Last updated: 2023-03-16.

Module: tf_quant_finance.rates.hagan_west.monotone_convex

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The monotone convex interpolation method.

The monotone convex method is a scheme devised by Hagan and West (Ref [1]). It is a commonly used method to interpolate interest rate yield curves. For more details see Refs [1, 2].

It is important to point out that the monotone convex method does not solve the standard interpolation problem but a modified one as described below.

Suppose we are given a strictly increasing sequence of scalars (which we will refer to as time) [t_1, t_2, ... t_n] and a set of values [f_1, f_2, ... f_n]. The aim is to find a function f(t) defined on the interval [0, t_n] which satisfies (in addition to continuity and positivity conditions, see Section 6 of Ref [2] for more details) the following

  Integral[f(u), t_{i-1} <= u <= t_i] = f_i,  with t_0 = 0

In the context of interest rate curve building, f(t) corresponds to the instantaneous forward rate at time t and the f_i correspond to the discrete forward rates that apply to the time period [t_{i-1}, t_i].

This implementation of the method currently supports batching along the interpolation times but not along the interpolated curves (i.e. it is possible to evaluate the f(t) for t as a vector of times but not build multiple curves at the same time).

References:

[1]: Patrick Hagan & Graeme West. Interpolation Methods for Curve Construction. Applied Mathematical Finance. Vol 13, No. 2, pp 89-129. June 2006. https://www.researchgate.net/publication/24071726_Interpolation_Methods_for_Curve_Construction [2]: Patrick Hagan & Graeme West. Methods for Constructing a Yield Curve. Wilmott Magazine, pp. 70-81. May 2008.

Functions

diff(...): Computes the difference between elements of an array at a regular interval.

interpolate(...): Performs the monotone convex interpolation.

interpolate_forward_rate(...): Interpolates instantaneous forward rate to supplied times .

interpolate_yields(...): Interpolates the yield curve to the supplied times.