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

ByThomas S. Y. Ho

AndSang Bin Lee

May 2005

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

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?

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

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

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

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

Ho-Lee 1-factor Constant Volatilities Model

P(T) discount fn

Forward price Convexity

term Uncertainty

term

n 1 n 2 iTni T n 1 T

(1 δ )(1 δ ) (1 δ) δP(T n)P (T) 2P(n) (1 δ) (1 δ )

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

1 21, 1,n 2

i,j 1 21 11, 1,

1 21, 1,

1 1( )P (T) 2( ) 1 1

n nn k n k

k kT n k T n k

i j

T n n T n n

d dP T nP n d d

d d

The Generalized Ho-Lee Model

1 110

11 00

(1 ( ))( 1)( ) (1 ( 1))

kn in ni jk

k j

n kP nPP n n k

11 1

1

1 ( 1)( ) ( 1)1 ( 1)

nn n n ii i i n

i

TT TT

3/ 2exp( 2 ( ) min( , ) )n ni in R R t

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

Market Observed Volatility Surface(%): An Example

Option Term

Swap tenorCap

volatility1 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

Estimated Average Errors in %70 swaptions observations/date;11/03-5/04 monthly data

  Generalized Ho-Lee

Ho-Lee (2004)

One factor 2.80 2.58

Two factor 1.54 1.75

Principal Yield Curve Movements98% parallel shift, 2% steepening

2 4 6 8 10 12 14 16 18 20

-0.2

0

0.2

0.4 level

steepness

Two yield curve movements implied by historical level data (1998-2004)

Rat

e S

hift

Time-to-Maturity (years)

2 4 6 8 10 12 14 16 18 20-0.2

0

0.2level

steepness

Two yield curve movements implied by the Two Factor Ho-Lee Model

Rat

e S

hift

Time-to-Maturity (years)

Davidson and MacKinnon C TestComparison of Alternative Models

(1 )i i i iY

( )i i i i iY

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

1 factor model vs 2 factor model

A B C D E

06/30/03 3.51% 4.34% 1.969 7.636 4607/30/03 3.73% 4.42% 2.776 8.119 4908/29/03 2.28% 2.93% 3.048 13.19 7209/30/03 3.49% 4.28% 3.215 11.02 6410/30/03 2.60% 3.34% 2.973 12.95 7111/28/03 1.50% 2.20% 1.981 13.08 7112/31/03 1.83% 2.65% 2.623 19.92 8501/31/04 1.72% 2.49% 2.42 16.52 8002/27/04 1.65% 2.33% 2.626 16.96 8103/31/04 2.22% 3.11% 2.944 21.6 8705/31/04 1.03% 1.44% 1.989 11.34 6506/30/04 2.09% 2.14% 1.877 2.062 6Average 2.30% 2.97% 2.537 12.866 65

* Note that the threshold rate is 3%.

R-square(%)

C test1&2 factor model of mean

1-factor model* Coefficientsdates

2-factor model* t-statistics

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

Normal vs Lognormal models

A B C D E

06/30/03 3.51% 3.79% 1.289 3.494 1507/30/03 3.73% 4.08% 2.558 4.888 2608/29/03 2.28% 2.67% 2.542 7.136 4209/30/03 3.49% 3.84% 2.129 4.74 2510/30/03 2.60% 2.97% 1.184 4.697 2411/28/03 1.50% 1.86% 1.067 6.132 3512/31/03 1.83% 2.25% 1.919 7.348 4401/31/04 1.72% 2.04% 1.733 6.15 3502/27/04 1.65% 1.89% 1.88 5.507 3103/31/04 2.22% 2.62% 2.048 6.172 3605/31/04 1.03% 1.36% 1.576 8.259 5006/30/04 2.09% 1.48% 0.0635 0.5661 0Average 2.30% 2.57% 1.666 5.424 30

R-square(%)

C test2 factor model mean

Dates3%

threshold rate

9% threshold

rateCoefficients t-statistics

1Factor Model Latticeintuitive results

0 20 40 60 80 100 120-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

10-year term

One Month Interest Rate on the Lattice of the Generalized Ho-Lee Model

In Contrast: Lognormal Model with Term Structure of Volatilities

0 20 40 60 80 100 1200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11000 Randomly Simulated Paths from Lattice of the one Factor BDT Model

10 year term

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

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

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

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

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

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?

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