Term Structure Dynamics of Interest Rates by Exponential-Affine Models

1. Term Structure Dynamics of Interest Rates by Exponential-Affine Models Master in Calcolo Scientifico Dipartimento di Matematica Università degli Studi “La Sapienza” Roma 23 Maggio 2005 Marco Papi m.papi@iac.cnr.it Università di Varese Dipartimento di Economia Istituto per le Applicazioni del Calcolo IAC - CNR

2. Term Structure Dynamics of Interest RatesMarco Papi The evolution of the interest rates with maturity from three months up to thirty years in the time frame June 1999-December 2002. The thick line is a linear interpolation of the ECB offcial rate (represented by cross points).

3. Term Structure Dynamics of Interest RatesMarco Papi The evolution of the term structure in the time frame January 1999-December 2002

4. Term Structure Dynamics of Interest RatesMarco Papi

5. Term Structure Dynamics of Interest RatesMarco Papi • Understanding and modelling the term structure of interest rates represents one of the most challenging topics of financial research. • There are many benefits from a better understanding of the term structure: pricing and hedging interest rate dependent assets or managing the risk of interest rates contingent portfolios. • Bond prices and interest rates derivatives are dependent on interest rates, which exhibit a complex stochastic behavior and are not directly tradable, which means that the dynamic replication strategy is more complex. • Thus each of existing models has its own advantages and drawbacks.

6. Term Structure Dynamics of Interest RatesMarco Papi Main problems: • How do you build a model to explain the yield curve. • How do you build a model in order to price derivatives? • How do you build a model to help you to hedge your positions? • What is a good (parsimonious?) way to describe the (partly observed) existing yield curve?

7. Term Structure Dynamics of Interest RatesMarco Papi Definition: • Bonds: T-bond = zero coupon bond, paying 1€ at the date of maturity T. Main Objectives: 1. Investigate the term structure, i.e. how prices of bonds with different dates of maturity are related to each other. 2. Compute arbitrage free prices of interest rate derivatives (bond options, swaps, caps, floors etc.)

8. Term Structure Dynamics of Interest RatesMarco Papi Definition: • Yield to Maturity: the continuously compounded rate of return that causes the bond price to rise to one at time T: For a fixed time t, the shape of R(t,T)as T increases determines the term structure of interest rates. In our framework, the yield curve is the same as the term structure of interest rates, as we only work with ZCBs.

9. Term Structure Dynamics of Interest RatesMarco Papi • Finance traditionally views bonds as contingent claims and interest rates as underlying assets. Definitions: • Instantaneous risk-free interest rate (short term rate): the yield on the currently maturing bond,

10. Term Structure Dynamics of Interest RatesMarco Papi Definitions: • Forward rate: the rate that can be agreed upon at time t for a risk-free loan starting at time T1 and finishing at time T2,

11. Term Structure Dynamics of Interest RatesMarco Papi No-Arbitrage Restrictions: • A bond price will never exceed its terminal value plus the outstanding coupon payments. • A zero-coupon price cannot exceed the price of any zero-coupon with a shorter maturity. • The value of a zero-coupon bond must be equal to a value of a replicating portfolio composed of zero-coupon bonds. • Interest rates should not be negative.

12. Term Structure Dynamics of Interest RatesMarco Papi Theories of term structure : • The expectation hypothesis: the term structure is driven by the investor’s expectations on future spot rates. The rate of return on a T-bond should be equal to the average of the expected short-term rate from t to T, There exist four continuous-time interpretations of the expectation hypothesis.

13. Term Structure Dynamics of Interest RatesMarco Papi Continuous versus discrete models: • Should we model the term structure in discrete or a continuous framework? • The power of continuous-time stochastic calculus allows more elegant derivations and proofs, and provides an adequate framework to produce more precise theoretical solutions and refined empirical hypothesis, unfortunately at the cost of a considerably higher degree of mathematical sophistication.

14. Term Structure Dynamics of Interest RatesMarco Papi Bond prices, interest rate Vs yield curve models: • Early models attempted to model the bond price dynamics. Their results did not allow for a better understanding of the term structure. • Many interest rate models describe the evolution of a given interest rate (often the short term i.r.). This translates the valuation problem into a partial differential equation that can be solved analytically or numerically. • An alternative is to specify the stochastic dynamics of the entire term structure, either by using all yields or all forward rates. The model complexity increases significantly.

15. Term Structure Dynamics of Interest RatesMarco Papi Single Vs multi-factor models: • Factor models assume that the structrure of interest rates is driven by a set of variables or factors. Empirical studies used Principal Component Analysis to decompose the motion of the interest rate term structure into 3 i.i.d factors: • Shift: it is parallel movement of all rates. It usually accounts for up 80-90 percent of the total variance. • Twist: it describes a situation in which long rates and short term rates move in opposite directions. It usually accounts for an additional 5-10 percent of the total variance. • Butterfly:the intermediate rate moves in the opposite direction of the short and long term rate. 1-2 percent of influence.

16. Term Structure Dynamics of Interest RatesMarco Papi The three most significant components computed from monthly yield changes, Jan 1999-Dec 2002

17. Term Structure Dynamics of Interest RatesMarco Papi Simulation of the term structure evolution based on the PCA

18. Term Structure Dynamics of Interest RatesMarco Papi Arbitrage-free versus Equilibrium models: • Arbitrage-free models start with assumptions about the stochastic behavior of one or many interest rates and a specific market price of risk and derive the price of all contingent claims. • Equilibrium models start from a description of the economy, including the utility function of a representative investor and derive the term structure of interest rates and the risk premium endogenously, assuming that the market is at equilibrium.

19. Term Structure Dynamics of Interest RatesMarco Papi • All our models will be set up in a given probability space and a filtration Ft generated by a st. Brownian motion W(t). Single factor models: • All the information can be summarized by one single specific factor, the short term rate r(t). For a Zcb maturing at time T (T ≥ t), we have B(t,T) = B(t,T,r(t)). • Short term rate dynamics:

20. Term Structure Dynamics of Interest RatesMarco Papi • Let us denote by V(t) the value at time t of an interest rate claim with maturity T. • From the one factor model assumption, we can write V(t) = V(t,T,r(t)) By Ito’s lemma, Dividing both sides by V(t) yields the rate of return:

21. Term Structure Dynamics of Interest RatesMarco Papi Now let us consider two distinct interest rate contingent claims V1 and V2 with maturities T1 and T2and let us form aportfolio of value The prices satisfy the equations The variations of the portfolio value are given by

22. Term Structure Dynamics of Interest RatesMarco Papi We can select x1 and x2 to cancel out the instantaneous risk of the position, i.e to reduce the volatility of P(t) to zero. This gives the following system of equations: Actually, in order to avoid arbitrage opportunities, the return must be equal to r(t). The system has a non trivial solution iff This common value λ(t,r(t)) is called the market risk-premium .

23. Term Structure Dynamics of Interest RatesMarco Papi This allows us to express the instantaneous return on the bond as We obtain the second order pde (called the Feynman-Kac equation) that must satisfy any interest rate contingent claim in a no-arbitrage one factor model: with one boundary condition. The term µr- σrλr is called the risk-adjusted drift .

24. Term Structure Dynamics of Interest RatesMarco Papi • A ZcbB(t,T) satisfies F-K equation with B(T,T) = 1. • A plain vanilla call option on B(t,T) with maturity TC < T, satisfies F-K equation • Foraswap of fixed rate δagainst a floating rate r with maturity date T, we have • with the boundary condition V(0) = 0.

25. Term Structure Dynamics of Interest RatesMarco Papi Theorem 1 . The solution toF-K equation for B(t,T) under the terminal condition B(T,T) = 1is given by Theorem 2[Harrison Kreps (1979)].Under some regularity conditions, a market is arbitrage-free if there exists an equivalent market measure Q, such that the discounted price process of any security is a Q-martingale. Therefore, we can write

26. Term Structure Dynamics of Interest RatesMarco Papi From one world to another • We start with the original risk-free interest rate dynamics, then we define a new process dW*(t) = dW(t) – λ(t) dt • Under technical conditions, using Girsanov’s Theorem , there exists a probability measure Q s.t. W*(t) is a Q-Brownian motion, where dQ = ρ(T,λ) dP where for any t ≤ T

27. Term Structure Dynamics of Interest RatesMarco Papi Model specification: P or Q ? • The specification of the dynamics of the rate r(t) under P causes problems as the equivalent probability measure Q may not be unique. • As in the B-S framework, we have one source of randomness and one state process, but r(t) is not the price of a traded asset. • The market is clearly incomplete and Q in not necessarily unique. • There are consequences on the parameter estimation: the set θof observable parameters enters in a pde collectively with λ. We need to use market traded assets to find the combination (θ, λ) that fits prices.

28. Term Structure Dynamics of Interest RatesMarco Papi Affine Models • Their popularity is due both to their tractability and flexibility. • In some cases explicit solutions to the F-K can be found, and it is relatively easy to price other instruments with this models. • They have sufficient free parameters so that they can fit term structures quite well. • Affine models were first investigated as a category by Brown and Schaefer (1994). • Duffie and Kan (1994,1996) developed a general theory. • Dai and Singleton (1998) provided a classification.

29. Term Structure Dynamics of Interest RatesMarco Papi Affine Models • Given n state variables X(t), spot rates take the following form • Taking the limit as τ → 0, we obtain an expression for the short rate r(t): r(t) = f + <g, X(t)> .

30. Term Structure Dynamics of Interest RatesMarco Papi Affine Models • If the model is affine, then price of a T-bond can be written in the form of an exponential-affine function of X: • Once the processes for the vector of state variables have been specified (under Q, say) is sufficient to establish prices in the model. • Duffie and Kan (1996) show that X(t) must be of the form

31. Term Structure Dynamics of Interest RatesMarco Papi Affine Models • To find bond prices we need to solve for a(.) and b(.) the F-K equation. It is not diffult to see that • There are fairly easy numerical solution methods available for this type of differential equations.

32. Term Structure Dynamics of Interest RatesMarco Papi Types of Affine Models • Commonly used affine models can be conveniently separated into three main types: • Gaussian affine models: all state variables have constant volatilities[Vasicek (‘77), Hull-White (’90), Babbs (’93)]. • CIR affine models: all state variables have square-root volatilities[CIR (’85), Longstaff (’90), El-Karoui (’92)]. • A three-factor affine family.[Sorensen (’94), Chen (’96)]. • In addition, an affine model may be “extended”, that is some of its parameters may be allowed to be deterministic functions of time.

33. Term Striature Dynamics of Interest RatesMarco Papi One Factor Affine Models • The term structure of interest rates is an affine function of the short rater(t): • Proposition. If underQ, µr (σr)2 are affine in r(t), then the model is affine.

34. Term Striature Dynamics of Interest RatesMarco Papi One Factor Gaussian Model • In a Gaussian model, dr(t) = [µ1(t)+µ2(t)]dt+σ2(t)dW(t), r(t) is normally distributed, and where • is normally distributed under Q, with a mean m anda variance v, and

35. Term Striature Dynamics of Interest RatesMarco Papi Some Specific Examples • Vasicek (1977) • when r goes over θ, the expected variation of r becomes negative and r tends to come back to its average long term level θ at an adjustment speed k. Vasicek postulates a constant risk-premium λ. • The explicit solution is • Interest rates can become negative, which is incompatible with no-arbitrage.

36. Term Striature Dynamics of Interest RatesMarco Papi Some Specific Examples (Vasicek) • Under the original measure P, the bond price dynamics is given by • This implies that both prices are lognormally distributed. Note that the volatility increases with T.

37. Term Striature Dynamics of Interest RatesMarco Papi Some Specific Examples (Vasicek) • The term structure can be positively shaped when r(t) < R(t,∞)-0.5(σr/k)2, negatively shaped for r(t) > R(t,∞)-0.5(σr/k)2. • Given the set of information at time s ≤ t,R(t,T) is normally distributed.

38. Term Striature Dynamics of Interest RatesMarco Papi Some Specific Examples (Vasicek) • Option prices [Jamshidian (’89)]: The option pricing formula has similarities with the Black & Scholes formula, since bond prices are lognormally distributed in the model: with

39. Term Structure Dynamics of Interest RatesMarco Papi Simulation of the term structure evolution based on the Vasicek model

40. Term Striature Dynamics of Interest RatesMarco Papi Some Specific Examples • CIR (1985).Cox, Ingersoll and Ross devoloped an equilibrium model in which interest rates are determined by the supply and demand of individuals having a logarithmic utility function. • The risk premium at equilibrium is • The short term process is known as the square-root process and has a variance proportional to the short rate rather than constant. • If r(0) > 0, k≥ 0, θ≥ 0, and kθ≥ 0.5(σ)2 , the SDE admits a unique solution, that is strictly positive, for all t > 0.

41. Term Striature Dynamics of Interest RatesMarco Papi Some Specific Examples (CIR) • The unique solution is • Given the information at time s, then the short term rate r(t) is distributed as a non central chi-squared [Feller, 1951]: with 2q+2 degrees of freedom and non central parameter 2u. • The distribution can be written explicitly as Iq is the modified Bessel function of the first type and order q.

42. Term Striature Dynamics of Interest RatesMarco Papi Some Specific Examples (CIR) • Bond prices solve • The solution is with

43. Term Striature Dynamics of Interest RatesMarco Papi Some Specific Examples (CIR) • Under the original measure P, bond price dynamics is given by • Term structure. The rate R(t,T) depends linearly on r(t) and R(t,∞), where • The value of r(t) determines the level of the term structure at time t, but not its shape. • Cox, Ingersoll and Ross provide formulas for the price of European call and put options.

44. Term Structure Dynamics of Interest RatesMarco Papi Simulation of the term structure evolution based on the CIR model

45. Term Striature Dynamics of Interest RatesMarco Papi Time-varying processes: Hull and White • Hull and White (1993) introduced a class of models which is consistent with a whole class of existing models. • with an exogeneously specified risk premium • The time-varying coefficients can be used to calibrate exactly the current market prices. • The price to be paid for this exact calibration is that bond and bond options prices are no longer analytically obtainable. • Using all the degrees of freedom in a model to fit the volatility constitutes an over-parametrization of the model. • In practice, the model is implemented with k and σ constant and θas time-varying.

46. Term Striature Dynamics of Interest RatesMarco Papi Other Models • Black, Derman, and Toy’s (1987, 1990) • The model is very popular among practitioners for various reasons. Unfortunately, the model lack analytical properties, and its implications and implicit assumptions are unknown. • Dothan (1978), Brennan and Schwartz (1980) • but there is no known distribution for r(t), and contingent claim prices must be computed using numerical procedures. • In particular Brennan and Schwartz used the model to price convertible bonds.

47. Term Striature Dynamics of Interest RatesMarco Papi Multi-Factor Models • Richard (1978), Cox, Ingersoll, Ross (1985) considered multiple factors: the real short term rate q(t) and the expected instantaneous inflation rate π(t), following indipendent processes: • Applying Ito’s formula, we obtain the pde solved by the price of a T-bond: • It is possible to express r as a function of πand q. • Richard obtained a complicated, but analytical, solution for the Zcb price.

48. Term Striature Dynamics of Interest RatesMarco Papi Multi-Factor Models • Longstaff and Schwartz (1992) developed a two factor model: • In their framework, there is only one good available in the economy • The factors can be related to observable quantities

49. Term Striature Dynamics of Interest RatesMarco Papi Multi-Factor Models • Chen (1996) proposed a three-factor model of the term structure: • r depends on its stochastic mean and stochastic volatility. • Closed form solutions for T-bonds and some interest rate derivatives are obtained in very specific cases.

50. Term Striature Dynamics of Interest RatesMarco Papi Estimation • Suppose that we have chosen a specific model, e.g. H-W . How do we estimate the parameters? • Naive answer: Use standard methods from statistical theory. • The parameters are Q-parameters. • Our observations are not under Q, but under P. Standard statistical techniques can not be used. • We need to know the market price of risk (λ). Who determines λ? The market. • We must get price information from the market in order to estimate parameters.