Hedging Nelson Siegel
Recalling Itos Lemma
{% dX(t) = \mu(X, t) dt + \sigma(X,t) dW %}
{% df(t, X(t)) = ( \frac{\partial f}{\partial t} + \mu \frac{\partial f}{\partial x} + \frac{1}{2} \sigma^2 \frac{\partial ^2 f}{\partial x^2} )dt
+ \sigma \frac{\partial f}{\partial x} dW(t)
%}
where {% W %} is a
Brownian Motion
Deriving the Hedging Equation
We assume that the value of a portfolio of bonds is driven by 4 factors, the factors specified in the Nelson Siegel Model. Then, we wish to be able to hedge three of the factors. That is, if we wanted to hedge all factors, we would simply invest in a variable rate instrument. Generally, we will want to take positions in one or more Nelson Siegel factors, and hedge the others. For this example, we assume that we want to take a position on the level factor, and hedge the other three.To do so, we set up a portfolio of 4 bonds. We take 1 unit of the first bond, and then take {% \Delta %} units of the other bonds (whose size will be determined)
{% \Pi = V_1 + \Delta_2 V_2 + \Delta_3 V_3 + \Delta_4 V_4 %}
Then we apply Itos Lemma and discard any terms of {% dt %} (that is, the non-random terms) We are left with terms like the following
{% d \Pi = \sigma_1 \frac{\partial V_1}{\partial X_1} dW_1 +
\sigma_2 \frac{\partial V_1}{\partial X_2} dW_2 +
\sigma_3 \frac{\partial V_1}{\partial X_3} dW_3 +
\sigma_4 \frac{\partial V_1}{\partial X_4} dW_4 +
%}
{% \Delta_2 \sigma_1 \frac{\partial V_2}{\partial X_1} dW_1 +
\Delta_2 \sigma_2 \frac{\partial V_2}{\partial X_2} dW_2 +
\Delta_2 \sigma_3 \frac{\partial V_2}{\partial X_3} dW_3 +
\Delta_2 \sigma_4 \frac{\partial V_2}{\partial X_4} dW_4 +
...
%}
Matrix Formulation
In order to calculate the bond weights, we set the problem up as a matrix equation
{%
\begin{bmatrix}
1 \\
0 \\
0 \\
0 \\
\end{bmatrix}
=
\begin{bmatrix}
\sigma_1 \frac{\partial V_1}{\partial X_1} & \sigma_2 \frac{\partial V_1}{\partial X_2} & \sigma_3 \frac{\partial V_1}{\partial X_3} & \sigma_4 \frac{\partial V_1}{\partial X_4} \\
\sigma_1 \frac{\partial V_2}{\partial X_1} & \sigma_2 \frac{\partial V_2}{\partial X_2} & \sigma_3 \frac{\partial V_2}{\partial X_3} & \sigma_4 \frac{\partial V_2}{\partial X_4} \\
\sigma_1 \frac{\partial V_3}{\partial X_1} & \sigma_2 \frac{\partial V_3}{\partial X_2} & \sigma_3 \frac{\partial V_3}{\partial X_3} & \sigma_4 \frac{\partial V_3}{\partial X_4} \\
\sigma_1 \frac{\partial V_4}{\partial X_1} & \sigma_2 \frac{\partial V_4}{\partial X_2} & \sigma_3 \frac{\partial V_4}{\partial X_3} & \sigma_4 \frac{\partial V_4}{\partial X_4} \\
\end{bmatrix}
\begin{bmatrix}
1 \\
\Delta_2 \\
\Delta_3 \\
\Delta_4 \\
\end{bmatrix}
%}
Then, we use the
matrix inverse
to calculate the unknown values
{%
\begin{bmatrix}
1 \\
\Delta_2 \\
\Delta_3 \\
\Delta_4 \\
\end{bmatrix}
=
\begin{bmatrix}
\sigma_1 \frac{\partial V_1}{\partial X_1} & \sigma_2 \frac{\partial V_1}{\partial X_2} & \sigma_3 \frac{\partial V_1}{\partial X_3} & \sigma_4 \frac{\partial V_1}{\partial X_4} \\
\sigma_1 \frac{\partial V_2}{\partial X_1} & \sigma_2 \frac{\partial V_2}{\partial X_2} & \sigma_3 \frac{\partial V_2}{\partial X_3} & \sigma_4 \frac{\partial V_2}{\partial X_4} \\
\sigma_1 \frac{\partial V_3}{\partial X_1} & \sigma_2 \frac{\partial V_3}{\partial X_2} & \sigma_3 \frac{\partial V_3}{\partial X_3} & \sigma_4 \frac{\partial V_3}{\partial X_4} \\
\sigma_1 \frac{\partial V_4}{\partial X_1} & \sigma_2 \frac{\partial V_4}{\partial X_2} & \sigma_3 \frac{\partial V_4}{\partial X_3} & \sigma_4 \frac{\partial V_4}{\partial X_4} \\
\end{bmatrix} ^{-1}
\begin{bmatrix}
1 \\
0 \\
0 \\
0 \\
\end{bmatrix}
%}