ABSTRACT
The business need for variable coverages δj , the backward construction technique of Section-3.2, together with a limited calibration set, leads to a situation in which the forwards K (t, T ) and their corresponding volatilities ξ (t, T ) will be known only at a discrete maturity set Tj (j = 1, .., n). To price an instrument depending on an intermediate maturity, for example, a caplet maturing between Tj and Tj+1, we therefore need to interpolate both volatility and forward. Our approach is to interpolate on deterministic functions like ξ (t, T ), and
then use properties of the model to derive interpolations for stochastic variables like K (t, T ) and discount functions B (t, T ). In contrast, direct interpolation on stochastic variables turns out to be inaccurate and unsatisfactory. Note that none of the methods described in this chapter are arbitrage-
free, though in practice they work fairly accurately. Moreover, they are also inconsistent in that Section-8.3 on interpolating discount factors ought to determine how the forwards in Section-8.1 are interpolated, but they don’t. The author suggests consulting Schlogl [113] for a more exacting analysis.
