Assume Xt is a multivariate Ornstein-Uhlenbeck process, i.e. dXt=σdBt−AXtdt
and the spot interest rate evolves by the following equation: rt=a+b⋅Xt.
After solving for Xt using etAXt and Ito and looking at ∫T0rsds, it turns out that ∫T0rsds∼N(aT+bT(I−e−TA)A−1X0,bTVTb)
where Vt is the covariance matrix of ∫T0(I−e−(T−u)A)A−1σdBu.
This gives us the yield curve y(t)=a+bT(I−e−tA)A−1X0t+bTVtb2t
and by plugging in A=(λ10λ) we finally arrive at y(t)=a+1−e−λtλtC0+e−λtC1+bTVtb2t.
The formula above without bTVtb2t is known as the Nelson-Siegel yield curve model. Could somebody clarify why neglecting bTVtb2t leads to arbitrage opportunities?
So I am essentially asking the following question:
Why is the above model (with bTVtb2t) arbitrage free?
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