Monday, February 9, 2015

fixed income - Yield curve fitting example in Wilmott on Quant Finance p.528



In Wilmott on Quantitative Finance Vol. 2, p. 528, Section 31.4.2, is given a power series expansion for a zero coupon bond


Z(r,t;T)=1+a(r)(Tt)+b(r)(Tt)2+c(r)(Tt)3+


then it says to substitute this into the bond pricing equation, which is, of course


Zt+12w2Zrr+(u+λw)ZrrZ=0


the result of which is given as


a2b(Tt)3c(Tt)2+12(w22(Tt)wwt)((Tt)2ar2+(Tt)22br2)  +((uλw)(Tt)(uλw)t)(Tt)(dadr+(Tt)2dbdr)r(1+a(Tt)+c(Tt)2)+=0


My conundrum, for starters, is that I don't know how the first parenthical grouping in each of


(w22(Tt)wwt)((Tt)2ar2+(Tt)22br2)


and


((uλw)(Tt)(uλw)t)(Tt)(dadr+(Tt)2dbdr)



terms came to be (and the latter line, it seems like errata for the partial in r, shouldn't it be ((Tt)dadr+(Tt)2dbdr) not preceded by the (Tt) term (because when it distributes it will give the wrong power for (Tt) on the dbdr term?


When I compute partial derivatives with respect to t, r, and rr of Z I don't get the same result for the partials of r and rr. Obviously, some steps are missing. I do get the same first three terms as that given above, due to Zt, but after that it diverges from the answer given until the rZ term at the end of the bond pricing equation (though, again, I do think it should have been a b(Tt)2 instead of c(Tt)2 in that last term.


I know this is asking a bit much perhaps, but any help is appreciated. Thanks in advance.



Answer



Your observations are pretty much correct.


The groupings are because of the fine print "Note how I have expanded the drift and volatility terms at t=T; in the above these are evaluated at r and T." on the same page (p.528).


Basically, w is a function of both r and t. Since we want to use w(r,T) instead of w(r,t), we taylor expand w(r,t) around w(r,T) with respect to t. The same is true for functions of w(r,t). Thus,


w(r,t)2=w(r,T)2+(tT)w(r,T)2t=w(r,T)22(Tt)w(r,T)w(r,T)t


The same is true for u(r,t)λw(r,t) where now u may also be a function of r and t: uλw(r,t)=(u(r,T)λw(r,T))+(tT)(u(r,T)λw(r,T))t=(u(r,T)λw(r,T))(Tt)(u(r,T)λw(r,T))t


I hope this solves your conundrum.



Other points




  1. Yes there is an errata: it should be ((Tt)dadr+(Tt)2dbdr).




  2. It should have been a b(Tt)2 instead of c(Tt)2 in the last term as you point out at the end of your question.




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