Bootstrapping discount curves

Definition Fix a valuation date $0$, a currency and a funding or collateral convention. A discount curve is a function $T \mapsto P(0, T)$ on an interval $[0, T_{\max}]$ that gives today’s price of 1 currency unit paid with certainty at $T$, the price of a default-free zero-coupon bond maturing at $T$, with $P(0, 0) = 1$. It is known today, so it is a deterministic function of the maturity, while bond prices $P(t, T)$ at later dates $t > 0$ can be random once interest-rate dynamics are introduced. In the frictionless, default-free model of this lesson, absence of arbitrage requires $P(0, T) > 0$, because a claim that pays 1 at $T$ and costs nothing or less than nothing would be an arbitrage. For $T > 0$ the continuously compounded zero rate is

A curve node is a maturity $T_j$ at which the value $P_j = P(0, T_j)$ is determined by the calibration instruments. Values between nodes, and beyond the last node, come from an interpolation or extrapolation rule chosen by the modeler.

Definition Let instruments $i = 1, \dots, n$ have prices $V_i$ at the valuation date and deterministic payments $C_{ij}$ at node dates $T_1 < \dots < T_n$, and suppose instrument $i$ pays nothing after $T_i$. Its pricing equation is

for $i = 1, \dots, n$, a linear system whose matrix $(C_{ij})$ is lower triangular. If every diagonal entry $C_{ii}$ is nonzero, the system has exactly one solution, found by forward substitution,

The determinant of a triangular matrix is the product of its diagonal entries, so a zero $C_{ii}$ leaves the system with no solution or with infinitely many. An instrument whose price introduces two new payment dates leaves two unknowns in one equation, and they are identified only by another price or by an interpolation rule. With more instruments than nodes, two different prices for the same cash flows leave the equations without any solution, and no single curve reprices them all. The unique solution need not be positive, so positivity has to be checked separately.

Definition Bootstrapping is the algorithm that solves the triangular system node by node in maturity order. It sets $P_1 = V_1 / C_{11}$, and for $i = 2, \dots, n$ it values the payments of instrument $i$ at the nodes already found, subtracts that value from $V_i$, and divides the remainder by the final payment $C_{ii}$. Its hypotheses are deterministic default-free payments, one valuation date, and inputs on common conventions for currency, settlement, day count and collateral, with quotes taken consistently from the bid, the ask or the mid. Under them the result is exact and reprices every input. When an instrument pays between nodes, those payment dates are tied to the nodes by an interpolation rule, and each step becomes one equation in the newest node, in general a nonlinear one. Calibrating several curves at once, or fitting more instruments than nodes by optimization, are different methods.

Definition An interpolation rule assigns values $P(0, t)$ at dates between nodes. Log-linear interpolation on $[T_j, T_{j+1}]$ sets

with weight $w = (t - T_j)/(T_{j+1} - T_j)$, so $P(0, t) = P_j^{\,1-w} P_{j+1}^{\,w}$. Positive nodes give positive values, and the instantaneous forward rate $f(0, t) = -\partial_t \ln P(0, t)$ is constant on each open interval between nodes,

with a jump at a node wherever the neighboring constants differ. Other common rules interpolate the factors $P(0, t)$ linearly or the zero rates $z$ linearly. All of them agree at the nodes and in general differ between them, so off-node prices, forward rates and sensitivities depend on the rule. The rule is a modeling assumption, not information contained in the quotes.

Definition For dates $T_1 < T_2$ on the curve, let $\delta > 0$ be the accrual fraction from $T_1$ to $T_2$ under the stated day count. The implied simple forward rate $F(0; T_1, T_2)$ is defined by

It is the rate $K$ that gives zero time-0 value to an agreement, made today, to deposit 1 at $T_1$ and receive $1 + \delta K$ at $T_2$, since that agreement is worth $-P(0, T_1) + (1 + \delta K) P(0, T_2)$. For a positive curve that is differentiable in $T$, the instantaneous forward rate and its inverse relation are

These are identities of the curve at time $0$. They are not forecasts of the spot rates that will prevail later.

Definition A curve is consistent when it reprices every calibration instrument exactly, all its nodes are positive, and every quantity computed from it, whether an off-node price, a forward rate or a sensitivity, uses the same nodes, the same interpolation rule and the same conventions for currency, settlement, day count and collateral. Repricing confirms the nodes at the payment dates. It cannot show that an off-node value is the right one, and it says nothing about future realized rates. A factor that increases from one node to the next means a negative forward rate, which the frictionless model allows. Factors that never increase follow only from an extra assumption, such as the possibility of holding cash from one date to the next at no cost. A bootstrap step whose remainder is zero or negative would give a factor that is not positive, so the quotes then admit an arbitrage in the frictionless model or the instruments are misspecified.