Financial Risk Management Term Structure Models Jan Annaert Ghent University Hull, Chapter 23

23-2 What is the problem? The standard model implies little for the interest rate process and its time path It is therefore difficult to handle American interest rate options, callable bonds, … This chapter deals with these problems in an internally consistent framework Two groups: –equilibrium models –no-arbitrage models

23-3 Model illustration Start with a process for the short term rate

23-4 stochastic model for r Expected value using model Discount at risk-free rate Estimate model Principle TSIR models

23-5 Parameter estimation Compute pricesCompare to market prices Parameter adjustment Estimation TSIR model

23-6 geometric Brownian motion binomial tree –build interest rate tree –build bond tree –build “derivative” tree Rendleman & Bartter

23-7 Mean Reversion Interest rate = stock price ??? Interest rates tend to a LT-equilibrium –high r: tendency to interest rate decreases –low r: tendency to interest rate increases volatility LT rate < volatility ST rate bond volatility is not proportional with duration

23-8 Mean reversion: Vasicek Interest rate model: Intuition:

23-9 Vasicek: interpretation b –LT-equilibrium a –speed with which disequilibria are “corrected”

23-10 formula: Analytical formula for European options on zero coupon bonds exist Vasicek (II)

23-11 Vasicek: Coupon bonds Idea: option on a coupon bond is the sum of options on zero coupon bonds Define:

23-12 Jamshidian Exercise call:

23-13 CIR-Process New: : the higher r, the higher its volatility Comparable formula available Cox Ingersoll & Ross Model

23-14 Two factor models Brennan & Schwartz –long rate and short rate Longstaff & Schwartz –short rate and volatility

23-15 No-Arbitrage models Problem in previous models is that often the prices of existing assets are not replicated, e.g. present term structure NA-models: start from the present term structure Here: only one factor models

23-16 Principle NA Models Assume a process for bond returns Derive the process for forward rates Derive the process for interest rates

23-17 Bond return process

23-18 Forward rate process

23-19 Instantaneous forward rate process

23-20 RN short term interest rate process

23-21 Heath Jarrow & Morton Specify volatility for the instantaneous forward rates at each moment The implied binomial tree may grow very large (exponential growth) Non-Markovian

23-22 Process: Markov-model Analytical expressions for bonds and European options are available Ho and Lee Model

23-23 Ho & Lee model (II) Disadvantages: all spot and forward rates share the same volatility no mean reversion

23-24 Hull & White model Extension of Vasicek’s model, but is able to replicate the initial TSIR Also the Ho & Lee model is a special case Process:

23-25 Hull & White model (II) Analytical formula available A wide(r) range of volatility structures are available Equivalent trinomial tree is available Problem:  (t) has to be determined simultaneously

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