SIAG/OPT Views-and-News A Forum for the SIAM Activity Group on Optimization
نویسنده
چکیده
The application considered in this article arises within nance and concerns approximation of a forward rate curve, given a set of bond prices. The simplest bond is a zero-coupon bond for which a given payment is obtained at a given time. This single payment may be normalized so that it is a unit payment. Consequently, for a given set of m zerocoupon bonds, it is known that bond i, i = 1; : : : ;m, gives unit value at time ti, and in addition its market price at time zero, vi, is known. Here, and throughout this article, the current time is set to zero and denoted by t0. For a more general discussion, see, e.g., Bjork [3, Chapter 15]. Without loss of generality, we assume that 0 = t0 < t1 < < tm. The purpose of the application is to extend these market prices to a smooth curve v(t) which is consistent with the information that is known, i.e., v(t0) = 1 and v(ti) = vi, i = 1; : : : ;m. Rather than working on v(t), the instantaneous forward rate f(t) is chosen. The instantaneous forward rate f(t) is the riskless interest rate that can be obtained over an innitesimal interval at time t for a decision made at time t0. (See, e.g., Bjork [3, Chapter 15] for a more precise de nition.) This means that for t 2 [t0; tm], the relationship between v(t) and the function f is given by
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