TechnicalNote1~TechnicalNote31
Technical Note No. 1*Options, Futures, and Other DerivativesJohn HullConvexity Adjustments to Eurodollar FuturesIn the Ho-Lee model the risk-neutral process for the short rate in the traditionalrisk-neutral world isdr = θ(t)dt + σ dzwhere r is the instantaneous short rate, θ is a function of time, a and σ are constants, anddz is a Wiener process. Define P(t, T) as the price of a bond paying $1 at time T as seenat time t. As explained in the text, the bond price has the form P(t, T) = A(t, T)e−r(T −t).From Itˆo’s lemma the process for the bond price in a traditional risk-neutral world isdP(t, T) = r(t)P(t, T)dt − (T − t)σP(t, T) dzDefine f(t, T1, T2) as the forward rate (continuously compounded) at time t for theperiod between T1 and T2.f(t, T1, T2) = ln[P(t, T1)] − ln[P(t, T2)]T2 − T1From Ito’s lemma the process followed by f(t, T1, T2) is