A Multi-Factor Binomial Interest Rate Model with State Time Dependent Volatilities

1. A Multi-Factor Binomial Interest Rate Model with State Time Dependent Volatilities By Thomas S. Y. Ho And Sang Bin Lee May 2005

2. Applications of Multi-factor Interest Rate Models • Valuation of interest rate options, mortgage-backed, corporate/municipal bonds,… • Balance sheet items: deposit accounts, annuities, pensions,… • Corporate management: risk management, VaR, asset/liability management… • Regulations: marking to market, Sarbane-Oxley… • Financial modeling of a firm: corporate finance

3. Interest Rate Models /Challenges • Interest rate models: Cox, Ingersoll and Ross, Vasicek • Binomial models: Ho-Lee, Black, Derman and Toy • Extensions of normal model: Hull-White • Generalized continuous time models: Heath, Jarrow, Morton approach • Market models: Brace, Gatarek, and Musiela/Jamshidian • Discrete time models: Das-Sundaram, Grant-Vora • What is a practical model?

4. Requirements of Interest Rate Models • Arbitrage-free conditions satisfied • Can be calibrated to a broad range of securities, not just swaptions/caps/floors • Multi-factor to capture the changing shape of the yield curve • Consistent with historical observations: mean reversion, no unreasonably high interest rate, no negative interest rates

5. Outline of the Presentation • Motivations of the model • Model assumptions: mathematical construct • Key ideas of the theory: Extending from Ho-Lee model (1986, 2004) • Model theoretical and empirical results • Practical applications of the model • Conclusion: challenges to mathematical finance

6. Model Assumptions • Binomial model: Cox Ross Rubinstein • Arbitrage-free condition: • Consistent with the spot curve • Expected risk free return at each node for all bonds • Recombining condition • General solution: risk neutral probabilities and time/state dependent solutions

7. Continuous Time Specification • dr = f(r,t)dt + σ(r, t) dz • σ(r, t) = σ(t) r for r < R • = σ(t) R for r > R

8. Ho-Lee 1-factor Constant Volatilities Model • P(T) discount fn • Forward price • Convexity term • Uncertainty term

9. The Ho-Lee n-Factor Time Dependent Modelforward/spot volatilities

10. The Generalized Ho-Lee Model

11. Calibration Procedure • Forward looking approach: implied market expectations, no historical data used • Specify the two term structures of volatilities by a set of parameters: a,b,…,e • Non-linear estimate the parameters such that the sum of the mean squared % errors in estimating the benchmark securities is minimize

12. Option Term Swap tenor Cap volatility 1 yr 3 yr 5 yr 7 yr 10 yr 1 yr 37.2 29.3 25.4 23.7 22.2 42.5 2 yr 28.3 24.8 22.7 21.7 20.5 40.5 3 yr 25.0 22.9 21.3 20.5 19.4 34.6 4 yr 22.7 21.3 20.0 19.4 18.3 31.1 5 yr 21.5 20.2 18.9 18.3 17.2 28.7 7 yr 19.2 18.0 16.9 16.2 15.5 25.5 10 yr 16.8 15.5 14.6 14.1 13.6 22.6 Market Observed Volatility Surface(%): An Example

13. Generalized Ho-Lee Ho-Lee (2004) One factor 2.80 2.58 Two factor 1.54 1.75 Estimated Average Errors in %70 swaptions observations/date;11/03-5/04 monthly data

14. Principal Yield Curve Movements98% parallel shift, 2% steepening

15. Davidson and MacKinnon C TestComparison of Alternative Models

16. 2-Factor Model vs 1-Factor Model • H0 : the one factor model is better than the two factor modelH1 : the two factor model is better than the one factor model •                    t-testcoefficient     std error  t-value     p-value       2.22         0.17        13.21         0.00 • Two factor model is accepted

17. 1 factor model vs 2 factor model

18. Lognormal vs Normal Model • H0: The threshhold rate is 9% • H1: The threshold rate is 3% • t-test: on 5/31/2004 • Coefficient std error t-value p-value • 2.648 0.661 4.007 0.004 • The results are mixed. Depends on the date

19. Normal vs Lognormal models

20. 1Factor Model Latticeintuitive results

21. In Contrast: Lognormal Model with Term Structure of Volatilities

22. Combining Two Risk Sources: Extended to Stock/Rate Recombined Lattice

23. Advantages of the Model: a Comparison • Arbitrage-free model: takes the market curve as given – relative valuation and use of key rate durations • Accepts volatility surface, contrasts with market model • Minimize model errors, contrasts with non-recombining interest rate models • Accurate calibration for a broad range of securities • A comparison with the continuous time model: specification of the instantaneous volatility

24. Applications of the Model • A consistent framework for pricing an interest rate contingent claims portfolio • Ho-Lee Journal of Fixed-Income 2004 • Portfolio strategies: static hedging… • Ho-Lee Financial Modeling Oxford University Press 2004 • Balance sheet management: • Ho Journal of Investment Management 2004 • Building structural models: credit risk • Ho-Lee Journal of Investment Management 2004 • Modeling a business: corporate finance • Ho-Lee working paper 2004 • Use of efficient sampling methods in the path space of the lattice: • Ho Journal of Derivatives LPS

25. Applications to Modeling a Firm • Financial statements • Fair value accounting, comprehensive income • Primitive Firm • Revenues determine the risk class • Correlations of revenues to the balance sheet risks • Firm is a contingent claim on the market prices and the primitive firm value

26. Applications of the Corporate Model • A relative valuation of the firm • A method to relative value equity and all debt claims • Risk transform from all business risks to the net income • Enterprise risk management • An integration of financial statements to financial modeling

27. Applications to Mathematical Finance • Lattice model offers a “co-ordinate system” for efficient sampling and new approaches to modeling • Information on each node is a fiber bundle • Lattice is a vector space, “Bond” is a vector • Arrow-Debreu securities defined at each node • Embedding a Euclidean metric in the manifold to measure risks • Can we approximate any derivatives by a set of benchmark securities? Replicate securities?

28. Conclusions • N-factor models are important to some of the applications of interest rate models in recent years • The model offers computational efficiency • The model provides better fit in the calibrating to the volatility surface when compared with some standard models • Avenues for future research