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We propose ARBITER, a risk-neutral neural operator for learning joint SPX-VIX term structures under no-arbitrage constraints. ARBITER maps market states to an operator that outputs implied volatility and variance curves while enforcing static arbitrage (calendar, vertical, butterfly), Lipschitz bounds, and monotonicity. The model couples operator learning with constrained decoders and is trained with extragradient-st…
The quintic Ornstein-Uhlenbeck volatility model is a stochastic volatility model where the volatility process is a polynomial function of degree five of a single Ornstein-Uhlenbeck process with fast mean reversion and large vol-of-vol. The model is able to achieve remarkable joint fits of the SPX-VIX smiles with only 6 effective parameters and an input curve that allows to match certain term structures. We provide se…
We propose a new approach for trading VIX futures. We assume that the term structure of VIX futures follows a Markov model. Our trading strategy selects a position in VIX futures by maximizing the expected utility for a day-ahead horizon given the current shape and level of the term structure. Computationally, we model the functional dependence between the VIX futures curve, the VIX futures positions, and the expecte…
This paper shows how to recover a stochastic volatility model (SVM) from a market model of the VIX futures term structure. Market models have more flexibility for fitting of curves than do SVMs, and therefore are better suited for pricing VIX futures and VIX derivatives. But the VIX itself is a derivative of the S&P500 (SPX) and it is common practice to price SPX derivatives using an SVM. Therefore, consistent modeli…
A new modelling approach that directly prescribes dynamics to the term structure of VIX futures is proposed in this paper. The approach is motivated by the tractability enjoyed by models that directly prescribe dynamics to the VIX, practices observed in interest-rate modelling, and the desire to develop a platform to better understand VIX option implied volatilities. The main contribution of the paper is the derivati…
We propose the VIX-derived volatility (VDV) model, a VIX-first framework for joint SPXVIX modeling. In the model, we define explicit dynamics for the VIX process to price VIX futures and options, yielding a VIX-side calibration that is independent of the SPX dynamics. Using the rolling-window definition of the VIX, we then derive a coupling function to obtain the SPX volatility process as a latent process consistent …
We present a study of the leading-order asymptotics for VIX option prices in Bergomi models in the short-maturity and small volatility-of-volatility regimes. Both out-of-the-money (OTM) and at-the-money (ATM) asymptotics are considered for one-factor, two-factor Bergomi and $N$-factor models. The leading-order asymptotics are obtained in closed-form, which are translated into predictions for the small-maturity asympt…
We present a study of the short-maturity asymptotics for VIX and European option prices in local-stochastic volatility models with compound Poisson jumps. Both out-of-the-money (OTM) and at-the-money (ATM) asymptotics are considered. The leading-order asymptotics are obtained in closed-form. We apply our results to three examples: the Eraker model, a Kou-type model, and a folded normal model. Numerical illustrations …
The Heston stochastic volatility model is arguably, the most popular stochastic volatility model used to price and risk manage exotic derivatives. In spite of this, it is not necessarily easy to calibrate to the market and obtain stable exotic option prices with this model. This paper focuses on the vol-of-vol parameter and its relation with the volatility of volatility index (VVIX) level. Four different approaches t…
We present a white-box, risk-sensitive framework for jointly hedging SPX and VIX exposures under transaction costs and regime shifts. The approach couples an arbitrage-free market teacher with a control layer that enforces safety as constraints. On the market side, we integrate an SSVI-based implied-volatility surface and a Cboe-compliant VIX computation (including wing pruning and 30-day interpolation), and connect …
This paper examines systematic put-writing strategies applied to S&P 500 Index options, with a focus on position sizing as a key determinant of long-term performance. Despite the well-documented volatility risk premium, where implied volatility exceeds realized volatility, the practical implementation of short-dated volatility-selling strategies remains underdeveloped in the literature. This study evaluates three pos…
We introduce the two-factor Quintic Ornstein-Uhlenbeck (OU) model, where volatility is modelled as a degree-five polynomial of the sum of two Ornstein-Uhlenbeck processes driven by the same Brownian motion, each mean-reverting at a different speed. We demonstrate that the model effectively captures the volatility surfaces of SPX and VIX while aligning with the skew-stickiness ratio (SSR) across maturities ranging fro…
We introduce a new identification strategy for uncertainty shocks to explain macroeconomic volatility in financial markets. The Chicago Board Options Exchange Volatility Index (VIX) measures market expectations of future volatility, but traditional methods based on second-moment shocks and time-varying volatility of the VIX often fail to capture the non-Gaussian, heavy-tailed nature of asset returns. To address this,…
We derive the short-maturity asymptotics for European and VIX option prices in local-stochastic volatility models where the volatility follows a continuous-path Markov process. Both out-of-the-money (OTM) and at-the-money (ATM) asymptotics are considered. Using large deviations theory methods, the asymptotics for the OTM options are expressed as a two-dimensional variational problem, which is reduced to an extremal p…
Predicting the S&P 500 index volatility is crucial for investors and financial analysts as it helps assess market risk and make informed investment decisions. Volatility represents the level of uncertainty or risk related to the size of changes in a security's value, making it an essential indicator for financial planning. This study explores four methods to improve the accuracy of volatility forecasts for the S&P 50…
The Chicago Board Options Exchange Volatility Index (VIX) is calculated from SPX options and derivatives of VIX are also traded in market, which leads to the so-called ``consistent modeling" problem. This paper proposes a time-changed Lévy model for log price with a composite change of time structure to capture both features of the implied SPX volatility and the implied volatility of volatility. Consistent modeling i…
This research mainly explores the characteristics of different strategies and whether VIX INDEX positively influences the investment portfolio in any period. Our portfolio has six significant cryptocurrencies, VIX INDEX and gold. We perform parameter estimation on all raw data and bring the two types into different investment strategies, complete them effectively according to other characteristics, and compare the re…
The class of Affine (Jump) Diffusion (AD) has, due to its closed form characteristic function (ChF), gained tremendous popularity among practitioners and researchers. However, there is clear evidence that a linearity constraint is insufficient for precise and consistent option pricing. Any non-affine model must pass the strict requirement of quick calibration -- which is often challenging. We focus here on Randomized…
Which expiry you hold and when you roll is a first-class P&L — not an operations footnote — especially in commodities and VIX.
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.
Term Structure of Volatility — How IV varies across expiries — front vs back month regimes.
VIX Futures Term Structure — Curve shape driving roll yield for vol ETNs and systematic short-vol carry.
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 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.
The 2s10s Treasury curve measures the spread between 10-year and 2-year Treasury yields and is a key indicator of growth expectations, policy path, and term structure dynamics.
Trade the VIX curve — short steep contango, respect backwardation — a roll-yield book in vol futures.
Which expiry you hold and when you roll is a first-class P&L — not an operations footnote — especially in commodities and VIX.
Trade the VIX curve — short steep contango, respect backwardation — a roll-yield book in vol futures.
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.
VIX Futures Term Structure — Curve shape driving roll yield for vol ETNs and systematic short-vol carry.
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 Carry Trade — Selling implied vol or rolling VIX futures in contango — crowded but regime-sensitive.