The Annals of Applied Probability

The pricing of contingent claims and optimal positions in asymptotically complete markets

Michail Anthropelos, Scott Robertson, and Konstantinos Spiliopoulos

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Abstract

We study utility indifference prices and optimal purchasing quantities for a contingent claim, in an incomplete semimartingale market, in the presence of vanishing hedging errors and/or risk aversion. Assuming that the average indifference price converges to a well-defined limit, we prove that optimally taken positions become large in absolute value at a specific rate. We draw motivation from and make connections to large deviations theory, and in particular, the celebrated Gärtner–Ellis theorem. We analyze a series of well studied examples where this limiting behavior occurs, such as fixed markets with vanishing risk aversion, the basis risk model with high correlation, models of large markets with vanishing trading restrictions and the Black–Scholes–Merton model with either vanishing default probabilities or vanishing transaction costs. Lastly, we show that the large claim regime could naturally arise in partial equilibrium models.

Article information

Source
Ann. Appl. Probab., Volume 27, Number 3 (2017), 1778-1830.

Dates
Received: September 2015
Revised: September 2016
First available in Project Euclid: 19 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1500451244

Digital Object Identifier
doi:10.1214/16-AAP1246

Mathematical Reviews number (MathSciNet)
MR3678485

Zentralblatt MATH identifier
06775362

Subjects
Primary: 91G99: None of the above, but in this section 60F10: Large deviations 60H10: Stochastic ordinary differential equations [See also 34F05]

Keywords
Indifference pricing incomplete markets utility functions large position size

Citation

Anthropelos, Michail; Robertson, Scott; Spiliopoulos, Konstantinos. The pricing of contingent claims and optimal positions in asymptotically complete markets. Ann. Appl. Probab. 27 (2017), no. 3, 1778--1830. doi:10.1214/16-AAP1246. https://projecteuclid.org/euclid.aoap/1500451244


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References

  • [1] Anthropelos, M. and Žitković, G. (2010). On agent’s agreement and partial-equilibrium pricing in incomplete markets. Math. Finance 20 411–446.
  • [2] Atkeson, G. A., Eisfeldt, L. A. and Weill, P.-O. (2013). The market of OTC derivatives. National Bureau of Economics Reseach. To appear.
  • [3] Barles, G. and Perthame, B. (1988). Exit time problems in optimal control and vanishing viscosity method. SIAM J. Control Optim. 26 1133–1148.
  • [4] Barles, G. and Soner, H. M. (1998). Option pricing with transaction costs and a nonlinear Black–Scholes equation. Finance Stoch. 2 369–397.
  • [5] Barrieu, P. and El Karoui, N. (2005). Inf-convolution of risk measures and optimal risk transfer. Finance Stoch. 9 269–298.
  • [6] Bichuch, M. (2012). Asymptotic analysis for optimal investment in finite time with transaction costs. SIAM J. Financial Math. 3 433–458.
  • [7] BIS (2014). Amounts outstanding of over-the-counter derivatives by risk category and instrument, Bank for International Settlements (BIS). Available at http://www.bis.org/about/index.htm..
  • [8] Bouchard, B., Elie, R. and Moreau, L. (2012). A note on utility based pricing and asymptotic risk diversification. Math. Financ. Econ. 6 59–74.
  • [9] Carmona, R. (2009). Indifference Pricing.Theory and Applications. Princeton Univ. Press, Princeton, NJ.
  • [10] Constantinides, G. M. (1986). Capital market equilibrium with transaction costs. J. Polit. Econ. 842–862.
  • [11] Czichowsky, C. and Schachermayer, W. (2016). Duality theory for portfolio optimisation under transaction costs. Ann. Appl. Probab. 26 1888–1941.
  • [12] Davis, M. (1997). Option pricing in incomplete markets. In Mathematics of Derivative Securities 216-226.
  • [13] Davis, M. H. A., Panas, V. G. and Zariphopoulou, T. (1993). European option pricing with transaction costs. SIAM J. Control Optim. 31 470–493.
  • [14] Delbaen, F., Grandits, P., Rheinländer, T., Samperi, D., Schweizer, M. and Stricker, C. (2002). Exponential hedging and entropic penalties. Math. Finance 12 99–123.
  • [15] Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd ed. Applications of Mathematics (New York) 38. Springer, New York.
  • [16] Fleming, W. H. and Soner, H. M. (2006). Controlled Markov Processes and Viscosity Solutions, 2nd ed. Stochastic Modelling and Applied Probability 25. Springer, New York.
  • [17] Föllmer, H. and Schied, A. (2004). Stochastic Finance, extended ed. de Gruyter Studies in Mathematics 27. de Gruyter, Berlin.
  • [18] Frittelli, M. (2000). The minimal entropy martingale measure and the valuation problem in incomplete markets. Math. Finance 10 39–52.
  • [19] Grandits, P. and Rheinländer, T. (2002). On the minimal entropy martingale measure. Ann. Probab. 30 1003–1038.
  • [20] Guasoni, P. and Muhle-Karbe, J. (2013). Portfolio choice with transaction costs: A user’s guide. In Paris-Princeton Lectures on Mathematical Finance 2013. Lecture Notes in Math. 2081 169–201. Springer, Cham.
  • [21] Henderson, V. (2002). Valuation of claims on nontraded assets using utility maximization. Math. Finance 12 351–373.
  • [22] Hodges, S. D. and Neuberger, A. (1989). Optimal replication of contingent claims under transactions costs. Review of Futures Markets 8 222–239.
  • [23] Hynd, R. (2014). Option pricing in the large risk aversion, small transaction cost limit. Comm. Partial Differential Equations 39 1998–2027.
  • [24] İlhan, A., Jonsson, M. and Sircar, R. (2005). Optimal investment with derivative securities. Finance Stoch. 9 585–595.
  • [25] Ishikawa, T. and Robertson, S. (2015). Contingent claim pricing in markets with defaultable assets. Working Paper.
  • [26] Kabanov, Y. M. and Stricker, C. (2002). On the optimal portfolio for the exponential utility maximization: Remarks to the six-author paper “Exponential hedging and entropic penalties” [Math. Finance 12 (2002), no. 2, 99–123; MR1891730 (2003b:91046)] by F. Delbaen, P. Grandits, T. Rheinländer, D. Samperi, M. Schweizer and C. Stricker. Math. Finance 12 125–134.
  • [27] Kallsen, J. and Muhle-Karbe, J. (2010). On using shadow prices in portfolio optimization with transaction costs. Ann. Appl. Probab. 20 1341–1358.
  • [28] Kallsen, J. and Muhle-Karbe, J. (2015). Option pricing and hedging with small transaction costs. Math. Finance 25 702–723.
  • [29] Lee, Y. and Rheinländer, T. (2012). Optimal martingale measures for defaultable assets. Stochastic Process. Appl. 122 2870–2884.
  • [30] Mania, M. and Schweizer, M. (2005). Dynamic exponential utility indifference valuation. Ann. Appl. Probab. 15 2113–2143.
  • [31] Musiela, M. and Zariphopoulou, T. (2004). An example of indifference prices under exponential preferences. Finance Stoch. 8 229–239.
  • [32] Owen, M. P. and Žitković, G. (2009). Optimal investment with an unbounded random endowment and utility-based pricing. Math. Finance 19 129–159.
  • [33] Robertson, S. (2012). Pricing for large positions in contingent claims. Math. Finance. To appear.
  • [34] Robertson, S. and Spiliopoulos, K. Indifference pricing for contingent claims: Large deviations effects. Mathematical Finance. To appear.
  • [35] Shreve, S. E. and Soner, H. M. (1994). Optimal investment and consumption with transaction costs. Ann. Appl. Probab. 4 609–692.
  • [36] Siorpaes, P. (2015). Optimal investment and price dependence in a semi-static market. Finance Stoch. 19 161–187.
  • [37] Stoikov, S. and Zariphopoulou, T. (2005). Optimal investments in the presence of unhedgeable risks and under cara preferences. IMA. To appear.
  • [38] Tehranchi, M. (2004). Explicit solutions of some utility maximization problems in incomplete markets. Stochastic Process. Appl. 114 109–125.
  • [39] van der Vaart, (1998). Asymptotic statistics. In Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge Univ. Press, New York.