April 25, 2024
The zero coupon bond D(u) pays 1 unit at time t We will also use D(t) for the price, or discount, of the bond. It is the cost of moving 1 unit at t back to today. Equivalently, 1 unit invested now can be redeemed at the realized return R(t) = 1/D(t) at time t. The forward realized return over the period from time t to time u is denoted R(t,u), where a units invested at t can be redeemed for aR(t,u) at time u. The forward discount over the period from t to u is D(t,u) = 1/R(t,u). It is the price of a zero coupon bond at time t paying 1 unit at u. Note R(t) = R(0,t) and D(t) = D(0,t).
What economists call the law of one price can be used to show how the realized return R(t,u) at time t of recieving one unit at u > t is determined by today’s realized return R(t) = R(0,t). We can invest one unit today and receive R(t) at time t < u, then reinvest that at time t to get R(t)R(t,u) at time u. Investing one unit today pays R(u) at time u. This shows R(u) = R(t)R(t,u) so R(t,u) = R(u)/R(t). In terms of discounts D(t,u) = D(u)/D(t).
Mathematicians like to work with discounts but traders prefer to work in terms of rates. There are various conventions for converting rates to discounts.
The spot rate, or yield, r(t) is determined by D(t) = \exp(-tr(t)). It is the constant rate that recovers the price of the zero coupon bond maturing at t.
The forward rate f(t) is determined by D(t) = \exp(-\int_0^t f(s)\,ds).
Exercise. Show r(t) = (1/t)\int_0^t f(s)\,ds.
The spot rate is the average of the forward rate.
Exercise. Show f(t) = r(t) + tr'(t).
Note f(t) = r(t) when r'(t) = 0. If r(t) has local bumps this formula magnifies those when its derivative is large. It is numerically more stable to work with forward rates when implementing these on a computer.
Exercise. Show any one of D(t), r(t), and f(t) determine the other two.
The forward spot r(t,u) is defined by D(t,u) = \exp(-(u - t)r(t,u)).
Exercise. Use D(t,u) = D(u)/D(t) to show r(t,u) = (ur(u) - tr(t))/(u - t).
The forward forward f(t,u) is defined by D(t,u) = \exp(-\int_t^u f(t, s)\,ds).
Exercise. Use D(t,u) = D(u)/D(t) to show f(t,u) = f(u), u\ge t.
Note f(t,t) = f(t).
It is simple to convert the forward today into the forward forward at t, just chop off the interval [0,t].
Model with stochastic interest rates are completly determined by the continuously compounded short rate, f_t. It is a random variable that corresponds to the unknown repurchase agreement rate at time t. A repurchase agreement specifies a time t, an interval \Delta t, and a rate r_t known at time t. One unit invested at time t pays 1 + r_t\Delta t at t + \Delta t. There exists f_t, known at time t, with 1 + r_t\Delta t = e^{f_t\Delta t}. Since 1 + x \approx e^x for small x, r_t\approx f_t.
Exercise. Show f_t = \log(1 + r_t\Delta t) /\Delta t.
Note since \log(1 + x)\approx x for small x this also shows f_t\approx r_t.
In the continuous time limit, one unit invested at time 0 and rolled over at the short/repo rate accrues to R_t = e^{\int_0^t f_s\,ds}. This instrument is the money market account. It can be used to borrow or lend money to fund trading strategies. The stochastic discount is D_t = 1/R_t.
Let D_t(u) be the price at time t of a zero coupon bond maturing at u. By the Fundamental Theorem of Arbitrage Pricing, D_t(u)D_t = E_t[D_u] so D_t(u) = E_t[D_u/D_t] = E_t[\exp(-\int_t^u f_s\,ds)].
The stochastic forward curve f_t(u) is defined by D_t(u) = \exp(-\int_t^u f_t(s)\,ds). It is the forward curve given information at time t.