Tom’s Question

Thank you for telling me exactly your definition of a constant maturity bond in the case of a zero coupon bond. If D_t(u) is the price at time t of a zero coupon bond maturing at u then the arbitrage-free value at time t of a constant maturity u zero is X_t = D_t(t + u).

Recall the value at time t of a zero coupon bond with maturity u is {D_t(u) = E_t[\exp(-\int_t^u f_s\,ds)]}, where f_t is the continuously compounded instantaneous forward rate and E_t denotes the conditional expectation given information at time t. We can write this as D_t(u) = \exp(-\int_t^u f_s(t)\,ds) where the implied forward as seen from time t is s\mapsto f_s(t), s \ge t.

Ho-Lee

The (one-factor) Ho-Lee model assumes the forward rate is f_t = \phi(t) + \sigma(t) B_t, where B_t is one-dimensional Brownian motion so we can find closed form solutions. In particular, if \phi and \sigma are constant we have D_t(u) = \exp(-(u - t)\phi + \sigma^2(u - t)^3/6 - \sigma(u - t)B_t). For a constant u maturity zero this gives D_t(t + u) = \exp(-u\phi + \sigma^2 u^3/6 - \sigma u B_t) Recall if N is normally distributed then {E[\exp(N)] = \exp(E[N] + \operatorname{Var}(N)/2)}. This implies \begin{aligned} E[D_t(t + u)] &= \exp(-u\phi + \sigma^2 u^3/6 + \sigma^2 u^2 t/2) \\ &= \exp(-u\phi + \sigma^2 u^2(u + 3t)/6) \\ \end{aligned} since \operatorname{Var}(B_t) = t.

Note {(\partial/\partial t)E[D_t(t + u)] = E[D_t(t + u)]\sigma^2 u^2/2} so the expected realized return, (\partial X_t/\partial t)/X_t = \partial \log X_t/\partial t, does not depend on t.

Exercise. Show E[f_t(t + u)] = \phi - \sigma^2 u^2/2.

This shows the convexity adjustment between forwards and futures is the expected realized return.

The realized return over the interval [t, t + \Delta t] is {R(t, \Delta t) = (X_{t + \Delta t} - X_t)/X_t = X_{t + \Delta t}/X_t - 1}.

\Delta\log X_t = \log X_{t + \Delta t} - \log X_t

Using X_t = D_t(t + u) we have R = X_{t + \Delta t}/X_t - 1 = \exp(-\sigma u\Delta B_t) - 1 so E[R] = \exp(\sigma^2 u^2\Delta t/2) - 1 \approx \sigma^2 u^2\Delta t/2.

Assume prices (X_t) and cash flows (C_t), t\ge 0, indexed by market tradeable instruments. Given a finite trading strategy (\tau_j,\Gamma_j) let \Gamma_t = \Gamma_j 1(t = \tau_j) so the position at time t is \sum_{s < t}\Gamma_s. The trade blotter will show mark-to-market value V_t = (\Delta_t + \Gamma_t)\cdot X_t and amounts A_t = \Delta_t\cdot C_t - \Gamma_t\cdot X_t.

t	X	C	\Gamma	\Delta	V	A
[0, \tau_0)	X_t	C_t	0	0	0	0
\tau_0	X_0	C_0	\Gamma_0	0	\Gamma_0\cdot X_0	-\Gamma_0\cdot X_0
(\tau_0, \tau_1)	X_t	C_t	0	\Gamma_0	\Gamma_0\cdot X_t	\Gamma_0\cdot C_t
\tau_1	X_1	C_1	\Gamma_1	\Gamma_0	(\Gamma_0+\Gamma_1)\cdot X_0	\Gamma_0\cdot C_1 - \Gamma_1\cdot X_1
(\tau_1, \tau_2)	X_t	C_t	0	\Gamma_0 + \Gamma_1	(\Gamma_0 + \Gamma_1)\cdot X_t	(\Gamma_0 + \Gamma_1)\cdot C_t
\cdots	\cdots	\cdots	\cdots	\cdots	\cdots	\cdots
\tau_n	X_n	C_n	\Gamma_n	\sum_{j < n}\Gamma_j	(\sum_{j\le n}\Gamma_j)\cdot X_0	(\sum_{j < n} \Gamma_j)\cdot C_n - \Gamma_n\cdot X_n
(\tau_n, \infty)	X_t	C_t	0	\sum_{j\le n}\Gamma_j	(\sum_{j\le n} \Gamma_j)\cdot X_t	(\sum_{j\le n})\cdot C_t

The strategy is closed out at \tau_n if \Gamma_n = -\sum_{j < n}\Gamma_j so \sum_{j\le n}\Gamma_j = 0. In this case the value V_{\tau_n} = 0 and the amount A_{\tau_n} = -\Gamma_n\cdot C_{\tau_n} - \Gamma_n\cdot X_{\tau_n}. We then have V_t = A_t = 0 for t > \tau_n.

This table can be simplified if we let \Delta_{+j} at \tau_j be the position over (\tau_j,\tau_{j+1}] so {\Delta_{+j} = \Delta_{+(j-1)} + \Gamma_j}. We now only need the t = \tau_j rows. We can also eliminate the non-informative X and C columns.

t	\Gamma	\Delta_+	V	A
\tau_0	\Gamma_0	\Gamma_0	\Delta_{+0}\cdot X_0	-\Gamma_0\cdot X_0
\tau_1	\Gamma_1	\Delta_{+0} + \Gamma_1	\Delta_{+1}\cdot X_1	\Delta_{+0}\cdot C_1 - \Gamma_1\cdot X_1
\cdots	\cdots	\cdots	\cdots	\cdots
\tau_n	\Gamma_n	0	0	-\Gamma_n\cdot(C_n + X_n)

In general, V_j = \Delta_{+j}\cdot X_j and A_j = \Delta_{+(j-1}}\cdot C_j - \Gamma_j\cdot X_j for 0 < j < n.

Ho-Lee

Replication