Abstract
This paper develops a semidefinite programming approach to computing bounds on the range of allowable absence of arbitrage prices for a European call option when option prices at other strikes and expirations are available and when moment related information on the underlying is known. The moment related information is incorporated in the problem through the fictitious prices of polynomial valued securities. The optimization then comes from relaxing a risk neutral pricing optimization problem in terms of moments of measures from a decomposition of the risk neutral pricing measure. We demonstrate this optimization formulation with computations using moment data from the standard Black-Scholes option pricing model and Merton’s jump diffusion model.
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References
Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81, 637–659 (1973)
Laurent, J.P., Leisen, D.: Building a consistent pricing model form observed option prices. In: Avellaneda, W.S.M. (ed.) Collected Papers of the New York University Mathematical Finance Seminar (2000)
Carr, P., Madan, D.: A note on sufficient conditions for no arbitrage. Finance Res. Lett. 2, 125–130 (2005)
Davis, M.H.A., Hobson, D.G.: The range of traded option prices. Math. Finance 17, 1–14 (2007)
Lo, A.: Semiparametric upper bounds for option prices and expected payoffs. J. Financ. Econ. 19, 373–388 (1987)
Bertsimas, D., Popescu, I.: On the relation between option and stock prices: A convex optimization approach. Oper. Res. 50(2), 358–374 (2002)
Bertsimas, D., Popescu, I.: Optimal inequalities in probability theory: A convex optimization approach. SIAM J. Optim. 15(3), 780–804 (2005)
Bertsimas, D., Popescu, I., Sethuraman, J.: Moment problems and semidefinite programming. In: Wolkovitz, H. (ed.) Semidefinite Programming, pp. 469–509 (2000)
Smith, J.E.: Generalized Chebychev inequalities: Theory and applications in decision analysis. Oper. Res. 43, 807–825 (1995)
d’Aspremont, A.: Shape constrained optimization with applications in finance and engineering. PhD thesis, Stanford University (2004)
d’Aspremont, A., Ghaoui, L.E.: Static arbitrage bounds on basket option prices. Math. Program., Ser. A (2005)
Laurence, P., Wang, T.: What’s a basket worth? Risk (2004)
Laurence, P., Wang, T.: Sharp upper and lower bounds for basket options. Appl. Math. Finance 12, 253–282 (2005)
Lasserre, J.B., Prieto-Rumeau, T., Zervos, M.: Pricing a class of exotic options via moments and SDP relaxations. Math. Finance 16(3), 469–494 (2006)
Primbs, J.A.: Piecewise polynomial replication strategies and moment matrices in convex optimization based option pricing bounds. MS&E Working Paper, no. 06-11-81654-28 (2006)
Lasserre, J.B.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11, 796–817 (2001)
Curto, R.E., Fialkow, L.A.: Recursiveness, positivity, and truncated moment problems. Houst. J. Math. 17, 603–635 (1991)
Curto, R.E., Fialkow, L.A.: The truncated complex k-moment problem. Trans. Am. Math. Soc. 352, 2825–2855 (2000)
Sturm, J.: SeDuMi: Version 1.05, October 2004
Peaucelle, D., Henrion, D., Labit, Y., Taitz, K.: User’s Guide for SeDuMi Interface 1.04, September 2002
Gotoh, J., Konno, H.: Bounding option prices by semidefinite programming: A cutting plane algorithm. Manag. Sci. 48, 665–678 (2002)
Merton, R.C.: Option pricing when the underlying stock returns are discontinuous. J. Financ. Econ. 5, 125–144 (1976)
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Communicated by D.G. Luanberger.
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Primbs, J.A. SDP Relaxation of Arbitrage Pricing Bounds Based on Option Prices and Moments. J Optim Theory Appl 144, 137–155 (2010). https://doi.org/10.1007/s10957-009-9605-5
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DOI: https://doi.org/10.1007/s10957-009-9605-5