Theory And Application Of Affine Term Structure Models

This article is an overview of interest rate term structure.Interest rate term structure reflects the relationship between different expire dates of term interest rate,it is an important fundamental research area in finance and economy,and interest rate term structure both play a vital part in pricing fixed bond income and interest risk management.Currently,the research and development of modern interest rate term structure are not irrelevant,actually,the pricing of interest rate derivatives is an engine of development of term structure research area.Therefore, the problem of pricing of interest rate derivatives is a key point of interest rate term structure model.The history of development of interest rate term structure models which generalized into four stages.The first stage:The creation of Black-Scholes(1973) model and Merton(1973) model.The second stage:Propose primary yield curve model.The third sage:Emerging of arbitrage model.The fourth stage:HJM model.Besides we introduce both advantage and disadvantage of every interest rate term structure model in each development stage,in the second chapter we provide basic knowledge of interest rate term structure.There are two main disadvantages of Black-Scholes model at primary time when the model applys into interest rate derivatives area.The one is when the expire date of the bond is approaching,the market price of bond will gradually revert to par value,which means the volatility of bond price will decrease when the expire date is approaching,this is contradict with the assumption that the volatility is constant in Black-Scholes model.The other one is swap rate is a weighted average which is basis of forward in Libor market,the weight is stochastic. However,Black-Scholes(1973) model only provides different term caplet value under the same presumed parameter of volatility,it can not give the relationship between the volatilities which belong to swaption and each caplet.These shortcomings propel the development of yield curve model.We give an concrete introduction of a general equilibrium models which developed after Black-Schole method in the third chapter.Some researchers such as Vasicek(1977) and Cox, Ingersoll and Ross(1985) discovered the vital essence of interest rate volatility, and used stochastic model to describe the evolution of interest rate.The character of the stochastic model assumed all yield curve only propelled by one variable that is short interest rate r(t),r(t) can be expressed by the stochastic process of mean reversion.Because of there is only one uncertain source,it is easy to compose risk free portfolio to price other derivatives.Vasicek(1977) firstly proposed an mean reversion term structure model,Cox,Ingersoll and Ross(1985) continuously promoted the term structure model theory into a general equilibrium economic circumstance.CIR model kept the characteristic of short interest rate which vary surround the mean.The model method which proposed by Vasicek and CIR was a main method untill HJM model raised in 1993.The Vasicek and CIR model method provided an equilibrium system,which made interest rate model method erect on the foundation of microeconomics theory.To some extent, the method simulated the current yield curve,and pointed out the change of yield curve in future,the difference between the market price and yield curve can be pointed out the potential opportunities of transaction.Generally,Vasicek and CIR models ascribe to equilibrium model,because these models definitely knocked the risk market price,in addition given an equilibrium economy status.Under the diverse stochastic models,Duffle and Kan(1996) pioneering proposed affine interest rate term structure model,not only it had a good characteristic of qualify tractability but also they precisely described dynamic change character of short interest rate curve.In the fourth chapter of this article we introduce affine term structure model.After Dai and Singleton(2000) standardized and generalized the affine interest rate term structure model,the model was becoming a main model in research of term structure dynamic modeling.So far,it have not yet appeared a model which has absolute advantage on dynamic change of interest rate term structure model,each model has their own advantage and disadvantage.After many abroad and national researchers' empirical analysis, the result shows affine tree factors interest rate model is an ideal model whatever it under theoretical or practical aspect.The development of interest rate term structure model and interest rate derivatives pricing theory are complementary,the former provides an enrich basic theory for later.In the fifth chapter of this article,we state a pricing method of swaption.Because of there are some problem in application of Black equation. First of all,the lognormal distribution assumption of bond price and interest rate is probably not right.The bond price is impossible subjects to geometrical Brown motion in whole term of bond.When the bond closes to expire date,the price of bond will approach par value,so the price of bond is impossible very high either,generally it has upper limitation.When the price of bond approaches its bond ceiling or expire date,the volatility of bond tends to be zero.Thus,the volatility of bond price depends on bond price level and expire date,lognormal distribution can not simulate real distribution very well.The pricing method which stated in the fifth chapter of this article is a very fast and accurate algorithm for pricing method at the present time,which takes advantage of risk neutral pricing theory,and the relationship of probability measure inversion and origin moment of random variable,and cumulant,characteristic function,probability function to research pricing swaption and bond option and other interest rate derivatives in the affine term structure model.