A Review of No Arbitrage Interest Rate Models

GERALD W. BUETOW, Jr., PhD, CFA

President and Founder, BFRC Services, LLC

FRANK J. FABOZZI, PhD, CFA, CPA

Professor of Finance, EDHEC Business School

JAMES SOCHACKI, PhD

Professor of Applied Mathematics, James Madison University

Abstract: Interest rates are commonly modeled using stochastic differential equations. One-factor models use a stochastic differential equation to represent the short rate and two-factor models use a stochastic differential equation for both the short rate and the long rate. The stochastic differential equations used to model interest rates must capture some of the market properties of interest rates such as mean reversion and/or a volatility that depends on the level of interest rates. There are two distinct approaches used to implement the stochastic differential equations into a term structure model: equilibrium and no arbitrage.

In modeling the behavior of interest rates, stochastic differential equations (SDEs) are commonly used. The SDEs used to model interest rates must capture some of the market properties of interest rates such as mean reversion and/or a volatility that depends on the level of interest rates. For a one-factor model, the SDE is used to model the behavior of the short-term rate, referred to simply as the “short rate.” The addition of another factor (i.e., a two-factor model) involves extending the SDE to represent the behavior of the short rate and a long-term rate (i.e., long rate). ...

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