APL2 implementation of numerical asset pricing models

ABSTRACT

The practice of modern finance theory depends on an ability to generate accurate and timely forecasts of asset returns. In this field, considerable effort has been expended to base the generation of asset returns on a set of state variables driven by the dynamics of the environment. This requires the solution of a fundamental parabolic partial differential equation, often with variable coefficient, and with a wide range of specification of boundary and initial value conditions. A major drawback in financial management of large, real-time problems of this sort is that they require numerical intensive computing. Approximations or simplifications are used. The one state variable model of Black and Scholes [Bla73] leads to a closed form solution of the value of a call option, as explored in an APL solution by Bogart [Bog87]. The two state variable model of Brennan and Schwartz [Bre79, Sch84] determines the value of an intermediate maturity bond whose value depends upon the dynamic evolution of: a short-term rate, such as the 3 month Treasury Bill rate, and a long term rate, such as the 30 year Treasury bond rate. This solution does not have a closed form and must be solved numerically or approximately. This paper describes a formulation of the Brennan and Schwartz model; develops a finite difference representation; describes the strategy for an APL2 implementation; and illustrates the results with the run of an application.

REFERENCES

• Ayb79: Ayres, H. R. and J. Y. Barry, "The Equilibrium Yield Curve for Government Securities," Financial Analysts Journal May/June 1979, 31-39.
• Bla73: Black, F. and M. Scholes, "The Pricing of Options and Corporate Liabilities," Journal of Political Economy 81 (1973), 637-659.
• Bog87: Barry Bogart, OPERA: options price evaluation and risk analysis system, Proceedings of the international conference on APL: APL in transition, p.478-486, May 10-14, 1987, Dallas, Texas, United States
• Bre79: Brennan, M. J. and E. S. Schwartz, "A Continuous Time Approach to the Pricing of Bonds," Journal of Banking and Finance 3 (1979), 133-155.
• Bre78: "Finite Difference Methods and Jump Processes Arising in the Pricing of Contingent Claims: A Synthesis," Journal of Financial and Quantitative Analysis (September 1978), 461-474.
• Cox78: Cox, J. C., J. E. Ingersoll, and S. A. Ross, "A Theory of the Term Structure of Interest Rates," Research Paper No. 468, Graduate School of Business, Stanford University, 1978.
• Mck70: McKee, S. and A. R. Mitchell, "Alternative Direction Methods for Parabolic Equations in Two Space Dimensions with Mixed Derivatives,*' Computer Journal 13.1 (February 1970), 81-86.
• Sch84: Schaeffer, S. M. and E.S. Schwartz, "A Two-Factor Model of the Term Structure: An Approximiate Analytical Solution," Mime0 , London Business School, 1984.

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.4 Quadrature and Numerical Differentiation
          Subjects: Finite difference methods

Additional Classification:
  D. Software
  D.3 PROGRAMMING LANGUAGES
      D.3.2 Language Classifications

          Nouns: APL

  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.0 General
          Subjects: Numerical algorithms
      G.1.8 Partial Differential Equations
          Subjects: Parabolic equations

  J. Computer Applications
  J.1 ADMINISTRATIVE DATA PROCESSING
      Subjects: Financial (e.g., EFTS)

General Terms:
Algorithms, Languages