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Finance and Stochastics

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01.10.2014

Optimal investment and contingent claim valuation in illiquid markets

verfasst von: Teemu Pennanen

Erschienen in: Finance and Stochastics | Ausgabe 4/2014

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Abstract

This paper extends basic results on arbitrage bounds and attainable claims to illiquid markets and general swap contracts where both claims and premiums may have multiple payout dates. Explicit consideration of swap contracts is essential in illiquid markets where the valuation of swaps cannot be reduced to the valuation of cumulative claims at maturity. We establish the existence of optimal trading strategies and the lower semicontinuity of the optimal value of optimal investment under conditions that extend the no-arbitrage condition in the classical linear market model. All results are derived with the “direct method” without resorting to duality arguments.

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An extended real-valued function is proper if it is not identically +∞ and never takes the value −∞.

Here and in what follows, δ _C denotes the indicator function of a set C in the sense of convex analysis: δ _C(x) equals 0 or +∞ depending on whether x∈C or not.

A set-valued mapping ω↦C(ω) is \(\mathcal{F}\)-measurable if \(\{\omega\, |\, C(\omega)\cap O\ne\emptyset\}\in\mathcal{F}\) for every open set O.

Theorem 2 of [46] gives lower semicontinuity with respect to a locally convex topology on a space of integrable functions, but the proof of [46, Theorem 2] establishes lower semicontinuity with respect to convergence in measure for f with an integrable lower bound.

Artzner, Ph., Delbaen, F., Eber, J.M., Heath, D.: Coherent measures of risk. Math. Finance 9, 203–228 (1999) CrossRefMATHMathSciNet

Barrieu, P., El Karoui, N.: Inf-convolution of risk measures and optimal risk transfer. Finance Stoch. 9, 269–298 (2005) CrossRefMATHMathSciNet

Biagini, S., Frittelli, M.: A unified framework for utility maximization problems: an Orlicz spaces approach. Ann. Appl. Probab. 18, 929–966 (2008) CrossRefMATHMathSciNet

Biagini, S., Frittelli, M., Grasselli, M.: Indifference price with general semimartingales. Math. Finance 21, 423–446 (2011) CrossRefMATHMathSciNet

Bielecki, T.R., Jeanblanc, M., Rutkowski, M.: Hedging of defaultable claims. In: Carmona, R.A., Çinlar, E., Ekeland, I., Jouini, E., Scheinkman, J.A., Touzi, N. (eds.) Paris–Princeton Lectures on Mathematical Finance 2003. Lecture Notes in Mathematics, vol. 1847, pp. 203–222. Springer, Berlin/Heidelberg (2004) CrossRef

Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81, 637–654 (1973) CrossRef

Bühlmann, H.: Mathematical Methods in Risk Theory. Grundlehren der Mathematischen Wissenschaften, vol. 172. Springer, New York (1970) MATH

Carmona, R. (ed.): Indifference Pricing: Theory and Applications. Princeton Series in Financial Engineering Princeton University Press, Princeton (2009)

Çetin, U., Rogers, L.C.G.: Modelling liquidity effects in discrete time. Math. Finance 17, 15–29 (2007) CrossRefMATHMathSciNet

10.

Cherny, A., Madan, D.B.: Markets as a counterparty: an introduction to conic finance. Int. J. Theor. Appl. Finance 13, 1149–1177 (2010) CrossRefMATHMathSciNet

11.

Courtault, J.-M., Delbaen, F., Kabanov, Y., Stricker, Ch.: On the law of one price. Finance Stoch. 8, 525–530 (2004) CrossRefMATHMathSciNet

12.

Cvitanić, J., Karatzas, I.: Convex duality in constrained portfolio optimization. Ann. Appl. Probab. 2, 767–818 (1992) CrossRefMATHMathSciNet

13.

Czichowsky, C., Schweizer, M.: Convex duality in mean-variance hedging under convex trading constraints. Adv. Appl. Probab. 44, 1084–1112 (2012) CrossRefMATHMathSciNet

14.

Davis, M.H.A.: Option pricing in incomplete markets. In: Dempster, M.A.H., Pliska, S.R. (eds.) Mathematics of Derivative Securities (Cambridge, 1995). Publ. Newton Inst., vol. 15, pp. 216–226. Cambridge University Press, Cambridge (1997)

15.

Davis, M.H.A.: Valuation, hedging and investment in incomplete financial markets. In: Hill, J.M., Moore, R. (eds.) Applied Mathematics Entering the 21st Century, pp. 49–70. SIAM, Philadelphia (2004)

16.

Davis, M.H.A., Panas, V.G., Zariphopoulou, T.: European option pricing with transaction costs. SIAM J. Control Optim. 31, 470–493 (1993) CrossRefMATHMathSciNet

17.

Delbaen, F., Grandits, P., Rheinländer, T., Samperi, D., Schweizer, M., Stricker, C.: Exponential hedging and entropic penalties. Math. Finance 12, 99–123 (2002) CrossRefMATHMathSciNet

18.

Delbaen, F., Schachermayer, W.: The Mathematics of Arbitrage. Springer, Berlin/Heidelberg (2006) MATH

19.

Dempster, M.A.H., Evstigneev, I.V., Taksar, M.I.: Asset pricing and hedging in financial markets with transaction costs: an approach based on the Von Neumann–Gale model. Ann. Finance 2, 327–355 (2006) CrossRefMATH

20.

Dermody, J.C., Rockafellar, R.T.: Cash stream valuation in the face of transaction costs and taxes. Math. Finance 1, 31–54 (1991) CrossRefMATH

21.

Dermody, J.C., Rockafellar, R.T.: Tax basis and nonlinearity in cash stream valuation. Math. Finance 5, 97–119 (1995) CrossRefMATH

22.

Dybvig, P.H., Marshall, W.J.: The new risk management: the good, the bad, and the ugly. Rev. Fed. Reserv. Bank. St. Louis 79, 9–22 (1997)

23.

Föllmer, H., Schied, A.: Stochastic Finance. An Introduction in Discrete Time, 3rd edn. de Gruyter, Berlin (2011) CrossRef

24.

Guasoni, P.: Optimal investment with transaction costs and without semimartingales. Ann. Appl. Probab. 12, 1227–1246 (2002) CrossRefMATHMathSciNet

25.

Guasoni, P.: Risk minimization under transaction costs. Finance Stoch. 6, 91–113 (2002) CrossRefMATHMathSciNet

26.

Guasoni, P., Rásonyi, M.: Hedging, arbitrage, and optimality with superlinear frictions. Ann. Appl. Probab. (2013, to appear). Available at: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2317344

27.

Hilli, P., Koivu, M., Pennanen, T.: Cash-flow based valuation of pension liabilities. Eur. Actuar. J. 1, 329–343 (2011) CrossRefMathSciNet

28.

Hodges, S.D., Neuberger, A.: Optimal replication of contingent claims under transaction costs. Rev. Futures Mark. 8, 222–239 (1989)

29.

Hugonnier, J., Kramkov, D., Schachermayer, W.: On utility-based pricing of contingent claims in incomplete markets. Math. Finance 15, 203–212 (2005) CrossRefMATHMathSciNet

30.

Ingersoll, J.E.: Theory of Financial Decision Making. Studies in Financial Economics. Rowman & Littlefield, Totowa (1987)

31.

Jaschke, S., Küchler, U.: Coherent risk measures and good-deal bounds. Finance Stoch. 5, 181–200 (2001) CrossRefMATHMathSciNet

32.

Jouini, E., Kallal, H.: Arbitrage in securities markets with short-sales constraints. Math. Finance 5, 197–232 (1995) CrossRefMATHMathSciNet

33.

Jouini, E., Kallal, H.: Martingales and arbitrage in securities markets with transaction costs. J. Econ. Theory 66, 178–197 (1995) CrossRefMATHMathSciNet

34.

Jouini, E., Napp, C.: Arbitrage and investment opportunities. Finance Stoch. 5, 305–325 (2001) CrossRefMATHMathSciNet

35.

Kabanov, Y.M., Safarian, M.: Markets with Transaction Costs. Springer, Berlin (2009) MATH

36.

Kramkov, D., Schachermayer, W.: The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Probab. 9, 904–950 (1999) CrossRefMATHMathSciNet

37.

Kramkov, D., Schachermayer, W.: Necessary and sufficient conditions in the problem of optimal investment in incomplete markets. Ann. Appl. Probab. 13, 1504–1516 (2003) CrossRefMATHMathSciNet

38.

LeRoy, S.F., Werner, J.: Principles of Financial Economics. Cambridge University Press, New York (2001)

39.

Madan, D.: Asset pricing theory for two price economies (2014). Preprint. Available at: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2414134

40.

Malo, P., Pennanen, T.: Reduced form modeling of limit order markets. Quant. Finance 12, 1025–1036 (2012) CrossRefMATHMathSciNet

41.

Napp, C.: The Dalang–Morton–Willinger theorem under cone constraints. J. Math. Econ. 39, 111–126 (2003) CrossRefMATHMathSciNet

42.

Owen, M.P., Žitković, G.: Optimal investment with an unbounded random endowment and utility-based pricing. Math. Finance 19, 129–159 (2009) CrossRefMATHMathSciNet

43.

Pennanen, T.: Arbitrage and deflators in illiquid markets. Finance Stoch. 15, 57–83 (2011) CrossRefMathSciNet

44.

Pennanen, T.: Dual representation of superhedging costs in illiquid markets. Math. Financ. Econ. 5, 233–248 (2011) CrossRefMATHMathSciNet

45.

Pennanen, T.: Superhedging in illiquid markets. Math. Finance 21, 519–540 (2011) CrossRefMATHMathSciNet

46.

Pennanen, T., Perkkiö, A.-P.: Stochastic programs without duality gaps. Math. Program. 136, 91–110 (2012) CrossRefMATHMathSciNet

47.

Rockafellar, R.T.: Convex Analysis. Princeton Mathematical Series, vol. 28. Princeton University Press, Princeton (1970) MATH

48.

Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 317. Springer, Berlin (1998) MATH

49.

Rokhlin, D.B.: An extended version of the Dalang–Morton–Willinger theorem under convex portfolio constraints. Theory Probab. Appl. 49, 429–443 (2005) CrossRefMATHMathSciNet

50.

Rouge, R., El Karoui, N.: Pricing via utility maximization and entropy. Math. Finance 10, 259–276 (2000) CrossRefMATHMathSciNet

51.

Schachermayer, W.: A Hilbert space proof of the fundamental theorem of asset pricing in finite discrete time. Insur. Math. Econ. 11, 249–257 (1992) CrossRefMATHMathSciNet

52.

Schachermayer, W.: The fundamental theorem of asset pricing under proportional transaction costs in finite discrete time. Math. Finance 14, 19–48 (2004) CrossRefMATHMathSciNet

Titel: Optimal investment and contingent claim valuation in illiquid markets
verfasst von: Teemu Pennanen
Publikationsdatum: 01.10.2014
Verlag: Springer Berlin Heidelberg
Erschienen in: Finance and Stochastics / Ausgabe 4/2014
Print ISSN: 0949-2984
Elektronische ISSN: 1432-1122
DOI: https://doi.org/10.1007/s00780-014-0240-0

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