Mental Accounting: A Closed-Form Alternative to the Black Scholes Model

Siddiqi, Hammad (2013): Mental Accounting: A Closed-Form Alternative to the Black Scholes Model.

Abstract

The principle of no arbitrage says that identical assets should offer the same returns. However, experimental and anecdotal evidence suggests that people often rely on analogy making while valuing assets. The principle of analogy making says that similar assets should offer the same returns. I show that the principle of analogy making generates a closed-form alternative to the Black Scholes formula that does not require a complete market. The new formula differs from the Black Scholes formula only due to the appearance of a parameter in the formula that captures the risk premium on the underlying. The new formula,called the analogy option pricing formula, provides a new explanation for the implied volatility skew puzzle in equity options. The key empirical predictions of the analogy formula are discussed. Existing Empirical evidence strongly supports these predictions.

• Item Type: MPRA Paper
• Language: English
• Keywords: Mental Accounting, Analogy Making, Option Pricing, Behavioral Finance, Implied Volatility Skew, Black Scholes
• Subjects: G - Financial Economics > G0 - General; G - Financial Economics > G0 - General > G02 - Behavioral Finance: Underlying Principles; G - Financial Economics > G1 - General Financial Markets; G - Financial Economics > G1 - General Financial Markets > G12 - Asset Pricing ; Trading Volume ; Bond Interest Rates; 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.

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.

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 R. Whaley. 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

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

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

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.

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

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

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

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/

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.

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

Miyahara, Y. (2001), “Geometric L´evy 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.

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. 28–46

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.

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.

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.