CHAPTER 4

One-Factor Short-Rate Models

4.1. Vasicek Model

Definition 4.1 (Short-rate dynamics in the Vasicek model). In the Vasicek model, the short rate is assumed to satisfy the stochastic differential equation

dr(t)=k(θ − r(t))dt + σdW (t), where k, θ, σ > 0andW is a Brownian motion under the risk-neutral measure.

Theorem 4.2 (Short rate in the Vasicek model). Let 0 ≤ s ≤ t ≤ T . The short rate in the Vasicek model is given by

   t r(t)=r(s)e−k(t−s) + θ 1 − e−k(t−s) + σ e−k(t−u)dW (u) s and is, conditionally on F(s), normally distributed with   E(r(t)|F(s)) = r(s)e−k(t−s) + θ 1 − e−k(t−s) and σ2   V(r(t)|F(s)) = 1 − e−2k(t−s) . 2k

Remark 4.3 (Short rate in the Vasicek model). The short rate r(t), for each time t, can be negative with positive probability. This is a major drawback of the Vasicek model. On the other hand, the short rate in the Vasicek model is mean reverting, i.e., rates revert to a long-time level, since

E(r(t)) → θ as t →∞.

Theorem 4.4 (Zero-coupon bond in the Vasicek model). In the Vasicek model, the price of a zero-coupon bond with maturity T at time t ∈ [0,T] is given by

P (t, T )=A(t, T )e−r(t)B(t,T ),

17 18 4. ONE-FACTOR SHORT-RATE MODELS where − e−k(T −t) B t, T 1 ( )= k and    σ2 σ2 A(t, T )=exp θ − (B(t, T ) − T + t) − B2(t, T ) . 2k2 4k

Theorem 4.5 (Bond-price dynamics in the Vasicek model). In the Vasicek model, the price of a zero-coupon bond with maturity T satisfies the stochastic dif- ferential equations

dP (t, T )=r(t)P (t, T )dt − σB(t, T )P (t, T )dW (t) and 1 σ2B2(t, T ) − r(t) σB(t, T ) d = dt + dW (t). P (t, T ) P (t, T ) P (t, T )

Theorem 4.6 (T -forward measure dynamics of the short rate in the Vasicek model). Under the T -forward measure QT , the short rate r in the Vasicek model satisfies

dr(t)= kθ − σ2B(t, T ) − kr(t) dt + σdW T (t), where the QT -Brownian motion W T is defined by

dW T (t)=dW (t)+σB(t, T )dt.

Let 0 ≤ s ≤ t ≤ T .Thenr is given by

 t r(t)=r(s)e−k(t−s) + M T (s, t)+σ e−k(t−u)dW (u), s where   σ2   σ2   M T (s, t)= θ − 1 − e−k(t−s) + e−k(T −t) − e−k(T +t−2s) , k2 2k2 and is, conditionally on F(s), normally distributed with

ET (r(t)|F(s)) = r(s)e−k(t−s) + M T (s, t) and σ2   VT (r(t)|F(s)) = 1 − e−2k(t−s) . 2k 4.1. VASICEK MODEL 19

Theorem 4.7 (Forward-rate dynamics in the Vasicek model). In the Vasicek model, the instantaneous forward interest rate with maturity T is given by   σ2 f(t, T )= θk − B(t, T ) B(t, T )+r(t)e−k(T −t) 2 and satisfies the stochastic differential equation

df(t, T )=σe−k(T −t)dW T (t).

Theorem 4.8 (Forward-rate dynamics in the Vasicek model). In the Vasicek model, the simply-compounded forward interest rate for the period [T,S] satisfies the stochastic differential equation   1 dF (t; T,S)=σ F (t; T,S)+ (B(t, S) − B(t, T ))dW S(t). τ(T,S)

Theorem 4.9 (Option on a zero-coupon bond in the Vasicek model). In the Vasicek model, the price of a European call option with strike K and maturity T and written on a zero-coupon bond with maturity S at time t ∈ [0,T] is given by

ZBC(t, T, S, K)=P (t, S)Φ(h) − KP(t, T )Φ(h − σ˜), where 1 − e−2k(T −t) σ˜ = σ B(T,S) 2k and   1 P (t, S) σ˜ h = ln + . σ˜ P (t, T )K 2 The price of a corresponding put option is given by

ZBP(t, T, S, K)=KP(t, T )Φ(−h +˜σ) − P (t, S)Φ(−h).

Theorem 4.10 (Caps and floors in the Vasicek model). In the Vasicek model, the price of a cap with notional value N,caprateK, and the set of times T ,is given by β Cap(t, T ,N,K)=N [P (t, Ti−1)Φ(−hi +˜σi) − (1 + τiK)P (t, Ti)Φ(−hi)] , i=α+1 while the price of a floor with notional value N,floorrateK, and the set of times T , is given by β Flr(t, T ,N,K)=N [(1 + τiK)P (t, Ti)Φ(hi) − P (t, Ti−1)Φ(hi − σ˜i)] , i=α+1 20 4. ONE-FACTOR SHORT-RATE MODELS where 1 − e−2k(Ti−1−t) σ˜i = σ B(Ti−1,Ti) 2k and   1 (1 + τiK)P (t, Ti) σ˜i hi = ln + . σ˜i P (t, Ti−1) 2 4.2. Exponential Vasicek Model

Definition 4.11 (Exponential Vasicek model). In the exponential Vasicek model, the short rate is given by

r(t)=ey(t) with dy(t)=k(θ − y(t))dt + σdW (t), where k, θ, σ > 0andW is a Brownian motion under the risk-neutral measure.

Theorem 4.12 (Short rate in the exponential Vasicek model). The short rate in the exponential Vasicek model satisfies the stochastic differential equation   σ2 dr(t)= kθ + − k ln(r(t)) r(t)dt + σr(t)dW (t). 2 Let 0 ≤ s ≤ t ≤ T .Thenr is given by     t  r(t)=exp ln(r(s))e−k(t−s) + θ 1 − e−k(t−s) + σ e−k(t−u)dW (u) s and is, conditionally on F(s), lognormally distributed with     σ2   E(r(t)|F(s)) = exp ln(r(s))e−k(t−s) + θ 1 − e−k(t−s) + 1 − e−2k(t−s) 4k and   V(r(t)|F(s)) = exp 2ln(r(s))e−k(t−s) +2θ 1 − e−k(t−s) ×      σ2   σ2   × exp 1 − e−2k(t−s) exp 1 − e−2k(t−s) − 1 . 2k 2k Remark 4.13 (Short rate in the exponential Vasicek model). Since the short rate r in the exponential Vasicek model is lognormally distributed, it is always pos- itive. A disadvantage is that P (t, T ) cannot be calculated explicitly. An advantage of the exponential Vasicek model is that r is always mean reverting with   σ2 E(r(t)|F(s)) → exp θ + as t →∞ 4k and      σ2 σ2 V(r(t)|F(s)) → exp 2θ + exp − 1 as t →∞. 2k 2k 4.4. COX–INGERSOLL–ROSS MODEL 21

4.3. Dothan Model

Definition 4.14 (Dothan model). In the Dothan model, the short rate is as- sumed to satisfy the stochastic differential equation

dr(t)=ar(t)dt + σr(t)dW (t), where σ>0, a ∈ R,andW is a Brownian motion under the risk-neutral measure.

Theorem 4.15 (Short rate in the Dothan model). Let 0 ≤ s ≤ t ≤ T .The short rate in the Dothan model is given by    σ2 r(t)=r(s)exp a − (t − s)+σ(W (t) − W (s)) 2 and is, conditionally on F(s), lognormally distributed with

E(r(t)|F(s)) = r(s)ea(t−s) and   2 V(r(t)|F(s)) = r2(s)e2a(t−s) eσ (t−s) − 1 .

Remark 4.16 (Short rate in the Dothan model). Since the short rate r in the Dothan model is lognormally distributed, it is always positive. Another advantage is that P (t, T ) can be calculated explicitly, although the formula is not as nice and involves the hyperbolic sine, the gamma function, and the modified Bessel function of the second kind. A disadvantage of the Dothan model is that r is mean reverting only iff a<0, and then the mean reversion level is zero.

4.4. Cox–Ingersoll–Ross Model

Definition 4.17 (CIR model). In the Cox–Ingersoll–Ross model,brieflyCIR model, the short rate is assumed to satisfy the stochastic differential equation  dr(t)=k(θ − r(t))dt + σ r(t)dW (t), where k, θ, σ > 0with2kθ > σ2 and W is a Brownian motion under the risk-neutral measure. 22 4. ONE-FACTOR SHORT-RATE MODELS

Theorem 4.18 (Short rate in the CIR model). Let 0 ≤ s ≤ t ≤ T . The short rate in the CIR model satisfies   E(r(t)|F(s)) = r(s)e−k(t−s) + θ 1 − e−k(t−s) and     σ2r(s) σ2θ 2 V(r(t)|F(s)) = e−k(t−s) − e−2k(t−s) + 1 − e−k(t−s) . k 2k

Remark 4.19 (Short rate in the CIR model). The prcess followed by the short rate in the CIR model is also sometimes called a square-root process. The nice mean reversion property in the Vasicek model is preserved in the CIR model. The bad property of possible negativity in the Vasicek model is removed in the CIR model by assuming 2kθ > σ2 and hence ensuring that the origin is inaccessible to the process. On the other hand, the distribution of the short rate in the CIR model is neither normal nor lognormal but it possesses a noncentral chi-squared distribution.

Theorem 4.20 (Zero-coupon bond in the CIR model). In the CIR model, the price of a zero-coupon bond with maturity T at time t ∈ [0,T] is given by

P (t, T )=A(t, T )e−r(t)B(t,T ), where  2kθ/σ2 2he(h+k)(T −t)/2 A(t, T )= 2h +(h + k)(eh(T −t) − 1) and 2(eh(T −t) − 1) B(t, T )= 2h +(h + k)(eh(T −t) − 1) with  h = k2 +2σ2.

Theorem 4.21 (Bond-price dynamics in the CIR model). In the CIR model, the price of a zero-coupon bond with maturity T satisfies the stochastic differential equations  dP (t, T )=r(t)P (t, T )dt − σ r(t)B(t, T )P (t, T )dW (t) and  1 (σ2B2(t, T ) − 1)r(t) σ r(t)B(t, T ) d = dt + dW (t). P (t, T ) P (t, T ) P (t, T ) 4.5. AFFINE TERM-STRUCTURE MODELS 23

Theorem 4.22 (T -forward measure dynamics of the short rate in the CIR model). Under the T -forward measure QT , the short rate r in the CIR model sat- isfies  dr(t)= kθ − (k + σ2B(t, T ))r(t) dt + σ r(t)dW T (t), where the QT -Brownian motion W T is defined by  dW T (t)=dW (t)+σB(t, T ) r(t)dt.

Theorem 4.23 (Forward-rate dynamics in the CIR model). In the CIR model, the instantaneous forward interest rate with maturity T is given by

f(t, T )=kθB(t, T )+r(t)BT (t, T ) and satisfies the stochastic differential equation  T df(t, T )=σ r(t)BT (t, T )dW (t).

Theorem 4.24 (Forward-rate dynamics in the CIR model). Let 0 ≤ t ≤ T ≤ S. In the CIR model, the forward rate F (t; T,S) satisfies   1 dF (t; T,S)=σ F (t; T,S)+ × τ(T,S)    A(t, S) × (B(t, S) − B(t, T )) ln (τ(T,S)F (t; T,S)+1) dW S(t). A(t, T )

4.5. Affine Term-Structure Models

Definition 4.25 (Affine term structure). If bond prices are given by

P (t, T )=A(t, T )e−r(t)B(t,T ) for all 0 ≤ t ≤ T, where A and B are deterministic functions, then we say that the model possesses an affine term structure.

Theorem 4.26 (Zero-coupon price in affine term-structure models). In an affine term-structure model in which the short rate satisfies the stochastic differential equa- tion  dr(t)=(α(t) − β(t)r(t))dt + γ(t)+δ(t)r(t)dW (t), we have that P is given by

P (t, T )=A(t, T )e−r(t)B(t,T ), 24 4. ONE-FACTOR SHORT-RATE MODELS where A and B are solutions of the system of ordinary differential equations

δ(t) 2 Bt(t, T )=β(t)B(t, T )+ B (t, T ) − 1,B(T,T)=0,  2  γ(t) At(t, T )=A(t, T )B(t, T ) α(t) − B(t, T ) ,A(T,T)=1. 2

Remark 4.27 (Zero-coupon price in affine term-structure models). The equa- tion for B does not involve A and is an ordinary differential equation, more precisely, a Riccati differential equation, which should be solved first. Then, using this solu- tion B, the equation for A is another ordinary differential equation, more precisely, a linear differential equation, which can be solved using the formula      T γ(u) A(t, T )=exp − B(u, T ) α(u) − B(u, T ) du . t 2

Theorem 4.28 (Affine term-structure models). In an affine term-structure model in which the short rate satisfies the stochastic differential equation

dr(t)=μ(t, r(t))dt + σ(t, r(t))dW (t) with μ and σ as in Theorem 4.26, we have the following formulas:

dP (t, T )=r(t)P (t, T )dt − σ(t, r(t))B(t, T )P (t, T )dW (t), 1 (σ2(t, r(t))B2(t, T ) − r(t) σ(t, r(t))B(t, T ) d = dt + dW (t), P (t, T ) P (t, T ) P (t, T ) dW T (t)=dW (t)+σ(t, r(t))B(t, T )dt,

dr(t)= μ(t, r(t)) − σ2(t, r(t))B(t, T ) dt + σ(t, r(t))dW T (t),

AT (t, T ) f(t, T )=r(t)BT (t, T ) − , A(t, T ) T df(t, T )=σ(t, r(t))BT (t, T )dW (t),   1 dF (t; T,S)=σ(t, r(t)) F (t; T,S)+ (B(t, S) − B(t, T ))dW S(t). τ(T,S)