A rough SABR formula

Masaaki Fukasawa; Jim Gatheral

doi:10.3934/fmf.2021003

Article Contents

2022, Volume 1, Issue 1: 81-97. Doi: 10.3934/fmf.2021003 open access

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A rough SABR formula

Masaaki Fukasawa^1, and
Jim Gatheral^2, ,

1.
Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, 560-8531, Japan
2.
Baruch College, City University of New York, One Bernard Baruch Way, New York, NY 10010, USA

^* Corresponding author: Jim Gatheral
^* Corresponding author: Jim Gatheral

Received: April 30, 2021

Published: March 2022

M. Fukasawa is supported by KAKENHI 21K03369

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Abstract

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.

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Mathematics Subject Classification: 91G30.

Citation:

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

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

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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) $

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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

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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

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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

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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)

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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

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