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# A rough SABR formula

• * Corresponding author: Jim Gatheral

M. Fukasawa is supported by KAKENHI 21K03369

• Following an approach originally suggested by Balland in the context of the SABR model, we derive an ODE that is satisfied by normalized volatility smiles for short maturities under a rough volatility extension of the SABR model that extends also the rough Bergomi model. We solve this ODE numerically and further present a very accurate approximation to the numerical solution that we dub the rough SABR formula.

Mathematics Subject Classification: 91G30.

 Citation: • • Figure 1.  The function $f$ for $H = 1/2$ (in red) and $H = 0$ (in blue)

Figure 2.  The function $f$: numerical solutions for various values of $H$

Figure 3.  The function $f$ and its approximation $f_A$ for two values of $\rho$. $H = 0.05$ is in blue, $H = 0.25$ is in red; solid line is the numerical solution $f$ and dashed, the approximation $f_A(y)$

Figure 4.  With $\beta(s) = s$ and parameters $H = 0.05$, $\eta = 1$, the dashed red line is the numerical solution $f$; Monte Carlo estimates of normalized implied volatility $\Sigma(k,\tau)/{\Sigma(0,\tau)}$ for $\tau = 1,\,3,\,6$, and $12$ months are as in the legend

Figure 5.  With $\beta(s) = s$ and parameters $H = 0.10$, $\eta = 1$, the dashed red line is the numerical solution $f$; Monte Carlo estimates of normalized implied volatility $\Sigma(k,\tau)/{\Sigma(0,\tau)}$ for $\tau = 1,\,3,\,6$, and $12$ months are as in the legend

Figure 6.  With $\beta(s) = s$ and parameters $H = 0.20$, $\eta = 1$, the dashed red line is the numerical solution $f$; Monte Carlo estimates of normalized implied volatility $\Sigma(k,\tau)/{\Sigma(0,\tau)}$ for $\tau = 1,\,3,\,6$, and $12$ months are as in the legend

Figure 7.  With $\beta(s) = \sqrt{s}$ and parameters $H = 0.05$, $\eta = 1$, Monte Carlo estimates of normalized implied volatility $\Sigma(k,\tau)/{\Sigma(0,\tau)}$ for $\tau = 1.5,\,3,\,6$, and $12$ months are as in the legend. Dashed lines are corresponding plots of the rough SABR formula (15)

Figure 8.  With parameters $H = 0.1525$, $\eta = 26.66$ and $\rho = 0.0339$, normalized implied volatilities $\Sigma(y,\tau)/{\Sigma(0,\tau)}$ (five of these per expiry) are plotted against $y = y(k,\tau)$. Here, the $U(\tau)$ are approximated by ATM forward volatilities. Blue crosses correspond to 1 week and 2 week expirations; red crosses correspond to expirations 1 year and over. The green curve is the rough SABR formula $f_A(y) = |y|/\sqrt{G_A(y)}$ with these parameters

•  Open Access Under a Creative Commons license

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