Dynamic Term Structure Modelling BDT & other One-factor Models Investments 2005 Agenda Motivation and quick review of static models The need for dynamic models Classical dynamic models and various specifications Drawbacks of classical models New insight and modern models The BDT Model in some detail BDT solution BDT examples After the BDT.... Prepared by Peter Lchte Jrgensen 2 of 61 Motivation and Quick Historical Background A simplified look at fixed income models is as follows: Static Models Dynamic Models

Equally important but different purpose. Prepared by Peter Lchte Jrgensen 3 of 61 Static Models Static models are models for the present, time zero only! Concerned with fitting observed bond prices or equivalently deriving todays term structure of zero-coupon yields zero-coupon interest rates pure discount rates spot rates Prepared by Peter Lchte Jrgensen equivalent! 4 of 61 Static Models II Typically assume some functional form for the R(T)curve, i.e. choose a model like

Nelson-Siegel, extended Nelson-Siegel (Svensson) Polynomial (cubic) spline exponential splines CIR (more later) etc. Estimate model parameters that provide the best fit to market prices Prepared by Peter Lchte Jrgensen 5 of 61 Zero-coupon interest rate curve estimation The Nelson-Siegel model Extended Nelson-Siegel: T y t a be T cTe

1 e T T T T d e e , T 2 Asymptotic interest rate (t yt a > Short term fast 0) decay (t 0 y0 = a + t T f Long term, initially zero, slow decay ~ T2, t 0 Medium term, initially zero, fast decay ~ T, t 0 b) 5 parameters

Robust model Very flexible Prepared by Peter Lchte Jrgensen 6 of 61 Static Models III Use model to price other fixed income securities today, e.g. bonds outside estimation sample standard swaps FRAs other with known future payments The models used contain no dynamic element and are not used for modeling (scenarios of) future prices or curves. On the morning of next trading day you re-fit. Prepared by Peter Lchte Jrgensen 7 of 61 Starting to think about uncertainty So far a quick review of static models. But we have started to look at factor modelling and looked into the Vasicek model in some detail.

Why? Because interest rates are uncertain and evolving/changing through time This insight is first step in the progression Static Models Dynamic Models It was a recognition of the necessity to model uncertainty and simple passage of time if you want analyze uncertainty surrounding bond prices and interest rate derivatives. Prepared by Peter Lchte Jrgensen 8 of 61 Modeling Uncertainty Why is it necessary to model uncertainty? Because there are many securities whose future payments depend on the evolution of interest rates in the future! E.g. $ callable bonds $ bond options $ caps/floors $ mortgage backed securities ! $ corporate bonds, etc. $ pension liabilities particularly hot in DK right now! $ swaptions, CMS

Prepared by Peter Lchte Jrgensen 9 of 61 Modeling Uncertainty II These instruments cannot be priced by static yield-curve models, la Nelson-Siegel, alone. Some analysts add a spread to some yield, but this approach is inconsistent. Particularly common in insurance. In finance we do not value interest-sensitive securities by discounting their cash flows by a Treasury yield plus a spread. Rather we use lattices or simulations to discount interest-sensitive cash flows. Those are the only ways that work. So all of these methods that just add spreads to a yield are not going to give you precision... On Wall Street, sometimes we talk about spreads - but that is only after we have determined price. We say, "This translates into a spread," but we would never use the spread to come up with what the price should be. David Babbel Prepared by Peter Lchte Jrgensen 10 of 61 Modelling Uncertainty III We must construct dynamic models that can generate future yield curve scenarios and associate probabilities to the different scenarios. This insight dates back to research in the mid to late 1970es Merton (1973)

Vasicek (1977) Cox, Ingersoll & Ross (1978, 1985) and others These are the classical models.... Prepared by Peter Lchte Jrgensen 11 of 61 The ideas of the Classic Models Step 1: We model pure discount bond prices: P( x t , t , T ) t T State of the world, (vector of factors, time) Maturity date Model prices must have the property that P( xT , T , T ) 1 T Prepared by Peter Lchte Jrgensen 12 of 61

The ideas of the Classic Models II Step 2: Name the factors and choose stochastic process for their evolution through time time t inflation level time t Consumer Confidence Index x(t ) time t Productivity Index time t " Interest Rate" etc. Process used is It-process/diffusion: Wiener process d x(t ) ( x t , t )dt ( x t , t )dWt Drift Volatility Prepared by Peter Lchte Jrgensen 13 of 61 The ideas of the Classic Models

III Step 3: Mathematical argument (Its lemma) shows bond dynamics must be (super short notation) P P 1 2 P 2 dP dx dt ( dx ) x t 2 x 2 where P() is the price functional we are looking for. P is also an It-process. In itself this is pretty useless.... Prepared by Peter Lchte Jrgensen 14 of 61 The ideas of the Classic Models IV Step 4: Economic argument: We want no dynamic arbitrage in the model, internal consistency. P(x,t,T) should solve the pde: 2

P P 1 T P ( ) rP 0 T t x 2 xx where is market price of interest rate risk.... Solve this subject to the terminal (maturity) condition... Prepared by Peter Lchte Jrgensen 15 of 61 Alternative Representation Alternatively the Feynman-Kac (probabilistic representation) is T t r (u , xu ) du P( x, t , T ) Et e Q where Q is risk-neutral measure. Can these relations actually be solved for P()?

Depends on how we specified the factor process. Prepared by Peter Lchte Jrgensen 16 of 61 Solutions ?? There is a chance of finding explicit/analytic solutions if we limit number of factors choose tractable processes The obvious first choice of factor in a 1-factor model for the bond market is the interest rate, r.....but which? Traditionally the instantaneous int. rate although a good case can be made that it is a bad choice. Prepared by Peter Lchte Jrgensen 17 of 61 A Battle of Specifications Some of the more famous specifications Merton (1973) dr dt dW (t ) Vasicek (1977) dr ( r )dt dW (t ) Dothan (1978)

dr rW (t ) Cox, Ingersoll & Ross (1985) dr ( r )dt r dW (t ) Closed form solution can be found in these cases Prepared by Peter Lchte Jrgensen 18 of 61 A framework for empirical work Unrestricted model dr=(+r)dt+rdz =0 = Vasicek dr=(+r)dt+dz =0 Merton dr=dt+dz Cox, Ingersoll & Ross dr=(+r)dt+r1/2dz =0

=1 Brennan & Schwartz dr=(+r)dt+rdz =0 CEV dr=rdt+rdz =0 GBM X-model dr=rdt+rdz dr=rdz =0 =3/2 Dothan CIR 2 dr=rdz dr=r3/2dz

Prepared by Peter Lchte Jrgensen 19 of 61 Example: The Vasicek model Zero-coupon bond price P(t , T ) A(t , T )e B (t ,T ) r ( t ) 1 e (T t ) B(t , T ) ( B(t , T ) T t )( 2 2 / 2) 2 B (t , T ) 2 A(t , T ) exp 2 4 Term structure 1 1 R(t , T ) ln A(t , T ) B(t , T )r (t ) Tt Tt Formulas for bond options can be derived. (Why?)

Prepared by Peter Lchte Jrgensen 20 of 61 Example: The CIR model Zero-coupon bond price P(t , T ) A(t , T )e B ( t ,T ) r ( t ) 2(e (T t ) 1) B(t , T ) ( )2(e (T t ) 1) 2 ( )(T t ) / 2 2e A(t , T ) (T t ) 1) 2 ( )(e 2 / 2 2 2 2 Term structure 1 1 R(t , T )

ln A(t , T ) B(t , T )r (t ) Tt Tt Again, formulas for bond options can be derived. Prepared by Peter Lchte Jrgensen 21 of 61 Observations and Critique Note: You actually also get the time zero curve! That is: You have a static model as the special case t=0! At the same time nice and the major problem with these models. The t=0 versions of these models rarely fit observed bond prices well! This is no surprise since no bond price information is taken into account in the estimation. Estimation is typically based on time series of rt. Prepared by Peter Lchte Jrgensen 22 of 61 Some curve-examples Vasicek Discount Function 0.06

1.2 0.05 1 Discount factor Zero coupon Interest Rate Vasicek Term Structure 0.04 0.03 0.02 0.01 0 0.8 0.6 0.4 0.2 0 0 5 10 15

20 0 Time to Maturity 5 10 15 20 Time to Maturity Prepared by Peter Lchte Jrgensen 23 of 61 Vasicek estimation example From a time series based estimation you might get mean reverison rate, 0.25 mean reversion level, 0.06 volatility

0.02 market price of risk 0.00 initial interest rate, r0, 0.03 time consistent But the Nelson-Siegel estimation based on prices is a different curve Zero coupon interest rate Vasicek Term Structure Curve 0.06 0.04 0.02 0 0 10 20 30 Time Prepared by Peter Lchte Jrgensen

24 of 61 Alternative estimation procedure Estimation of a classic model such as the Vasicek can be based on prices best fit. You are likely to obtain a good fit! Recall Nobel Laureate Richard Feynmans opinion: Give me three parameters and I can fit an elephant. Give me five and I can make it wave its trunk! But....estimates are likely to vary a lot from day to day and estimates may make no economic sense e.g. negative or very high mean reversion level and volatility Prepared by Peter Lchte Jrgensen 25 of 61 Conclusion The classic one-factor models have a problem with the real world which they often do not fit very well. The models are internally consistent ...but not externally consistent Prepared by Peter Lchte Jrgensen 26 of 61 New Insight Around early to mid 1980es these weaknesses were realized. In particular it was realized that

if we want to model the dynamics of the yield curve it makes no sense to ignore the information contained in the current, observed curve The model for the present curve and the observed/fitted curve should coincide they should be externally consistent Pioneers were Ho & Lee (1986), Heath, Jarrow & Morton (1987,1988,1992) Black, Derman &Toy (1990) Prepared by Peter Lchte Jrgensen 27 of 61 The Ho & Lee model Unfortunately the Ho & Lee model was quickly labelled the first no-arbitrage free model of term structure movements. This has created a lot of confusion as if the classic models were not arbitrage-free... In fact the Ho & Lee model n 1 describes price evolutions Pi 1 (T 1) Pi n (T ) Pi n 1 (T 1) and is in fact not even free of arbitrage since interest rates can become negative in the model! Prepared by Peter Lchte Jrgensen

28 of 61 Ho & Lee Properties So there are many different opinions on what arbitrage free really means But is is safe to say that Ho & Lees model was the first that obeyed the external consistency criterion no static arbitrage. The Ho & Lee model was not really operational and very difficult to estimate. But the idea was out.... Prepared by Peter Lchte Jrgensen 29 of 61 The Black, Derman & Toy model The BDT quickly became a cult model, especially in Denmark Goldman Sachs working paper was difficult to get hold of A lot of details about the model were left out in the paper. Few people knew what the model was really about ScanRate/Rio implemented the model in the systems you had to know the model! Prepared by Peter Lchte Jrgensen 30 of 61

A Closer Look at The BDT Model BDT is a one-factor model using the short rate as the factor In its original form it is a discrete time model The uncertainty structure is binomial, i.e. (i,n) (state i, time n) Prepared by Peter Lchte Jrgensen 31 of 61 Some notation Let us denote T-period zero-coupon price in state i and at time n as n Pi (T ) Absence of dynamic arbitrage (internal consistency) implies Pi n (T 1) Pi n (1) qPi n11 (T ) (1 q ) Pi n 1 (T ) where q is the risk-neutral probability. In basic version of BDT this is assumed to equal !

Any future state-contingent claim can be priced if all short rates are known... Prepared by Peter Lchte Jrgensen 32 of 61 General Pricing The pricing relation Vi n 1 n 1 1 n 1 Pi (1) Vi 1 Vi 2 2 1 Vi n11 Vi n 1 1 ri n 2 n and if you have interim, state independent payments (coupons)

Vi n n 1 1 n 1 1 n 1 Pi (1)c Vi 1 Vi 2 2 1 n 1 Vi n11 Vi n 1 c n 1 ri 2 n Prepared by Peter Lchte Jrgensen 33 of 61 Pricing Bonds 3 year 10% Bullet Binominal tree Bond Prices 10

0 98.00 12.25 97.9 7 11 10.50 10 10 0 99.55 99.5 9 10 0 101.1 3 9 9.0 0

98 99.55 1 97.97 10 2 1.11 100.9 2 10 0 Bond prices are found by discounting one period at a time, backwards, beginning at maturity. Prepared by Peter Lchte Jrgensen 34 of 61 Implementation These algorithms are very easily programmed backwards recursion. All you need is the short interest rate in every node of the lattice 2 r11 0 0

r r01 r2 r12 r02 Prepared by Peter Lchte Jrgensen 35 of 61 But this is really the hard part... and initially we do not have these short rates. To begin with the lattice looks as follows ? ? r00 ? ? ? Before we can do anything the model must be solved Prepared by Peter Lchte Jrgensen 36 of 61

Solving the model Solving the BDT model is a complicated task since we must make sure that the lattice of short rates is consistent with an observed/estimated initial term structure curve (external) an observed/estimated initial volatility curve These the arbitrage pricing relation (internal) are required inputs hence the BDT model is automatically externally consistent! Prepared by Peter Lchte Jrgensen 37 of 61 Solving the BDT model The initial discount function or equivalently the term strucure curve is assumed known/observed. Note the relation (discrete compounding) 1 P (T ) P (T ) (1 R(T ))T

0 0 known/observed/estimated Example T 1 2 3 4 5 P(T) 0.909 0.812 0.712 0.624 0.543 R(T)

10% 11% 12% 12.5% 13% Prepared by Peter Lchte Jrgensen 38 of 61 What volatility curve? The volatility curve which must be provided as input concerns the 1-period-ahead volatilities of zero-coupon rates as a function of time to maturity, T 0.12 Zero coupon interest rate 0.1 Tomorrow two TS-curves are possible: 0.1 0.06 0.04 0.02 0

0 0.08 1 2 3 4 5 3 4 5 Time 0.06 0.04 0.02 0 0.12 0 1

2 3 4 5 Time Zero coupon interest rate Zero coupon interest rate 0.12 0.08 0.1 0.08 0.06 0.04 0.02 0 0 1 2 Time

Prepared by Peter Lchte Jrgensen 39 of 61 Volatilities For example R11 (7) This vol is denoted (8)! R01 (7) Volatility is defined and calculated as ~ (T ) ln R 1 (T 1) R 11 (T 1) ln 1 R 0 (T 1) 2 Estimates are relatively easily obtained.... Prepared by Peter Lchte Jrgensen

40 of 61 BDTs example T 1 2 3 4 5 (T) 20% 19% 18% 17% 16% (meaningless) Otherwise the decreasing pattern is typical we often estimate smaller volatilities for longer maturities.

One additional asumption: (n) (n) stdev(ln r ) is const. given n Now the model can be solved! Note: A lot of preparatory work here as opposed to the classic models. That is the price for external consistency. Prepared by Peter Lchte Jrgensen 41 of 61 Solving the BDT example 1st step determining r11 and r01 We have r11 ln 1 r (2) 0 19% 2 and 1 1

P (2) P (1) P11 (1) P01 (1) 2 2 P (1) 1 1 2 1 r11 1 r01 Two equations in two unknowns. Substitute and reduce: Prepared by Peter Lchte Jrgensen 42 of 61 Solving the BDT example II 1 1 1 1 2 1 20.19 (1.11) 2(1.10) 1 r0 e

1 r01 r01 9.79% r11 9.79% e 0.38 14.31% and we have completed the first step 14.31% 10% 9.79% ? ? ? Prepared by Peter Lchte Jrgensen 43 of 61 Solving the BDT example III 2nd step: Determiningr02 , r12 and r22 R11 (2) ln 1 R0 (2) (3) 18% 2 and 1 1

P (1) P11 (2) P01 (2) 2 2 P (1) 1 1 2 (1 R11 (2)) 2 (1 R01 (2)) 2 P (1) 1 1 2 (1 R01 (2)e 0.36 ) 2 (1 R01 (2)) 2 P(3) One equation in one unknown.... Prepared by Peter Lchte Jrgensen 44 of 61

Solving the BDT example IV We find R01 (2) 10.7553% P01 (2) 0.8152 R11 (2) R01 (2)e 0.36 15.4159% P11 (2) 0.7507 Bringing back the arbitrage relation.. 1 1 P11 (2) P11 (1) P22 (1) P12 (1) 2 2 1 1 P01 (2) P01 (1) P12 (1) P02 (1) 2 2 1 1 1 1 P11 (2) P11 (1) 2 2 2 1 r

2 1 r 2 1 1 1 1 1 P01 (2) P01 (1) 2 2 2 1 r1 2 1 r0 Prepared by Peter Lchte Jrgensen 45 of 61 Solving the BDT example V Two equations in three unknowns.... but recall the final assumption r12 r22 ln 2 ln 2 r r ( 2) 1 0 constant 2 2 e

2 (2) r22 r12 2 2 r1 r0 r12 r02 e 2 (2) r22 r12 e 2 (2) r02 e 4 (2) Prepared by Peter Lchte Jrgensen 46 of 61 Solving the BDT example VI The earlier equation system is now 1 1 1

1 P11 (2) P11 (1) 2 4 2 2 2 ( 1 r e ) 2 ( 1 r ) 0 0e 1 1 1 1 P01 (2) P01 (1) 2 2 2 2 ( 1

r e ) 2 ( 1 r 0 0 ) This is two equations in two unknowns. Solve numerically r02 9.76% ( 2) 17.21% r12 13.77% r22 19.42% Prepared by Peter Lchte Jrgensen 47 of 61 Two year lattice 19.42% 14.31% 10% 9.79%

13.77% 9.76% Complexity does not increase as we look further out! Alternative method of forward induction (Jamshidian 1991) Prepared by Peter Lchte Jrgensen 48 of 61 The BDT Model Mean Reversion TSOI: 3->5%, TSOV: 25->11% 40% 35% 30% 25% 20% 15% 10% 5% 0% 1 3 5 7

9 11 13 15 17 19 21 23 25 27 29 Prepared by Peter Lchte Jrgensen 49 of 61 The BDT Model Mean fleeing 3,5% 350% 3,0% 300% 2,5% 250% 2,0% 200% 1,5% 150%

1,0% 100% 0,5% 50% 0,0% 0% Upper limit Lower limit TSOI: 3->5%, TSOV: 10% flat 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 Prepared by Peter Lchte Jrgensen 50 of 61 The BDT Model Evolution of local volatility TSOI: 3->5% 30% 25% 20% 15%

10% 5% 0% 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 TSOV Local volatility Prepared by Peter Lchte Jrgensen 51 of 61 The BDT Model Log normality of short rates after 15 years TSOI: 3->5%; TSOV: 25->11%;0-30Y 25% 20% 15% 10% 5% 0%

0% 5% 10% 15% 20% Prepared by Peter Lchte Jrgensen 25% 30% 35% 52 of 61 Further developments of BDT ideas Depending on the particular application the BDT model can be too hard to implement in practice and a nicer/more flexible formulation might be warranted. A marriage of classical models and new ideas can be arranged! Ho & Lee: Hull & White: BDT: Black & Karasinski:

dr (t )dt dW (t ) dr ( (t ) ar )dt dW (t ) ' (t ) d ln r [ (t ) ln r ] (t )dW (t ) (t ) ln r ( (t ) a(t ) ln r )dt (t ) dW (t ) Prepared by Peter Lchte Jrgensen 53 of 61 Exercise In the first spread sheet lattice: Check for the first three years that the model is calibrated, ie. determine P(1), P(2), P(3) and find volatilities (2) and (3). The initial term structure is calibrated on Aug 15, 2004. Check the pricing of st.ln 5%2005 in BOTH LATTICES. Determine the first zero-coupon rates (e.g. five years out) of the two possible curves and show the curves in the same graph. The two lattices assign identical prices to fixed income securities at time 0 because the models are calibrated to the same initial curve, but what about interest rate derivatives? Prepared by Peter Lchte Jrgensen

54 of 61 BDT Model Applications SimCorp Financial Training A/S www.simcorp.com Pricing Bonds 3 year 10% Bullet Binominal tree Bond Prices 10 0 98.00 12.25 97.9 7 11 10.50 10 10 0

99.55 99.5 9 10 0 101.1 3 9 9.0 0 98 99.55 1 97.97 10 2 1.11 100.9 2 10 0 Bond prices are found by discounting one period at a time, backwards, beginning at maturity.

Prepared by Peter Lchte Jrgensen 56 of 61 Pricing Bond Options 2 year American call on 3 year 10% bullet, strike 99 Binominal tree Bond Prices 98.00 12.25 97.9 7 11 10.50 10 9.0 0 0.00 10 0 99.55

99.5 9 10 0 101.1 3 9 10 0 American Call 100.9 2 10 0 0.59 1.08 0.0 0 0.2 5 2.1 3*

1.1 3 0.55 1.9 2 * The option is exercised immediately Using the BDT model the price of the American call option can be found to be 1.08. Value of Callable Bond is: NonCall-CallOption = 99.59 1.08 = 98.51 Prepared by Peter Lchte Jrgensen 57 of 61 Pricing Mortgage Bonds Danish Mortgage-Backed Securities Bond Pool of underlying Loans Callable Bond Model Prepayment Risk of Call PDK =PNON PCALL Option i PiCALL Whomogenous: , W , 1.0 PDK =PNON are Debtors

not Several Call options Other Features: Cost of Prepaying Premium required Debtor Prepayment behaviour (first, optimal) Model Prepayment Model Tax DK Cash flows Path Dependency? Prepared by Peter Lchte Jrgensen 58 of 61 Pricing Danish Mortgage-Backed Securities Zero Yields Short Rates Debtor Model Volatilit y Short rate model, e.g. BDT

Price MBS Rentability Calculation s Price/Risk/ Return Prepared by Peter Lchte Jrgensen 59 of 61 Caps/Floors Product description Long term options based on a money market rate at future dates (often 3M or 6M LIBOR). Caps ensure a maximum funding rate compared to floors which ensure a minimum deposit rate. A purchased collar is a combination of a long cap and a short floor. Libor Libor Compensation from purchased cap Strike Strike

Libor Libor Compensation from purchased floor 3 6 9 12 15 1 2 24 . 8 1 Time (months) 3 Prepared by Peter Lchte Jrgensen 6 9 12 15 1 2 24 . 8 1 Time (months) 60 of 61 Pricing an Interest Rate Cap 3Y Cap on 1Y rate, strike 10% 3Y Cap (1Y) = 1Y Call IRG (1Y) + 2Y Call IRG (1Y) 1Y Call IRG Binomial

tree 2Y Call IRG 2.25 2.00 1.1225 12.25 11 10 0.90 1.11 11 10.50 10 0.41 9 1.11 0.50 0.45 1.1050 0.60 0.00 0.21

9 0.00 1/ 2 0.90 0 1.1 1/ 2 2.00 0.45 1.11 1.11 Value 3Y Cap = 0.41 + 0.60 = 1.01 Tree is in Bond yields, strike is Money Market Rate (here is no difference) Also beware of Day Counts Prepared by Peter Lchte Jrgensen 61 of 61