CHAPTER 5
Extended One-Factor Short-Rate Models
5.1. Ho–Le Model
Definition 5.1 (Ho–Le model). In the Ho–Le model, the short rate is assumed to satisfy the stochastic differential equation
dr(t)=θ(t)dt + σdW (t), where σ>0, θ is deterministic, and W is a Brownian motion under the risk-neutral measure.
Theorem 5.2 (Ho–Le model). In the Ho–Le model, we have the following for- mulas: t r(t)=r(s)+ θ(u)du + σ(W (t) − W (s)), s t E(r(t)|F(s)) = r(s)+ θ(u)du and V(r(t)|F(s)) = σ2(t − s), s P (t, T )=A(t, T )e−r(t)(T −t), σ2 T where A(t, T )=exp (T − t)3 − (T − u)θ(u)du , 6 t
dP (t, T )=r(t)P (t, T )dt − σ(T − t)P (t, T )dW (t), 1 σ2(T − t)2 − r(t) σ(T − t) d = dt + dW (t), P (t, T ) P (t, T ) P (t, T ) dW T (t)=dW (t)+σ(T − t)dt, dr(t)= θ(t) − σ2(T − t) dt + σdW T (t), σ2 T f(t, T )=r(t) − (T − t)2 + θ(u)du and df(t, T )=σdW T (t), 2 t 1 dF (t; T,S)=σ F (t; T,S)+ (S − T )dW S(t), τ(T,S) ZBC(t, T, S, K)=P (t, S)Φ(h) − KP(t, T )Φ(h − σ˜),
25 26 5. EXTENDED ONE-FACTOR SHORT-RATE MODELS
ZBP(t, T, S, K)=KP(t, T )Φ(−h +˜σ) − P (t, S)Φ(−h), √ 1 P (t, S) σ˜ where σ˜ = σ(S − T ) T − t and h = ln + , σ˜ P (t, T )K 2 β Cap(t, T ,N,K)=N [P (t, Ti−1)Φ(−hi +˜σi) − (1 + τiK)P (t, Ti)Φ(−hi)] , i=α+1 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 1 P (t, Ti) σ˜i where σ˜i = σ(Ti − Ti−1) Ti−1 − t and hi = ln + . σ˜i P (t, Ti−1)K 2
Theorem 5.3 (Calibration in the Ho–Le model). IftheHo–Lemodeliscali- brated to a given interest rate structure {f M(0,t):t ≥ 0}, i.e.,
f(0,t)=f M(0,t) for all t ≥ 0, then ∂fM ,t θ t (0 ) σ2t t ≥ . ( )= ∂t + for all 0
Theorem 5.4 (Zero-coupon bond price in the calibrated Ho–Le model). If the Ho–Le model is calibrated to a given interest rate structure {f M(0,t):t ≥ 0},then P M(0,T) σ2 P (t, T )=e−r(t)(T −t) exp (T − t)f M(0,t) − t(T − t)2 , P M(0,t) 2 where t P M(0,t)=exp − f M(0,u)du for all t ≥ 0. 0
5.2. Hull–White Model (Extended Vasicek Model)
Definition 5.5 (Short-rate dynamics in the Hull–White model). In the Hull– White model, the short rate is assumed to satisfy the stochastic differential equation
dr(t)=k(θ(t) − r(t))dt + σdW (t), where k, σ > 0, θ is deterministic, and W is a Brownian motion under the risk- neutral measure. 5.2. HULL–WHITE MODEL (EXTENDED VASICEK MODEL) 27
Remark 5.6 (Hull–White model). The Hull–White model is also called the extended Vasicek model or the G++ model and can be considered, more generally, with the constants k and σ replaced by deterministic functions.
Theorem 5.7 (Short rate in the Hull–White model). Let 0 ≤ s ≤ t ≤ T .The short rate in the Hull–White model is given by t t r(t)=r(s)e−k(t−s) + k θ(u)e−k(t−u)du + σ e−k(t−u)dW (u) s s and is, conditionally on F(s), normally distributed with t E(r(t)|F(s)) = r(s)e−k(t−s) + k θ(u)e−k(t−u)du s and σ2 V(r(t)|F(s)) = 1 − e−2k(t−s) . 2k
Remark 5.8 (Short rate in the Hull–White model). As in the Vasicek model, the short rate r(t) in the extended Vasicek model, for each time t, can be negative with positive probability, namely, with probability ⎛ ⎞ t r(0)e−kt + k θ(u)e−k(t−u)du Φ ⎝− 0 ⎠ , σ2 −2kt 2k (1 − e ) which is often “negligible in practice”. On the other hand, the short rate in the Vasicek model is mean reverting provided t ϕ∗ = lim k θ(u)e−k(t−u)du t→∞ 0 exists, and then E(r(t)) → ϕ∗ as t →∞.
Theorem 5.9 (Zero-coupon bond in the Hull–White model). In the Hull–White 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 T A¯(t, T )=A(t, T )exp −k θ(u)B(u, T )du t and A and B are as in the Vasicek model, Theorem 4.4 with θ =0. 28 5. EXTENDED ONE-FACTOR SHORT-RATE MODELS
Theorem 5.10 (Forward rate in the Hull–White model). In the Hull–White model, the instantaneous forward interest rate with maturity T is given by T σ2 f(t, T )=k θ(u)e−k(T −u)du − B2(t, T )+r(t)e−k(T −t). t 2
Theorem 5.11 (Calibration in the Hull–White model). IftheHull–Whitemodel is calibrated to a given interest rate structure {f M(0,t):t ≥ 0},then
1 ∂fM(0,t) σ2 θ(t)=f M(0,t)+ + 1 − e−2kt for all t ≥ 0. k ∂t 2k2
Theorem 5.12 (Zero-coupon bond in the calibrated Hull–White model). If the Hull–White model is calibrated to a given interest rate structure, then P M(0,T) σ2 P (t, T )=e−r(t)B(t,T ) exp B(t, T )f M(0,t) − 1 − e−2kt B2(t, T ) . P M(0,t) 4k
Theorem 5.13 (Option on a zero-coupon bond in the Hull–White model). In the Hull–White 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 σ˜ and h are as in the Vasicek model, Theorem 4.9.
ZBP(t, T, S, K)=KP(t, T )Φ(−h +˜σ) − P (t, S)Φ(−h).
Theorem 5.14 (Caps and floors in the Hull–White model). In the Hull–White model, the price of a cap with notional value N,caprateK, and the set of times T , is given by