Term Structure Models 1. Zero-coupon bond prices and yields 2. Vasicek model 3. Cox-Ingersoll-Ross model 4. Multifactor Cox-Ingersoll-Ross models 5. Affine models 6. Completely affine models 7. Bond risk premia 8. Inflation and nominal asset prices 1 Readings and References Back, chapter 17. Duffie, chapter 7. Vasicek, O., 1977, An equilibrium characterization of the term structure, Journal of Financial Economics 5, 177-188. Cox, J., J. Ingersoll, and S. Ross, 1985, A theory of the term structure of interest rates, Econometrica 53, 385-407. Longsta↵, F., and E. Schwartz, 1992, Interest rate volatility and the term structure: A two-factor general equilibrium model, Journal of Finance 47, 1259-1282. Duffie, D., and R. Kan, 1996, A yield-factor model of interest rates, Mathematical Finance 6, 379-406. Du↵ee, G., 2002, Term premia and interest rate forecasts in affine models, Journal of Finance 57, 405-443. Drechsler, I., A. Savov, and P. Schnabl, A Model of Monetary Policy and Risk Premia, Journal of Finance forthcoming. 2 Summary of the Continuous-Time Financial Market dS0,t dSk,t I Security prices satisfy = rt dt and =(µk,t δk,t) dt + σk,tdBt. S0,t Sk,t − I Given tight tr. strat. ⇡t and consumption ct, portfolio value Xt satisfies the WEE: dXt = rtXt dt + ⇡t(µt rt1) dt + ⇡tσt dBt ct dt. • − − No arbitrage if ⇡tσt =0then ⇡t(µt rt1) = 0 ✓t s.t. σt✓t = µt rt1 I ) − )9 − dXt = rtXt dt + ⇡tσt(✓t dt + dBt) ct dt. ) − t 1 t 2 d ✓ dBs ✓s ds ⇤ = = − 0 s0 −2 0 | | I Under emm ⇤ given by dP ZT where Zt e , P P t R R B = Bt + ✓s ds is Brownian motion. • t⇤ 0 t rsRds Let βt = e− 0 and sdf process Mt = βtZt.ThentheWEEcanalsobewritten: R WEE*: dβtXt + βtct dt = βt⇡tσt dB • t⇤ WEE-M: dMtXt + Mtct dt = Mt[⇡tσt ✓tXt] dBt • − T βu βT T Mu MT I So Xt = Et⇤ t cu du + XT = Et t cu du + XT if ⇡ is mtgale-gen. { βt βt } { Mt Mt } R R I If σ is nonsingular, every c.plan (c, XT ) can be generated by a mtgale-gen. tr.strat. 3 Zero-Coupon Bond Prices and Yields I Now let’s go through some classic models of the prices of default-free bonds. T T I Let Pt be the time t price of the zero maturing at time T and yt be its yield. I Suppose we’re in a complete market. We can represent zero prices in various ways: T T yT (T t) ru du MT P e− t − = E⇤ e− t = Et . (1) t ⌘ t { } { M } R t I For the purpose of modeling bond prices and yields, we can work entirely under the risk-neutral measure . P⇤ I In order to characterize expected bond returns, we need to work under the true measure . P 4 The Vasicek Model I In the Vasicek (1977) model, the instantaneous riskless rate rt is modeled as a constant-volatility mean-reverting Ornstein-Uhlenbeck process, drt = (¯r rt) dt + σ dB⇤ , (2) − where r¯ is the “long-run mean” of rt and > 0 is its “speed of mean-reversion.” I The Ornstein-Uhlenbeck process is the continuous-time analogue of the discrete- time AR(1) process. The solution of (2)foranyu t 0 is I ≥ ≥ u (u t) (u t) (u s) ru = e− − rt +(1 e− − )¯r + σ e− − dBs⇤, (3) − Zt so ru is normally distributed with conditional mean and variance (u t) (u t) E⇤ ru = e− − rt +(1 e− − )¯r and (4) t { } − u 2 2 2(u s) σ 2(u t) vart⇤ ru = σ e− − ds = (1 e− − ) . (5) { } Zt 2 − I Normality of ru means negative interest rates are possible in this model. 5 T T yT (T t) ru du It follows from P e− t − = E e− t that zero yields are “affine” in I t ⌘ t⇤{ } rt,andthusalsonormal: R T y = a(T t)+b(T t)rt, where (6) t − − σ2 σ2 r¯2 1 e ⌧ σ2 1 e 2⌧ a(⌧)=¯r +( − )( − − ) ( − − ) and (7) − 22 3 ⌧ − 43 ⌧ 1 1 e ⌧ b(⌧)= ( − − ) . (8) ⌧ I The derivation of this result from Back (2010) is as follows: T T T u (u t) (u s) ru du =(T t)¯r (¯r rt) e− − du + σ e− − dBs⇤ du t − − − t t t Z Z Z TZ u 1 (T t) (u s) =(T t)¯r (¯r rt)(1 e− − )+σ e− − dBs⇤ du . − − − − Zt Zt I Switching the order of integration in the last integral gives T u T T T (u s) (u s) σ (T s) σ e− − dBs⇤ du = σ e− − du dBs⇤ = (1 e− − ) dBs⇤ . Zt Zt Zt Zs Zt − 6 I The integrand of this stochastic integral is nonrandom, so the integral is normal with mean zero and variance 2 T 2 2 2 σ (T s) 2 σ 2σ (T t) σ 2(T t) (1 e− − ) ds =(T t) (1 e− − )+ (1 e− − ) . 2 2 3 3 Zt − − − − 2 − T Thus ru du is normal with I − t R T 1 (T t) Et⇤ ru du = (T t)¯r + (¯r rt)(1 e− − ) and (9) {− Zt } − − − − T 2 2 2 σ 2σ (T t) σ 2(T t) var⇤ r du =(T t) (1 e− − )+ (1 e− − ) , t u 2 3 3 {− Zt } − − − 2 − (10) I and it follows from the usual rule for expectations of exponentials of normals that T T ru du 1 (T t) log P = log E⇤ e− t = (T t)¯r + (¯r rt)(1 e− − ) t t { } − − − − R 2 2 2 1 σ 2σ (T t) σ 2(T t) + [(T t) (1 e− − )) + (1 e− − ))] . (11) 2 − 2 − 3 − 23 − T I Homework: Verify that Pt satisfies the fundamental PDE for derivative prices. 7 The Cox-Ingersoll-Ross Model I In the Cox-Ingersoll-Ross (1985) model, the instantaneous riskless rate rt is modeled as a mean-reverting square-root process, drt = (¯r rt) dt + σprt dB⇤ , (12) − where r>¯ 0 is the long-run mean of rt, > 0 is its speed of mean-reversion, and the solution to (12) is nonnegative for all t, so it is possible to take the square 2 root. If r¯ σ /2,thenrt is a.s. strictly positive for all t. ≥ I The exact solution to (12) is more complicated than the solution to the O-U process in Vasicek, so we derive zero prices by solving their fundamental PDE. I Since the square-root process (12) is Markov, zero prices are functions of rt: T T ru du T P = E⇤ e− t = p (rt,t) . (13) t t { } R I Assuming the price function pT is smooth enough for an application of Itˆo’s lemma, 1 @P T (¯r r)pT + σ2rpT + = rpT s.t. pT (r, T )=1. (14) − r 2 rr @t 8 I The solution to (14)hasyieldsaffine in rt: T ↵(T t) β(T t)rt T p (rt,t)=e− − − − and y = a(T t)+b(T t)rt . (15) t − − where a(⌧)=↵(⌧)/⌧ and b(⌧)=β(⌧)/⌧ and 2r¯ ( + γ)⌧ 2γ 2(e⌧ 1) ↵(⌧)= [ + log ] and β(⌧)= − , (16) − σ2 2 c(⌧) c(⌧) where c(⌧)=2γ +( + γ)(e⌧ 1) and γ = p2 +2σ2. − 2 To verify this solution, note that (15)impliespr = βp, prr = β p,and I − pt =(↵0 + rβ0)p, so the fundamental PDE holds if and only if 1 2 2 (¯r r)β + σ β r + ↵0 + β0r = r. (17) − − 2 This holds for all values of r if and only if 1 2 2 + σ β + β0 =1 and ↵0 = r¯β . (18) 2 The ODE for β above is said to be of the Riccati type. 9 I The terminal condition pT (r, T )=1is equivalent to ↵(0) = β(0) = 0. I Di↵erentiating β in (16) shows it is a solution to (18). I Given β, integrating ↵0 in (18) gives ↵ in (16). 10 The Multifactor CIR Model We can get a multifactor version of the CIR model by taking r to be the sum of square-root processes: rt = X1t + X2t where dXit = i(X¯i Xit) dt + σipXit dB⇤ . (19) − i I Independence of the Xi implies T T T T ru du X1 du X2 du p = E⇤ e− t = E⇤ e− t u E⇤ e− t u (20) t t { } t { } t { } ↵ (T R t) ↵ (T t) β (T t)RX β (T t)X R = e− 1 − − 2 − − 1 − 1t− 2 − 2t (21) where the ↵i and βi are defined as in the single-factor CIR model. T = T + T T = ↵i(⌧) + βi(⌧) I This implies yt y1t y2t where yit ⌧ ⌧ Xit. Other Multifactor Models Similarly, we could construct other multifactor interest rate processes as sums of independent processes, and get product pricing formulas if price formulas exist for each independent process. 11 Affine Term Structure Models In an n-factor affine term structure model, the short rate r is affine in n Markov state variables X whose drift and instantaneous covariance matrix are also affine in X: rt = δ0 + δ0Xt and dXt =(φ + KXt) dt + σ(Xt) dB⇤, where (22) B is a vector of n independent Brownian motions under , I ⇤ P⇤ δ0 is a constant, δ and φ are a constant n-vectors, K is a constant n n matrix, I ⇥ σ is an n n matrix-valued function s.t. each element of the covariance matrix I ⇥ σ(x)σ(x)0 is affine in x.