Compound Rate Variance
The following is a basic derivation of the so-called 1/3 adjustment industry standard for compound rates. This formula became important as LIBOR gets replaced with overnight compound rates, following regulatory action against banks fixing LIBOR during the Great Financial Crisis.
HJM Model Setup
These forward rates are defined from the zero coupon bond $P(t,T)$ by the equation: $$P(t,T) = \exp\left( -\int_{t}^{T} f(t,u)du\right)$$
The Heath-Jarrow-Morton (HJM) framework prescribes the evolution of the forward rate curve under the risk-neutral measure. The HJM 1-factor SDE is: $$df(t,T) = \sigma(t,T)v(t,T) dt + \sigma(t,T) dW_t$$ where $v(t,T) = \int_t^T \sigma(t,u) du$.
Notations: log-libor and log-compound rate
We define a log libor rate $L_t$ observed at $t$ for the period from $T_s$ to $T_e$, where $t $$L_t = \frac{1}{T_e-T_s} \int_{T_s}^{T_e} f(t,u)du$$ A log compound rate $R_t$ can be estimated for any $t \leq T_e$. The forward rate $f(t,u)$ is $\mathcal{F}_t$ adapted, and diffuses up to time $t=u$, where $f(u,u)=r_u$ is the short rate, which is $\mathcal{F}_u$ measurable. Unlike libor, diffusion happens even after $t>T_s$. To reflect that rate diffusion happens from $T_0$ until $t=u$ for a compound rate, the notation $f(t \wedge u ,u)$ need to be used: $$R_t = \frac{1}{T_e-T_s} \int_{T_s}^{T_e} f(t \wedge u ,u)du$$ 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} As $R_t(T_e-T_s)$ is often small, $\tilde{R}_t$ and $R_t$ usually have similar values and range. From a theoretical point of view, 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 $R_t$ which is more tractable. As we are interested in the variance of $R_t$ at the exercise date $t=T_x=T_e$, starting the diffusion at a date $T_0$, we will only track the diffusion terms and collect the drift terms into a term $C$ that is deterministic when the forward volatility $\sigma(t,u)$ is deterministic. We have \begin{eqnarray} R_{T_x} &=& C + \frac{1}{T_e-T_s} \int_{u=T_s}^{T_e} \int_{t=T_0}^{u \wedge T_x} \sigma(t,u) dW_t du \end{eqnarray} We can distinguish 2 cases for a given calculation date $T_0$: The second case is relevant for pricing options such as caps or floors after they have been traded. if the rate period is future: $T_0 \begin{eqnarray} R_{T_e} &=& 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 \\\\ &=& 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} We used the stochastic Fubini theorem to express the compound rate as an Ito integral, this enables us to compute the compound rate variance under a single factor, 0 mean reversion, constant volatility assumption, $\sigma(t,u) := \sigma$, we get: \begin{eqnarray} R_{T_e} &=& 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: $$ Var(R_{T_e}) = \frac{1}{(T_e-T_s)^2} \int_{t=T_0}^{T_s} \sigma^2 (T_e-T_s)^2 dt + \frac{1}{(T_e-T_s)^2} \int_{t=T_s}^{T_e} \sigma^2 (T_e-t)^2 dt $$ $$Var(R_{T_e}) = \sigma^2 (T_s-T_0) + \sigma^2 \frac{1}{3} (T_e-T_s)$$ The 1/3 factor arises from integrating a squared linear term over time, reflecting the average variance contribution over the period. if the period is current: $T_s \begin{eqnarray} R_{T_e} &=& 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} We also used the stochastic Fubini theorem and the constant variance assumption as in the previous sections. Using Ito's isometry: $$ Var(R_{T_e}) = \int_{t=T_0}^{T_e} \sigma^2 \left[\frac{T_e-t}{T_e-T_s}\right]^2 dt $$ $$Var(R_{T_e}) = \sigma^2 \frac{1}{3} (T_e-T_0) \left(\frac{T_e-T_0}{T_e-T_s} \right)^2$$ The variance adjustment is an adjustment of $\sigma^2 T$ to be used in the Black normal formula. It 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&=& T_{0s} + \frac{1}{3} (T_{0e} - T_{0s}) \left[ \frac{T_{0e} - T_{0s}}{T_e-T_s} \right]^2 \end{eqnarray} where $(x)^+ = \max(x,0)$ denotes the positive part. LIBOR-based options had a thriving market, while compound options were seldom traded. This was causing the rates volatility surface to be priced from LIBOR instruments while compound rate options used the same volatility input with the 1/3 adjustment. Once compound options became the standard, the 1/3 formula is used both in option pricing and calibration. The 1/3 adjustment bridges the variance of compound rates to the Black model, facilitating the shift from LIBOR to overnight compound rates in option pricing and volatility calibration.Derivation of the Log Rate Variance
Forward Period
Current Period
Expression as a Time Adjustment