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We revisit the problem of pricing options with historical volatility estimators. We do this in the context of a generalized GARCH model with multiple time scales and asymmetry. It is argued that the reason for the observed volatility risk premium is tail risk aversion. We parametrize such risk aversion in terms of three coefficients: convexity, skew and kurtosis risk premium. We propose that option prices under the r…
We investigate whether the volatility risk premium is negative by examining the statistical properties of delta-hedged option portfolios (buy the option and hedge with stock). Within a stochastic volatility framework, we demonstrate a correspondence between the sign and magnitude of the volatility risk premium and the mean delta-hedged portfolio returns. Using a sample of S&P 500 index options, we provide empirical t…
The accumulation of trading experience and empirical evidence since the original Black-Scholes (BS) model was developed, have made it increasingly evident that volatility is not a constant parameter, as BS assumed, but stochastic. With a second random factor associated with volatility affecting security returns, it would not be surprising if investors cared about bearing risk related to that factor. And there is cons…
One key stylized fact in the empirical option pricing literature is the existence of an implied volatility surface (IVS). The usual approach consists of Þtting a linear model linking the implied volatility to the time to maturity and the moneyness, for each cross section of options data. However, recent empirical evidence suggests that the parameters characterizing the IVS change over time. In this paper we study whe…
Abstract This study employs a non-parametric approach to investigate the volatility risk premium in the over-the-counter currency option market. Using a large database of daily delta-neutral straddle quotes in four major currencies—the British pound, the euro, the Japanese yen, and the Swiss franc—we find that volatility risk is priced in all four currencies across different option maturities. We find that the volati…
In this study, we constructed daily high-frequency sentiment data and used the VAR method to attempt to predict the next day's implied volatility surface. We utilized 630,000 text data entries from the East Money Stock Forum from 2014 to 2023 and employed deep learning methods such as BERT and LSTM to build daily market sentiment indicators. By applying FFT and EMD methods for sentiment decomposition, we found that h…
We propose a new model for the forecasting of both the implied volatility surfaces and the underlying asset price. In the spirit of Guyon and Lekeufack (2023) who are interested in the dependence of volatility indices (e.g. the VIX) on the paths of the associated equity indices (e.g. the S\&P 500), we first study how vanilla options implied volatility can be predicted using the past trajectory of the underlying asset…
We show that the Realized GARCH model yields close-form expression for both the Volatility Index (VIX) and the volatility risk premium (VRP). The Realized GARCH model is driven by two shocks, a return shock and a volatility shock, and these are natural state variables in the stochastic discount factor (SDF). The volatility shock endows the exponentially affine SDF with a compensation for volatility risk. This leads t…
This paper examines how shocks to currency volatilities predict exchange rates. Using option-implied volatilities, we construct a dynamic, directed network of volatility connections. Currencies that transmit more volatility shocks, which control for common correlation, earn lower excess returns. Buying the weakest and selling the strongest transmitters delivers high risk-adjusted performance, driven by spot exchange …
We test various volatility models using the Bitcoin spot price series. Our models include HIST, EMA ARCH, GARCH, and EGARCH, models. Both of our in-sample-fit and out-of-sample-forecast results suggest that GARCH and EGARCH models perform much better than other models. Moreover, the EGARCH model's asymmetric term is positive and insignificant, which suggests that Bitcoin prices lack the asymmetric volatility response…
Low-frequency historical data, high-frequency historical data and option data are three major sources, which can be used to forecast the underlying security's volatility. In this paper, we propose two econometric models, which integrate three information sources. In GARCH-Itô-OI model, we assume that the option-implied volatility can influence the security's future volatility, and the option-implied volatility is tre…
We derive representations of local risk-minimization of call and put options for Barndorff-Nielsen and Shephard models: jump type stochastic volatility models whose squared volatility process is given by a non-Gaussian rnstein-Uhlenbeck process. The general form of Barndorff-Nielsen and Shephard models includes two parameters: volatility risk premium $β$ and leverage effect $ρ$. Arai and Suzuki (2015, arxiv:1503.0858…
We propose a microstructural model for the order flow in financial markets that distinguishes between {\it core orders} and {\it reaction flow}, both modeled as Hawkes processes. This model has a natural scaling limit that reconciles a number of salient empirical properties: persistent signed order flow, rough trading volume and volatility, and power-law market impact. In our framework, all these quantities are pinne…
An automated market maker (AMM) provides a method for creating a decentralized exchange on the blockchain. For this purpose, individual investors lend liquidity to the AMM pool in exchange for a stream of fees earned from its operations as a market maker. Within this work, we reinterpret the loss-versus-rebalancing as the implied fee stream generated by an AMM so that a risk-neutral investor is indifferent in the dec…
The Capital Asset Pricing Model (CAPM) relates a well-diversified stock portfolio to a benchmark portfolio. We insert size effect in the CAPM, capturing the observation that small stocks have higher risk and return than large stocks, on average. Our goal is to make the resulting linear regressions have independent identically distributed Gaussian residuals. In some cases, we find that including the Volatility Index a…
We develop a liquidity-sensitive multivariate volatility framework to improve the estimation of time-varying covariance structures under market frictions. We introduce two novel portfolio-level liquidity measures, liquidity jump and liquidity diffusion, which capture magnitude and volatility of liquidity fluctuation, respectively, and construct liquidity-adjusted return and volatility that reflect real-time liquidity…
We establish innovative liquidity premium measures, and construct liquidity-adjusted return and volatility to model assets with extreme liquidity, represented by a portfolio of selected crypto assets, and upon which we develop a set of liquidity-adjusted ARMA-GARCH/EGARCH models. We demonstrate that these models produce superior predictability at extreme liquidity to their traditional counterparts. We provide empiric…
Introduction: The paper addresses the challenging problem of predicting the short-term realized volatility of the Bitcoin price using order flow information. The inherent stochastic nature and anti-persistence of price pose difficulties in accurate prediction. Methods: To address this, we propose a method that transforms order flow data over a fixed time interval (snapshots) into images. The order flow includes trade…
Crypto Realized Vol Regime — Shifts in realized volatility that redefine sizing and carry.
Enter on a trend signal, then trail a stop at k × ATR behind the favorable extreme — Wilder volatility as the exit engine.
A CTA that is structurally short implied volatility — harvesting VRP with futures options, and owning a jump left tail.
The core CTA recipe: in each futures market, go long if the trend is up and short if it is down, size by volatility, and let the stop or the signal flip you out.
Scale the whole book (or each market) so forecast σ hits a target — the reason a 15% vol CTA is not ‘more leveraged crude’ in a quiet month.
A managed-futures book that is structurally long options or long VIX-curve convexity — pays carry, aims to print in jumps and persistent stress.
Trade the VIX curve as a first-class market — trend on VIX, carry on contango, and a respect for inversion — not just an equity hedge overlay.
GARCH Volatility Model — Conditional heteroskedasticity framework for forecasting volatility clusters.
Implied volatility is the σ you plug into Black-Scholes to match the market price — a quote of the option, not a forecast you must believe.
Implied Volatility Surface — Strike and tenor structure of implied vol, the core object for vol trading and risk.
Local Volatility Model — Strike-dependent diffusion used to fit vanilla surfaces consistently.
The MOVE Index tracks implied volatility in the U.S. Treasury market and serves as a benchmark for rates uncertainty and macro stress.
Realized Volatility — Historical return variation that determines PnL for delta-hedged option positions.
Skew measures the relative richness of downside versus upside implied volatility, helping track hedging demand and asymmetry in market risk pricing.
Stochastic Volatility Model — Models where volatility itself is random, capturing smile dynamics and VRP.
SVI Parameterization — Arbitrage-aware parameterization of volatility smiles for interpolation and trading.
Term Structure of Volatility — How IV varies across expiries — front vs back month regimes.
Variance Risk Premium — Gap between implied and realized volatility that systematic vol sellers harvest.
Vega is the sensitivity of option value to implied volatility — the vol-dollar you are long or short.
Vega Exposure — Sensitivity to implied volatility changes — core risk for vol books and structured products.
VIX Index measures implied volatility in S&P 500 options and is widely used as a shorthand for equity market fear and risk aversion.
VIX term structure tracks the shape of volatility futures across maturities and helps identify whether the market is pricing stable conditions or near-term stress.
Volatility Arbitrage — Trading discrepancies between implied, realized, and cross-asset volatility.
Volatility Carry Trade — Selling implied vol or rolling VIX futures in contango — crowded but regime-sensitive.
Volatility of Volatility — Uncertainty about future volatility, critical for tail hedges and vol-of-vol products.
Volatility Smile — Strike-dependent implied vol pattern reflecting crash and demand premia.
Nonfarm Payrolls — The headline US jobs report that routinely moves rates, FX, and equity index volatility.
FX Implied Volatility — Option-implied uncertainty for currency pairs, key for hedging and risk budgeting.
Variance is the expected squared deviation from the mean, Var(X) = E[(X − μ)²]. It is the second moment that becomes volatility after a square root and a convention.
A circuit breaker is an exchange halt when prices move too far too fast — a pause so the book can rebuild, not a valuation.
Intraday Volatility — Within-day return variation informing execution timing and gamma scalping.
The efficient frontier is the set of mean-variance-optimal portfolios — maximum expected return for each volatility, given the inputs.
Low Volatility Anomaly — Empirical outperformance of low-beta stocks, crowded in risk-off regimes.
The Sharpe ratio is excess return per unit of total volatility — a ranking statistic that hates fat tails and loves the last sample.
Target Volatility — Dynamic scaling of exposure to maintain constant portfolio volatility.
Tracking error is the volatility of active return versus a benchmark — how much the book is allowed to be not-the-index.
Enter on a trend signal, then trail a stop at k × ATR behind the favorable extreme — Wilder volatility as the exit engine.
Crypto Realized Vol Regime — Shifts in realized volatility that redefine sizing and carry.
A CTA that is structurally short implied volatility — harvesting VRP with futures options, and owning a jump left tail.
The core CTA recipe: in each futures market, go long if the trend is up and short if it is down, size by volatility, and let the stop or the signal flip you out.
Scale the whole book (or each market) so forecast σ hits a target — the reason a 15% vol CTA is not ‘more leveraged crude’ in a quiet month.
The efficient frontier is the set of mean-variance-optimal portfolios — maximum expected return for each volatility, given the inputs.
FX Implied Volatility — Option-implied uncertainty for currency pairs, key for hedging and risk budgeting.
GARCH Volatility Model — Conditional heteroskedasticity framework for forecasting volatility clusters.
Short high residual-vol names and long low residual-vol names — Ang et al.’s IVOL puzzle as a book.
Implied volatility is the σ you plug into Black-Scholes to match the market price — a quote of the option, not a forecast you must believe.
Implied Volatility Surface — Strike and tenor structure of implied vol, the core object for vol trading and risk.
Intraday Volatility — Within-day return variation informing execution timing and gamma scalping.
Local Volatility Model — Strike-dependent diffusion used to fit vanilla surfaces consistently.
A managed-futures book that is structurally long options or long VIX-curve convexity — pays carry, aims to print in jumps and persistent stress.
Low Volatility Anomaly — Empirical outperformance of low-beta stocks, crowded in risk-off regimes.
Overweight low-realized-vol (or low-beta) stocks and underweight high-vol names — the low-risk anomaly as a long-short or defensive long-only.
Blend intermediate momentum, short-term reversal, and a volatility filter — a multi-horizon equity recipe.
The MOVE Index tracks implied volatility in the U.S. Treasury market and serves as a benchmark for rates uncertainty and macro stress.
Nonfarm Payrolls — The headline US jobs report that routinely moves rates, FX, and equity index volatility.
Realized Volatility — Historical return variation that determines PnL for delta-hedged option positions.
Run a frequent rebalance across a crypto basket to harvest volatility and dispersion — a diversity/rebalance premium, not a coin pick.
The Sharpe ratio is excess return per unit of total volatility — a ranking statistic that hates fat tails and loves the last sample.
Skew measures the relative richness of downside versus upside implied volatility, helping track hedging demand and asymmetry in market risk pricing.
Stochastic Volatility Model — Models where volatility itself is random, capturing smile dynamics and VRP.
A funding-stress monitor can warn before equity volatility reprices.
A liquidity premium monitor can separate default risk from liquidity-driven spread widening.
A card for handling fast markets, wide spreads, liquidity air pockets, and broken execution assumptions.