View source for Smith–Wilson method

The '''Smith–Wilson method''' is a method for extrapolating [[Forward_rate|forward rates]]. It is recommended by [[European_Insurance_and_Occupational_Pensions_Authority|EIOPA]] to extrapolate interest rates. It was introduced in 2000 by A. Smith and T. Wilson for [[Bacon & Woodrow]].

== Mathematical formulation ==
Let UFR be some ultimate forward rate and <math>u_i</math> be the time to the i'th maturity. Then <math>P(t)</math> defines the price of a zero-coupon bond at time t.

<math>P(t) = e^{-UFR\cdot t} + \sum_{j=1}^N \xi_j \cdot W(t, u_j)</math>

Where
<math>W(t, u_j) = e^{-UFR\cdot (t+u_j)} \cdot (\alpha\cdot \min(t, u_j) - 0.5e^{-\alpha\cdot \max(t, u_j)}\cdot (e^{\alpha\cdot \min(t, u_j)} - e^{-\alpha\cdot \min(t, u_j)}))</math>

and the symmetric W matrix is
<math>W = (W(u_i, u_j))_{i=1,...,N:j=1,...,N}</math>

and
<math>p = (P(u_1), ..., P(u_N))^T</math>,
<math>\mu = (e^{-UFR\cdot u_1}, ..., e^{-UFR\cdot u_N})^T</math>,
<math>\xi = W^{-1}(p-\mu)</math>.

== References ==
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{{reflist}}
* A Technical Note on the Smith-Wilson Method, The Financial Supervisory Authority of Norway, (1 July 2010)
* Lagerås, Andreas & Lindholm, Mathias. (2016). Issues with the Smith-Wilson method. Insurance: Mathematics and Economics. 71. 10.1016/j.insmatheco.2016.08.009. 
* Smith, A. and Wilson, T. (2000). Fitting Yield Curves with Long Term Constraints. Research report, Bacon & Woodrow.
* [https://www.eiopa.europa.eu/system/files/2022-09/eiopa-bos-22-409-technical-documentation.pdf Technical documentation of the methodology to derive EIOPA's risk-free interest rate term structures]

[[Category:Financial models]]
[[Category:Fixed income analysis]]
[[Category:Insurance]]
[[Category:Mathematical finance]]