I'm trying to understand bond pricing with the Vasicek interest rate model. I'm using McDonald's book for this purpose (not homework).
Recall that Vasicek dynamics are drt=a(b−rt)dt+σdZt.
Now, Macdonald introduces the exponential affine formulas to price a unit zero:
P(r,t,T)=A(t,T)exp(rB(t,T))A(t,T)=exp(ˉr(B(t,T)−T+t)+B(t,T)2σ24a)B(t,T)=1−e−a(T−t)aˉr=b+σϕa−σ2a2
In the course of deriving these expressions, Macdonald asks us to assume that ϕ, which is the Sharpe ratio for the motion, is constant. But we can see that it is only constant when a=0.
Later, Macdonald talks about the Sharpe ratio for the "interest rate risk", a phrasing I find very obscure. Is that the bond price process? The Vasicek process? In either case, they're driven by the same Brownian motion and should have the same (non-constant) Sharpe ratio.
Can somebody explain how to apply these formulas? Just a sketch would do -- but I'm stymied by the presentation.
Bond Prices
Assume that the short rate rt follows the Ito process as described by the following stochastic differential equation drt=μ(rt,t)dt+σ(rt,t)dWPt
we assume the bond price to be dependent on
rt only, independent of default risk, liquidity and other factors. If we write the bond price as
P(rt,t)=V(t,rt,T) such that
V(t,rt,t)=1 then
dV=(Vt+μVr+12σ2Vrr)dt+σVrdWt
for simplicity let
μV=Vt+μVr+12σ2VrrVσV=σVrV
thus we have
dV=μVVdt+σVVdWt
The following portfolio is constructed: we buy a bond of dollar value V1 with maturity
T1 and sell another bond of dollar value
V2 with maturity
T2. The portfolio value
Π is given by
Π=V1−V2
According to the bond price dynamics,we have
Π=(μV1V1−μV2V2)dt+(σV1V1−σV2V2)dWt
Suppose
V1 and
V2 are chosen such that
V1=σV2σV2−σV1ΠV2=σV1σV2−σV1Π
then the stochastic term in
dΠ vanishes and the equation becomes
dΠ=(μV1σV2−μV2σV1σV2−σV1)Πdt
Since the portfolio is instantaneously riskless, in order to avoid arbitrage opportunities,it must earn the riskless short rate so that
dΠ=r(t)Πdt ,then
μV1−r(t)σV1=μV2−r(t)σV2
The above relation is valid for arbitrary maturity dates
T1 and
T2, so the ratio should be independent of maturity
T.Let the common ratio be defined by
λ, that is,
μV−r(t)σV=λ(rt,t)
The quantity
λ is called the market price of risk of the short rate.If we substitute
μV(r,t) and
σV(r,t) into above Equation, we obtain the following governing differential equation for the price of a zero-coupon bond
Vt+(μ−λσ)Vr+12σ2Vrr−rtV=0
Change Measure
we assume Q be a martingale measure such that dWPt=−λ(r,t)dt+dWQt
thus we have
drt=μ∗(rt,t)dt+σ(rt,t)dWQt
where
μ∗(rt,t)=μ(rt,t)−λ(rt,t)σ(rt,t)
Affine Term Structure Models
A short rate model that generates the bond price solution of the form P(t,T)=V(t,rt,T)=eA(t,T)−B(t,T)rt
Suppose the dynamics of the short rate
rt under the risk neutral measure
Q is governed by
drt=μ∗(rt,t)dt+σ(rt,t)dWQt
where
μ∗(rt,T)=α(t)rt+β(t)σ2(rt,T)=γ(t)rt+δ(t)
We show the governing equation for
P(t,T)=V(t,r,T) is given by
Vt+μ∗Vr+12σ2Vrr−rtV=0
Substituting the assumed affine solution of bond price into this equation, we obtain Bt(t,T)+α(t)B(t,T)−12γ(t)B2(t,T)=−1B(T,T)=0
and
At(t,T)=β(t)B(t,T)−12δ(t)B2(t,T)A(T,T)=0
Vasicek Model
Vasicek (1977) proposed the stochastic process for the short rate rt under the Martingle measure to be governed by the Ornstein–Uhlenbeck process: drt=a(b−rt)dt+σdWQt
hence
α(t)=−a,β(t)=abγ(t)=0,δ(t)=σ2
thus we have
B(t,T)=1−e−a(T−t)a
and
A(t,T)=exp((b+σϕa−σ2a2)(B(t,T)−T+t)+B(t,T)2σ24a)
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