SPX Options Research Articles

A general valuation framework for rough stochastic local volatility models and applications

Rough volatility models are a new class of stochastic volatility models that have been shown to provide a consistently good fit to implied volatility smiles of SPX options. They are continuous-time stochastic volatility models, whose volatility process is driven by a fractional Brownian motion with the corresponding Hurst parameter less than a half. Albeit the empirical success, the valuation of derivative securities under rough volatility models is challenging. The reason is that it is neither a semi-martingale nor a Markov process. This paper proposes a novel valuation framework for rough stochastic local volatility (RSLV) models. In particular, we introduce the perturbed stochastic local volatility (PSLV) model as the semi-martingale approximation for the RSLV model and establish its existence, uniqueness, Markovian representation and convergence. Then we propose a fast continuous-time Markov chain (CTMC) approximation algorithm to the PSLV model and establish its convergence. Numerical experiments demonstrate the convergence of our approximation method to the true prices, and also the remarkable accuracy and efficiency of the method in pricing European, barrier and American options. Comparing with existing literature, a significant reduction in the CPU time to arrive at the same level of accuracy is observed.

Joint calibration of local volatility models with stochastic interest rates using semimartingale optimal transport

We develop and implement a non-parametric method for joint exact calibration of a local volatility model and a correlated stochastic short rate model using semimartingale optimal transport. The method relies on the duality results established in Joseph et al. (Calibration of local volatility models with stochastic interest rates using optimal transport, 2023). and jointly calibrates the whole equity-rate dynamics. It uses an iterative approach which starts with a parametric model and tries to stay close to it, until a perfect calibration is obtained. We demonstrate the performance of our approach on market data using European SPX options and European cap interest rate options. Finally, we compare the joint calibration approach with the sequential calibration, in which the short rate model is calibrated first and frozen.

Joint calibration to SPX and VIX options with signature‐based models

AbstractWe consider a stochastic volatility model where the dynamics of the volatility are described by a linear function of the (time extended) signature of a primary process which is supposed to be a polynomial diffusion. We obtain closed form expressions for the VIX squared, exploiting the fact that the truncated signature of a polynomial diffusion is again a polynomial diffusion. Adding to such a primary process the Brownian motion driving the stock price, allows then to express both the log‐price and the VIX squared as linear functions of the signature of the corresponding augmented process. This feature can then be efficiently used for pricing and calibration purposes. Indeed, as the signature samples can be easily precomputed, the calibration task can be split into an offline sampling and a standard optimization. We also propose a Fourier pricing approach for both VIX and SPX options exploiting that the signature of the augmented primary process is an infinite dimensional affine process. For both the SPX and VIX options we obtain highly accurate calibration results, showing that this model class allows to solve the joint calibration problem without adding jumps or rough volatility.

Pricing VIX options based on mean-reverting models driven by information

Financial time series are dynamic and influenced by different types of information from the market. In this study, we propose new models for SPX and VIX options using the Hawkes process, jump process with stochastic intensity, and tempered stable process to capture these changes in financial time series based on three distinct characteristics of market information. We calculate the VIX option pricing formula using these models and find that the simplified VIX model based on VIX characteristics has significantly less pricing error than the consistent VIX model derived from the SPX model. Additionally, our findings suggest that the tempered stable process effectively models the volatility of VIX, sparse large jumps, and infinitesimal jumps. It also shows potential as an alternative to Brownian motion for representing volatility. Conversely, jump processes with stochastic jump intensities adeptly describe asymmetric jumps, and their integration with Brownian motion provides a more accurate depiction of the VIX’s volatility and jump dynamics. Finally, the introduction of jump processes into mean-reverting models for the VIX indicates a relatively low correlation between volatility magnitudes and the current VIX levels. This research contributes to the theory of SPX and VIX options and offers guidance for the development of other information-driven economic models.

The role of intermediaries in derivatives markets: Evidence from VIX options

Consistent with models of intermediaries who absorb demand pressure from end-users and respond by changing prices, net option demand is positively related to option prices in the market for VIX puts and calls. VIX and SPX option markets are integrated: market-makers absorb end-users’ net long volatility positions and net demand affects prices across markets. Net demand changes in the most liquid markets result in spillover effects between the VIX and SPX markets. A dynamic characterization of prices and net demand suggests that VIX option markets and the SPX put market are integrated, while the SPX call market is more segregated.

How Integrated are Credit and Equity Markets? Evidence from Index Options

ABSTRACTWe study the extent to which credit index (CDX) options are priced consistent with S&P 500 (SPX) equity index options. We derive analytical expressions for CDX and SPX options within a structural credit‐risk model with stochastic volatility and jumps using new results for pricing compound options via multivariate affine transform analysis. The model captures many aspects of the joint dynamics of CDX and SPX options. However, it cannot reconcile the relative levels of option prices, suggesting that credit and equity markets are not fully integrated. A strategy of selling CDX volatility yields significantly higher excess returns than selling SPX volatility.

Counter-cyclical Margins for Option Portfolios

We propose a counter-cyclical initial margin model for option portfolios. Our model explores the intrinsic netting within a given portfolio of European options and outputs a constant upper bound of the maximum possible loss. This feature would allow option clearinghouses and regulators to gauge the tightest margin levels that are stable. We compare our model with the scenario-based SPAN model and the sensitivity-based SIMM model in terms of the netting efficiency and the procyclical property. Using the SPX options and the interest rate swaptions as examples, we quantify the minimum amount of additional margins needed to make them fully counter-cyclical. We then show how to strike a balance between risk-sensitivity and counter-cyclicality if needed by mixing our model flexibly with a prevailing risk-sensitive margin model.

The Price of Higher Order Catastrophe Insurance: The Case of VIX Options

ABSTRACTWe develop a tractable equilibrium pricing model to explain observed characteristics in equity returns, VIX futures, S&P 500 options, and VIX options data based on affine jump‐diffusive state dynamics and representative agents endowed with Duffie‐Epstein recursive preferences. Our calibrated model replicates consumption, dividends, and asset market data, including VIX futures returns, the average implied volatilities in SPX and VIX options, and first‐ and higher‐order moments of VIX options returns. We document a time variation in the shape of VIX‐option‐implied volatility and a time‐varying hedging relationship between VIX and SPX options that our model both captures.

Joint calibration of S&P 500 and VIX options under local stochastic volatility models

AbstractIt is extremely challenging to design a model calibrating both SPX and VIX option prices. A long‐standing conjecture due to Julien Guyon is that it may not be possible to calibrate these two quantities with a continuous model. So far, most studied continuous time models are affine, so we investigate the conjecture among 14 well‐known non‐affine local stochastic volatility models in this article. First, we propose a unified efficient willow tree method for S&P500 and VIX option pricing under non‐affine models. Second, we compare the joint calibration performance on these 14 models on the S&P500 and VIX option prices data from 2006 to 2019. We find that the VIX option price data can provide extra information of the variance dynamics of the models, and the non‐affine structure and the volatility with linear, rather than square‐root, diffusion process provide a better fit for the both data sets than the affine counterparts. Among the 14 stochastic models, the SABR model provides the best in‐ and out‐of‐sample performance (in terms of mean square error) regardless of the state of the economy. Nevertheless, even for the best‐fitted SABR model, the relative error on the VIX option is still around 18%, still quite significant. Therefore, we found the non‐affine and local volatility structure improve the joint calibration but are still far from satisfactory.

Empirical analysis of rough and classical stochastic volatility models to the SPX and VIX markets

We conduct an empirical analysis of rough and classical stochastic volatility models to the SPX and VIX options markets. Our analysis focusses primarily on calibration quality and is split in two parts. In part one, we perform a historical calibration to SPX options over the years 2004–2019 of a selection of models that include the one-factor rough Bergomi and rough Heston models. In part two, we consider three calibration dates with low, typical, and high volatility, examine a wide selection of models, and calibrate to both SPX options as well as jointly to SPX and VIX options. The key results are as follows: The rough Bergomi and rough Heston models fail to create a term structure of smile effect that is sufficiently pronounced for SPX options. Moreover, we discover that short-expiry SPX smiles generally are more symmetric than long-expiry smiles, a feature we neither find that these models can reproduce. We propose an alternative volatility model driven by two Ornstein-Uhlenbeck processes that uses a non-standard transformation function. Calibrating it to SPX options, we obtain almost perfect fits, and calibrating it jointly to SPX and VIX options, we obtain very decent fits. This suggests, contrary to what one might be led to believe based on much of the existing literature, that the joint SPX-VIX calibration problem is largely solvable with classical two-factor volatility, all without roughness and jumps.

Multistep forecast of the implied volatility surface using deep learning

AbstractModeling implied volatility surface (IVS) is of paramount importance to price and hedge an option. We contribute to the literature by modeling the entire IVS using convolutional long‐short‐term memory (ConvLSTM) and long‐short‐term memory (LSTM) neural networks to produce multivariate and multistep forecasts of the S&P 500 IVS. Using daily SPX options data (2002–2019), we find that both LSTM and ConvLSTM fit the training data extremely well with mean absolute percentage error (MAPE) being 3.56% and 3.88%, respectively. The ConvLSTM (8.26% MAPE) model significantly outperforms LSTM and traditional time series models in predicting the IVS out of sample.

SPX Calibration of Option Approximations under Rough Heston Model

The volatility of stock return does not follow the classical Brownian motion, but instead it follows a form that is closely related to fractional Brownian motion. Taking advantage of this information, the rough version of classical Heston model also known as rough Heston model has been derived as the macroscopic level of microscopic Hawkes process where it acts as a high-frequency price process. Unlike the pricing of options under the classical Heston model, it is significantly harder to price options under rough Heston model due to the large computational cost needed. Previously, some studies have proposed a few approximation methods to speed up the option computation. In this study, we calibrate five different approximation methods for pricing options under rough Heston model to SPX options, namely a third-order Padé approximant, three variants of fourth-order Padé approximant, and an approximation formula made from decomposing the option price. The main purpose of this study is to fill in the gap on lack of numerical study on real market options. The numerical experiment includes calibration of the mentioned methods to SPX options before and after the Lehman Brothers collapse.

Consistent and efficient pricing of SPX and VIX options under multiscale stochastic volatility

AbstractWe provide consistent and efficient pricing for both Standard & Poor's 500 Index options and the Chicago Board Options Exchange's Volatility Index options under a multiscale stochastic volatility model. To capture the multiscale, our model adds a fast scale factor to Heston's volatility and we derive approximate analytic formulas for the options under the model. The analytic tractability can greatly improve the efficiency of calibration compared to fitting procedures using a numerical scheme. Our experiment using options data for 3 years shows that the model reduces about 20% of the errors for a single‐scale model.

Semimartingale and continuous-time Markov chain approximation for rough stochastic local volatility models

Rough volatility models have recently been empirically shown to provide a good fit to historical volatility time series and implied volatility smiles of SPX options. They are continuous-time stochastic volatility models, whose volatility process is driven by a fractional Brownian motion with Hurst parameter less than half. Due to the challenge that it is neither a semimartingale nor a Markov process, there is no unified method that not only applies to all rough volatility models, but also is computationally efficient. This paper proposes a semimartingale and continuous-time Markov chain (CTMC) approximation approach for the general class of rough stochastic local volatility (RSLV) models. In particular, we introduce the perturbed stochastic local volatility (PSLV) model as the semimartingale approximation for the RSLV model and establish its existence , uniqueness and Markovian representation. We propose a fast CTMC algorithm and prove its weak convergence. Numerical experiments demonstrate the accuracy and high efficiency of the method in pricing European, barrier and American options. Comparing with existing literature, a significant reduction in the CPU time to arrive at the same level of accuracy is observed.

Historical Analysis of Rough Volatility Models to the SPX Market

We perform a historical analysis of selected rough volatility models to the SPX market. Tailoring the neural network pricing method of [27] to our needs, we train neural networks for the rough Heston model from [14], the rough Bergomi model from [4] as well as an extended version of the latter. As a benchmark we include also the classical Heston model from [24]. Calibrating the models across 15 years of historical SPX options prices we first and foremost document consistently superior results using rough volatility. Comparing rough Heston and rough Bergomi we also find that while the former model calibrates slightly better, the latter model produces more robust predictions. Our calibration results also illuminate a structural problem in that both of these models (on average) produces too little curvature at short expirations, too little skew at long expirations. Using an extended rough Bergomi model where the explosion rates of smile and skew are decoupled, did not resolve this problem.

Valuation of VIX and target volatility options with affine GARCH models

AbstractIn this paper we propose semiclosed‐form solutions, subject to an inversion of the Fourier transform, for the price of VIX options and target volatility options under affine GARCH models based on Gaussian and Inverse Gaussian distributions. We illustrate the advantage of the proposed analytic expressions by comparing them with those obtained from benchmark Monte–Carlo simulations. The empirical performance of the two affine GARCH models is tested using different calibration exercises based on historical returns and market quotes on VIX and SPX options.

The Price of Higher Order Catastrophe Insurance: The Case of VIX Options

We develop a tractable equilibrium pricing model to explain observed characteristics in equity returns, VIX futures, S&P 500 options, and VIX options data based on affine jump- diffusive state dynamics and representative agents endowed with Duffie-Epstein recursive preferences. A specific model aimed at capturing VIX options prices and other asset market data is shown to successfully replicate the salient features of consumption, dividends, and asset market data, including the first two moments of VIX futures returns, the average implied volatilities in SPX and VIX options, and first and higher-order moments of VIX options returns. We document a time variation in the shape of VIX option implied volatility and a time-varying hedging relationship between VIX and SPX options which our model both captures.

How Integrated Are Credit and Equity Markets? Evidence From Index Options

In recent years, a liquid market for options on a broad credit default swap index (CDX) has developed. We study the extent to which these options are priced consistently with options on a broad equity index (SPX). We consider a rich structural credit risk model in which firm assets follow a jump-diffusion process with idiosyncratic and systematic risk, and we derive analytical expressions for CDX and SPX options. Calibrating the model, we find that it captures many aspects of the joint dynamics of CDX and SPX options. However, it cannot reconcile the relative levels of option implied volatilities, suggesting that credit and equity markets are not fully integrated. A strategy of selling CDX options yields significantly higher average excess returns and Sharpe ratios than selling SPX options.

Valuation of VIX and Target Volatility Options with Affine GARCH Models

In this paper we propose semi-closed-form solutions, subject to an inversion of the Fourier transform, for the price of VIX options and target volatility options (TVOs) under affine GARCH models based on Gaussian and Inverse Gaussian distributions. We illustrate the advantage of the proposed analytic expressions by comparing them with those obtained from benchmark Monte-Carlo simulations. The empirical performance of the two affine GARCH models is tested using different calibration exercises based on historical returns and market quotes on VIX and SPX options.

Joint Modelling and Calibration of SPX and VIX by Optimal Transport

This paper addresses the joint calibration problem of SPX options and VIX options or futures. We show that the problem can be formulated as a semimartingale optimal transport problem under a finite number of discrete constraints, in the spirit of [arXiv:1906.06478]. We introduce a PDE formulation along with its dual counterpart. The solution, a calibrated diffusion process, can be represented via the solutions of Hamilton-Jacobi-Bellman equations arising from the dual formulation. The method is tested on both simulated data and market data. Numerical examples show that the model can be accurately calibrated to SPX options, VIX options and VIX futures simultaneously.