based on the Monty Hall problem”. Journal of Finance, pp. 969–997.

Lakoff, G. (1987), Women, Fire, and Dangerous Things, Chicago, IL: The University of Chicago Press.

Lakonishok, J., I. Lee, N. D. Pearson, and A. M. Poteshman, 2007, “Option Market Activity,” Review of Financial Studies, 20, 813-857.

Lettau, M., and S. Ludvigson (2010): Measuring and Modelling Variation in the Risk-Return Trade-off" in Handbook of Financial Econometrics, ed. by Y. Ait-Sahalia, and L. P. Hansen, chap. 11,pp. 617-690. Elsevier Science B.V., Amsterdam, North Holland.

Lustig, H., and A. Verdelhan (2010), “Business Cycle Variation in the Risk-Return Trade-off". Working paper, UCLA.

Melino A., Turnbull, S., 1990. “Pricing foreign currency options with stochastic volatility”. Journal of Econometrics 45, 239-265

Merton, R. C., 1989, “On the Application of the Continuous-Time Theory of Finance to Financial Intermediation and Insurance,” The Geneva Papers on Risk and Insurance 14, 225-261.

Miyahara, Y. (2001), “Geometric Levy Process & MEMM Pricing Model and Related Estimation Problems”, Asia-Pacific Financial Markets 8, 45–60 (2001)

Mullainathan, S., Schwartzstein, J., & Shleifer, A. (2008) “Coarse Thinking and Persuasion”. The Quarterly Journal of Economics, Vol. 123, Issue 2 (05), pp. 577-619.

Pan, J. (2002), “The jump-risk premia implicit in options: Evidence from an integrated time-series Study”, Journal of Financial Economics, 63, pp. 3-50.

Poterba, J.M. and Summers, L.H. (1988). “Mean Reversion in Stock-Prices: Evidence and Implications”. Journal of Financial Economics 22: 27-59

Rendleman, R. (2002), Applied Derivatives: Options, Futures, and Swaps. Wiley-Blackwell.

Rennison, G., and Pedersen, N. (2012) “The Volatility Risk Premium”, PIMCO, September.

Rockenbach, B. (2004), “The Behavioral Relevance of Mental Accounting for the Pricing of Financial Options”. Journal of Economic Behavior and Organization, Vol. 53, pp. 513-527.

Rubinstein, M., 1985. Nonparametric tests of alternative option pricing models using all reported trades and quotes on the 30 most active CBOE option classes from August 23,1976 through August 31, 1978. Journal of Finance 40, 455-480.

Rubinstein, M., 1994. Implied binomial trees. Journal of Finance 49, 771-818.

Schwert W. G. (1989), “Why does stock market volatility change over time?” Journal of Finance 44, 5 (1989), pp. 28–6

Schwert W. G. (1990), “Stock volatility and the crash of 87”. Review of Financial Studies 3, 1 (1990), pp. 77–102

Selten, R.: 1978, 'The Chain-Store Paradox', Theory and Decision 9, 127-159.

Shefrin, H. (2008), “A Behavioral Approach to Asset Pricing”, Published by Academic Press, June.

Shefrin, H. (2010), “Behavioralizing Finance” Foundations and Trends in Finance Published by Now Publishers Inc.

Siddiqi, H. (2009), “Is the Lure of Choice Reflected in Market Prices? Experimental Evidence based on the 4-Door Monty Hall Problem”. Journal of Economic Psychology, April.

Siddiqi, H. (2011), “Does Coarse Thinking Matter for Option Pricing? Evidence from an Experiment” IUP Journal of Behavioral Finance, Vol. VIII, No.2. pp. 58-69

Siddiqi, H. (2012), “The Relevance of Thinking by Analogy for Investors’ Willingness to Pay: An Experimental Study”, Journal of Economic Psychology, Vol. 33, Issue 1, pp. 19-29.

Soner, H.M., S. Shreve, and J. Cvitanic, (1995), “There is no nontrivial hedging portfolio for option pricing with transaction costs”, Annals of Applied Probability 5, 327–355.

Stein, E. M., and Stein, J. C (1991) “Stock price distributions with stochastic volatility: An analytic approach” Review of Financial Studies, 4(4):727–752.

Summers, L. H. (1986). “Does the Stock-Market Rationally Reflect Fundamental Values?” Journal of Finance 41: 591-601 Thaler, R. (1999), “Mental Accounting Matters”, Journal of Behavioral Decision Making, 12, pp. 183-206.

Thaler, R. H. "Toward a positive theory of consumer choice" (1980) Journal of Economic Behavior and Organization, 1, 39-60

Wiggins, J., 1987. Option values under stochastic volatility: theory and empirical estimates. Journal of Financial Economics 19, 351-372.

Zaltman, G. (1997), “Rethinking Market Research: Putting People Back In,” Journal of Marketing Research 34, pp. 424-437.