Stochastic Volatility Process Research Articles

On Pricing Options Under Two Stochastic Volatility Processes

On Pricing Options Under Two Stochastic Volatility Processes

An Econometric Analysis of Volatility Discovery

We investigate information processing in the stochastic process driving stock’s volatility (volatility discovery). We apply fractionally cointegration techniques to decompose the estimates of the market-specific integrated variances into an estimate of the common integrated variance of the efficient price and a transitory component. The market weights on the common integrated variance of the efficient price are the volatility discovery measures. We relate the volatility discovery measure to the price discovery framework and formally show their roles on the identification of the integrated variance of the efficient price. We establish the limiting distribution of the volatility discovery measures by resorting to both long span and in-fill asymptotics. The empirical application is in line with our theoretical results, as it reveals that trading venues incorporate new information into the stochastic volatility process in an individual manner and that the volatility discovery analysis identifies a distinct information process than that based on the price discovery analysis.

On the Implied Volatility of Asian Options Under Stochastic Volatility Models

In this paper, we study the short-time behaviour of the at-the-money implied volatility for arithmetic Asian options with fixed strike price. The asset price is assumed to follow the Black–Scholes model with a general stochastic volatility process. Using techniques of the Malliavin calculus developed in Alòs, García-Lorite, and Muguruza [2022. On Smile Properties of Volatility Derivatives: Understanding the VIX Skew. SIAM Journal on Financial Mathematics. 13(1): 32–69. https://doi.org/10.1137/19M1269981], we give sufficient conditions on the stochastic volatility in order to compute the level of the implied volatility of the option when the maturity converges to zero. Then, we find a short maturity asymptotic formula for the skew slope of the implied volatility that depends on the correlation between prices and volatilities and the Hurst parameter of the volatility model. We apply our general results to the SABR and fractional Bergomi models, and provide numerical simulations that confirm the accurateness of the asymptotic formulas.

CBI-time-changed Lévy processes

We introduce and study the class of CBI-time-changed Lévy processes (CBITCL), obtained by time-changing a Lévy process with respect to an integrated continuous-state branching process with immigration (CBI). We characterize CBITCL processes as solutions to a certain stochastic integral equation and relate them to affine stochastic volatility processes. We provide a complete analysis of the time of explosion of exponential moments of CBITCL processes and study their asymptotic behavior. In addition, we show that CBITCL processes are stable with respect to a suitable class of equivalent changes of measure. As illustrated by some examples, CBITCL processes are flexible and tractable processes with a significant potential for applications in finance.

Estimating and testing skewness in a stochastic volatility model

In this paper we propose a novel approach to estimating and testing skewness in a stochastic volatility (SV) model. Our key idea is to replace a normal return error in the standard SV model with a split normal error. We show that this simple variation in the model brings about two large computational advantages. First, the stochastic volatility process can be simulated fast and efficiently using a one-block Gibbs sampling technique. Second, more importantly, this is the first to provide a marginal likelihood calculation method to formally test the coexistence of stochastic volatility and skewness in return errors within a Bayesian framework. We demonstrate the efficiency and reliability of our posterior sampling and model comparison methods through a simulation study. The simulation results show that neglecting skewness leads to inaccurate estimates on both the volatility process and conditional expected returns. Our empirical applications to daily stock return data provide a strong evidence of negative skewness.

A novel stochastic modeling framework for coal production and logistics through options pricing analysis

We propose a novel stochastic modeling framework for coal production and logistics using option pricing theory. The problem of valuing the inherent real optionality a coal producer has when mining and processing thermal coal is modelled as pricing spread options of three assets under the stochastic volatility model. We derive a three-dimensional Fast Fourier Transform (“FFT”) lower bound approximation to value the inherent real optionality and for robustness check, we compare the semi-analytical pricing accuracy with the Monte Carlo simulation. Model parameters are estimated from the historical monthly data, and stochastic volatility parameters are obtained by matching the Kurtosis of the low-ash diff data to the Kurtosis of the stochastic volatility process which is assumed to follow Cox–Ingersoll–Ross (“CIR”) model.

Stochastic volatility modeling of high-frequency CSI 300 index and dynamic jump prediction driven by machine learning

<abstract><p>This paper models stochastic process of price time series of $ CSI $ $ 300 $ index in Chinese financial market, analyzes volatility characteristics of intraday high-frequency price data. In the new generalized Barndorff-Nielsen and Shephard model, the lag caused by asynchrony of market information and market microstructure noises are considered, and the problem of lack of long-term dependence is solved. To speed up the valuation process, several machine learning and deep learning algorithms are used to estimate parameter and evaluate forecast results. Tracking historical jumps of different magnitudes offers promising avenues for simulating dynamic price processes and predicting future jumps. Numerical results show that the deterministic component of stochastic volatility processes would always be captured over short and longer-term windows. Research finding could be suitable for influence investors and regulators interested in predicting market dynamics based on high-frequency realized volatility.</p></abstract>

An Ornstein–Uhlenbeck Model with the Stochastic Volatility Process and Tempered Stable Process for VIX Option Pricing

To effectively fit the dynamics and structure of frequent small jumps and sparse large jumps in the VIX time series, we introduce the tempered stable process (the CTS process and CGMY process) into the Ornstein–Uhlenbeck (OU) stochastic volatility model to build an OU model with the stochastic volatility process and tempered stable process. Based on two different assumptions for the underlying assets, we derive the formula of pricing models via two methods. Empirical studies are conducted to prove that our pricing models have a better performance in matching the VIX options. Furthermore, we find that the pricing model via the infinitesimal value method yields better results than pricing models with a measure of change. Overall, our proposed models enrich the derivative pricing theory and help investors understand and hedge risks.

A class of portfolio optimization solvable problems

This short paper reveals the largest class of stochastic volatility processes solvable in closed form within expected utility theory for a hyperbolic absolute risk aversion investor. The risky-asset setting considers a framework outside the seminal work of Liu (2007), and highlights applications not yet studied in the literature. The work also demonstrates that analytical solutions for ambiguity-aversion analyses within the framework of Maenhout (2004) are feasible.

Valuation of guaranteed minimum maturity benefits under generalised regime-switching models using the Fourier Cosine method

This paper presents a flexible valuation approach for variable annuity (VA) contracts embedded with guaranteed minimum maturity benefit (GMMB) riders written on an underlying fund that evolves according to a general regime-switching framework. Unlike the classical regime-switching models which only allow model parameters to change upon regime switches, our framework allows, more importantly, model structures to vary. With mild assumptions on the characteristic function of the log-stock price, our model settings enable the study of fundamental features of the market dynamics, such as stochastic volatility and jumps, on the underlying fund value of GMMB in a unified framework. This novel idea is illustrated by a three-regime model whose environments can be characterised by either the geometric Brownian motion process, the Heston (1993) stochastic volatility process, or double exponential process. Three versions of the GMMB riders are considered; a fixed or roll-up guarantee, a geometric average guarantee and a ratchet guarantee. With the Fourier Cosine (COS) method which utilises characteristic functions, explicit valuation expressions for various contracts are derived, and numerical illustrations are performed to analyse the efficiency of the approach in terms of computational speed and accuracy. The paper makes a unique contribution by presenting regime-dependent bounds and an algorithm for determining the optimal grid points required for the COS method to achieve a specific level of accuracy. Numerical experiments for the valuation framework reveal that i) as the likelihood of regime shifts increases, the price difference of VA contracts with different initial regimes diminishes, which is consistent with financial intuition; ii) the generalised regime-switching models are more suitable for VA pricing when the time-to-maturity spans over several decades, during which structural changes, although infrequent, are inevitable.

Modeling the volatilities of globally listed private equity markets

PurposeThis paper aims to analyze the characteristics of stochastic volatility processes in globally listed private equity (LPE) markets, which are represented by nine global, regional and style indices, and reveals transmissions in the conditional variances between the different markets, based on weekly data covering the period January 2011 to December 2020.Design/methodology/approachThe study uses the generalized autoregressive conditional heteroscedasticity [GARCH(p, q)] model and its exponential GARCH (EGARCH) and GARCH-in-mean extensions.FindingsThe estimates of the volatility models GARCH, EGARCH and GARCH-in-mean GARCH-M for testing the stylized properties persistence, asymmetry, mean reversion and risk premium lead to very different results, depending on the respective LPE index.Practical implicationsThe knowledge of conditional volatilities of LPE returns as well as the detection of volatility transmissions between the different LPE markets under investigation serve to support asset allocation decisions with respect to risk management or portfolio allocation. Hence, the findings are important for all kinds of investors and asset managers who consider investments in LPE.Originality/valueThe authors present a novel study that examines the conditional variance for globally LPE markets by using LPX indices, offering valuable insight into this growing asset class.

Time Discretized Variational Iteration Method for the Stochastic Volatility Process with Jumps

A model for both stochastic jumps and volatility for equity returns in the area of option pricing is the stochastic volatility process with jumps (SVPJ). A major advantage of this model lies in the area of mean reversion and volatility clustering between returns and volatility with uphill movements in price asserts. Thus, in this article, we propose to solve the SVPJ model numerically through a discretized variational iteration method (DVIM) to obtain sample paths for the state variable and variance process at various timesteps and replications in order to estimate the expected jump times at various iterates resulting from executing the DVIM as n increases. These jumps help in estimating the degree of randomness in the financial market. It was observed that the average computed expected jump times for the state variable and variance process is moderated by the parameters (variance process through mean reversion), Θ (long-run mean of the variance process), σ (volatility variance process) and λ (constant intensity of the Poisson process) at each iterate. For instance, when = 0.0, Θ = 0.0, σ = 0.0 and λ = 1.0, the state variable cluttered maximally compared to the variance process with less volatility cluttering with an average computed expected jump times of 52.40607869 as n increases in the DVIM scheme. Similarly, when = 3.99, Θ = 0.014, σ = 0.27 and λ = 0.11, the stochastic jumps for the state variable are less cluttered compared to the variance process with maximum volatility cluttering as n increases in the DVIM scheme. In terms of option pricing, the value 52.40607869 suggest a better bargain compared to the value 20.40344029 due to the fact that it yields less volatility rate. MAPLE 18 software was used for all computations in this research.

Goal Achieving Probabilities of Mean-Variance Strategies in a Market with Regime-Switching Volatility

In this paper, we establish properties for the switch-when-safe mean-variance strategies in the context of a Black-Scholes market model with stochastic volatility processes driven by a continuous-time Markov chain with a finite number of states. More precisely, expressions for the goal-achieving probabilities of the terminal wealth are obtained and numerical comparisons of lower bounds for these probabilities are shown for various market parameters. We conclude with asymptotic results when the Markovian changes in the volatility parameters appear with either higher or lower frequencies.

On the Relationship between Uhlig Extended and beta‐Bartlett Processes

Stochastic volatility processes are used in multi‐variate time series analysis to track time‐varying patterns in covariance matrices. Uhlig extended (UE) and beta‐Bartlett (BB) processes are especially convenient for analyzing high‐dimensional time series because they are conjugate with Wishart likelihoods. In this article, we show that UE and BB are closely related, but not equivalent: their hyperparameters can be matched so that they have the same forward‐filtered posteriors and one‐step ahead forecasts, but different joint (smoothed) posterior distributions. Under this circumstance, Bayes factors cannot discriminate the models and alternative approaches to model comparison are needed. We illustrate these issues in a retrospective analysis of volatilities of returns of foreign exchange rates. Additionally, we provide a backward sampling algorithm for the BB process, for which retrospective analysis had not been developed.

Singular spectrum analysis for value at risk in stochastic volatility models

AbstractEstimation of the value at risk (VaR) requires prediction of the future volatility. Whereas this is a simple task in ARCH and related models, it becomes much more complicated in stochastic volatility (SV) processes where the volatility is a function of a latent variable that is not observable. In‐sample (present and past values) and out‐of‐sample (future values) predictions of that unobservable variable are thus necessary. This paper proposes singular spectrum analysis (SSA), which is a fully nonparametric technique that can be used for both purposes. A combination of traditional forecasting techniques and SSA is also considered to estimate the VaR. Their performance is assessed in an extensive Monte Carlo and with an application to a daily series of S&P500 returns.

Analyzing Commodity Futures Using Factor State-Space Models with Wishart Stochastic Volatility

A factor state-space approach with stochastic volatility is proposed for modeling and forecasting the maturity structure of future commodity contracts. The proposed approach builds upon the dynamic 3-factor Nelson-Siegel model and its 4-factor Svensson extension and assumes for the latent level, slope and curvature factors a Gaussian vector autoregression with a multivariate Wishart stochastic volatility process. A computationally fast and easy to implement MCMC algorithm for the Bayesian posterior analysis is developed, which exploits the conjugacy of the Wishart and the Gaussian distribution. An empirical application to daily prices for contracts on crude oil with stipulated delivery dates ranging from one to 24 months ahead show that the estimated 4-factor Svensson model with two curvature factors provides a good parsimonious representation of the serial correlation in the individual prices and their volatility. It also shows that this model has a good out-of-sample forecast performance.

Efficient State Preparation for Quantum Amplitude Estimation

Quantum Amplitude Estimation (QAE) can achieve a quadratic speed-up for applications classically solved by Monte Carlo simulation. A key requirement to realize this advantage is efficient state preparation. If state preparation is too expensive, it can diminish the quantum advantage. Preparing arbitrary quantum states has exponential complexity with respect to the number of qubits, thus, is not applicable. Currently known efficient techniques require problems based on log-concave probability distributions, involve learning an unknown distribution from empirical data, or fully rely on quantum arithmetic. In this paper, we introduce an approach to simplify state preparation, together with a circuit optimization technique, both of which can help reduce the circuit complexity for QAE state preparation significantly. We demonstrate the introduced techniques for a numerical integration example on real quantum hardware, as well as for option pricing under the Heston model, i.e., based on a stochastic volatility process, using simulation.

Deep Hedging of Derivatives Using Reinforcement Learning

This article shows how reinforcement learning can be used to derive optimal hedging strategies for derivatives when there are transaction costs. The article illustrates the approach by showing the difference between using delta hedging and optimal hedging for a short position in a call option when the objective is to minimize a function equal to the mean hedging cost plus a constant times the standard deviation of the hedging cost. Two situations are considered. In the first, the asset price follows a geometric Brownian motion. In the second, the asset price follows a stochastic volatility process. The article extends the basic reinforcement learning approach in several ways. First, it uses two different Q-functions to track both the expected value of the cost and the expected value of the square of the cost. This approach increases the range of objective functions that can be used. Second, it uses a learning algorithm that allows for continuous state and action space. Third, it compares the accounting profit and loss (P&amp;L) approach and the cash flow approach. The authors find that a hybrid approach involving the use of an accounting P&amp;L approach that incorporates a relatively simple valuation model works well. <b>TOPICS:</b>Big data/machine learning, derivatives, simulations <b>Key Findings</b> ▪ The authors show how reinforcement learning can be used to derive optimal hedging strategies for derivatives in the presence of transaction costs. ▪ They extend the standard reinforcement learning approach by using multiple Q-functions to increase the range of objective functions that can be used and by using algorithms that allow the state space and action space to be continuous. ▪ For valuing the hedging position at each step, the authors compare three approaches: the accounting P&amp;L, cash flow, and a hybrid involving the use of an accounting P&amp;L approach that incorporates a relatively simple valuation model.

Do investors gain from forecasting the asymmetric return co‐movements of financial and real assets?

AbstractRecent research on asset allocation emphasizes the importance of considering non‐traditional asset classes such as commodities and real estate—the former for their diversification properties, and the latter due to its importance in the average investor's portfolio. However, modelling and forecasting asset return co‐movements is challenging because the dependence structure is dynamic, regime‐specific, and non‐elliptical. Moreover, little is known about the economic source of this time‐varying dependence or how to use this information to improve investor portfolios. We use a flexible framework to assess the economic value to investors of incorporating better forecasting information about return co‐movements between equities, bonds, commodities, and real estate. The dependence structure is allowed to be dynamic and non‐elliptical, while the state variables follow Markov‐switching stochastic volatility processes. We find that the predictability of return co‐movements is significantly improved by incorporating macro and non‐macroeconomic variables, in particular inflation uncertainty and bond illiquidity. The economic value added to investors is significant across levels of risk aversion, and the model outperforms traditional multivariate GARCH frameworks.

The continuous limit of weak GARCH

GARCH models are called ‘strong’ or ‘weak’ depending on the presence of parametric distributional assumptions for the innovations. The symmetric weak GARCH(1, 1) is the only model in the GARCH class that has been proved to be closed under the temporal aggregation property . This property is fundamental in two respects: (a) for a time-series model to be invariant to the data frequency; and (b) for a unique option-pricing model to exist as a continuous-time limit. While the symmetric weak GARCH(1, 1) is temporally aggregating precisely because it makes no parametric distributional assumptions, the lack of these also makes it harder to derive theoretical results. Rising to this challenge, we prove that its continuous-time limit is a geometric mean-reverting stochastic volatility process with diffusion coefficient governed by a time-varying kurtosis of log returns. When log returns are normal the limit coincides with Nelson’s strong GARCH(1, 1) limit. But unlike strong GARCH models, the weak GARCH(1, 1) has a unique limit because it makes no assumptions about the convergence of model parameters. The convergence of each parameter is uniquely determined by the temporal aggregation property. Empirical results show that the additional time-varying kurtosis parameter enhances both term-structure and smile effects in implied volatilities, thereby affording greater flexibility for the weak GARCH limit to fit real-world data from option prices.