SABR Brownian Motions Under the Forward Measure
Summary
The document asks whether the Brownian motions in the SABR stochastic differential equations are specified under the physical measure or a risk-neutral measure. Its answer explains that a forward rate is modeled as a martingale under the measure associated with the bond maturing at the rate’s end date. The corresponding numeraire is that zero-coupon bond, and the SABR dynamics should be written with Brownian motions under this forward measure.
This clarifies that the measure is tied to the modeled forward rate and its numeraire, rather than being universally P or Q. The discussion is brief and provides no derivation or broader reference beyond its stated forward-measure setup. It also flags that the volatility and rate dynamics should be written with the appropriate measure-indexed Brownian motions.
Key ideas
- A forward rate is modeled as a martingale under its associated forward measure.
- The associated numeraire is the zero-coupon bond maturing at the rate’s end date.
- SABR Brownian motions should be labeled with the measure under which the dynamics are defined.
- The document does not establish a universal measure convention for every SABR application.
Tags
Full text
# Under what measure is the SABR stochastic differential equations
# Under what measure is the SABR stochastic differential equations
The SABR Model is a CEV (constant elasticity of variance) Cox asset process with correlated lognormal stochastic volatility. A forward rate $F(t,T)$ to time $T$, observed at $t$, and the instantaneous volatility, $\sigma(t)$ follow the stochastic differential equations \begin{align} &dF(t,T)=\sigma(t)F(t,T)^\beta dW_F(t) \label{eq:true_sabr_model1} \\ &d\sigma(t)=\xi\sigma(t)dW_\sigma(t) \label{eq:true_sabr_model2} \end{align} where as the parameter $\rho$ represents the instantaneous correlation between the standard Brownian motions $W_F(t)$ and $W_\sigma(t)$ ($\langle dW_F(t)dW_\sigma(t)\rangle=\rho dt$).
My question is, are the Brownian motions in the SABR model under the physical measure $P$ or the risk-neutral measure $Q$? I can not find anything about it in the original paper. Can anyone help me with a reference to where it is stated explicitly?
## Answer by rvignolo (score 3, accepted)
https://quant.stackexchange.com/a/58261
The simple forward rate $F_n(t) = F(t, T_n, T_{n+1})$ is a martingale under the measure $Q^{T_{n+1}}$, which means that the associated numeraire is the zero coupon bond $P(t, T_{n+1})$.
In the SABR model, the forward rate $F_n(t)$ is assumed to evolve under the associated measure $Q^{T_{n+1}}$ according to:
\begin{aligned} dF_n(t) &= \sigma(t) \cdot F_n(t)^{\beta} \cdot dW^{Q^{T_{n+1}}}_n(t),\\ d\sigma(t) &= \xi \cdot \sigma(t) \cdot dZ^{Q^{T_{n+1}}}(t) \end{aligned}
Please, notice the differences with your equations.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.