Smith–Wilson method

The Smith–Wilson method is a method for extrapolating forward rates. It is recommended by 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 $$u_i$$ be the time to the i'th maturity. Then $$P(t)$$ defines the price of a zero-coupon bond at time t.

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

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

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

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