Conditions for FX Triangles to Preserve SABR Dynamics
Summary
The document asks whether currency forwards that satisfy SABR dynamics individually will retain that form when combined through an FX triangle. Treating the cross rate as the product of two diffusion processes, the answer derives restrictions from matching their diffusion and stochastic-volatility terms. Under the stated setup, the three rates must have beta equal to one and share the same alpha and proportional volatility evolution; the correlation between the component rates must also be constant. The cross rate’s correlation parameter is then determined from the component volatilities and their correlations, and must remain within its valid range.
The response says these conditions are both necessary and sufficient, but highly restrictive. It distinguishes using SABR as a dynamic model from using it as a sparse parameterization of implied-volatility smiles. Separate SABR fits for each currency pair can therefore still be used for smile description even when they do not form a consistent joint dynamic model. The answer also notes that another smile parameterization may be more common in FX practice.
Key ideas
- A cross rate formed as the product of two diffusion processes inherits restrictions if all three rates are to follow SABR dynamics.
- The stated compatibility conditions require beta equal to one and matching stochastic-volatility behavior across the rates.
- The cross-rate correlation depends on component volatilities and correlations and must be within its valid range.
- The compatibility conditions are restrictive, while separate SABR fits can still parameterize individual FX volatility smiles.
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# Cocycle Condition for FX and SABR
# 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).
## Answer by Antoine Conze (score 2)
https://quant.stackexchange.com/a/63366
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 $|\rho^Z| \leq 1$.
It's easily checked that put together, these necessary conditions are also sufficient, but it's all rather restrictive.
That being said, SABR is usually not used as a model but rather as a sparse parameterization of the implied volatility smile, so there is nothing wrong with fitting distinct SABR to each of the currency pair in the triangle, although the vanna-volga parameterization might be more popular for FX smiles.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.