*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(...)`](../../../tf_quant_finance/math/diff.md): Computes the difference between elements of an array at a regular interval.
[`interpolate(...)`](../../../tf_quant_finance/rates/hagan_west/monotone_convex/interpolate.md): Performs the monotone convex interpolation.
[`interpolate_forward_rate(...)`](../../../tf_quant_finance/rates/hagan_west/monotone_convex/interpolate_forward_rate.md): Interpolates instantaneous forward rate to supplied times .
[`interpolate_yields(...)`](../../../tf_quant_finance/rates/hagan_west/monotone_convex/interpolate_yields.md): Interpolates the yield curve to the supplied times.