Dynamic Term Structure Modelling BDT & other One-factor Models

1. Dynamic Term Structure Modelling BDT & other One-factor Models SimCorp Financial Training A/S www.simcorp.com

2. Agenda • 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 Løchte Jørgensen for SimCorp Financial Training

3. Motivation and Quick Historical Background A simplified look at fixed income models is as follows: Equally important but different purpose. Static Models Dynamic Models Prepared by Peter Løchte Jørgensen for SimCorp Financial Training

4. Static Models • Models for the present, time zero only! • Concerned with fitting observed bond prices or equivalently deriving today’s term structure of • zero-coupon yields • zero-coupon interest rates • pure discount rates • spot rates equivalent! Prepared by Peter Løchte Jørgensen for SimCorp Financial Training

5. Static Models II • Typically assume a functional form for the R(T)-curve, i.e. choose a model like • Nelson-Siegel • Polynomial (cubic) spline • exponential splines • CIR (more later) • etc. • Estimate model parameters that provide the best fit Prepared by Peter Løchte Jørgensen for SimCorp Financial Training

6. Static Models III • Use model to price other fixed income securities today, e.g. • bonds outside estimation sample • standard swaps • FRA’s • 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 Løchte Jørgensen for SimCorp Financial Training

7. Starting to think about uncertainty • Yesterday you considered static modelling techniques... • Earlier today you have studied interest rate volatility... This was a first step in the progression It was a recognition of the necessity to model uncertainty! Static Models Dynamic Models Prepared by Peter Løchte Jørgensen for SimCorp Financial Training

8. 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. Prepared by Peter Løchte Jørgensen for SimCorp Financial Training

9. Modeling Uncertainty II These instruments cannot be priced by static yield-curve models, à la Nelson-Siegel, alone. 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 1970’es • Merton (1973) • Vasicek (1977) • Cox, Ingersoll & Ross (1978, 1985) • and others These are the ”classical models”.... Prepared by Peter Løchte Jørgensen for SimCorp Financial Training

10. The ideas of the Classic Models Step 1: We model pure discount bond prices: State of the world, (vector of factors, time) Maturity date Model prices must have the property that Prepared by Peter Løchte Jørgensen for SimCorp Financial Training

11. The ideas of the Classic Models II Step 2: Name the factors and choose stochastic process for their evolution through time Process used is Itô-process/diffusion: Wiener process ”Drift” ”Volatility” Prepared by Peter Løchte Jørgensen for SimCorp Financial Training

12. The ideas of the Classic Models III Step 3: Mathematical argument (Itô’s lemma) shows bond dynamics must be (super short notation) where P() is the price functional we are looking for. P is also an Itô-process. In itself this is pretty useless.... Prepared by Peter Løchte Jørgensen for SimCorp Financial Training

13. 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: where  is market price of interest rate risk.... Solve this subject to the terminal (maturity) condition... Prepared by Peter Løchte Jørgensen for SimCorp Financial Training

14. Alternative Representation Alternatively the Feynman-Kac (probabilistic representation) is 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 Løchte Jørgensen for SimCorp Financial Training

15. 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. ratealthough a good case can be made that it is a bad choice. Prepared by Peter Løchte Jørgensen for SimCorp Financial Training

16. A Battle of Specifications • Some of the more ”famous” specifications • Merton (1973) • Vasicek (1977) • Dothan (1978) • Cox, Ingersoll & Ross (1985) Closed form solution can be found in these cases Prepared by Peter Løchte Jørgensen for SimCorp Financial Training

17. Unrestricted model dr=(+r)dt+rdz =0 =0 =½ =1 Vasicek dr=(+r)dt+dz Cox, Ingersoll & Ross dr=(+r)dt+r1/2dz Brennan & Schwartz dr=(+r)dt+rdz CEV dr=rdt+rdz =0 =0 =0 Merton dr=dt+dz GBM dr=rdt+rdz ”X-model” dr=rdz =0 =3/2 Dothan dr=rdz CIR 2 dr=r3/2dz Prepared by Peter Løchte Jørgensen for SimCorp Financial Training

18. Example: The Vasicek model Zero-coupon bond price Term structure Formulas for bond options can be derived. Prepared by Peter Løchte Jørgensen for SimCorp Financial Training

19. Example: The CIR model Zero-coupon bond price Term structure Again, formulas for bond options can be derived. Prepared by Peter Løchte Jørgensen for SimCorp Financial Training

20. 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 Løchte Jørgensen for SimCorp Financial Training

21. 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 Prepared by Peter Løchte Jørgensen for SimCorp Financial Training

22. 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 Feynman’s opinion: ”Give me three parameters and I can fit an elephant. Give me five and I can make it wave it’s 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 Løchte Jørgensen for SimCorp Financial Training

23. 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 notexternally consistent Prepared by Peter Løchte Jørgensen for SimCorp Financial Training

24. New Insight • Around early to mid 1980’es 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 Løchte Jørgensen for SimCorp Financial Training

25. 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 describes price evolutions and is in fact not even free of arbitrage since interest rates can become negative in the model! Prepared by Peter Løchte Jørgensen for SimCorp Financial Training

26. Ho & Lee Properties • So there are many different opinions on what ”arbitrage free” really means • But is is safe to say that Ho & Lee’s 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 Løchte Jørgensen for SimCorp Financial Training

27. 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 Løchte Jørgensen for SimCorp Financial Training

28. 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 Løchte Jørgensen for SimCorp Financial Training

29. Some notation Let us denote T-period zero-coupon price in state i and at time n as Absence of dynamic arbitrage (internal consistency) implies where q is the risk-neutral probability. In basic version of BDT this is assumed to equal ½! Any future state-contingent claim can be priced if all short rates are known... Prepared by Peter Løchte Jørgensen for SimCorp Financial Training

30. General Pricing • The pricing relation and if you have interim, state independent payments (coupons) Prepared by Peter Løchte Jørgensen for SimCorp Financial Training

31. Implementation • These algorithms are very easily programmed – backwards recursion. • All you need is the short interest rate in every node of the lattice Prepared by Peter Løchte Jørgensen for SimCorp Financial Training

32. But initially we do not have these short rates. To begin with the lattice looks as follows • Before we can do anything the model must be solved ? ? ? ? ? Prepared by Peter Løchte Jørgensen for SimCorp Financial Training

33. 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 • the arbitrage pricing relation (internal) These are required inputs – hence the BDT model is automatically externally consistent! Prepared by Peter Løchte Jørgensen for SimCorp Financial Training

34. Solving the BDT model • The initial discount function or equivalently the term strucure curve is assumed known. • Note the relation (discrete compounding) • Example known/observed/estimated Prepared by Peter Løchte Jørgensen for SimCorp Financial Training

35. 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 • Tomorrow two curves are possible: Prepared by Peter Løchte Jørgensen for SimCorp Financial Training

36. Volatilities • For example • Volatility is defined and calculated as • Estimates are relatively easily obtained (cf. yesterday).... Prepared by Peter Løchte Jørgensen for SimCorp Financial Training

37. BDT’s example (meaningless) Otherwise the decreasing pattern is typical – we often estimate smaller volatilities for longer maturities. One additional asumption: 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 Løchte Jørgensen for SimCorp Financial Training

38. Solving the BDT example 1st step – determining We have Two equations in two unknowns. Substitute and reduce: Prepared by Peter Løchte Jørgensen for SimCorp Financial Training

39. Solving the BDT example II ? and we have completed the first step 14.31% 10% ? 9.79% ? Prepared by Peter Løchte Jørgensen for SimCorp Financial Training

40. Solving the BDT example III 2nd step: Determining One equation in one unknown.... Prepared by Peter Løchte Jørgensen for SimCorp Financial Training

41. Solving the BDT example IV We find Bringing back the arbitrage relation.. Prepared by Peter Løchte Jørgensen for SimCorp Financial Training

42. Solving the BDT example V • Two equations in three unknowns.... but recall the final assumption Prepared by Peter Løchte Jørgensen for SimCorp Financial Training

43. Solving the BDT example VI The earlier equation system is now This is two equations in two unknowns. Solve numerically Prepared by Peter Løchte Jørgensen for SimCorp Financial Training

44. Two year lattice 19.42% 14.31% 10% 13.77% 9.79% 9.76% Complexity does not increase as we look further out! Alternative method of forward induction (Jamshidian 1991) Prepared by Peter Løchte Jørgensen for SimCorp Financial Training