Abstract
In the years following the publication of Black and Scholes (J Political Econ, 81(3), 637–654, 1973), numerous alternative models have been proposed for pricing and hedging equity derivatives. Prominent examples include stochastic volatility models, jump-diffusion models, and models based on Lévy processes. These all have their own shortcomings, and evidence suggests that none is up to the task of satisfactorily pricing and hedging extremely long-dated claims. Since they all fall within the ambit of risk-neutral valuation, it is natural to speculate that the deficiencies of these models are (at least in part) attributable to the constraints imposed by the risk-neutral approach itself. To investigate this idea, we present a simple two-parameter model for a diversified equity accumulation index. Although our model does not admit an equivalent risk-neutral probability measure, it nevertheless fulfils a minimal no-arbitrage condition for an economically viable financial market. Furthermore, we demonstrate that contingent claims can be priced and hedged, without the need for an equivalent change of probability measure. Convenient formulae for the prices and hedge ratios of a number of standard European claims are derived, and a series of hedge experiments for extremely long-dated claims on the S&P 500 total return index are conducted. Our model serves also as a convenient medium for illustrating and clarifying several points on asset price bubbles and the economics of arbitrage.
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References
Abramowitz, M., Stegun, I.A. (eds): Handbook of Mathematical Functions. Dover, New York (1972)
Aït-Sahalia Y.: Maximum likelihood estimation of discretely sampled diffusions: a closed-form approximation approach. Econometrica 70(1), 223–262 (2002)
Andersen L.B.G., Piterbarg V.V.: Moment explosions in stochastic volatility models. Financ. Stoch. 11(1), 29–50 (2007)
Barndorff-Nielsen O.E.: Processes of normal inverse Gaussian type. Financ. Stoch. 2(1), 41–68 (1998)
Bates D.S.: Jumps and stochastic volatility: exchange rate processes implicit in Deutsche mark options. Rev. Financ. Stud. 9(1), 69–107 (1996)
Black, F.: Studies in stock price volatility changes. In: Proceedings of the 1976 Business Meeting of the Business and Economic Statistics Section, pp. 177–181. American Statistical Association, Atlanta (1976)
Black F., Scholes M.: The pricing of options and corporate liabilities. J. Political Econ. 81(3), 637–654 (1973)
Borodin A.N., Salminen P.: Handbook of Brownian Motion, second edn. Birkhäuser, Basel (2002)
Carr P., Geman H., Madan D.B., Yor M.: The fine structure of asset returns: an empirical investigation. J. Bus. 75(2), 305–332 (2002)
Carr P., Geman H., Madan D.B., Yor M.: Stochastic volatility for Lévy processes. Math. Financ. 13(3), 345–382 (2003)
Cox A.M.G., Hobson D.G.: Local martingales, bubbles and option prices. Financ. Stoch. 9(4), 477–492 (2005)
Cox J.C., Ross S.A.: The valuation of options for alternative stochastic processes. J. Finan. Econ. 3(1–2), 145–166 (1976)
Delbaen F., Schachermayer W.: A general version of the fundamental theorem of asset pricing. Math. Ann. 300(3), 463–520 (1994)
Duffie D., Pan J., Singleton K.: Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68(6), 1343–1376 (2000)
Durham G.B., Gallant A.R.: Numerical techniques for maximum likelihood estimation of continuous-time diffusion processes. J. Bus. Econ. Statist. 20(3), 297–316 (2002)
Dybvig P.H., Huang C.: Non-negative wealth, absence of arbitrage, and feasible consumption plans. Rev. Finan. Stud. 1(4), 377–401 (1988)
Eberlein E., Keller U., Prause K.: New insights into smile, mispricing and value at risk: the hyperbolic model. J. Bus. 71(3), 371–405 (1998)
Ekström E., Tysk J.: Bubbles, convexity and the Black–Scholes equation. Ann. Appl. Probab. 19(4), 1369–1384 (2009)
Geman H., El Karoui N., Rochet J.C.: Changes of numéraire, changes of probability measure and option pricing. J. Appl. Probab. 32(2), 443–458 (1995)
Heston S.L.: A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Finan. Stud. 6(2), 327–343 (1993)
Heston S.L., Loewenstein M., Willard G.A.: Options and bubbles. Rev. Finan. Stud. 20(2), 359–389 (2007)
Hull J., White A.: The pricing of options on assets with stochastic volatilities. J. Financ. 42(2), 281–300 (1987)
Hulley H.: The economic plausibility of strict local martingales in financial modelling. In: Chiarella, C., Novikov, A. (eds) Contemporary Quantitative Finance, pp. 53–75. Springer, Berlin (2010)
Johnson N.L., Kotz S., Balakrishnan N.: Continuous Univariate Distributions, vol. 1. 2nd edn. Wiley, New York (1994)
Johnson N.L., Kotz S., Balakrishnan N.: Continuous Univariate Distributions, vol. 2. 2nd edn. Wiley, New York (1995)
Karatzas I., Kardaras C.: The numéraire portfolio in semimartingale financial models. Financ. Stoch. 11(4), 447–493 (2007)
Kou S.G.: A jump-diffusion model for option pricing. Manage. Sci. 48(8), 1086–1101 (2002)
Liu J., Longstaff F.A.: Losing money on arbitrage: optimal dynamic portfolio choice in markets with arbitrage opportunities. Rev. Finan. Stud. 17(3), 611–641 (2004)
Loewenstein M., Willard G.A.: Local martingales, arbitrage, and viability: free snacks and cheap thrills. Econ. Theory 16(1), 135–161 (2000)
Lowenstein R.: When Genius Failed: The Rise and Fall of Long-Term Capital Management. Random House, New York (2000)
Madan D.B., Seneta E.: The variance gamma (V.G.) model for share market returns. J. Bus. 63(4), 511–524 (1990)
Madan, D.B., Yor, M.: Ito’s integrated formula for strict local martingales. In: Séminaire de Probabilités XXXIX, Lecture Notes in Mathematics, vol. 1874, pp. 157–170. Springer, Berlin (2006)
Merton R.C.: Option pricing when underlying stock returns are discontinuous. J. Finan. Econ. 3(1–2), 125–144 (1976)
Miller S.M., Platen E.: Analytic pricing of contingent claims under the real-world measure. Int. J. Theor. Appl. Financ. 11(8), 841–867 (2008)
Platen, E.: A minimal financial market model. In: Mathematical Finance (Konstanz, 2000): Trends in Mathematics, pp. 293–301. Birkhäuser, Basel (2001)
Platen E.: Arbitrage in continuous complete markets. Adv. Appl. Probab. 34(3), 540–558 (2002)
Platen E.: An alternative interest rate term structure model. Int. J. Theor. Appl. Financ. 8(6), 717–735 (2005)
Platen E., Heath D.: A Benchmark Approach to Quantitative Finance. Springer, Berlin (2006)
Revuz D., Yor M.: Continuous Martingales and Brownian Motion. 3nd edn. Springer, Berlin (1999)
Schroder M.: Computing the constant elasticity of variance option pricing formula. J. Financ. 44(1), 211–219 (1989)
Schutz D.: Der Fall der UBS, Bilanz. Pyramid Media Group, New York (2000)
Shleifer A., Vishny R.W.: The limits of arbitrage. J. Financ. 52(1), 35–55 (1997)
Siegel A.F.: The noncentral chi-squared distribution with zero degrees of freedom and testing for uniformity. Biometrika 66(2), 381–386 (1979)
Sin C.A.: Complications with stochastic volatility models. Adv. Appl. Probab. 30(1), 256–268 (1998)
Stein E.M., Stein J.C.: Stock price distributions with stochastic volatility: an analytic approach. Rev. Finan. Stud. 4(4), 727–752 (1991)
Strasser E.: Characterization of arbitrage-free markets. Ann. Appl. Probab. 15(1A), 116–124 (2005)
Yor M.: On some exponential functionals of Brownian motion. Adv. Appl. Probab. 24(3), 509–531 (1992)
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Hulley, H., Platen, E. Hedging for the long run. Math Finan Econ 6, 105–124 (2012). https://doi.org/10.1007/s11579-012-0072-7
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DOI: https://doi.org/10.1007/s11579-012-0072-7
Keywords
- Long-dated claims
- Risk-neutral valuation
- Real-world valuation
- Arbitrage
- Minimal market model
- Squared Bessel processes
- Hedge simulations
- Asset price bubbles