March  2022, 1(1): 81-97. doi: 10.3934/fmf.2021003

A rough SABR formula

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

Received  May 2021 Published  March 2022 Early access  June 2021

Fund Project: 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.

Citation: Masaaki Fukasawa, Jim Gatheral. A rough SABR formula. Frontiers of Mathematical Finance, 2022, 1 (1) : 81-97. doi: 10.3934/fmf.2021003
References:
[1]

E. AlòsJ. A. León and J. Vives, On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility, Finance Stoch., 11 (2007), 571-589.  doi: 10.1007/s00780-007-0049-1.

[2]

J. Andreasen and B. Huge, Expanded forward volatility., Risk, 1 (2013), 104-107. 

[3]

P. Balland, Forward Smile, Presentation at Global Derivatives, Paris, 2006.

[4]

C. BayerP. Friz and J. Gatheral, Pricing under rough volatility, Quant. Finance, 16 (2016), 887-904.  doi: 10.1080/14697688.2015.1099717.

[5]

M. BennedsenA. Lunde and M. S. Pakkanen, Hybrid scheme for Brownian semistationary processes, Finance Stoch., 21 (2017), 931-965.  doi: 10.1007/s00780-017-0335-5.

[6]

H. BerestyckiJ. Busca and I. Florent, Asymptotics and calibration of local volatility models, Quant. Finance, 2 (2002), 61-69.  doi: 10.1088/1469-7688/2/1/305.

[7]

H. BerestyckiJ. Busca and I. Florent, Computing the implied volatility in stochastic volatility models, Comm. Pure Appl. Math., 57 (2004), 1352-1373.  doi: 10.1002/cpa.20039.

[8]

O. El EuchM. FukasawaJ. Gatheral and M. Rosenbaum, Short-term at-the-money asymptotics under stochastic volatility models, SIAM J. Financial Math., 10 (2019), 491-511.  doi: 10.1137/18M1167565.

[9]

M. Forde and H. Zhang, Asymptotics for rough stochastic volatility models, SIAM J. Financial Math., 8 (2017), 114-145.  doi: 10.1137/15M1009330.

[10]

M. Fukasawa, Short-time at-the-money skew and rough fractional volatility, Quant. Finance, 17 (2017), 189-198.  doi: 10.1080/14697688.2016.1197410.

[11]

M. Fukasawa and A. Hirano, Refinement by reducing and reusing random numbers of the Hybrid scheme for Brownian semistationary processes, Quant. Finance, 21 (2021), 1127-1146.  doi: 10.1080/14697688.2020.1866209.

[12]

M. Fukasawa, B. Horvath and P. Tankov, Hedging under rough volatility, preprint, arXiv: 2105.04073.

[13]

P. S. HaganD. KumarA. S. Lesniewski and D. E. Woodward, Managing smile risk, Wilmott Magazine, 1 (2002), 84-108. 

[14]

Y. Osajima, The asymptotic expansion formula of implied volatility for dynamic SABR model and FX hybrid model., Available from: https://ssrn.com/abstract=965265. doi: 10.2139/ssrn. 965265.

[15]

D. Reiswich and U. Wystup, FXvolatility smile construction, Wilmott Magazine, 60 (2012), 58-69. 

show all references

References:
[1]

E. AlòsJ. A. León and J. Vives, On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility, Finance Stoch., 11 (2007), 571-589.  doi: 10.1007/s00780-007-0049-1.

[2]

J. Andreasen and B. Huge, Expanded forward volatility., Risk, 1 (2013), 104-107. 

[3]

P. Balland, Forward Smile, Presentation at Global Derivatives, Paris, 2006.

[4]

C. BayerP. Friz and J. Gatheral, Pricing under rough volatility, Quant. Finance, 16 (2016), 887-904.  doi: 10.1080/14697688.2015.1099717.

[5]

M. BennedsenA. Lunde and M. S. Pakkanen, Hybrid scheme for Brownian semistationary processes, Finance Stoch., 21 (2017), 931-965.  doi: 10.1007/s00780-017-0335-5.

[6]

H. BerestyckiJ. Busca and I. Florent, Asymptotics and calibration of local volatility models, Quant. Finance, 2 (2002), 61-69.  doi: 10.1088/1469-7688/2/1/305.

[7]

H. BerestyckiJ. Busca and I. Florent, Computing the implied volatility in stochastic volatility models, Comm. Pure Appl. Math., 57 (2004), 1352-1373.  doi: 10.1002/cpa.20039.

[8]

O. El EuchM. FukasawaJ. Gatheral and M. Rosenbaum, Short-term at-the-money asymptotics under stochastic volatility models, SIAM J. Financial Math., 10 (2019), 491-511.  doi: 10.1137/18M1167565.

[9]

M. Forde and H. Zhang, Asymptotics for rough stochastic volatility models, SIAM J. Financial Math., 8 (2017), 114-145.  doi: 10.1137/15M1009330.

[10]

M. Fukasawa, Short-time at-the-money skew and rough fractional volatility, Quant. Finance, 17 (2017), 189-198.  doi: 10.1080/14697688.2016.1197410.

[11]

M. Fukasawa and A. Hirano, Refinement by reducing and reusing random numbers of the Hybrid scheme for Brownian semistationary processes, Quant. Finance, 21 (2021), 1127-1146.  doi: 10.1080/14697688.2020.1866209.

[12]

M. Fukasawa, B. Horvath and P. Tankov, Hedging under rough volatility, preprint, arXiv: 2105.04073.

[13]

P. S. HaganD. KumarA. S. Lesniewski and D. E. Woodward, Managing smile risk, Wilmott Magazine, 1 (2002), 84-108. 

[14]

Y. Osajima, The asymptotic expansion formula of implied volatility for dynamic SABR model and FX hybrid model., Available from: https://ssrn.com/abstract=965265. doi: 10.2139/ssrn. 965265.

[15]

D. Reiswich and U. Wystup, FXvolatility smile construction, Wilmott Magazine, 60 (2012), 58-69. 

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
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