Two new rational formulae for normal implied volatility are presented.
problem Calculating normal implied volatility using iterative methods.
method Two explicit rational formulae that avoid iteration and logarithms.
result Accurate and fast formulae for normal implied volatility.
First, we show that implied normal volatility is intimately linked with the incomplete Gamma function. Then, we deduce an expansion on implied normal volatility in terms of the time-value of a European call option. Then, we formulate an equivalence between the implied normal volatility and the lognormal implied volatil…
Monotonicity of normalized implied-volatility coordinates under no-arbitrage
problem Monotonicity of normalized implied-volatility coordinates under no-arbitrage
method Elementary discrete no-arbitrage proof
result Monotonicity principle extended to Bachelier implied volatility
We study the dynamics of the normal implied volatility in a local volatility model, using a small-time expansion in powers of maturity T. At leading order in this expansion, the asymptotics of the normal implied volatility is similar, up to a different definition of the moneyness, to that of the log-normal volatility. …
This paper examines Bachelier implied volatility at extreme strikes.
problem Investigates appropriate implied volatility extrapolation at extreme strikes.
method Compares Bachelier and Black-Scholes models, focusing on normal distribution and vanilla options.
result Bachelier implied variance grows at most linearly in log-moneyness, similar to Black-Scholes.
We study specific nonlinear transformations of the Black-Scholes implied volatility to show remarkable properties of the volatility surface. Model-free bounds on the implied volatility skew are given. Pricing formulas for the European options which are written in terms of the implied volatility are given. In particular…
Proposes a new way to represent uncertainty using implied volatility.
problem Uncertainty in financial markets and biological systems.
method Mathematical analysis of various probability distributions.
result Representation of different probability distributions using BSM implied volatility.
Using classical Taylor series techniques, we develop a unified approach to pricing and implied volatility for European-style options in a general local-stochastic volatility setting. Our price approximations require only a normal CDF and our implied volatility approximations are fully explicit (ie, they require no spec…
We extend the model-free formula of [Fukasawa 2012] for E[Ψ(XT)], where XT=logST/F is the log-price of an asset, to functions Ψ of exponential growth. The resulting integral representation is written in terms of normalized implied volatilities. Just as Fukasawa's work provides rigourous ground for Ch…
This paper explores the harmonic mean of implied volatility and its relation to local volatility.
problem Understanding the relationship between implied volatility and local volatility.
method Investigates the harmonic mean of a positive function for any fixed maturity, linking it to Fukasawa's invertible map.
result The short-dated implied volatility approaches the arithmetic mean of the local volatility in a new coordinate system.
The paper models Gasoil options using Brent benchmarks, improving volatility estimation.
problem Inability to directly model illiquid Gasoil options market.
method Jointly models Brent and Gasoil futures prices with a correlated Bachelier model, estimating volatility spread.
result The proposed framework accurately maps Brent implied volatilities to Gasoil implied volatilities.
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 certai…
New formula for implied volatility from Black-Scholes model.
problem Computing implied volatility from Black-Scholes model.
method Analytical solution using inverse Gaussian distribution.
result Explicit formulas for implied volatility with high precision.
The paper examines the short-time implied volatility of additive processes and finds key parameters.
problem Characterizing the short-time implied volatility of equity markets.
method Examined pure jump exponential additive processes with power-law scaling parameters.
result The implied volatility is consistent with equity market characteristics if and only if β=1 and δ=-1/2.
In this paper, we derive the price of a European call option of an asset following a normal process assuming stochastic volatility. The volatility is assumed to follow the Cox Ingersoll Ross (CIR) process. We then use the fast Fourier transform (FFT) to evaluate the option price given we know the characteristic functio…
Develops a martingale expansion for stochastic volatility models.
problem Approximating marginal distributions of stochastic volatility models.
method Martingale expansion framework for continuous stochastic volatility models.
result First-order perturbation expansions for small volatility-of-volatility and fast mean-reversion models.
The rough Bergomi model, introduced by Bayer, Friz and Gatheral [Quant. Finance 16(6), 887-904, 2016], is one of the recent rough volatility models that are consistent with the stylised fact of implied volatility surfaces being essentially time-invariant, and are able to capture the term structure of skew observed in e…
We study the pricing problem for a European call option when the volatility of the underlying asset is random and follows the exponential Ornstein-Uhlenbeck model. The random diffusion model proposed is a two-dimensional market process that takes a log-Brownian motion to describe price dynamics and an Ornstein-Uhlenbec…
This work introduces a geometric approach to probability representation and option pricing.
problem Representing probability distributions geometrically for better understanding and approximation.
method Introducing a geometric representation of probability using implied volatility and geometric transformations.
result Any probability distribution on positive reals can be represented by a planar curve, facilitating approximation and analysis.
Exact relationships found between ATM slope, volatility swap, and zero vanna.
problem Understanding relationships between implied volatilities and swaps.
method Analyzes exact relationships between ATM slope, volatility swap, and zero vanna.
result Exact relationships between ATM slope, volatility swap, and zero vanna.
The model outperforms other models in option pricing, especially for short-term implied volatility.
problem Improper calibration and pricing of exotic options in financial models.
method Stochastic volatility model with double-exponential jumps, Fourier pricing techniques.
result The model outperforms other models in fitting the short-term implied volatility smile and pricing exotic options.
Innovative extensions to option pricing models using asymmetric Brownian motion and random walk approaches.
problem Capturing empirical phenomena like return skewness, heavy tails, and volatility asymmetry in option pricing models.
method Developing the Geometric Asymmetric Brownian Motion (GABM) within the Bachelier--Black--Scholes--Merton framework.
result Deriving closed-form option pricing formulas and a discrete-time binomial tree algorithm that converges to the GABM limit.
Derives token price process for AMM tokens, finds leverage effect and pricing discrepancies.
problem Derives token price process for AMM tokens.
method Derives CEV process for token price, derives closed-form option prices, introduces liquidity-adjusted Greeks.
result Token price process is CEV, with leverage effect and pricing discrepancies.
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 v…
Study on skew and curvature of implied and local volatilities using Malliavin calculus.
problem Relationship between short-end of local and implied volatility surfaces.
method Malliavin calculus techniques
result Recover the $ rac{1}{H+3/2}$ rule for rough volatilities and relationships between skew and curvature.
Study on implied volatility of Asian options with stochastic volatility.
problem Understanding the implied volatility of Asian options under stochastic volatility models.
method Using Malliavin calculus and anticipating Ito's formula, the paper computes and finds asymptotic formulas for the implied volatility and skew.
result Developed short-maturity asymptotic formulas for the skew of the implied volatility, which depends on the roughness of the volatility model.
Paper estimates Hurst parameter from implied volatilities.
problem Estimating Hurst parameter from implied volatilities.
method Uses covariance between asset return and realized volatility, and applies limit theorems for stochastic volatility models.
result Direct relation between covariance and slope of at-the-money implied volatility established.
Measures of implied volatility roughness corrected for bias.
problem Bias in measuring implied volatility roughness.
method Examined implied volatility of short-term options and VIX index, corrected for bias.
result Corrected measures indicate appropriate proxies for underlying volatility.
The paper analyzes implied volatility for European and Asian options under stochastic volatility Bachelier model.
problem Analyzing implied volatility for European and Asian options under stochastic volatility.
method Using Malliavin calculus and anticipating Ito's formula, the paper computes and finds asymptotic formulas for implied volatility and skew.
result The paper provides a short maturity asymptotic formula for the skew of implied volatility that depends on the roughness of the volatility model.
Study uses sentiment analysis to predict implied volatility surface, improving prediction accuracy.
problem Improving prediction accuracy of implied volatility surface.
method Constructed daily high-frequency sentiment data, used VAR method, deep learning (BERT, LSTM), FFT, EMD for sentiment decomposition.
result High-frequency sentiment correlates with ATM options' implied volatility, low-frequency with DOTM options.
Study finds option volume imbalance predicts equity market returns.
problem Predicting equity market returns using option volume imbalance.
method Nonlinear analysis of option volumes decomposed into five market participant classes.
result Strong signals of predictability of excess market returns from Market-Maker volumes.
New FX option interpolations impact implied volatilities.
problem Different interpolations of FX option quotes lead to varying implied volatilities.
method Analysis of various exact interpolations of broker quotes.
result Different interpolations result in different implied volatilities.
This paper provides a neural approach to represent option implied information.
problem Link between implied density and volatility for arbitrage-free modeling.
method Minimalist perspective on implied volatility, neural representation with arbitrage constraints.
result Shallow feedforward network with a single hidden layer effectively approximates implied density and volatility.
For nearly every major stock market there exist equity and implied volatility indices. These play important roles within finance: be it as a benchmark, a measure of general uncertainty or a way of investing or hedging. It is well known in the academic literature, that correlations and higher moments between different i…
We consider an asset whose risk-neutral dynamics are described by a general class of local-stochastic volatility models and derive a family of asymptotic expansions for European-style option prices and implied volatilities. Our implied volatility expansions are explicit; they do not require any special functions nor do…
We propose a probabilistic framework for pricing derivatives, which acknowledges that information and beliefs are subjective. Market prices can be translated into implied probabilities. In particular, futures imply returns for these implied probability distributions. We argue that volatility is not risk, but uncertaint…
Study on implied volatility of Inverse options under stochastic volatility models.
problem Short-time behavior and skew of implied volatility for Inverse European options.
method Malliavin calculus, anticipating Itô's formula, asymptotic analysis.
result Asymptotic formula for skew of implied volatility, extending to Quanto-Inverse options.
Study shows non-convergence of short-maturity expansion in SABR model.
problem Analyzing convergence of SABR model option prices.
method Analyzing analyticity properties of the payoff function and using the McKean kernel.
result Short-maturity expansion for SABR model is asymptotic and non-convergent.
In informationally efficient financial markets, option prices and this implied volatility should immediately be adjusted to new information that arrives along with a jump in underlying's return, whereas gradual changes in implied volatility would indicate market inefficiency. Using minute-by-minute data on S&P 500 inde…
Study local volatility from rough volatility models, finding new skew rule.
problem Understanding local volatility from rough volatility models.
method Analyzing asymptotic behavior of local volatility surface generated by rough stochastic volatility models.
result New skew rule: ratio of implied and local vol skews tends to 1/(H + 3/2).
A new method calculates implied volatilities without using option prices.
problem Calculating implied volatilities without option prices.
method Conic finance approach to uniquely strip volatilities from bid and ask quotes.
result Allows joint calculation of implied liquidity from bid and ask quotes.
New model predicts implied volatility using past asset price paths.
problem Forecasting implied volatility surfaces and asset prices.
method Proposes a new model using past asset price trajectories to predict implied volatility.
result Large part of implied volatility movements can be explained by past returns and squares.
The paper studies estimation of parameters of diffusion market models from historical data. The standard definition of implied volatility for these models presents its value as an implicit function of several parameters, including the risk-free interest rate. In reality, the risk free interest rate is unknown and need …
It is "well known" that there is no explicit expression for the Black-Scholes implied volatility. We prove that, as a function of underlying, strike, and call price, implied volatility does not belong to the class of D-finite functions. This does not rule out all explicit expressions, but shows that implied volatility …
The BBF, SABR, and rough SABR formulas provide nearly arbitrage-free implied vol approximations.
problem Arbitrage in implied volatility calculations.
method Analytical proofs for BBF, SABR, and rough SABR formulas under specific models.
result These formulas offer asymptotically arbitrage-free approximations of implied volatility.
Proposes a flexible framework for implied volatility surfaces with random parameters.
problem Inconsistent calibration of parametric implied volatility models when market volatility deviates from the model's regime.
method Introduces random coefficients for parametric implied volatility formulas, preserving analytic flexibility and efficiency.
result Demonstrates improved modeling of implied volatility curves, especially for short-term options and earnings announcements.
We create precise formulas for VIX option implied volatility.
problem Calibrating VIX option prices in forward variance models.
method Developed closed-form expansions using weak-approximation techniques.
result Explicit formulas for implied volatility with computable correction terms.
We enhance short-rate models to control implied volatility analytically.
problem Controlling implied volatility in short-rate models.
method Randomized Affine Diffusion (RAnD) method applied to Heath-Jarrow-Morton framework.
result Randomized short-rate models improve calibration and control implied volatility shapes.