×

Risk measure pricing and hedging in the presence of transaction costs. (English) Zbl 1210.91059

Summary: Recently a risk measure pricing and hedging is replacing a utility-based maximization problem in the literature. In this paper, we treat the optimal problem of risk measure pricing and hedging in the friction market, i.e. in the presence of transaction costs. The risk measure pricing is also verified with the contexts in the literature.

MSC:

91G20 Derivative securities (option pricing, hedging, etc.)
91B30 Risk theory, insurance (MSC2010)
60G48 Generalizations of martingales
91B24 Microeconomic theory (price theory and economic markets)
Full Text: DOI

References:

[1] P. Barrieu and N. El Karoui,Optimal derivatives design under dynamic risk measures, Mathematics of Finance, Contemporary Mathematics(A.M.S. Proceedings 351), 2004. · Zbl 1070.91019
[2] P. Barrieu and N. El Karoui,Inf-convolution of risk-measures and optimal risk transfer, Finance Stochastics 9, 269–298, 2005. · Zbl 1088.60037 · doi:10.1007/s00780-005-0152-0
[3] F. Belini and M. Fritelli,On the existence of minimax martingale measures, Mathematical Finance 12, 1–21, 2002. · Zbl 1014.91031 · doi:10.1111/1467-9965.00001
[4] B. Bensaid, P. Lesne and J. Scheinkman,Derivative Asset Pricing with Transaction Costs, Mathematical Finance 2, 63–86, 1992. · Zbl 0900.90100 · doi:10.1111/j.1467-9965.1992.tb00039.x
[5] F. Black and M. Scholes,The Pricing of Options and Corporate Liabilities, Journal of Political Economy 81, 673–654, 1973. · Zbl 1092.91524 · doi:10.1086/260062
[6] P. Boyle and T. Vorst,Option replication in discrete time with transaction costs, J. Finance 47, 271–293, 1992. · doi:10.2307/2329098
[7] P. Carr, H. German and D. Mardan,Pricing and hedging in incomplete markets, Journal of Financial Economics 62, 131–167, 2001. · doi:10.1016/S0304-405X(01)00075-7
[8] G. M. Constantinides and S. Perrakis,Stochastic dominance bounds on derivatives prices in a multiperiod economy with proportional transaction costs, Journal of Econom. Dynam. Control 26, 1323–1352, 2002. · Zbl 1131.91332 · doi:10.1016/S0165-1889(01)00047-1
[9] G. M. Constantinides and T. Zariphopoulou,Bounds on prices of contingent claims in an intertemporal economy with proportional transaction costs and general preferences, Finance Stochastics 3, 345–369, 1999. · Zbl 0935.91014 · doi:10.1007/s007800050066
[10] G. M. Constantinides and T. Zariphopoulou,Bounds on derivative prices in an intertemporal setting with proportional transaction costs and multiple securities, Mathematical Finance 11, 331–346, 2001. · Zbl 0980.91019 · doi:10.1111/1467-9965.00118
[11] J. Cvitanic and I. Karatzas,Hedging and portfolio optimization under transaction costs: a martingale approach, Mathematical Finance 6, 133–166, 1996. · Zbl 0919.90007 · doi:10.1111/j.1467-9965.1996.tb00075.x
[12] J. Cvitanic, H. Pham and N. Touzi,A closed-form solution to the problem of superreplication under transaction costs, Finance and Stochastics 3, 335–354, 1999.
[13] M. H. A. Davis. Option pricing in incomplete markets,Mathematics of derivative securities(M. A. H. Dempster andS. R. Pliska eds), Cambridge University Press, Cambridge, UK, 216–226, 1997. · Zbl 0914.90017
[14] M. H. A. Davis, V. G. Panas and T. Zariphopoulou,European Option Pricing with Transaction Costs, SIAM J. Control Optim. 31, 470–493, 1993. · Zbl 0779.90011 · doi:10.1137/0331022
[15] F. Delbaen. Coherent risk measures on general probability spaces,Advances in finance and stochastics: Essays in honor of Dieter Sondermann, Springer, 1–37, 2002.
[16] F. Delbaen and W. Schachermayer,A General Version of the Fundamental Theorem of Asset Pricing, Mathematische Annalen 300, 463–520, 1994. · Zbl 0865.90014 · doi:10.1007/BF01450498
[17] Dellacherie and Meyer,Probabilities and potential B, North-Holland Publishing Co., Amsterdam, 1982.
[18] C. Edirshinghe, V. Naik and U. Uppal,Optimal replication of options with transactions costs and general preferences, Rev. Futures Markets 8, 222–239, 1989.
[19] H. Follmer and D. Kramkov,Optimal decompositions under constraints, Probability Theory Related Fields 109, 1–25, 1997. · Zbl 0882.60063 · doi:10.1007/s004400050122
[20] H. Follmer and A. Schied,Stochastic Finance, An Introduction in Discrete Time, 2nd Edition, de Gruyter Studies in Mathematics 27, 2004. · Zbl 1126.91028
[21] H. Follmer and M. Schweizer,Hedging of contingent claims under incomplete information, Applied Stochastic Analysis(M. H. A. Davis and R. J. Elliot eds), New York:Gordon and Breach, 389–414, 1991.
[22] M. Fritelli,The minimal entropy martingale measure and the valuation in incomplete markets, Mathematical Finance 10, 39–52, 2002. · Zbl 1034.91041 · doi:10.1111/1467-9965.00079
[23] P. Guasoni,Optimal investment with transaction costs and without semimartingales, Ann. Appl. Probab. 12, 1227–1246, 2002. · Zbl 1016.60065 · doi:10.1214/aoap/1037125861
[24] P. Guasoni,Risk minimization under transaction costs, Finance Stoch. 6, 91–113, 2002. · Zbl 1007.91021 · doi:10.1007/s780-002-8402-0
[25] V. Henderson and D. G. Hobson,Utility indifference pricing-an overview, Volume on Indifference Pricing (R. Carmona, ed.), Princeton University Press, to appear. · Zbl 1158.91379
[26] S. D. Hodges, A. Neuberger,Stochastic Finance, A. Review of Futures Market 8, 222–239, 1989.
[27] S. Jaschke and U. Kuchler,Coherent risk measures and good-deal bounds, Finance Stochastics 5, 181–200, 2001. · Zbl 0993.91023 · doi:10.1007/PL00013530
[28] E. Jouini and H. Kappal,Martingales and arbitrage in securities markets with transaction costs, Journal of Economic Theory 66, 178–197, 1995. · Zbl 0830.90020 · doi:10.1006/jeth.1995.1037
[29] Yu. M. Kabanov,Hedging and liquidation under transaction costs in currency markets, Finance Stochastics 3, 237–248, 1999. · Zbl 0926.60036 · doi:10.1007/s007800050061
[30] Yu. M. Kabanov and Ch. Stricker,The Harrison-Pliska arbitrage pricing under transaction costs, Journal of Mathematical Economics 35, 185–196, 2001. · Zbl 0986.91012 · doi:10.1016/S0304-4068(00)00064-1
[31] R. E. Kalman,The theory of optimal control and the calculus of variations, Proc. of Conf. on Mathematical Optimization Techniques (Santa Monica, 1960), R. Bellan (ed.), Univ. of California Press, California, 1963. · Zbl 0112.06303
[32] S. Kloppel and M. Schweizer,Dynamic utility indifference valuation via risk measures, preprint, ETH 2006.
[33] D. Lamberton, H. Pham, and M. Schweizer,Local risk-minimization under transaction costs, Mathematics of Operations Research 23, 585–612, 1998. · Zbl 0994.91024 · doi:10.1287/moor.23.3.585
[34] K. Larsen, T. Pirvu, S. E. Shreve and R. Tutuncu,Satisfying convex risk limits by trading, Finance Stochastics 9, 177–195, 2005. · Zbl 1092.91048 · doi:10.1007/s00780-004-0137-4
[35] J. Memin,Espaces de semi martingales et changement de probabilite, Probability Theory and Related Fields, 52, No. 1, 9–39, 1980. · Zbl 0407.60046
[36] F. Mercurio and T. C. F. Vorst,Options pricing and hedging in discrete time with transaction costs, Mathematics of Derivative Securities(M. A. H. Dempster and S. R. Pliska eds.), Cambridge University Press, Cambridge, UK, 190–215, 1997. · Zbl 0947.91038
[37] R. C. Merton,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, 1989.
[38] M. Musiela and T. Zariphopoulou,An example of indifference prices under exponential preferences, Finance Stochastics 8, 229–239, 2004a. · Zbl 1062.93048 · doi:10.1007/s00780-003-0112-5
[39] M. Musiela and T. Zariphopoulou,A valuation algorithm for indifference prices in incomplete markets, Finance Stochastics 8, 399–414, 2004b. · Zbl 1097.91046 · doi:10.1007/s00780-003-0117-0
[40] H. Pham,Minimal shortfall risk and applications to finance and insurance problems, The Annals of Applied Probability 12, 143–172, 2002. · Zbl 1015.93071 · doi:10.1214/aoap/1015961159
[41] R. Rouge and N. E. Karoui,Pricing via Utility Maximization and Entropy, Mathematical Finance, Vol. 10, No. 2, 259–276, 2000. · Zbl 1052.91512 · doi:10.1111/1467-9965.00093
[42] W. Schachermayer,The Fundamental Theorem of Asset Pricing under Proportional Transaction Costs in Finite Discrete Time, Mathematical Finance, Vol. 14, No. 1, 19–48, 2004. · Zbl 1119.91046 · doi:10.1111/j.0960-1627.2004.00180.x
[43] Mingxin Xu,Risk measure pricing and hedging in incomplete markets, Annals of Finance 2, 51–71, 2006. · Zbl 1233.91291 · doi:10.1007/s10436-005-0023-x
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.