Smoothing splines and interest rate curves

Yield curves are important in Economics and used by finance professionals to analyze bonds and look for trading opportunities and by economists, to try to understand economic conditions.

The yield is the amount in cash that returns to the owners of a security, for example, if a sovereign bond with maturity at time $T$ is bought at time $t$ for a price $P(t,T)$, the yield would be high if $P(t,T)$ is much less than the maturity price $P(T,T)$ (which would obviously be the risk-free amount received by the lender). A higher yield allows the owner to make more profit from the investment, but it may also mean that the security is being sold at a discount as a result of some risk related to the security. For a recent example of this, Greek bonds, which were sold at such discounts because the market anticipated a default, this made yields soar, with risk-averting investors unloading bonds at the lower prices that risk-taking investors were comfortable buying in. Ultimately everybody took a haircut.

For fixed income mathematics, a set of axioms must be assumed, namely:

1. The market trades continuously over its trading horizon: it extends from the current time to some distant future time such that the maturities of all the instruments to be valued fall between now and the trading horizon..
2. The market is efficient: information is available to all traders simultaneously, and every trader makes use of all the available information, also related to the axiom that
   
3. There are no arbitrage opportunities: the price of a portfolio is the sum of its constituent parts.
   
4. The market is complete: any desired cash flow can be obtained from a suitable self-financing strategy based on a portfolio of discount bonds.

It is assumed further that there are no legal barriers to trading and that the market is rational (traders try to maximize their profits).

Now, define the zero yield in terms of the bond price above. The zero yield, as seen at time $t$, of a bond that matures at time $T$, is denoted by $y(t,T)$ and is defined by
$$
P(t,T)=e^{-(T-t) y(t,T) }
$$
for every $t<T$, which means that the current price of the bond is equal to discounted price at maturity (using compound interest).

Suppose that at time $t$ we enter into a forward contract to deliver at time $T_1$ a bond that will mature at time $T_2$ (obviously $T_2>T_1$). Let the forward price of the bond be denoted by $P(t,T_1,T_2)$. At the same time, a bond that matures at time $T_1$ is purchased at a price of $P(t,T_1)$. Again at time $t$, a bond that matures at time $T_2$ is bought with a price of $P(t,T_2)$. Notice that the axiom specifying that there are no arbitrage opportunities implies that the price of the bond maturing at time $T_2$ must be equal to the product of the price of the bond maturing at time $T_1$ and the forward price $P(t,T_1) P(t,T_1,T_2)$. Let the implied forward rate $f(t,T_1,T_2)$, valued at time $t$, for the period $(T_1,T_2)$ as
$$
P(t,T_1,T_2)=e^{-(T_2-T_1) f(t,T_1,T_2) }
$$
Notice that $P(t,T_2) =P(t,T_1) P(t,T_1,T_2)=e^{-(T_1-t) y(t,T_1) } e^{-(T_2-T_1) f(t,T_1,T_2) }$ and, summing the exponents, we get
$$
f(t,T_1,T_2)= \frac{(T_2-t) y(t,T_2) - (T_1 - t) y(t,T_1)}{T_2 - T_1}
$$
which is the period forward rate. However, the instantaneous forward rate is of much greater importance in the theory of the term structure. The instantaneous forward rate for time $T$, as seen at time $t$, is denoted by $f(t,T)$ and is the continuously compounded rate defined by
$$
f(t,T)= \lim_{h \rightarrow 0} {f(t,T,T+h) } = y(t,T) + (T-t) y_T (t,T)
$$
where $y_T = \frac{\partial}{\partial T} y(t,T)$. We can interpret the previous equation as the current  yield value plus the instantaneous change of the yield with the maturity time.

To interpolate curves, and thus have values for all the points within the interval and not only the data points that are made available to us by the problem, we can use a number of different methods. One needs some complexity to be able to capture what the nature is saying, but at the same time this complexity might be cause by some "noise", or perturbations that we want to avoid (market panics, manipulations or poorly registered data). We can choose a parametric family, such as polynomials, fix the order and fit the coefficients. However, this appears as somehow arbitrary. One can choose to interpolate with a non-parametric method such as splines. When using some functional space, one must restrict himself to functions that meet some criteria, not only that approximate well (or maybe the best approximation in that space). This is to avoid overfitting the said noise.

Now we can choose to fit an approximating yield or the forward curve that approximates the true yield or forward curves, with the information given by actual bond prices as of time $t$, by minimizing the following functional (assuming we fit the yield curve $\varphi$):
$$
R(\varphi,t) =  \sum_{i=1}^n (P(t,T_i) - P(t,T_i,\varphi))^2 + \lambda \int \left( \frac{\partial^2}{\partial s^2} \varphi(s) \right)^2 ds.
$$
Where $\lambda$ is a regularization parameter, and $P(t,T_i,\varphi)$ is a functional that prices a bond of maturity $T_i$ at time $t$ with the yield curve $\varphi$. We change formulation, for practical derivation purposes (we now compute the observed yield $y(t,T_i)$ from the observed bond price)
$$
R(\varphi,t) = \sum_{i=1}^n (y(t,T_i) - \varphi(T_i))^2 + \lambda \int \left( \frac{\partial^2}{\partial s^2} \varphi(s) \right)^2 ds.
$$
And then we apply the standard smoothing splines theory to compute the solution, which is:
$$
\mathbf{w} = (I+\lambda K)^{-1} \mathbf{y}
$$
Where $\mathbf{y}$ is a vector with elements $y(t,T_i)$, $K$ is a matrix whose elements are of the form $\int \frac{\partial^2}{\partial s^2} \phi_i (s)  \frac{\partial^2}{\partial s^2} \phi_j (s) ds$, with $\phi$ is a choosen spline basis (notice the dot-product structure) and $\mathbf{w}$ is a vector with the coefficients on that basis, so that the estimated function is
$$
\varphi = \sum_i w_i \phi_i
$$