Yield Curve Models

Models

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.

Zero Coupon Bond

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.

LIBOR Market Model

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.

Ho-Lee

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.