Masaaki Fukasawa
Following-up Fukasawa and Gatheral (Frontiers of Mathematical Finance, 2022), we prove that the BBF formula, the SABR formula, and the rough SABR formula provide asymptotically arbitrage-free approximations of the implied volatility under, respectively, the local volatility model, the SABR model, and the rough SABR model.
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The article describes a global and arbitrage-free parametrization of the eSSVI surfaces introduced by Hendriks and Martini in 2019. A robust calibration of such surfaces has already been proposed by the quantitative research team at Zeliade in 2019, but it is sequential in expiries and lacks of a global view on the surface. The alternative calibration suggested in this article is faster and always guarantees an arbitrage-free fit of market data.
Monotonicity of Normalized Implied-Volatility Coordinates under No-Arbitrage
本文通过离散无套利证明，在Black-Scholes和Bachelier模型中，归一化隐含波动率坐标关于执行价格单调递减，并推导了正态方差恒等式。
WSVI: A Dimensionless Shape Family for Implied Volatility and Its Static No-Arbitrage Structure
本文提出一种新的隐含波动率参数族WSVI，解决了eSSVI无法生成二元事件临近到期期权W型微笑的问题，构建了其静态无套利结构并明确了相关条件。
In this paper, Malliavin calculus is applied to arrive at exact formulas for the difference between the volatility swap strike and the zero vanna implied volatility for volatilities driven by fractional noise. To the best of our knowledge, our estimate is the first to derive the rigorous relationship between the zero vanna implied volatility and the volatility swap strike. In particular, we will see that the zero vanna implied volatility is a better approximation for the volatility swap strike than the ATMI.
An Optimal Transport approach to arbitrage correction: application to Volatility Stress-Tests
本文提出一种基于最优传输的套利修正方法，用于监管压力测试中的波动率曲面扭曲问题，通过正则化技术与Sinkhorn算法实现高效收敛。
The measures of roughness of the volatility in the litterature are based on the realized volatility of high frequency data. Some authors show that this leads to a biased estimate, and does not necessarily indicate roughness of the underlying volatility process. Here, we attempt to measure the roughness of the implied volatility of short term options, as well as of the VIX index, and evaluate whether they may be more appropriate proxies of the underlying instant volatility.
In this note, Black--Scholes implied volatility is expressed in terms of various optimisation problems. From these representations, upper and lower bounds are derived which hold uniformly across moneyness and call price. Various symmetries of the Black--Scholes formula are exploited to derive new bounds from old. These bounds are used to reprove asymptotic formulae for implied volatility at extreme strikes and/or maturities.
On the implied volatility of Asian options under stochastic volatility models
本文利用 Malliavin 分析推导固定执行价算术亚式期权在短到期下的平价隐含波动率水平及偏斜渐近公式，并在 SABR 与 rough Bergomi 模型中通过数值模拟验证其准确性。
Market impact is the link between the volume of a (large) order and the price move during and after the execution of this order. We show that under no-arbitrage assumption, the market impact function can only be of power-law type. Furthermore, we prove that this implies that the macroscopic price is diffusive with rough volatility, with a one-to-one correspondence between the exponent of the impact function and the Hurst parameter of the volatility. Hence we simply explain the universal rough behavior of the volatility as a consequence of the no-arbitrage property. From a mathematical viewpoin…
Exact relationships between the short time-to-maturity ATM implied volatility slope, the (dual) volatility swap, and the (dual) zero vanna implied volatility are given.
On the implied volatility of European and Asian call options under the stochastic volatility Bachelier model
本文针对随机波动率 Bachelier 模型，运用 Malliavin 分析推导欧式与算术亚式看涨期权短期平值隐含波动率水平及偏斜渐近公式，并在 SABR 与 fractional Bergomi 模型中验证。
Arbitrage-Aware Probabilistic Multi-Step Forecasting of Implied Volatility Surfaces: Joint Generation of Surface-Return Trajectories Using Latent Diffusion
针对隐含波动率曲面的多步预测问题，提出条件潜在扩散框架，结合考虑套利的自编码器与扩散模型，在SPX数据集上生成逼真多步场景且点预测性能优于持续性基准。
On the implied volatility of Inverse options under stochastic volatility models
本文研究随机波动率模型下反向期权平值隐含波动率的短期行为，利用Malliavin微积分推导出隐含波动率水平与偏斜的短到期渐近公式，并将其应用于SABR、分数Bergomi模型及比特币期权实证分析。
Estimating the Hurst parameter from the zero vanna implied volatility and its dual
本文针对短到期期限下资产收益率与已实现波动率的协方差近似问题，推导了适用于H∈(0,1)随机波动率模型的极限定理，建立相关关联，提出仅用离散隐含波动率估计Hurst参数的方法，并给出粗糙Bergomi模型下的数值验证。
Latent Flow Matching for Arbitrage-Aware Implied Volatility Surface Generation
该研究提出套利感知潜在流匹配框架，经正则化变分自编码器压缩波动率曲面，结合流匹配模型生成符合无套利条件的隐含波动率曲面，在分布相似性、尾部保留等方面表现良好。
Discovering parametrizations of implied volatility with symbolic regression
本文通过符号回归方法，从市场数据中发现隐含波动率切片的参数化表示，与SVI参数化相比，展示了简洁且拟合性能优异的公式。
Volatility Parametrizations with Random Coefficients: Analytic Flexibility for Implied Volatility Surfaces
本文提出一种无套利框架，使参数化隐含波动率公式中的参数成为随机变量，从而增强隐含波动率曲面的建模灵活性。
We design three continuous--time models in finite horizon of a commodity price, whose dynamics can be affected by the actions of a representative risk--neutral producer and a representative risk--neutral trader. Depending on the model, the producer can control the drift and/or the volatility of the price whereas the trader can at most affect the volatility. The producer can affect the volatility in two ways: either by randomizing her production rate or, as the trader, using other means such as spreading false information. Moreover, the producer contracts at time zero a fixed position in a Euro…
The Price of Liquidity: Implied Volatility of Automated Market Maker Fees
本文将AMM的再平衡损失重构为隐含费用流，并设计费用固定换浮动互换以提取数字资产的隐含波动率与相关性，再用多市场已实现费用实证验证。
Several asymptotic results for the implied volatility generated by a rough volatility model have been obtained in recent years (notably in the small-maturity regime), providing a better understanding of the shapes of the volatility surface induced by rough volatility models, and supporting their calibration power to S&P500 option data. Rough volatility models also generate a local volatility surface, via the so-called Markovian projection of the stochastic volatility. We complement the existing results on the implied volatility by studying the asymptotic behavior of the local volatility surfac…
We explore the abilities of two machine learning approaches for no-arbitrage interpolation of European vanilla option prices, which jointly yield the corresponding local volatility surface: a finite dimensional Gaussian process (GP) regression approach under no-arbitrage constraints based on prices, and a neural net (NN) approach with penalization of arbitrages based on implied volatilities. We demonstrate the performance of these approaches relative to the SSVI industry standard. The GP approach is proven arbitrage-free, whereas arbitrages are only penalized under the SSVI and NN approaches. …
What kind of implied volatility extrapolation is appropriate? Roger Lee proved that the Black-Scholes implied variance can not grow faster than linearly in log-moneyness. This paper investigates what happens in the Bachelier (or Normal) implied volatility world, making sure to cover the various aspects of vanilla option arbitrages.
Forecasting implied volatility surface with generative diffusion models
本文提出基于扩散概率模型的生成模型，通过条件化市场变量预测无套利隐含波动率面，采用信号噪声比无参数加权方案解决训练数据中的套利问题，实验显示其在波动率预测上优于现有方法。
Fee Implied Volatility on Uniswap v3: A DEX Native Proxy and Its Limits
该研究推导了Uniswap v3的费用隐含波动率代理σ_fee，证明仅靠链上数据无法恢复Black-Scholes一致的隐含波动率，仅能得到费用隐含活动指数。
The no Butterfly arbitrage domain of Gatheral SVI 5-parameters formula for the volatility smile has been recently described. It requires in general a numerical minimization of 2 functions altogether with a few root finding procedures. We study here the case of some sub-SVIs (all with 3 parameters): the Symmetric SVI, the Vanishing Upward/Downward SVI, and SSVI, for which we provide an explicit domain, with no numerical procedure required.
On the skew and curvature of implied and local volatilities
本文基于Malliavin calculus技术，研究了局部与隐含波动率曲面短端间的关系，恢复了粗糙波动率的$\ rac{1}{H+3/2}$规则，并揭示了两者平价短期偏斜与曲率间依赖参数$H$的相互转换关系。
The Mathematics of Volatility Surfaces
本文构建了隐含、局部与学习型波动率曲面的统一数学理论，推导了相关恒等式、对冲公式等，形成涵盖多方面的框架及实证方案。
We propose a hybrid method for generating arbitrage-free implied volatility (IV) surfaces consistent with historical data by combining model-free Variational Autoencoders (VAEs) with continuous time stochastic differential equation (SDE) driven models. We focus on two classes of SDE models: regime switching models and Lévy additive processes. By projecting historical surfaces onto the space of SDE model parameters, we obtain a distribution on the parameter subspace faithful to the data on which we then train a VAE. Arbitrage-free IV surfaces are then generated by sampling from the posterior di…
We consider a general local-stochastic volatility model and an investor with exponential utility. For a European-style contingent claim, whose payoff may depend on either a traded or non-traded asset, we derive an explicit approximation for both the buyer's and seller's indifference price. For European calls on a traded asset, we translate indifference prices into an explicit approximation of the buyer's and seller's implied volatility surface. For European claims on a non-traded asset, we establish rigorous error bounds for the indifference price approximation. Finally, we implement our indif…
First, we give an asymptotic expansion of short-dated at-the-money implied volatility that refines the preceding works and proves in particular that non-rough volatility models are inconsistent to a power law of volatility skew. Second, we show that given a power law of volatility skew in an option market, a continuous price dynamics of the underlying asset with non-rough volatility admits an arbitrage opportunity. The volatility therefore has to be rough in a viable market of the underlying asset of which the volatility skew obeys a power law.
We develop a method to study the implied volatility for exotic options and volatility derivatives with European payoffs such as VIX options. Our approach, based on Malliavin calculus techniques, allows us to describe the properties of the at-the-money implied volatility (ATMI) in terms of the Malliavin derivatives of the underlying process. More precisely, we study the short-time behaviour of the ATMI level and skew. As an application, we describe the short-term behavior of the ATMI of VIX and realized variance options in terms of the Hurst parameter of the model, and most importantly we descr…
The implied volatility skew has received relatively little attention in the literature on short-term asymptotics for financial models with jumps, despite its importance in model selection and calibration. We rectify this by providing high-order asymptotic expansions for the at-the-money implied volatility skew, under a rich class of stochastic volatility models with independent stable-like jumps of infinite variation. The case of a pure-jump stable-like Lévy model is also considered under the minimal possible conditions for the resulting expansion to be well defined. Unlike recent results for …
A lower bound for the volatility swap in the lognormal SABR model
本文证明了在短期到期极限下，条件对数正态SABR模型的零Vanna隐含波动率是波动率掉期执行价的更紧下界，适用于所有相关性参数且在相关性非正时优于平值隐含波动率。
We consider a strictly pathwise setting for Delta hedging exotic options, based on Föllmer's pathwise Itō calculus. Price trajectories are $d$-dimensional continuous functions whose pathwise quadratic variations and covariations are determined by a given local volatility matrix. The existence of Delta hedging strategies in this pathwise setting is established via existence results for recursive schemes of parabolic Cauchy problems and via the existence of functional Cauchy problems on path space. Our main results establish the nonexistence of pathwise arbitrage opportunities in classes of stra…
In liquid option markets, W-shaped implied volatility curves have occasionally be observed. We show that such shapes can be reproduced in a mixture of two variance-gamma models. This is in contrast to lognormal models, where at least three different distributions have to be mixed in order to produce a W-shape, as recently shown by Glasserman and Pirjol.
The capitalization-weighted total relative variation $\sum_{i=1}^d \int_0^\cdot μ_i (t) \mathrm{d} \langle \log μ_i \rangle (t)$ in an equity market consisting of a fixed number $d$ of assets with capitalization weights $μ_i (\cdot)$ is an observable and nondecreasing function of time. If this observable of the market is not just nondecreasing, but actually grows at a rate which is bounded away from zero, then strong arbitrage can be constructed relative to the market over sufficiently long time horizons. It has been an open issue for more than ten years, whether such strong outperformance of …
Empirical studies have emphasized that the equity implied volatility is characterized by a negative skew inversely proportional to the square root of the time-to-maturity. We examine the short-time-to-maturity behavior of the implied volatility smile for pure jump exponential additive processes. An excellent calibration of the equity volatility surfaces has been achieved by a class of these additive processes with power-law scaling. The two power-law scaling parameters are $β$, related to the variance of jumps, and $δ$, related to the smile asymmetry. It has been observed, in option market dat…
We present small-time implied volatility asymptotics for Realised Variance (RV) and VIX options for a number of (rough) stochastic volatility models via large deviations principle. We provide numerical results along with efficient and robust numerical recipes to compute the rate function; the backbone of our theoretical framework. Based on our results, we further develop approximation schemes for the density of RV, which in turn allows to express the volatility swap in close-form. Lastly, we investigate different constructions of multi-factor models and how each of them affects the convexity o…
Sparse modeling approach to the arbitrage-free interpolation of plain-vanilla option prices and implied volatilities
本研究提出一种基于积分方程离散化与SVD稀疏建模的普通香草期权无套利插值方法，结合L₁正则化提升稳定性，无需指定随机过程等模型，在多类测试案例中验证了有效性。
Implied Volatility Expansions for VIX Options in Forward Variance Models
本文在远期方差模型框架下，利用弱近似技术推导出VIX期权隐含波动率的闭式展开，实现快速准确校准。
We consider implied volatilities in asset pricing models, where the discounted underlying is a strict local martingale under the pricing measure. Our main result gives an asymptotic expansion of the right wing of the implied volatility smile and shows that the strict local martingale property can be determined from this expansion. This result complements the well-known asymptotic results of Lee and Benaim-Friz, which apply only to true martingales. This also shows that `price bubbles' in the sense of strict local martingale behaviour can in principle be detected by an analysis of implied volat…
We undertake a systematic comparison between implied volatility, as represented by VIX (new methodology) and VXO (old methodology), and realized volatility. We compare visually and statistically distributions of realized and implied variance (volatility squared) and study the distribution of their ratio. We find that the ratio is best fitted by heavy-tailed -- lognormal and fat-tailed (power-law) -- distributions, depending on whether preceding or concurrent month of realized variance is used. We do not find substantial difference in accuracy between VIX and VXO. Additionally, we study the var…
Yet another asymptotic formula for implied volatility
本文在连续局部鞅模型中推导隐含方差的一阶渐近公式，将总隐含方差表示为对数价格二次变差的条件期望，并统一涵盖多种渐近情形。
We propose a new static parameterization of the implied volatility surface which is constructed by using polynomials of sigmoid functions combined with some other terms. This parameterization is flexible enough to fit market implied volatilities which demonstrate smile or skew. An arbitrage-free calibration algorithm is considered that constructs the implied volatility surface as a grid in the strike-expiration space and guarantees a lack of arbitrage at every node of this grid. We also demonstrate how to construct an arbitrage-free interpolation and extrapolation in time, as well as build a l…
Implied and Realized Volatility: A Study of Distributions and the Distribution of Difference
该研究通过拟合VIX、VXO指数对应的已实现方差与隐含波动率平方的分布，发现广义贝塔分布拟合最优，且隐含波动率对已实现波动率或存在反馈效应，差值分布与线性回归的皮尔逊相关系数一致。
We analyze correlations between squared volatility indices, VIX and VXO, and realized variances -- the known one, for the current month, and the predicted one, for the following month. We show that the ratio of the two is best fitted by a Beta Prime distribution, whose shape parameters depend strongly on which of the two months is used.
We propose a novel time discretization for the log-normal SABR model which is a popular stochastic volatility model that is widely used in financial practice. Our time discretization is a variant of the Euler-Maruyama scheme. We study its asymptotic properties in the limit of a large number of time steps under a certain asymptotic regime which includes the case of finite maturity, small vol-of-vol and large initial volatility with fixed product of vol-of-vol and initial volatility. We derive an almost sure limit and a large deviations result for the log-asset price in the limit of large number…
Finding the nonnegative minimal solutions of Cauchy PDEs in a volatility-stabilized market
本文研究波动率稳定市场中的强相对套利问题，利用时间变换的 Bessel 桥提出求解 Cauchy PDE 非负最小解的数值方法与实用算法，并通过示例验证了其有效性。
Smile asymptotics for Bachelier implied volatility
本文研究了Bachelier模型中隐含波动率的渐近行为，通过正则变化理论推导了微笑边缘的显式表达式，并建立了波动率微笑斜率与特征函数解析带的关系。
Short-time behavior of the At-The-Money implied volatility for the jump-diffusion stochastic volatility Bachelier model
本文用Malliavin微积分研究跳跃扩散Bachelier模型下平值隐含波动率的短期水平与斜率，证明水平对所有纯跳跃Lévy过程一致，并给出斜率存在条件及数值验证。
Implied volatility is at the very core of modern finance, notwithstanding standard option pricing models continue to derive option prices starting from the joint dynamics of the underlying asset price and the spot volatility. These models often cause difficulties: no closed formulas for prices, demanding calibration techniques, unclear maps between spot and implied volatility. Inspired by the practice of using implied volatility as quoting system for option prices, models for the joint dynamics of the underlying asset price and the implied volatility have been proposed to replace standard opti…
We derive a backward and forward nonlinear PDEs that govern the implied volatility of a contingent claim whenever the latter is well-defined. This would include at least any contingent claim written on a positive stock price whose payoff at a possibly random time is convex. We also discuss suitable initial and boundary conditions for those PDEs. Finally, we demonstrate how to solve them numerically by using an iterative finite-difference approach.
From Volatility to Variance: A Skew-Enhanced SABR Model and Its Empirical Study in the Chinese Financial Options Market
本文提出一种增强偏斜的SABR模型，通过改进资产价格和方差过程的随机动态，提升对复杂波动率微笑模式的拟合能力，并在中国期权市场中进行了实证分析。
Basket implied volatility skew and stickiness
本研究推导了篮子期权短期隐含波动率展开式，分离成分方差与权重波动影响，并证明偏斜粘性比率收敛于H+3/2的普适极限。
A Wiener Chaos Approach to Martingale Modelling and Implied Volatility Calibration
本文提出了一种基于维纳混沌的martingale建模方法，用于高效校准隐含波动率表面。
An Explicit Solution to Black-Scholes Implied Volatility
本文提出了一种基于量化函数的显式公式，用于计算Black-Scholes隐含波动率，通过将波动率视为左端点并仅使用可观察的期权输入来实现精确的解析解，同时重新组织了希腊字母和无套利限制。
Heath-Jarrow-Morton meet lifted Heston in energy markets for joint historical and implied calibration
本文提出一种多因子 Heath-Jarrow-Morton 模型与提升的 Heston 模型结合的方法，用于能源市场中的联合历史与隐含校准，通过快速校准流程实现更精确的波动率曲面拟合。
Understanding Short-Term Implied Volatility Dynamics: A Model-Independent Approach Beyond Stochastic Volatility
本文采用基于分布的无模型方法，通过累积量条件推导短期隐含波动率的渐近展开式，刻画不同分布下的平值偏度与曲率特性，提出可解释的校准方法，为期权定价与模型评估提供工具。
In the regime switching extension of Black-Scholes-Merton model of asset price dynamics, one assumes that the volatility coefficient evolves as a hidden pure jump process. Under the assumption of Markov regime switching, we have considered the locally risk minimizing price of European vanilla options. By pretending these prices or their noisy versions as traded prices, we have first computed the implied volatility (IV) of the underlying asset. Then by performing several numerical experiments we have investigated the dependence of IV on the time to maturity (TTM) and strike price of the vanilla…
SANOS Smooth strictly Arbitrage-free Non-parametric Option Surfaces
提出一种简单、数值高效且高度灵活的非参数方法，构建平滑且严格无套利的期权价格曲面，通过线性规划校准市场报价，并引入离散局部波动率参数化。
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