Stochastic Volatility Framework Research Articles

A model‐free approximation for barrier options in a general stochastic volatility framework

AbstractFor a general stochastic volatility framework with correlation between the spot price and the instantaneous volatility, an analytical approximation for single barrier options with continuous monitoring is given. The approximation is expressed only in terms of market observable implied volatilities and prices. As such the approximation is independent of the specific form and number of parameters of the skew‐generating stochastic volatility model.

Quantile-based spillover connectedness among stochastic volatilities of ESG equities, Islamic and conventional stocks with implications for portfolio management

PurposeThis study aims to investigate the relationship between stock markets, environmental, social and governance (ESG) factors and Shariah-compliant in an integrated framework.Design/methodology/approachThe authors employ the multivariate factor stochastic volatility (mvFSV) framework to extract the volatility of the different sectoral indices. Based on this evidence, the authors employ the quantile vector autoregressive (QVAR) approach to examine the dynamic spillover connectedness among the aforementioned indices.FindingsThe study emphasizes the following major findings: (1) significant time-varying spillover connectedness across quantiles, (2) bidirectional and asymmetric spillover effect among the ESG index and the other sectoral indices, (3) the strength of spillover connectedness is time-varying across quantiles, (4) based on the perspective of portfolio optimization, ESG market is a significant strong forecasting contributor to conventional and Shariah-compliant markets, (5) overall, the findings point out serious quantile pass-through effect among ESG index and the other sectoral indices during the COVID-19 health crisis.Originality/valueThis study extends the previous literature in the following ways. First, to the best of the researchers’ knowledge, none of the existing studies have investigated the relationship between stock markets, ESG factors and Shariah-compliant in an integrated framework. Second, this study extends the previous scholarships by applying the mvFSV. Third, the authors propose a new rolling version to estimate dynamic spillovers, namely the rolling-window quantile VAR method. This approach provides a great advantage in computing the dynamics of return and variance spillover between variables in terms not only of the overall factor but also of the net (pairwise) aspect.

A Model-Free Approximation for Barrier Options in a General Stochastic Volatility Framework

A Model-Free Approximation for Barrier Options in a General Stochastic Volatility Framework

CBI-time-changed Lévy processes for multi-currency modeling

We develop a stochastic volatility framework for modeling multiple currencies based on CBI-time-changed Lévy processes. The proposed framework captures the typical risk characteristics of FX markets and is coherent with the symmetries of FX rates. Moreover, due to the self-exciting behavior of CBI processes, the volatilities of FX rates exhibit self-exciting dynamics. By relying on the theory of affine processes, we show that our approach is analytically tractable and that the model structure is invariant under a suitable class of risk-neutral measures. A semi-closed pricing formula for currency options is obtained by Fourier methods. We propose two calibration methods, also by relying on deep-learning techniques, and show that a simple specification of the model can achieve a good fit to market data on a currency triangle.

Optimal Limit Order Book Trading Strategies with Stochastic Volatility in the Underlying Asset

In quantitative finance, there have been numerous new aspects and developments related with the stochastic control and optimization problems which handle the controlled variables of performing the behavior of a dynamical system to achieve certain objectives. In this paper, we address the optimal trading strategies via price impact models using Heston stochastic volatility framework including jump processes either in price or in volatility of the price dynamics with the aim of maximizing expected return of the trader by controlling the inventories. Two types of utility functions are considered: quadratic and exponential. In both cases, the remaining inventories of the market maker are charged with a liquidation cost. In order to achieve the optimal quotes, we control the inventory risk and follow the influence of each parameter in the model to the best bid and ask prices. We show that the risk metrics including profit and loss distribution (PnL), standard deviation and Sharpe ratio play important roles for the trader to make decisions on the strategies. We apply finite differences and linear interpolation as well as extrapolation techniques to obtain a solution of the nonlinear Hamilton-Jacobi-Bellman (HJB) equation. Moreover, we consider different cases on the modeling to carry out the numerical simulations.

Pricing of the geometric Asian options under a multifactor stochastic volatility model

This paper focuses on the pricing of continuous geometric Asian options (GAOs) under a new multifactor stochastic volatility model. The model considers fast and slow mean reverting factors of volatility, where slow volatility factor is approximated by a quadratic arc. The asymptotic expansion of the price function is assumed, and the first order price approximation is derived using the perturbation techniques for both floating and fixed strike GAOs. Much simplified pricing formulae for the GAOs are obtained in this multifactor stochastic volatility framework. The zeroth order term in the price approximation is the modified Black–Scholes price for the GAOs. This modified price is expressed in terms of the Black–Scholes price for the GAOs. The accuracy of the approximate option pricing formulae is established, and also verified numerically by comparing the model prices with the Monte Carlo simulation prices and the Black–Scholes prices for the GAOs. The model parameter is estimated by capturing the volatility smiles. The sensitivity analysis is also performed to investigate the effect of underlying parameters on the approximated prices.

Performance measurement for option portfolios in a stochastic volatility framework

Measuring the performance of stock portfolios that include options is challenging due to options' nonlinearity in the underlying, their exposure to volatility risk, and their time decay. Our contribution to the literature is twofold: First, we provide a theoretically rigorous derivation of the time-variable factor loadings in a two-factor model under stochastic volatility according to [Heston, S.L., A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud., 1993, 6, 327–343.]. Within this setting, the portfolio returns are explained by the market and an additional option factor, i.e. a portfolio of standard options exposed to volatility risk. We show that (i) any option factor is suitable to perfectly explain the portfolio behavior if simple returns are considered in instantaneous time and that (ii) the option factor's loading equals the fraction of the volatility elasticities of the portfolio and of the option factor while the option factor's underlying elasticity enters the factor loading of the underlying. Second, we analyze the behavior of option factors in practical applications, where time is discrete and factor loadings are estimated in a single regression over a certain time horizon. We show how the bias of (Jensen) alpha and its sign depend on the skewness of market returns. For several option factors from the literature, we conduct a simulation study to analyze their suitability to reduce this bias. As the results are disappointing, we propose a two-step procedure for choosing an adequate factor.

Convergence analysis for continuous-time Markov chain approximation of stochastic local volatility models: Option pricing and Greeks

This paper establishes the precise second order convergence rates of the continuous-time Markov chain (CTMC) approximation method for pricing options under the general framework of stochastic local volatility (SLV) models. The stochastic local volatility models studied in this paper include Heston model, 4/2 model, α-Hypergeometric model, stochastic alpha beta rho (SABR) model, Heston-SABR model and quadratic SLV model. Using the stochastic flow theorem, the closed-form CTMC approximation formula for the Greeks are obtained and the second order convergence rates are proved. Numerical examples confirm the theoretical findings.

A CLOSED-FORM PRICING FORMULA FOR EUROPEAN EXCHANGE OPTIONS WITH STOCHASTIC VOLATILITY

This article derives a closed-form pricing formula for the European exchange option in a stochastic volatility framework. Firstly, with the Feynman–Kac theorem's application, we obtain a relation between the price of the European exchange option and a European vanilla call option with unit strike price under a doubly stochastic volatility model. Then, we obtain the closed-form solution for the vanilla option using the characteristic function. A key distinguishing feature of the proposed simplified approach is that it does not require a change of numeraire in contrast with the usual methods to price exchange options. Finally, through numerical experiments, the accuracy of the newly derived formula is verified by comparing with the results obtained using Monte Carlo simulations.

Pricing multiple barrier derivatives under stochastic volatility

This work generalizes existing one- and two-dimensional pricing formulas with an equal number of barriers to a setting of n dimensions and up to two barriers in the presence of stochastic volatility. This allows for the consideration of multidimensional single-barrier derivatives with, for example, a collateral triggered by a barrier default of the issuing company. We introduce stochastic volatility to a multidimensional Black–Scholes framework via the common Cox–Ingersoll–Ross process and present semianalytical solutions for collateralized structured products with two barriers, one representing default and the other a market-related option. Our model accommodates implied volatility skew along the lines of displaced diffusions. We show that our semianalytical formulas are more efficient in terms of computational speed than Monte Carlo simulations, particularly for tail scenarios. Moreover, our proposed analytical simplifications permit a twenty-fivefold gain in time savings compared with the results given by the main theorem. These multidimensional structured products gained increasing popularity after the subprime and financial crises. Therefore, we perform comprehensive sensitivity analyses with respect to stochastic volatility parameters and contribute to a better understanding of multidimensional barrier derivatives in a stochastic volatility framework.

A Bayesian panel stochastic volatility measure of financial stability

AbstractWe propose to model financial stability, opting for an alternative bank profit function whose volatility is measured within a framework of panel stochastic volatility. Within this model financial stability and volatility are latent variables. To observe financial stability and volatility we employ Bayesian inference procedures organized around Sequential Monte Carlo (SMC) technique and particle filtering. We do so in a single stage that controls also for non‐linearities, while we also allow for some key bank and country‐specific variables to impact upon financial stability and volatility. Thus, we provide a new measure of financial stability by country, over time and also at a global level. In an empirical application, we derive financial stability indexes for a plethora of countries, as well as the global financial stability index that acts as an early warning index. Our results suggest that the financial cycle is subject to non‐linearities. We argue that the global financial system should closely monitor large, systemic, banks as key to support financial stability.

RBF methods in a Stochastic Volatility framework for Greeks computation

In this paper we aim to apply a new, proposed meshless approach for Heston PDE resolution. In Mathematical Finance, most classical models may be reformulated as partial differential equations (PDE), which are commonly solved by finite difference and Monte Carlo methods. Alternatively, bio-inspired methods may be considered to select parameters and to solve said PDEs. In particular, we focus on Heston model PDE and we solve it via Radial Basis Functions (RBF) methods. Moreover some considerations on the computation of model parameters through learning approaches have been discussed. We aim to price a vanilla call option, focusing on the derivation of the formulae for its Greeks profiles. Eventually, we conclude RBFs are an accurate alternative method to price derivatives in a stochastic volatility framework, especially since they allow for a fast and precise computation of Greeks.

Volatility forecasts using stochastic volatility models with nonlinear leverage effects

AbstractThe leverage effect—the correlation between an asset's return and its volatility—has played a key role in forecasting and understanding volatility and risk. While it is a long standing consensus that leverage effects exist and improve forecasts, empirical evidence puzzlingly does not show that this effect exists for many individual stocks, mischaracterizing risk, and therefore leading to poor predictive performance. We examine this puzzle, with the goal to improve density forecasts, by relaxing the assumption of linearity of the leverage effect. Nonlinear generalizations of the leverage effect are proposed within the Bayesian stochastic volatility framework in order to capture flexible leverage structures. Efficient Bayesian sequential computation is developed and implemented to estimate this effect in a practical, on‐line manner. Examining 615 stocks that comprise the S&P500 and Nikkei 225, we find that our proposed nonlinear leverage effect model improves predictive performances for 89% of all stocks compared to the conventional stochastic volatility model.

Inflation Volatility with Regime Switching

AbstractThis paper presents a new approach to model U.S. inflation dynamics by allowing regime switching in an unobserved components stochastic volatility framework. We use a modified particle filter to construct likelihood and estimate the model using MLE. The number of regimes is determined based on a bootstrap. We find that a model with three regimes and regime‐dependent constant volatilities has superior performance. In addition, we show that since 2000:II, U.S. inflation has entered a regime with moderate volatility where most of the volatility comes from transitory shocks.

A Full Asymptotic Series of European Call Option Prices in the SABR Model with Beta = 1

We develop two new pricing formulae for European options. The purpose of these formulae is to better understand the impact of each term of the model, as well as improve the speed of the calculations. We consider the SABR model (with β=1) of stochastic volatility, which we analyze by tools from Malliavin Calculus. We follow the approach of Alos et al. (2006) who showed that under stochastic volatility framework, the option prices can be written as the sum of the classic Hull-White (1987) term and a correction due to correlation. We derive the Hull-White term, by using the conditional density of the average volatility, and write it as a two-dimensional integral. For the correction part, we use two different approaches. Both approaches rely on the pairing of the exponential formula developed by Jin, Peng, and Schellhorn (2016) with analytical calculations. The first approach, which we call “Dyson series on the return’s idiosyncratic noise” yields a complete series expansion but necessitates the calculation of a 7-dimensional integral. Two of these dimensions come from the use of Yor’s (1992) formula for the joint density of a Brownian motion and the time-integral of geometric Brownian motion. The second approach, which we call “Dyson series on the common noise” necessitates the calculation of only a one-dimensional integral, but the formula is more complex. This research consisted of both analytical derivations and numerical calculations. The latter show that our formulae are in general more exact, yet more time-consuming to calculate, than the first order expansion of Hagan et al. (2002).

Volatility Discovery across Interlinked Securities

Where does new volatility enter the volatility of securities listed in many countries? While literature has focused on where information enters the price, I develop a framework to study how each market’s volatility contributes to the permanent volatility of the Asset. I build a VECM with an Autoregressive Stochastic Volatility (ASV) framework estimated using the MCMC method and Bayesian inference. This specification allows defining the measures of a market’s contribution to volatility discovery. In the application, I study cash and 3-months futures markets of some metals traded on the London Metals Exchange. I also study the EURO STOXX 50 Index and its futures. I find that for most the securities, while price discovery happens on the cash market, the volatility discovery mostly happens in the futures market. Overall, the results suggest that information discovery and volatility discovery do not necessarily have the same determinants.

Option Pricing in Stochastic Volatility Models Driven by Fractional Jump-Diffusion Processes

In this paper, we propose a fractional stochastic volatility jump-diffusion model which extends the Bates(1996) model. Where we model the volatility as a fractional process. Extensive empirical studies show that the distributions of the logarithmic returns of financial asset usually exhibit properties of self-similarity and long-range dependence and since the fractional Brownian motion has these two important properties, it has the ability to capture the behavior of underlying asset price. Further incorporating jumps into the stochastic volatility framework gives further freedom to financial mathematicians to fit both the short and long end of the implied volatility surface. We propose a stochastic model which contains both fractional and jump process. Then we price options using Monte Carlo simulations along with a variance reduction technique(antithetic variates). We use market data from the S&P 500 index and we compare our results with the Heston and Bates model using error measures. The results show our model greatly outperforms previous models in terms of estimation accuracy.

The short‐time behavior of VIX‐implied volatilities in a multifactor stochastic volatility framework

AbstractWe consider a modeling setup where the volatility index (VIX) dynamics are explicitly computable as a smooth transformation of a purely diffusive, multidimensional Markov process. The framework is general enough to embed many popular stochastic volatility models. We develop closed‐form expansions and sharp error bounds for VIX futures, options, and implied volatilities. In particular, we derive exact asymptotic results for VIX‐implied volatilities, and their sensitivities, in the joint limit of short time‐to‐maturity and small log‐moneyness. The expansions obtained are explicit based on elementary functions and they neatly uncover how the VIX skew depends on the specific choice of the volatility and the vol‐of‐vol processes. Our results are based on perturbation techniques applied to the infinitesimal generator of the underlying process. This methodology has previously been adopted to derive approximations of equity (SPX) options. However, the generalizations needed to cover the case of VIX options are by no means straightforward as the dynamics of the underlying VIX futures are not explicitly known. To illustrate the accuracy of our technique, we provide numerical implementations for a selection of model specifications.

Asset liability management for an ordinary insurance system with proportional reinsurance in a CIR stochastic interest rate and Heston stochastic volatility framework

<p style='text-indent:20px;'>This paper investigates the asset liability management problem for an ordinary insurance system incorporating the standard concept of proportional reinsurance coverage in a stochastic interest rate and stochastic volatility framework. The goal of the insurer is to maximize the expectation of the constant relative risk aversion (CRRA) of the terminal value of the wealth, while the goal of the reinsurer is to maximize the expected exponential utility (CARA) of the terminal wealth held by the reinsurer. We assume that the financial market consists of risk-free assets and risky assets, and both the insurer and the reinsurer invest on one risk-free asset and one risky asset. By using the stochastic optimal control method, analytical expressions are derived for the optimal reinsurance control strategy and the optimal investment strategies for both the insurer and the reinsurer in terms of the solutions to the underlying Hamilton-Jacobi-Bellman equations and stochastic differential equations for the wealths. Subsequently, a semi-analytical method has been developed to solve the Hamilton-Jacobi-Bellman equation. Finally, we present numerical examples to illustrate the theoretical results obtained in this paper, followed by sensitivity tests to investigate the impact of reinsurance, risk aversion, and the key parameters on the optimal strategies.

A general framework for pricing Asian options under stochastic volatility on parallel architectures

In this paper, we present a transform-based algorithm for pricing discretely monitored arithmetic Asian options with remarkable accuracy in a general stochastic volatility framework, including affine models and time-changed Lévy processes. The accuracy is justified both theoretically and experimentally. In addition, to speed up the valuation process, we employ high-performance computing technologies. More specifically, we develop a parallel option pricing system that can be easily reproduced on parallel computers, also realized as a cluster of personal computers. Numerical results showing the accuracy, speed and efficiency of the procedure are reported in the paper.