April 25, 2024
A fixed income instrument is specified by cash flows (c_j) at times (u_j). It is a portfolio of zero coupon bonds and its present value is P = \sum_j c_j D(u_j) where D(u) is the discount to time u. The present value at time t is P_t = \sum_{u_j > t} c_j D_t(u_j) where D_t(u) is the price at time t of a zero coupon bond maturing at u.
A zero coupon bond, D(u), pays one unit at maturity u so C^{D(u)}_u = 1 is the only cash flow. We write D_t(u) for the price X_t^{D(u)} of the zero coupon bond at time t. An arbitrage free model with stochastic discount D_t requires the price at time t to satisfy D_t(u)D_t = E_t[D_u] so D_t(u) = E_t[D_u]/D_t.
We can write D_t = \exp(-\int_0^t f_s\,ds) where f_t is the continuously compounded forward rate, or short rate process. It corresponds to repurchase agreement rates.
Exercise. Show D_t(u) = E_t[\exp(-\int_t^u f(s)\,ds)].
An interesting feature of fixed income is that the short rate determines the price dynamics of all instruments, assuming there are no defaults.
A risky zero coupon bond with recovery R and default time T has a single cash flow C_u = 1 if default occurs after maturity or C_T = R if T \le u. It is customary to assume R is constant. We must expand the sample space to Ω\times (0,\infty] where (ω,t)\in Ω\times (0,\infty] indicates default occured at time t. The partition of (0\infty] representing information available at time t for the default time is \{(t,\infty]\} \cup \{\{s\}:s \le t\}. If default has not occured prior to t we only know T > t. If default occured prior to time t we know exactly when it happened.
We write D_t^{R,T}(u) for the price X_t^{D^{R,T}(u)} of the risky zero coupon bond at time t. The dynamics of a risky zero are determined by D_t^{R,T}(u) D_t = E_t[R 1(T \le u) D_T + 1(T > u) D_u]. The credit spread s_t = s_t^{R,T}(u) defined by D_t^{R,T}(u) = D_t(u) e^{-u s_t} incorporates both recovery and default.
If rates are zero then D_t = 1 for all t and this simplifies to D_0^{R,T}(u) = R P(T \le u) + P(T > u) when t = 0. If T is exponentially distributed with hazard rate λ then P(T > t) = e^{-λ t} and D_0^{R,T}(u) = R + (1 - R)e^{-λu}. When λ = 0 the right hand side is 1. When R = 0 the credit spread equals the hazard rate. If λu is small then the approximation e^x \approx 1 + x for small x gives the rule of thumb s = λ(1 - R) where s = s_0 = s_0^{R,T}(u) is the credit spread.
For general t we have D_t^{R,T}(u) = R P(T \le u | T > t) 1(t < T \le u) + P(T > u | T > t) 1(T > u).
Unlike in the credit default swap market, mathematical finance literture likes to assume recovery is delayed until maturity. It is also popular to make the unrealistic assumption that default time is independent of the stochastic discount. Under these assumptions we have D_t^{R,T}(u) = D_t(u)\bigl(R P(T \le u | T > t) 1(t < T \le u) + P(T > u | T > t) 1(T > u)\bigr). In principal, R could be random and joint distributions involving the default time and stochastic discount could be specified.
Suppose a fixed income instrument pays cash flows (c_k) at times (u_k). The yield, y(p), given a price p is determined by p = \sum_k c_k e^{-y(p)u_k}. It is the constant forward curve that reprices the fixed income instrument. It is a convenient proxy for price, just like Black-Scholes/Merton implied volatility.
It is not the case zero coupon bonds of all maturities are traded. The discount curve D(t) is used to interpolate a discount for all maturities. An instrument with cash flows (c_k) at times (u_k) and price p fits the curve if p = \sum_k c_k D(u_k). Typically a collection of such intruments and prices are given and we wish to find a discount curve that fits all of them. This is a highly underdetermined problem and there is a vast literature on various methods of interpolation. The simplest approach is to use market data directly and avoid non-financial artifacts introduced by various splining methods.
Given a collection of fixed income instruments ordered by increasing maturity and corresponding prices we can bootstrap a discount curve having a piecewise constant forward curve that matches each price. The first forward is the yield of the first instrument. Given a discount to time t and a forward rate f we can extend the discount for u > t by D(u) = D(t)e^{-f(u - t)}.
The bootstrap method is deterministic. It assumes the forward curve is piecewise constant with jumps at maturities of instruments used to build the curve. As instruments of increasing maturity are added, the initial part of the curve is fixed and the new constant segment is chosen to match the price of the instrument being added. It is important that no two instruments have nearly equal maturity since the forward between those dates may require a large adjustment to fit the price.
The vast literature on various methods of interpolating discount curves should be ignored. Splining introduces mathematical artifacts into the discount. A cubic Hermite tension spline can produce a forward curve that is pleasing to the eye, but makes it difficult to explain to a trader why their rho bucketing is off. It is better to add synthetic instruments at intermediate maturities with prices determined by an interpolation method that traders can understand. Curves should be bootstrapped with the instruments traders are using to hedge their position.
A piecewise constant curve is determined by times (t_j), 0\le j\le n, and forwards (f_j), 0 < j \le n, where f(t) = f_j for t_{j-1} < t \le t_j. Note f(t_j) = f_j and the curve is undefined for t > t_n. We assume t_0 = 0 so (t_j, f_j), 1\le j\le n, determine the curve.
Given a forward curve to time t_n and an instrument with maturity t > t_n we must find f such that p = \sum_{u_k \le t_n} c_k D(u_k) + \sum_{u_k > t_n} c_k D(t_n)e^{-f(u_k - t_n)} where p is the instrument price. The discount D(u) is determined for u \le t_n and the forward f is constant for u > t_n. This can be solved using one-dimensional root finding to produce the next point (t_{n+1}, f_{n+1}) = (t, f) of the piecewise constant forward curve where t is the maturity of the added instument.
If there is exactly one cash flow past t_n, (c, u), then this equation has a closed form solution since the second sum has only one term c D(t_n)e^{-f(u - t_n)}. Denoting the first sum by p_n we have f = \frac{-\log((p - p_n)/c D(t_n))}{u - t_n} given price p to produce the next point (t_{n+1}, f_{n+1}) = (u, f).
If we extend the curve with an instrument having exactly two cash flows (c_0, u_0) and (c_1, u_1) then there are also closed form solutions. Since u_1 > t_n we have two cases, u_0 \le t_n and u_0 > t_n. If u_0 \le t_n then D(u_0) is known and we have p = p_n + c_0 D(u_0) + c_1 D(t_n)e^{-f(u_1 - t_n)} so f = \frac{-\log((p - p_n - c_0 D(u_0))/c_1 D(t_n))}{u_1 - t_n}. If u_0 > t_n we have p = p_n + c_0 D(t_n)e^{-f(u_0 - t_n)} + c_1 D(t_n)e^{-f(u_1 - t_n)}.
Exercise. Find an explicit formula for f.
A forward rate agreement F^{f,δ}(u,v) over the period [u, v] with coupon f and day count basis δ pays -1 unit at the effective date u, and 1 + fδ(u,v) at the termination date v, where δ(u,v) is the day count fraction for the period. The day count fraction is approximately equal to the time in years between u and v for any day count basis.
The forward par coupon at time t, F_t^{f,δ}(u,v) is the coupon that makes the price at time t\le u equal to 0. Since 0 = E_t[-D_u + (1 + F_t^δ(u,v)δ(u,v))D_v] the par coupon is F_t^δ(u,v) = (D_t(u)/D_t(v) - 1)/δ(u,v). Writing F_t = F_t^δ(u,v) and δ = δ(u,v) we have E_t[F_tδ D_v] = F_tδ E_t[D_v] = E_t[D_u - D_v] = D_t(u) - D_t(v)
There are also forward rate agreements not involving the exchange of notional. A (fixed rate) payer FRA has the single cash flow (f - F_u^δ(u,v))δ(u,v) at time v. A receiver FRA has the negative of this cash flow. The value at any time t \le u is determined by
\begin{aligned} V_t D_t &= E_t[(f - F_u(u,v;δ))δ(u,v) D_v] \\ &= E_t[fδ(u,v) D_v - E_u[D_u - D_v]] \\ &= E_t[fδ(u,v) D_v - D_u + D_v] \\ &= E_t[-D_u + (1 + fδ(u,v)) D_v] \\ \end{aligned}
which is the same as for a forward rate agreement that does exchange notional. These two types of FRAS’s have very different risk characteristics. If either counter-party defaults during the time notionals are exchanged the loss can be much larger than when the payment is only the difference of the fixed and floating rate.
An interest rate swap F^{c,δ}(t_0, \ldots, t_n) with calculation dates (t_j), coupon c, and day count basis δ pays -1 unit at the effective date t_0, cδ(t_{j-1},t_j) at t_j, 0 < j < n, and 1 + cδ(t_{n-1},t_n) at termination t_n.
The swap par coupon at time t, F_t(t_0,\ldots,t_n;δ), is the coupon that makes the price at time t\le {t_0} equal to 0: 0 = E_t[-D_{t_0} + \sum_{0<j<n} F_tδ(t_{j-1},t_j) D_{t_j} + (1 + F_tδ(t_{n-1},t_n) D_{t_n}]. Hence the par coupon, F_t(t_0,\ldots,t_n;δ) = (D_t(t_0) - D_t(t_n))/\sum_{0<j\le n}δ(t_{j-1},t_j) D_t(t_j), is determined by zero coupon bond prices.
Note that if n = 1 this is identical to a forward rate agreement.
There are also interest rate swaps not involving the exchange of notional. A (fixed rate) payer has the cash flows (c - F_{t_j}(t_{j-1},t_j;δ))δ(t_{j-1},t_j) at times t_j, 0 < j \le n. A receiver has the negative of these cash flow.
As with forward rate agreements, the coupon making the value at time t equal to zero is the swap par coupon.
Options on FRA’s are called floorlets or caplets.
A floorlet is a put option on an at-the-money forward rate agreement. It pays \max\{k - F_u(u,v),0\}δ(u,v) at time v. Its value at time t < u is determined by V_t D_t = E_t \max\{k - F_u(u,v),0\}δ(u,v) D_u. Writing F_u = F_u(u,v) and δ = δ(u,v) we have
\begin{align*} V_t D_t &= E_t[\max\{k - F_u,0\}δ D_v] \\ &= E_t[\max\{kδ - (1/D_u(v) - 1),0\} D_v] \\ &= E_t[\max\{1 + kδ - 1/D_u(v),0\} D_v] \\ &= E^*_t[\max\{1 + kδ - 1/D_u(v),0\}] E_tD_v \\ &= E^*_t[\max\{1 + kδ - 1/D_u(v),0\}] D_t(v)D_t \\ \end{align*} where E_t^* is the expectation under the forward measure P^* defined by dP_t^*/dP_t = D_v/E_t D_v. This shows the value at t of a floorlet is V_t = E^*_t[\max\{1 + kδ - 1/D_u(v),0\}] D_t(v).
A caplet is a call option on an at-the-money forward rate agreement. It pays \max\{F_u(u,v) - k,0\}δ(u,v) at time v. Its value at time t < u is determined by V_t D_t = E_t \max\{F_u(u,v) - k,0\}δ(u,v) D_u. Similar to floorlets, the value at t of a caplet is V_t = E^*_t[\max\{1/D_u(v) - (1 + kδ),0\}] D_t(v)
A floor and a cap are just a sequence of back-to-back floorlets or caplets.
A swaption is an option on a swap. It has a single cash flow \max\{k - F_{t_0}(t_0,\ldots,t_n;δ), 0\} at the effective date of the swap, t_0.