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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?

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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.

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    $\begingroup$ Today I learn. Could you add the references you gave to this answer? $\endgroup$
    – Bob Jansen
    Sep 26 '20 at 14:59
  • $\begingroup$ Yes of course. A paper from the authors, here, in page 2, eq. (2.1.c). Damiano Brigo and Fabio Mercurio book: Interest Rate Models - Theory and Practice, page 508. Also, Andersen and Piterbarg book: Interest Rate Modeling, Lemma 4.2.3 in page 174. Thank you! $\endgroup$
    – rvignolo
    Sep 26 '20 at 15:08
  • $\begingroup$ Also shown for equity options in section 1.3.1 here $\endgroup$
    – Klein
    Sep 26 '20 at 15:38

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