Saturday, December 19, 2015

interest rates - Pricing a zero with Vasicek model


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(brt)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)=1ea(Tt)aˉr=b+σϕaσ2a2



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.



Answer



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=V1V2
According to the bond price dynamics,we have Π=(μV1V1μV2V2)dt+(σV1V1σV2V2)dWt
Suppose V1 and V2 are chosen such that V1=σV2σV2σV1ΠV2=σV1σV2σV1Π
then the stochastic term in dΠ vanishes and the equation becomes dΠ=(μV1σV2μV2σV1σV2σV1)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 μV1r(t)σV1=μV2r(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, μVr(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+(μλσ)Vr+12σ2VrrrtV=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+μVr+12σ2VrrrtV=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(brt)dt+σdWQt

hence α(t)=a,β(t)=abγ(t)=0,δ(t)=σ2
thus we have B(t,T)=1ea(Tt)a
and A(t,T)=exp((b+σϕaσ2a2)(B(t,T)T+t)+B(t,T)2σ24a)


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