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Semi-analytic pricing of double barrier options with time-dependent barriers and rebates at hit
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 |
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.
References:
[1] |
E. Alòs, J. 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. Bayer, P. Friz and J. Gatheral,
Pricing under rough volatility, Quant. Finance, 16 (2016), 887-904.
doi: 10.1080/14697688.2015.1099717. |
[5] |
M. Bennedsen, A. 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. Berestycki, J. 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. Berestycki, J. 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 Euch, M. Fukasawa, J. 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. Hagan, D. Kumar, A. 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. Alòs, J. 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. Bayer, P. Friz and J. Gatheral,
Pricing under rough volatility, Quant. Finance, 16 (2016), 887-904.
doi: 10.1080/14697688.2015.1099717. |
[5] |
M. Bennedsen, A. 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. Berestycki, J. 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. Berestycki, J. 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 Euch, M. Fukasawa, J. 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. Hagan, D. Kumar, A. 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.
|






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