Continuous equilibrium in affine and information-based capital asset pricing models. (English) Zbl 1298.91090
Summary: We consider a class of generalized capital asset pricing models in continuous time with a finite number of agents and tradable securities. The securities may not be sufficient to span all sources of uncertainty. If the agents have exponential utility functions and the individual endowments are spanned by the securities, an equilibrium exists and the agents’ optimal trading strategies are constant. Affine processes, and the theory of information-based asset pricing are used to model the endogenous asset price dynamics and the terminal payoff. The derived semi-explicit pricing formulae are applied to numerically analyze the impact of the agents’ risk aversion on the implied volatility of simultaneously-traded European-style options.
MSC:
| 91B25 | Asset pricing models (MSC2010) |
| 91G20 | Derivative securities (option pricing, hedging, etc.) |
Keywords:
continuous-time equilibrium; exponential utility; CAPM; affine processes; information-based asset pricing; implied volatilityReferences:
| [1] | Bain, A., Crisan, D.: Fundamentals of Stochastic Filtering. Berlin: Springer (2009) · Zbl 1176.62091 |
| [2] | Biagini, S., Fritelli, M.: A unified framework for utility maximization problems: an Orlicz space approach. Ann Probab 18(3), 929–966 (2008) · Zbl 1151.60019 · doi:10.1214/07-AAP469 |
| [3] | Brody, D.C., Hughston, L.P., Macrina, A.: Beyond Hazard Rates: a New Framework for Credit-Risk Modelling. In: Elliott, R., Fu, M., Jarrow, R., Yen, J.Y. (eds.) Advances in Mathematical Finance: Festschrift Volume in honour of Dilip Madan. Birkhäuser/Springer, Berlin (2007) |
| [4] | Brody, D.C., Hughston, L.P., Macrina, A.: Information-based asset pricing. Int J Theor Appl Financ 11, 107–142 (2008) · Zbl 1152.91487 |
| [5] | Carmona, R., Fehr, M., Hinz, J., Porchet, A.: Market design for emission trading schemes. SIAM Rev 52(3), 403–452 (2010) · Zbl 1198.91166 · doi:10.1137/080722813 |
| [6] | Cheridito, P., Horst, U., Kupper, M., Pirvu, T.: Equilibrium Pricing in Incomplete Markets Under Translation Invariant Preferences. arXiv e-prints (2011) · Zbl 1410.91439 |
| [7] | Delbaen, F., Grandits, P., Rheinländer, T., Samperi, D., Schweizer, M., Stricker, C.: Exponential hedging and entropic penalties. Math Financ 12(2), 99–123 (2002) · Zbl 1072.91019 · doi:10.1111/1467-9965.02001 |
| [8] | Delbaen, F., Schachermayer, W.: The Mathematics of Arbitrage. Berlin: Springer (2006) · Zbl 1106.91031 |
| [9] | Dudley, R.M.: Real Analysis and Probability. Belmont: Wadsworth& Brooks/Cole (1989) |
| [10] | Duffie, D., Filipovic, D., Schachermayer, W.: Affine processes and applications in finance. Ann Appl Probab 13(3), 984–1053 (2003) · Zbl 1048.60059 · doi:10.1214/aoap/1060202833 |
| [11] | Duffie, D., Huang, C.: Implementing Arrow-Debreu equilibria by continuous trading of few long-lived securities. Econometrica 53, 1337–1356 (1985) · Zbl 0576.90014 · doi:10.2307/1913211 |
| [12] | Duffie, D., Singleton, K.: Credit Risk: Pricing, Measurement, and Management. Princeton: Princeton Univeristy Press (2003) |
| [13] | Filipović, D.: Term Structure Models–A Graduate Course. Berlin: Springer (2009) · Zbl 1184.91002 |
| [14] | Friz, P., Keller-Ressel, M.: Moment Explosions in Stochastic Volatility Models. In: Cont, R. (ed.) Encycl Quantit Financ IV, pp. 1247–1253. Wiley, Chichester (2010) |
| [15] | Gârleanu, N., Pedersen, L.H., Poteshman, A.M.: Demand-based option pricing. Rev Financ Stud 22(10), 4259–4299 (2009) · doi:10.1093/rfs/hhp005 |
| [16] | He, H., Leland, H.: On equilibrium asset price processes. Rev Financ Stud 6(3), 593–617 (1993) · doi:10.1093/rfs/6.3.593 |
| [17] | Heston, S.: A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev Financ Stud 6(2), 327–343 (1993) · Zbl 1384.35131 · doi:10.1093/rfs/6.2.327 |
| [18] | Horst, U., dos Reis, G., Pirvu, T.: On securitization, market completion and equilibrium risk transfer. Math Financ Econ 2(4), 211–252 (2010) · Zbl 1255.91401 · doi:10.1007/s11579-010-0022-1 |
| [19] | Hoyle, E., Hughston, L.P., Macrina, A.: Lévy random bridges and the modelling of financial information. Stoch Process Appl 121(4), 856–884 (2011) · Zbl 1225.60079 |
| [20] | Jofre, A., Rockafellar, R.T., Wets, R.J.B.: General economic equilibrium with incomplete markets and money. Preprint (2011) |
| [21] | Karatzas, I., Lehoczky, J.P., Shreve, S.E.: Existence and uniqueness of multi-agent equilibrium in a stochastic, dynamic consumption/investment model. Math Oper Res 15, 80–128 (1990) · Zbl 0707.90018 · doi:10.1287/moor.15.1.80 |
| [22] | Keller-Ressel, M.: Affine Processes–Theory and Applications in Finance. Ph.D. thesis, Technical University of Vienna (2009) |
| [23] | Keller-Ressel, M., Schachermayer, W., Teichmann, J.: Affine processes are regular. Probab Theory Relat Fields 151, 591–611 (2011) · Zbl 1235.60093 · doi:10.1007/s00440-010-0309-4 |
| [24] | Liptser, R., Shiryaev, A.: Statistics of Random Processes. I General Theory, 2nd edn. Berlin: Springer (2001) · Zbl 1008.62072 |
| [25] | Mayerhofer, E., Keller-Ressel, M.: On Exponential Moments of Affine Processes. ArXiv e-prints (2011) · Zbl 1332.60115 |
| [26] | Protter, P.E.: Stochastic Integration and Differential Equations, 2nd edn. Berlin: Springer (2005) |
| [27] | Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, Fundamental Principles of Mathematical Sciences, vol. 293, 3rd edn. Berlin: Springer (1999) · Zbl 0917.60006 |
| [28] | Schachermayer, W., Teichmann, J.: How close are the option pricing formulas of Bachelier and Black–Merton–Scholes? Math Financ 18(1), 155–170 (2008) · Zbl 1138.91479 · doi:10.1111/j.1467-9965.2007.00326.x |
| [29] | Sircar, R., Sturm, S.: From Smile Asymptotics to Market Risk Measures. Math Financ (2011, to appear) · Zbl 1314.91215 |
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.
