Cocycle Condition for FX and SABR

I was wondering whether SABR model (or some of its modifications) is actually used by practionarers.

Also, if one models the FX forward with SABR, would the cocycle condition be satisfied? That is, if $$FXF^{ccy1ccy2}$$ and $$FXF^{ccy2ccy3}$$ follow a SABR dynamics, does the same hold for $$FXF^{ccy1ccy3}$$? And how would the $$\alpha, \beta, \rho$$ parameters of the third forward read in terms of the ones of the other two? (here, I denote by $$FXF^{ccy1ccy2}$$ the FX forward rate wrt ccy1 and ccy2).

• It seems to me the forward condition tying the three currencies together has to hold at each moment in time (due to no arbitrage) regardless of the dynamics. Tha last part I don't know. Apr 10, 2021 at 17:17
• @noob2 thanks for your reply. Could you please expand on your intuition? It is clear to me that the cocycle condition must hold, what it is not clear is if SABR dynamics is "cocycle-closed", if you want... I do not see why also the third forward should follow a SABR dynamics (if it does, of which I am not sure) provided the first two do. Apr 10, 2021 at 20:21

Assuming $$X_t$$ and $$Y_t$$ are diffusion processes and $$Z_t=X_t Y_t$$, then $$dZ_t/Z_t = dX_t/X_t+dY_t/Y_t + ... dt$$ So if the diffusive parts of $$X_t$$, $$Y_t$$ and $$Z_t$$ are SABR, then $$\sigma^Z_t Z_t^{\beta^Z-1} dW^Z_t + ... dt = \sigma^X_t X_t^{\beta^X-1} dW^X_t + \sigma^Y_t Y_t^{\beta^Y-1} dW^Y_t + ... dt$$ and a first necessary condition is $$\beta^X = \beta^Y = \beta^Z = 1$$.
Next, you have $$\sigma^Z_t=\sqrt{(\sigma^X_t)^2 + (\sigma^Y_t)^2 + 2 \rho_{XY}\sigma^X_t \sigma^Y_t}$$ and
$$d\sigma^Z_t/\sigma^Z_t = ((\sigma^X_t)^2 d\sigma^X_t/\sigma^X_t + (\sigma^Y_t)^2 d\sigma^Y_t/\sigma^Y_t+\rho_{XY}\sigma^X_t \sigma^Y_t(d\sigma^X_t/\sigma^X_t+d\sigma^Y_t/\sigma^Y_t))/(\sigma^Z_t)^2 \\ + ... dt$$ and another necessary condition is that $$d\sigma^X_t/\sigma^X_t = d\sigma^Y_t/\sigma^Y_t$$, which in turns implies that if $$X_t$$, $$Y_t$$ and $$Z_t$$ are SABR, then $$\alpha^X=\alpha^Y=\alpha^Z$$, $$\rho_{XY}$$ is constant, and $$\sigma^X_t/\sigma^X_0 = \sigma^Y_t/\sigma^Y_0 = \sigma^Z_t/\sigma^Z_0$$ (all three stochastic volatilities are driven by the same brownian motion and have the same volvol).
The SABR parameters $$\rho^Z$$ is then computed as
$$\rho^Z = (\sigma^X_0 \rho^X + \sigma^Y_0 \rho^Y)/\sqrt{(\sigma^X_0)^2 + (\sigma^Y_0)^2 + 2 \rho_{XY}\sigma^X_0 \sigma^Y_0}$$ and an additional necessary condition is that $$|\rho^Z| \leq 1$$.