title: Fixed Income
author: Keith A. Lewis
institution: KALX, LLC
email: [email protected]
classoption: fleqn
abstract: Fixed cash flows
...
\newcommand\RR{\mathbf{R}}
\renewcommand\AA{\mathcal{A}}
\newcommand{\Cov}{\operatorname{Cov}}
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, $t_0$.