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Compound Rate Variance

The following is a basic rederivation of the so called 1/3 adjustment industry standard for compound rates.

Notations and Model Setup

Using HJM notations, we define a log libor rate $L_t$ observed at $t$ for the period from $T_s$ to $T_e$, where $t<T_s$ from instantaneous forward $f(t,u)$:

$$L_t = \frac{1}{T_e-T_s} \int_{T_s}^{T_e} f(t,u)du$$

A log compound rate $R_t$ has observation extending to $t<T_e$

$$R_t = \frac{1}{T_e-T_s} \int_{T_s}^{T_e} f(t \wedge u ,u)du$$

Where the risk neutral diffusion for $f(t,u)$ is given by HJM equation:

$$df(t,u) = \sigma(t,u)v(t,u) dt + \sigma(t,u) dW_t$$

The actual Libor $\tilde{L}$ and actual compound rate $\tilde{R}$ are linked to the log rates by the relation: \begin{eqnarray} \ln (1+ (T_e-T_s) \tilde{R}_t) &=& R_t (T_e-T_s) \\ \ln (1+ (T_e-T_s) \tilde{L}_t) &=& L_t (T_e-T_s) \end{eqnarray} A normal rate diffusion of $f(t,u)$ leads to a normal diffusion of $R_t$ and a shifted lognormal diffusion of $\tilde{R}_t$. We will focus on the distribution of $R_t$.

Derivation of the Log Rate Variance

As we are interested in variance, we will only track the diffusion terms and collect the drift terms into a a term $C$ that is deterministic when the forward volatility $\sigma(t,u)$ is deterministic. We have \begin{eqnarray} R_t &=& C + \frac{1}{T_e-T_s} \int_{u=T_s}^{T_e} \int_{t=T_0}^u \sigma(t,u) dW_t du \end{eqnarray}

We can distinguish 2 cases for a given calculate date $T_0$:

  • forward period $T_0<T_s$
  • current period $T_s<T_0<T_e$

Forward Period

if the period is future: $T_0<T_s$

\begin{eqnarray} R_t &=& C + \frac{1}{T_e-T_s} \int_{u=T_s}^{T_e} \int_{t=T_0}^u \sigma(t,u) dW_t du \\ &=& C + \frac{1}{T_e-T_s} \int_{u=T_s}^{T_e} \int_{t=T_0}^{T_s} \sigma(t,u) dW_t du + \frac{1}{T_e-T_s} \int_{u=T_s}^{T_e} \int_{t=T_s}^u \sigma(t,u) dW_t du \\ R_t &=& C + \frac{1}{T_e-T_s} \int_{t=T_0}^{T_s} \int_{u=T_s}^{T_e} \sigma(t,u) du dW_t + \frac{1}{T_e-T_s} \int_{t=T_s}^{T_e} \int_{u=t}^{T_e} \sigma(t,u) du dW_t
\end{eqnarray} assuming single factor, 0 mean reversion, constant volatility, $\sigma(t,u) := \sigma$, we get: \begin{eqnarray}R_t &=& C + \frac{1}{T_e-T_s} \int_{t=T_0}^{T_s} \sigma (T_e-T_s) dW_t + \frac{1}{T_e-T_s} \int_{t=T_s}^{T_e} \sigma (T_e-t) dW_t
\end{eqnarray} Using Ito's isometry: $$Var(R_t) = \sigma^2 (T_s-T_0) + \sigma^2 \frac{1}{3} (T_e-T_s)$$

Current Period

if the period is current: $T_s<T_0<T_e$

\begin{eqnarray} R_t &=& C + \frac{1}{T_e-T_s} \int_{u=T_0}^{T_e} \int_{t=T_0}^u \sigma(t,u) dW_t du \\ &=& C + \frac{1}{T_e-T_s} \int_{t=T_0}^{T_e} \int_{u=t}^{T_e} \sigma(t,u) du dW_t \\ &=& C + \int_{t=T_0}^{T_e} \sigma \frac{T_e-t}{T_e-T_s} dW_t
\end{eqnarray} Using Ito's isometry: $$Var(R_t) = \sigma^2 \frac{1}{3} (T_e-T_0) \left(\frac{T_e-T_0}{T_e-T_s} \right)^2$$

Expression as a Time Adjustment

The variance adjustment can be reexpressed as a time to expiry where the parameter $T$ to be put in black formula invoked with a calculation date $T_0$ for a period $[T_s, T_e]$ is given by: \begin{eqnarray} T_{0s} &=& (T_s-T_0)^+ \\ T_{0e} &=& (T_e-T_0)^+ = T_e - T_0 \\ T&=& T_{0s} + \frac{1}{3} (T_{0e} - T_{0s}) \left( \frac{T_{0e} - T_{0s}}{T_e-T_s} \right)^2 \end{eqnarray}