Masaaki Fukasawa / 深澤 正彰

Professor / 教授

Graduate School of Engineering Science / 基礎工学研究科
Osaka University / 大阪大学

560-8531 JAPAN

Co-Editor of Finance and Stochastics
Managing Editor of Quantitative Finance


Dr. Fukasawa or : how I learned to stop worrying and love the skew

I am a Professor of Mathematics at Osaka University. Looking back, I notice that "skew" has been a keyword underlying my research so far. My first project was concerned with the Edgeworth expansion for ergodic diffusions [1]. The Edgeworth expansion is a combination of the central limit thereom and the Taylor expansion. It aims at improving a normal approximation to a probability distribution based on the central limit theorem. The Edgeworth correction to the approximation amounts to incorporating the skewness of the distribution. The skewness is a measure of asymmetry and defined as the normalized third moment.

When I got a position at Center for the Study of Finance and Insurance, Osaka University and touched state of the art in Mathematical Finance, soon I noticed that my Edgeworth result could provide a rigorous justification to a singular perturbation method for option pricing and calibration [6]. The calibration is a procedure to make a model match the market prices. The market prices are translated to the Black-Scholes implied volatility surface. The surface is flat if and only if the underlying asset price of the options is log-normally distributed. Actually it is not flat and the slope is called the volatility skew. Some of my works [6,7,17,21,27] have related the volatility skew to the skewness of the price distribution under a pricing measure through asymptotic expansions.

The skewness is significant not only in the price distribution under the pricing measure but also in the historical return distribution. This is due to the correlation between price movements and transaction times. I found that this skewness (and the kurtosis) determines the asymptotic distribution of volatility estimators [2,4,5,12] under high frequency observations.

The limit theorem of volatility estimators was extended to that of discretization errors in stochastic integration [8]. I showed that the skewness and kurtosis of the increments of the integrand determine the asymptotic error distribution. Then, an interesting question arises: what is the most efficient way of discretizing a stochastic integral in the sense that the limit distribution is the most concentrated around zero ? I solved this by using Kurtosis-Skewness inequalities for general random variables : one of them is classical and due to Pearson. I obtained a family of inequalities in [8,16], which in particular generalizes Pearson's inequality.


A stochastic integral represents the Profit-and-Loss of a trading strategy in the context of Mathematical Finance. The optimal discretization problem of stochastic integrals is then translated to the optimal discrete hedging problem in financial engineering. This answers the question how the optimal continuous-time strategy in theory is efficiently realized in practice by a finite number of transactions [3,8,9,16,19].

More recently I leaned to love the rough [7,21,23,26,27,28,31,33,38,39,40,42,43]. The figure below represents a volatility surface generated by the rough SABR formula [38].



[1] Edgeworth expansion for ergodic diffusions, Probab. Theory Relat. Fields 142 (2008), 1-20. minor typo

[2] 実現ボラティリティの漸近分布について, 統計数理 Proc. Inst. Stat. Math. 57 (2009), no.1, 3-16.

[3] 離散ヘッジ戦略の漸近有効性, 金融工学研究所懸賞論文集 (2009).

[4] Central limit theorem for the realized volatility based on tick time sampling, Finance Stoch. 14 (2010), 209-233.

[5] Realized volatility with stochastic sampling, Stochastic Process. Appl. 120 (2010), 829-852.

[6] Asymptotic analysis for stochastic volatility: Edgeworth expansion, Electronic J. Probab. 16 (2011), 764-791.

[7] Asymptotic analysis for stochastic volatility: martingale expansion, Finance Stoch. 15 (2011), 635-654.

[8] Discretization error of stochastic integrals, Ann. Appl. Probab. 21 (2011), 1436-1465.

[9] Asymptotically efficient discrete hedging, Stochastic Analysis with Financial Applications, Progress in Probability 65 (2011), 331-346.

[10] (with I. Ishida, N. Maghrebi, K. Oya, M. Ubukata and K. Yamazaki) Model-free implied volatility: from surface to index, IJTAF 14 (2011), no.4, 433-463.

[11] The normalizing transformation of the implied volatility smile, Math. Finance 22 (2012) 753-762.
-- a preliminary version: Normalization for implied volatility, arXiv:1008.5055

[12] (with M. Rosenbaum) Central limit theorems for realized volatility under hitting times of an irregular grid, Stochastic Process. Appl. 122 (2012), 3901-3920.

[13] Conservative delta hedging under transaction costs, Recent Advances In Financial Engineering 2011, 55-72, World Scientific (2012).

[14] Limit theorems for random walks under irregular conductance, Proc. Japan Acad. Ser. A Math. Sci. 89 (2013), no. 8, 87–91.

[15] (with T. Arai) Convex risk measures for good deal bounds, Math. Finance 24 (2014), 464-484.

[16] Efficient discretization of stochastic integrals, Finance Stoch. 18 (2014), 175-208.

[17] Volatility derivatives and model-free implied leverage, IJTAF 17 (2014), no.1, 1450002.

[18] (with A. Brouste, H. Hino, S. Iacus, K. Kamatani, Y. Koike, H. Masuda, R. Nomura, T. Ogihara, Y. Shimizu, M. Uchida and N. Yoshida) The YUIMA project: a computational framework for simulation and inference of stochastic differential equations, Journal of Statistical Software 57 (2014), Issue 4.

[19] (with J. Cai, M. Rosenbaum and P. Tankov) Optimal discretization of hedging strategies with directional views, SIAM J. Finan. Math. 7 (2016), 34-69.

[20] (with J. Cai) Asymptotic replication with modified volatility under small transaction costs, Finance Stoch. 20 (2016), 381-431.

[21] Short-time at-the-money skew and rough fractional volatility, Quant. Finance 17 (2017), No. 2, 189–198.

[22] 高頻度データに対する Whittle 推定, 統計数理 Proc. Inst. Stat. Math. 65 (2017), no.1, 71-85.

[23] (with O. El Euch and M. Rosenbaum) The microstructual foundations of leverage effect and rough volatility, Finance Stoch. 22 (2018), 241-280.

[24] (with M. Stadje) Perfect hedging under endogenous permanent market impacts, Finance Stoch. 22 (2018), 417-442.

[25] (with B. Bouchard, M. Herdegen and J. Muhle-Karbe) Equilibrium returns with transaction costs, Finance Stoch. 22 (2018), 569-601.

[26] (with A. Brouste) Local asymptotic normality property for fractional Gaussian noise under high-frequency observations, Ann. Statist. 46 (2018), 2045-2061.

[27] (with O. El Euch, J. Gatheral and M. Rosenbaum) Short-term at-the-money asymptotics under stochastic volatility models, SIAM J. Finan. Math. 10 (2019), 491-511.

[28] (with T. Takabatake) Asymptotically efficient estimators for self-similar stationary Gaussian noises under high frequency observations, Bernoulli 25 (2019), 1870-1900.

[29] (with J. Obloj) Efficient discretisation of stochastic differential equations, Stochastics 92 (2020), 833-851.

[30] (with E. Alos) The asymptotic expansion of the regular discretization error of Itô integrals, Math. Finance 31 (2021), 323-365.

[31] Volatility has to be rough, Quant. Finance 21 (2021), 1-8.

[32] EM algorithm for stochastic hybrid systems, Stat. Inference Stoch. Process. 24 (2021), 223-239.

[33] (with A. Hirano) Refinement by reducing and reusing random numbers of the Hybrid scheme for Brownian semistationary processes, Quant. Finance 21 (2021), 1127-1146.

[34] (with K. Matsushita) Realized cumulants for martingales, Electron. Commun. Probab. 26 (2021), article no. 12, 1-10.

[35] (with M. Ikeda) A new discretization scheme for one dimensional stochastic differential equations using time change method, Electron. Commun. Probab. 26 (2021), article no. 51, 1-12.

[36] (with M. Ohnishi and M. Shimoshimizu) Discrete–time optimal execution under a generalized price impact model with Markovian exogenous orders, IJTAF 24 (2021), no. 5, 2150025.

[37] (with H. Maeda and J. Sekine) On optimal thresholds for pairs trading in a one-dimensional diffusion model, ANZIAM J. 63 (2021), 104-122.

[38] (with J. Gatheral) A rough SABR formula, Frontiers of Mathematical Finance, 1 (2022), 81-97.

[39] (with M. Forde, S. Gerhold and B. Smith) The Riemann-Liouville field and its GMC as H -> 0, and skew flattening for the rough Bergomi model, Statistics & Probability Letters 181 (2022), 109265.

[40] (with C. Bayer and S. Nakahara) On the weak convergence rate in the discretization of rough volatility models, SIAM J. Finan. Math. 13 (2022), SC66-SC73.

[41] (with H. Chau and M. Rasonyi) Super-replication with transaction costs under model uncertainty for continuous processes, Math. Finance 32 (2022), 1066-1085.

[42] (with T. Takabatake and R. Westphal) Consistent estimation for fractional stochastic volatility model under high-frequency asymptotics (Is Volatility Rough ?), Math. Finance. 32 (2022), 1086-1132. Online suppplement

[43] On asymptotically arbitrage-free approximations of the implied volatility, to appear in Frontiers of Mathematical Finance.


[1] Volatility Index Japan - Model-free implied volatility of NIKKEI stock average

[2] YUIMA Project - R package for simulation and estimation of SDE

Curriculum Vitae


1997 Apr - 2000 Mar / Kunitachi Senior High School / 東京都立国立高校

2000 Apr - 2002 Mar / College of Arts and Sciences, the University of Tokyo / 東京大学教養学部理科II類

2002 Apr - 2004 Mar / Department of Mathematics, the University of Tokyo / 東京大学理学部数学科

2004 Apr - 2007 Dec / Graduate School of Mathematical Sciences, the University of Tokyo / 東京大学大学院数理科学研究科


2006 Apr - 2007 Dec / JSPS research fellow (DC1), the University of Tokyo

2007 Dec - 2010 Dec / Assistant Professor, Center for the Study of Finance and Insurance, Osaka University

2010 Dec - 2011 May / Researcher (Visiting Professor), FIM, ETH Zurich

2011 Jun - 2016 Mar / Associate Professor, Department of Mathematics, Osaka University

2016 Apr - present / Professor, Graduate School of Engineering Science, Osaka University


2011 Jun - 2018 Jun / Associate Editor of Finance and Stochastics, Springer

2013 Jan - present / Managing Editor of Quantitative Finance, Taylor and Francis

2014 Apr - 2018 Dec / Associate Editor of Japan Journal of Industrial and Applied Mathematics, Springer

2016 Dec - present / Editorial board member of Asymptotic Analysis, IOP Press

2017 Jan - 2019 Dec / Associate Editor of SIAM Journal of Financial Mathematics, SIAM

2017 Oct - present / Associate Editor of Osaka Journal of Mathematics, Osaka University and Osaka City University

2018 Jul - present / Co-Editor of Finance and Stochastics, Springer


2009 Dec / Ph.D in Mathematical Sciences, the University of Tokyo / 博士(数理科学)東京大学

2010 Sep / Takebe prize, the Mathematical Society of Japan / 日本数学会賞建部賢弘賞奨励賞

M. FUKASAWA WEB, last updated on 28 Sep 2022