The Annals of Applied Probability

Affine processes and applications in finance

D. Duffie, D. Filipović, and W. Schachermayer

Full-text: Open access


We provide the definition and a complete characterization of regular affine processes. This type of process unifies the concepts of continuous-state branching processes with immigration and Ornstein--Uhlenbeck type processes. We show, and provide foundations for, a wide range of financial applications for regular affine processes.

Article information

Ann. Appl. Probab., Volume 13, Number 3 (2003), 984-1053.

First available in Project Euclid: 6 August 2003

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 91B28

Affine process characteristic function continuous-state branching with immigration default risk infinitely decomposable interest rates option pricing Ornstein-Uhlenbeck type


Duffie, D.; Filipović, D.; Schachermayer, W. Affine processes and applications in finance. Ann. Appl. Probab. 13 (2003), no. 3, 984--1053. doi:10.1214/aoap/1060202833.

Export citation


  • [1] AMANN, H. (1990). Ordinary Differential Equations. An Introduction to Nonlinear Analy sis. de Gruy ter, Berlin.
  • [2] BACKUS, D., FORESI, S. and TELMER, C. (2001). Affine term structure models and the forward premium anomoly. J. Finance 56 279-304.
  • [3] BAKSHI, G., CAO, C. and CHEN, Z. (1997). Empirical performance of alternative option pricing models. J. Finance 52 2003-2049.
  • [4] BAKSHI, G. and MADAN, D. (2000). Spanning and derivative security valuation. J. Finan. Econom. 55 205-238.
  • [5] BALDUZZI, P., DAS, S. and FORESI, S. (1998). The central tendency: A second factor in bond yields. Rev. Econ. Statist. 80 62-72.
  • [6] BALDUZZI, P., DAS, S. R., FORESI, S. and SUNDARAM, R. (1996). A simple approach to three-factor affine term structure models. J. Fixed Income 6 43-53.
  • [7] BARNDORFF-NIELSEN, O. E. and SHEPHARD, N. (2001). Non-Gaussian Ornstein- Uhlenbeck-based models and some of their uses in financial economics. J. R. Stat. Soc. Ser. B Stat. Methodol. 63 167-241.
  • [8] BATES, D. (1996). Jumps and stochastic volatility: Exchange rate processes implicit in deutsche mark option. Rev. Financial Studies 9 69-107.
  • [9] BATES, D. (1997). Post-87' crash fears in S-and-P 500 futures options. J. Econometrics 94 181-238.
  • [10] BAUER, H. (1996). Probability Theory. de Gruy ter, Berlin.
  • [11] BERARDI, A. and ESPOSITO, M. (1999). A base model for multifactor specifications of the term structure. Economic Notes 28 145-170.
  • [12] BIRKHOFF, G. and ROTA, G. C. (1989). Ordinary Differential Equations, 4th ed. Wiley, New York.
  • [13] BJÖRK, T., KABANOV, Y. and RUNGGALDIER, W. (1997). Bond market structure in the presence of marked point processes. Math. Finance 7 211-239.
  • [14] BJÖRK, T. and LANDÉN, C. (2002). On the term structure of forward and futures prices. In Mathematical Finance (H. Geman, D. Madan, S. R. Pliska and T. Vorst, eds.) 111-150. Springer, New York.
  • [15] BLACK, F. and SCHOLES, M. (1973). The pricing of options and corporate liabilities. J. Political Economy 81 637-654.
  • [16] BRÉMAUD, P. (1981). Point Processes and Queues, Martingale Dy namics. Springer, New York.
  • [17] BRENNAN, M. and XIA, Y. (2000). Stochastic interest rates and bond-stock mix. European Finance Rev. 4 197-210.
  • [18] BROWN, R. and SCHAEFER, S. (1994). Interest rate volatility and the shape of the term structure. Philos. Trans. Roy. Soc. London Ser. A 347 449-598.
  • [19] BROWN, R. and SCHAEFER, S. (1994). The term structure of real interest rates and the Cox, Ingersoll, and Ross model. J. Finan. Econom. 35 3-42.
  • [20] CARVERHILL, A. (1988). The Ho and Lee term structure theory: A continuous time version. Technical report, Financial Options Research Centre, Univ. Warwick.
  • [21] CHACKO, G. (1999). Continuous-time estimation of exponential separable term structure models: A general approach. Technical report, School of Business, Harvard Univ.
  • [22] CHACKO, G. and VICEIRA, L. (1999). Dy namic consumption and portfolio choice with stochastic volatility. Technical report, School of Business, Harvard Univ.
  • [23] CHEN, L. (1996). Stochastic Mean and Stochastic Volatility: A Three-factor Model of the Term Structure of Interest Rates and Its Application to the Pricing of Interest Rate Derivatives, I. Blackwell, Oxford.
  • [24] CHEN, R. and SCOTT, L. (1995). Interest rate options in multifactor Cox-Ingersoll-Ross models of the term structure. J. Derivatives 3 53-72.
  • [25] CHEN, R.-R. and SCOTT, L. (1993). Multifactor Cox-Ingersoll-Ross models of the term structure: Estimates and tests from a state-space model using a Kalman filter. Technical report, Dept. Finance, Rutgers Univ.
  • [26] COLLIN-DUFRESNE, P. and GOLDSTEIN, R. (2001). Efficient pricing of swaptions in the affine framework. Technical report, Carnegie-Mellon Univ.
  • [27] COLLIN-DUFRESNE, P. and GOLDSTEIN, R. (2001). Generalizing the affine framework to HJM and random fields. Technical report, Carnegie-Mellon Univ.
  • [28] COLLIN-DUFRESNE, P. and GOLDSTEIN, R. (2002). Unspanned stochastic volatility: Empirical evidence and affine representations. J. Finance 57 1685-1730.
  • [29] COURANT, R. and HILBERT, D. (1993). Methoden der Mathematischen physik, 4th ed. Springer, Berlin.
  • [30] COX, J., INGERSOLL, J. and ROSS, S. (1985). A theory of the term structure of interest rates. Econometrica 53 385-408.
  • [31] DAI, Q. and SINGLETON, K. (2000). Specification analysis of affine term structure models. J. Finance 55 1943-1978.
  • [32] DELBAEN, F. and SCHACHERMAy ER, W. (1998). The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann. 312 215-250.
  • [33] DIEUDONNÉ, J. (1960). Foundations of Modern Analy sis. Academic Press, New York.
  • [34] DUAN, J.-C. and SIMONATO, J.-G. (1993). Estimating exponential-affine term structure models. Technical report, Dept. Finance, McGill Univ. and CIRANO.
  • [35] DUFFEE, G. (1999). Estimating the price of default risk. Rev. Financial Studies 12 197-226.
  • [36] DUFFEE, G. (2002). Term premia and interest rate forecasts in affine models. J. Finance 57 405-444.
  • [37] DUFFIE, D. (1998). First to default valuation. Technical report, Graduate School of Business, Stanford Univ.
  • [38] DUFFIE, D. and GÂRLEANU, N. (2001). Risk and valuation of collateralized debt valuation. Financial Analy sts Journal 57 41-62.
  • [39] DUFFIE, D. and KAN, R. (1996). A yield-factor model of interest rates. Math. Finance 6 379- 406.
  • [40] DUFFIE, D., PAN, J. and SINGLETON, K. (2000). Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68 1343-1376.
  • [41] DUFFIE, D. and SINGLETON, K. (1999). Modeling term structures of defaultable bonds. Rev. Financial Studies 12 687-720.
  • [42] ETHIER, S. N. and KURTZ, T. G. (1986). Markov Processes. Characterization and Convergence. Wiley, New York.
  • [43] FELLER, W. (1951). Two singular diffusion problems. Ann. Math. 54 173-182.
  • [44] FELLER, W. (1971). An Introduction to Probability Theory and Its Applications, 2nd ed. Wiley, New York.
  • [45] FILIPOVI ´C, D. (2001). Consistency Problems for Heath-Jarrow-Morton Interest Rate Models. Springer, Berlin.
  • [46] FILIPOVI ´C, D. (2001). A general characterization of one-factor affine term structure models. Finance Stoch. 5 389-412.
  • [47] FILIPOVI ´C, D. (2002). Time-inhomogeneous affine processes. Working Paper ORFE, Princeton Univ.
  • [48] FISHER, M. and GILLES, C. (1996). Estimating exponential affine models of the term structure. Technical report, Borad of Governors of the Federal Reserve, Washington, DC.
  • [49] FRACHOT, A. (1996). A reexamination of the uncovered interest rate parity hy pothesis. J. Int. Money and Finance 15 419-437.
  • [50] GIBBONS, M. and RAMASWAMY, K. (1993). A test of the Cox-Ingersoll-Ross model of the term structure of interest rates. Rev. Financial Studies 6 619-658.
  • [51] GOLDSTEIN, R. (2000). The term structure of interest rates as a random field. Rev. Financial Studies 13 365-384.
  • [52] HARRISON, M. and KREPS, D. (1979). Martingales and arbitrage in multiperiod securities markets. J. Econom. Theory 20 381-408.
  • [53] HARRISON, M. and PLISKA, S. (1981). Martingales and stochastic integrals in the theory of continuous trading. Stochastic Process. Appl. 11 215-260.
  • [54] HEIDARI, M. and WU, L. (2002). Term structure of interest rates, yield curve residuals, and the consistent pricing of interest rates and interest rate derivatives. Working paper, Graduate School of Business, Fordham Univ.
  • [55] HESTON, S. (1989). Discrete time versions of continuous time interest rate models. Technical report, Graduate School of Industrial Administration, Carnegie-Mellon Univ.
  • [56] HESTON, S. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financial Studies 6 327-344.
  • [57] JACOD, J. and SHIRy AEV, A. N. (1987). Limit Theorems for Stochastic Processes. Springer, Berlin.
  • [58] JAMSHIDIAN, F. (1989). An exact bond option formula. J. Finance 44 205-209.
  • [59] JAMSHIDIAN, F. (1991). Bond and option evaluation in the gaussian interest rate model. Research in Finance 9 131-170.
  • [60] JARROW, R., LANDO, D. and TURNBULL, S. (1997). A Markov model for the term structure of credit spreads. Rev. Financial Studies 10 481-523.
  • [61] JIANG, G. and KNIGHT, J. (1999). Efficient estimation of the continuous time stochastic volatility model via the empirical characteristic function. Technical report, Univ. Western Ontario.
  • [62] KAWAZU, K. and WATANABE, S. (1971). Branching processes with immigration and related limit theorems. Theory Probab. Appl. 16 36-54.
  • [63] LANDO, D. (1998). On Cox processes and credit risky securities. Review of Derivatives Research 2 99-120.
  • [64] LANGETIEG, T. (1980). A multivariate model of the term structure. J. Finance 35 71-97.
  • [65] LESNE, J.-F. (1995). Indirect inference estimation of yield curve factor models. Technical report, Univ. Cergy-Pontoise, THEMA.
  • [66] LIU, J. (2001). Portfolio selection in affine environments. Technical report, Anderson Graduate School of Business, Univ. California, Los Angeles.
  • [67] LIU, J., LONGSTAFF, F. and PAN, J. (1999). Dy namic asset allocation with event risk. J. Finance. To appear.
  • [68] LIU, J., PAN, J. and PEDERSEN, L. H. (1999). Density-based inference of affine jumpdiffusions. Technical report, Graduate School of Business, Stanford Univ.
  • [69] LONGSTAFF, F. and SCHWARZ, E. (1992). Interest rate volatility and the term structure: A two-factor general equilibrium model. J. Finance 47 1259-1282.
  • [70] LUKACS, E. (1970). Characteristic Functions, 2nd ed. Hafner, New York.
  • [71] MADAN, D. and UNAL, H. (1998). Pricing the risks of default. Review of Derivatives Research 2 121-160.
  • [72] MAGHSOODI, Y. (1996). Solution of the extended CIR term structure and bond option valuation. Math. Finance 6 89-109.
  • [73] PAN, J. (2002). The jump-risk premia implicit in options: Evidence from an integrated timeseries study. Journal of Financial Economics 63 3-50.
  • [74] PEARSON, N. and SUN, T.-S. (1994). An empirical examination of the Cox, Ingersoll, and Ross model of the term structure of interest rates using the method of maximum likelihood. J. Finance 54 929-959.
  • [75] PIAZZESI, M. (2003). Affine term structure models. In Hanbook of Financial Econometrics (Y. Ait-Sahalia and L. P. Hansen, eds.). North-Holland, Amsterdam.
  • [76] REVUZ, D. and YOR, M. (1994). Continuous Martingales and Brownian Motion. Springer, Berlin.
  • [77] ROGERS, L. C. G. and WILLIAMS, D. (1994). Diffusions, Markov Processes and Martingales 1, 2nd ed. Wiley, Chichester.
  • [78] RUDIN, W. (1991). Functional Analy sis, 2nd ed. McGraw-Hill.
  • [79] SATO, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press.
  • [80] SCHRODER, M. and SKIADAS, C. (1999). Optimal consumption and portfolio selection with stochastic differential utility. J. Economic Theory 89 68-126.
  • [81] SCHRODER, M. and SKIADAS, C. (2002). An isomorphism between asset pricing models with and without linear habit formation. Review of Financial Studies 15 1189-1221.
  • [82] SCHRODER, M. and SKIADAS, C. (2002). Optimal lifetime consumption-portfolio strategies under trading cone constraints and recursive preferences. Technical report, Eli Broad Graduate School of Management, Michigan State Univ.
  • [83] SCOTT, L. (1996). The valuation of interest rate derivatives in a multifactor Cox-Ingersoll- Ross model that matches the initial term structure. Technical report, Univ. Georgia.
  • [84] SCOTT, L. (1997). Pricing stock options in a jump-diffusion model with stochastic volatility and interest rates: Applications of Fourier inversion methods. Math. Finance 7 413-426.
  • [85] SHIGA, T. and WATANABE, S. (1973). Bessel diffusions as a one-parameter family of diffusion processes. Z. Wahrsch. Verw. Gebiete 27 37-46.
  • [86] SINGLETON, K. (2001). Estimation of affine asset pricing models using the empirical characteristic function. J. Econometrics 102 111-141.
  • [87] STEIN, E. and STEIN, J. (1991). Stock price distributions with stochastic volatility: An analytic approach. Review of Financial Studies 4 725-752.
  • [88] TAIRA, K. (1988). Diffusion Processes and Partial Differential Equations. Academic Press, London.
  • [89] VAN STEENKISTE, R. and FORESI, S. (1999). Arrow-Debreu prices for affine models. Technical report, Salomon Smith Barney, Inc., Goldman Sachs Asset Management.
  • [90] VASICEK, O. (1977). An equilibrium characterization of the term structure. Journal of Financial Economics 5 177-188.
  • [91] VASICEK, O. (1995). The finite factor model of bond prices. Technical report, KMV Corporation, San Francisco.
  • [92] VENTTSEL', A. D. (1959). On boundary conditions for multidimensional diffusion processes. Theory Probab. Appl. 4 164-177.
  • [93] WANG, N. (2001). Optimal consumption and the joint distribution of income and wealth. Technical report, Graduate School of Business, Stanford Univ.
  • [94] WATANABE, S. (1969). On two-dimensional Markov processes with branching property. Trans. Amer. Math. Soc. 136 447-466.