April 25, 2024
Unlike the equity world where the menagerie of stock prices are like bees coming out of a hive, there is a single thread that ties together all fixed income instrument prices. The continuously compounded stochastic forward rate f_t corresponds quite closely to the SOFR rate. The Secured Overnight Financing Rate is based on the large volume of overnight repurchase agreements that actually trade in the market1. LIBOR was a term rate based on calling up traders and asking for their opinions.
Using the mathematical fiction of continuous time trading, the continuously compounded stochastic forward rate f_t is the rate at which money can be borrowed or lent over an infinitesimal time period. Any amount a\in\boldsymbol{R} can be invested at time t and return a(1 + f_t\,dt) = a\exp(f_t dt) at time t + dt. Rolling over 1 unit invested at time 0 results in realized return R_t = \exp(\int_0^t f_s\,ds) at time t. It is the price of a money market account at time t. The stochastic discount is D_t = 1/R_t.
Holding an instrument entails cash flows that, unlike prices, are almost always 0. Stocks have dividends, bonds have coupons, futures have periodic cash flows in the margin account. The price of a futures is always 0. Futures are naturally occuring martingales. Money market accounts have no cash flows.
In every arbitrage-free model prices X_t and cash flows C_t must satisfy X_t D_t = E_t[X_u D_u + \sum_{t < s \le u} C_s D_s],\quad t \le u. If an instrument has no cash flows then X_t D_t is a martingale, as Black, Scholes, and Merton demonstrated. If X_u D_u goes to 0 as u\to\infty then the price is the expected value of discounted future cash flows, as Graham and Todd taught us.
A zero coupon bond maturing at u has a single non-zero cash flow C_u^{D(u)} = 1. In an arbitrage-free model X_t^{D(u)}D_t = E_t[D_u]. We write {X_t^{D(u)} = D_t(u) = E_t[D_u]/D_t} for the price at time t of a zero coupon bond maturing at u.
Exercise. Show D_t(u) = E_t[\exp(-\int_t^u f_s\,ds)].
Every fixed income product is a portfolio of zero coupon bonds. For example, a forward rate agreement over the period [u,v] has two cash flows C_u = -1 and C_v = 1 + \delta(u,v) f where \delta(u,v) is the day count fraction for the interval and f is a fixed coupon rate. The par coupon makes the price 0.
Exercise. If a forward rate agreement has price 0 then f = (D(u)/D(v) - 1)/\delta(u, v).
Hint: Use 0 = E[-D_u + (1 + \delta(u,v) f)D_v].
Mathematical analytics uses the dimensionless discount but traders like to use rates. Letting D(t) = D_0(t), the forward curve f(t) is defined by D(t) = \exp(-\int_0^t f(s)\,ds).
Exercise. Show f(t) = -(d/dt)\log D(t).
Exercise. Show 0 = E[(f_t - f(t))D_t].
Hint: Compute dD(t)/dt two ways using E[\exp(-\int_0^t f_s\,ds)] = \exp(-\int_0^t f(s)\,ds)
This shows f(t) is the par coupon for a forward rate agreement with cash flow f_t - f(t) at time t.
Futures quotes are naturally occuring martingales. The futures quote at time 0 on f_t expiring at t is \phi(t) = E[f_t].
Since {0 = E[(f_t - f(t))D_t] = E[f_t D_t] - f(t) D(t) = E[f_t] D(t) + \operatorname{Cov}(f_t, D_t) - f(t) D(t)} we have \phi(t) - f(t) = -\operatorname{Cov}(f_t, D_t)/D(t). In general f_t and D_t are negatively correlated so the difference between futures and forwards (convexity) is positive.
Every fixed income model is determined by the stochastic forward rate.
The LIBOR market model assumes f_t = \phi(t)\exp(\sigma(t)\cdot B_t - \|\sigma(t)\|^2t/2) where B_t is vector-valued Brownian motion and \sigma(t) is a vector valued function of time. The futures quotes \phi(t) can be observed in the market. If we have at-the-money call options with prices {E[\max\{f_t - f(t), 0\}D_t]} then the implied volatility gives us \|\sigma(t)\|. Swaption prices can be used to determine the convexity. A typical assumption is \sigma(t) = s(t)(\cos\alpha(t), \sin\alpha(t)) for some functions s(t) and \alpha(t). One felicitous feature of this parameterization is the futures quotes and options prices are independent of \alpha(t) since \|\sigma(t)\| = s(t).
There is no closed-form formula for calculating \operatorname{Cov}(f_t, D_t) in the LIBOR market model but there is for the Ho-Lee model.
The Ho-Lee model specifies a stochastic forward rate f_t = \phi(t) + \sigma(t)\cdot B_t where \phi(t) is the futures quote and \sigma(t) is the vector-valued volatility.
Exercise. Show for the Ho-Lee model with constant scalar volatility \sigma = \sigma(t) D(t) = \exp(-\int_0^t \phi(s)\,ds + \sigma^2 t^3/6)
Hint: Use E[\exp(N)] = \exp(E[N] + \operatorname{Var}(N)/2) for any normally distributed random variable and show {\operatorname{Var}(\int_0^t B_s\,ds) = \int_0^t \int_0^t \operatorname{Cov}(B_u, B_v)\,du\,dv = t^3/3}.
Since D(t) = \exp(-\int_0^t f(s)\,ds) we have f(t) = \phi(t) - \sigma^2 t^2/2.
Exercise. Show \operatorname{Cov}(N, f(M)) = E[f'(M)] \operatorname{Cov}(N, M) if N and M are jointly normal.
Hint: Use E[\exp(N) f(M)] = E[\exp(N)] E[f(M + \operatorname{Cov}(N,M))] if N and M are jointly normal, differentiate E[\exp(\alpha N) f(M)] with respect to \alpha, and set \alpha = 0.
Exercise. Show \operatorname{Cov}(f_t, D_t) = -D(t)\sigma^2 t^2/2.
Hint: Use the previous exercise.
Note this agrees with the previous convexity calculation.
On September 17, 2019 SOFR went from 2.43% to 5.25% and spiked to 10% at one point during the trading day. This was very disconcerting.↩︎