# Term Structure (Polynomial)

## Contents

## Definition

A polynomial term structure representation assumes that the instantaneous forward rate $f(t,T)$ can be described by an n-order polynomial of the time-to-settlement/time-to-maturity, i.e.

$\displaystyle{ f(t,T) = \sum_{i=0}^{n} a_{i}(T-t)^{i} }$

## Interrelation between spot and forward rates

Assume that the instantaneous continuously compounded forward rate can be described by a n-order polynomial. Then the continuously compounded spot rate is given by

$\displaystyle{ \begin{eqnarray} r_{c}(t,T) &=& \frac{1}{T-t} \int_{t}^{T} f_{c}(t,u) du \\ &=& \frac{1}{T-t} \sum_{i=0}^{n} \frac{a_{i}}{i+1}(T-t)^{i+1} = \sum_{i=0}^{n} \frac{a_{i}}{i+1}(T-t)^{i} \\ &=& \sum_{i=0}^{n} b_{i}(T-t)^{i} \end{eqnarray} }$

i.e. the spot rate is also an n-order polynomial with adapted coefficients $b_{i} = \frac{a_{i}}{i+1}$.

## QFF implementation

The internal compounding of the `PolynomialTermStructure` class is the continuous compounding, i.e. if a polynomial is specified during the creation process, then it refers to the intantaneous continuously compounded forward rate.

## Codebook

double[] coefficients = {0.05, -0.01, 0.01}; ILevelTS polynomial_ts = PolynomialTS.create(coefficients);