Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-12-02T23:57:11.008Z Has data issue: false hasContentIssue false
  • English
  • Français

Asset Pricing under the Quadratic Class

Published online by Cambridge University Press:  06 April 2009

Markus Leippold  and
Liuren Wu
Markus Leippold
Affiliation:
[email protected], Swiss Banking Institute, University of Zürich, Plattenstr. 14, 8032 Zürich, Switzerland
Liuren Wu
Affiliation:
[email protected], Graduate School of Business, Fordham University, 113 West 60th Street, New York, NY 10023.
Get access Rights & Permissions [Opens in a new window]

Abstract

We identify and characterize a class of term structure models where bond yields are quadratic functions of the state vector. We label this class the quadratic class and aim to lay a solid theoretical foundation for its future empirical application. We consider asset pricing in general and derivative pricing in particular under the quadratic class. We provide two general transform methods in pricing a wide variety of fixed income derivatives in closed or semi-closed form. We further illustrate how the quadratic model and the transform methods can be applied to more general settings.

Type
Research Article
Information
Journal of Financial and Quantitative Analysis , Volume 37 , Issue 2 , June 2002 , pp. 271 - 295
Copyright
Copyright © School of Business Administration, University of Washington 2002

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ahn, D.-H.; Dittmar, R. F.; and Gallant, A. R.. “Quadratic Term Structure Models: Theory and Evidence.” Review of Financial Studies, 15 (2001), 243–288.CrossRefGoogle Scholar
Ait-Sahalia, Y. “Testing Continuous-Time Models of the Spot Interest Rate.” Review of Financial Studies, 9 (1996), 385–426.CrossRefGoogle Scholar
Alan, S., and Ord, J. K.. Kendall's Advanced Theory of Statistics, 5th ed. New York, NY: Oxford Univ. Press, vol. 1 (1987).Google Scholar
Backus, D.; Foresi, S.; Mozumdar, A.; and Wu, L.. “Predictable Changes in Yields and Forward Rates.” Journal of Financial Economics, 59 (2001), 281–311.CrossRefGoogle Scholar
Backus, D.; Foresi, S.; and Telmer, C.. “Affine Term Structure Models and the Forward Premium Anomaly.” Journal of Finance, 56 (2001), 279–304.CrossRefGoogle Scholar
Backus, D.; Telmer, C.; and Wu, L.. “Design and Estimation of Affine Yield Models.” Unpubl. Manuscript, New York Univ. (1999).Google Scholar
Bakshi, G.; Cao, C.; and Chen, Z.. “Empirical Performance of Alternative Option Pricing Models.” Journal of Finance, 52 (1997), 2003–2049.CrossRefGoogle Scholar
Bakshi, G., and Madan, D.. “Spanning and Derivative-Security Valuation.” Journal of Financial Economics, 55 (2000), 205–238.CrossRefGoogle Scholar
Bates, D. “Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options.” Review of Financial Studies, 9 (1996), 69–107.CrossRefGoogle Scholar
Bates, D. “Post-'87 Crash Fears in the S” Journal of Econometrics, 94 (2000), 181–238.CrossRefGoogle Scholar
Beaglehole, D. R., and Tenney, M.. “General Solution of Some Interest Rate-Contingent Claim Pricing Equations.” Journal of Fixed Income, 1 (1991), 69–83.CrossRefGoogle Scholar
Beaglehole, D. R., and Tenney, M.. “A Nonlinear Equilibrium Model of Term Structures of Interest Rates: Corrections and Additions.” Journal of Financial Economics, 32 (1992), 345–454.CrossRefGoogle Scholar
Björk, T., and Christensen, B. J.. “Interest Rate Dynamics and Consistent Forward Rate Curves.” Mathematical Finance, 9 (1999), 323–348.CrossRefGoogle Scholar
Brandt, M., and Yaron, A.. “Time-Consistent No-Arbitrage Models of the Term Structure.” Unpubl. Manuscript, Univ. of Pennsylvania (2001).CrossRefGoogle Scholar
Carr, P., and Madan, D. B.. “Option Valuation Using the Fast Fourier Transform.” Journal of Computational Finance, 2 (1999), 61–73.CrossRefGoogle Scholar
Chacko, G. “Continuous-Time Estimation of Exponential Separable Term Structure Models: A General Approach.” Unpubl. Manuscript, Harvard Univ. (1999).Google Scholar
Chacko, G., and Viceira, L.. “Spectral GMM Estimation of Continuous-Time Processes.” Unpubl. Manuscript, Harvard Univ. (2000).CrossRefGoogle Scholar
Chapman, D. A., and Pearson, N. D.. “Is the Short Rate Drift Actually Nonlinear?” Journal of Finance, 55 (2000), 355–388.CrossRefGoogle Scholar
Conley, T. G.; Hansen, L. P.; Luttmer, E. G. J.; and J. A. Scheinkman. “Short-Term Interest Rates as Subordinated Diffusions.” Review of Financial Studies, 10 (1997), 525–577.CrossRefGoogle Scholar
Constantinides, G. M. “A Theory of the Nominal Term Structure of Interest Rates.” Review of Financial Studies, 5 (1992), 531–552.CrossRefGoogle Scholar
Dai, Q., and Singleton, K.. “Specification Analysis of Affine Term Structure Models.” Journal of Finance, 55 (2000), 1943–1978.CrossRefGoogle Scholar
Dai, Q., and Singleton, K.. “Expectation Puzzles, Time-Varying Risk Premia, and Affine Models of the Term Structure.” Journal of Financial Economics, (forthcoming 2002).CrossRefGoogle Scholar
Duffee, G. R. “Term Premia and Interest Rate Forecasts in Affine Models.” Journal of Finance, 57 (2002), 405–443.CrossRefGoogle Scholar
Duffee, G. R., and Stanton, R. H.. “Estimation of Dynamic Term Structure Models.” Unpubl. Manuscript, Univ. of California, Berkeley (2001).Google Scholar
Duffie, D.Dynamic Asset Pricing Theory, 2nd ed.Princeton, NJ: Princeton Univ. Press (1992).Google Scholar
Duffie, D., and Kan, R.. “A Yield-Factor Model of Interest Rates.” Mathematical Finance, 6 (1996), 379–406.CrossRefGoogle Scholar
Duffie, D.; Pan, J.; and Singleton, K.. “Transform Analysis and Asset Pricing for Affine Jump Diffusions.” Econometrica, 68 (2000), 1343–1376.CrossRefGoogle Scholar
El Karoui, N.; Myneni, R.; and Viswanathan, R.. “Arbitrage Pricing and Hedging of Interest Rate Claims with State Variables: I Theory.” Working Paper, Univ. of Paris (1992).Google Scholar
Filipović, D. “Exponential-Polynomial Families and the Term Structure of Interest Rates.” Bernoulli, 6 (2000), 1–27.CrossRefGoogle Scholar
Filipović, D. “Separable Term Structures and the Maximal Degree Problems.” Unpubl. Manuscript, ETH Zürich, Switzerland (2001).Google Scholar
Fisher, M., and Gilles, C.. “Modeling the State-Price Deflator and the Term Structure of Interest Rates.” Unpubl. Manuscript, Federal Reserve Bank of Atlanta/Bear and Stearns & Co. (1999).Google Scholar
Foresi, S., and Steenkiste, R. V.. “Arrow-Debreu Prices for Affine Models.” Unpubl. Manuscript, Goldman Sachs and Salomon Smith Barney (1999).CrossRefGoogle Scholar
Heston, S. “Closed-Form Solution for Options with Stochastic Volatility, with Application to Bond and Currency Options.” Review of Financial Studies, 6 (1993), 327–343.CrossRefGoogle Scholar
Holmquist, B. “Expectations of Products of Quadratic Forms in Normal Variates.” Stochastic Analysis and Applications, 14 (1996), 149–164.CrossRefGoogle Scholar
Jacod, J., and Shiryaev, A. N.. Limit Theorems for Stochastic Processes. Berlin: Springer-Verlag (1987).CrossRefGoogle Scholar
Jamshidian, F. “Bond, Futures and Option Valuation in the Quadratic Interest Rate Model.” Applied Mathematical Finance, 3 (1996), 93–115.CrossRefGoogle Scholar
Johnson, N. L., and Kotz, S.. Distributions in Statistics: Continuous Univariate Distributions 2. Boston, MA: Houghton Miffin Co. (1970), ch. 29.Google Scholar
Kathri, C. G. “Quadratic Forms in Normal Variables.” In Handbook of Statistics, Krishnaiah, P. R., ed. Amsterdam: North-Holland (1980), vol. 1, 443–469.Google Scholar
Leippold, M., and Wu, L.. “The Potential Approach to Bond and Currency Pricing.” Unpubl. Manuscript, Univ. of St. Gallen/Fordham Univ. (1999).CrossRefGoogle Scholar
Leippold, M., and Wu, L.. “Design and Estimation of Quadratic Term Structure Models.” Unpubl. Manuscript, Univ. of St. Gallen and Fordham Univ. (2000).Google Scholar
Longstaff, F. A. “A Nonlinear General Equilibrium Model of the Term Structure of Interest Rates.” Journal of Financial Economics, 23 (1989), 195–224.CrossRefGoogle Scholar
Mathai, A. M., and Provost, S. B.. Quadratic Forms in Random Variables. New York, NY: Marcel Dekker (1992).Google Scholar
Pan, E. “Collapse of Detail.” International Journal of Theoretical and Applied Finance, 1 (1998), 247–282.CrossRefGoogle Scholar
Pfann, G. A.; Schotman, P. C.; and R. Tschernig. “Nonlinear Interest Rate Dynamics and Implications for the Term Structure.” Journal of Econometrics, 74 (1996), 149–176.CrossRefGoogle Scholar
Rogers, L. C. G. “The Potential Approach to the Term Structure of Interest Rates and Foreign Exchange Rates.” Mathematical Finance, 7 (1997), 157–176.CrossRefGoogle Scholar
Singleton, K. “Estimation of Affine Asset Pricing Models Using the Empirical Characteristic Function.” Journal of Econometrics, 102 (2001), 111–141.CrossRefGoogle Scholar
Stanton, R. “A Nonparametric Model of Term Structure Dynamics and the Market Price of Interest Rate Risk.” Journal of Finance, 52 (1997), 1973–2002.CrossRefGoogle Scholar
Titchmarsh, E. C.Theory of Fourier Integrals, 2nd ed.London: Oxford Univ. Press (1975).Google Scholar