Volatility Option Research Articles

Pricing of stochastic volatility stock index option based on Feynman path integral

<sec>Under the background that stock index options urgently need launching in China, the research on stock option pricing model has important theoretical and practical significance. In the traditional B-S-M model it is assumed that the volatility remains unchanged, which differs tremendously from the market’s reality. When the market fluctuates drastically, it is difficult to realize the risk management function of stock index options. Although in the Heston model, as one of the traditional stochastic volatility option pricing models, the correlation risk between the volatility and underlying price is taken into consideration, its pricing accuracy is still to be improved. From the quantum finance perspective, in this paper we use the Feynman path integral method to explore a more practical stock index option pricing model.</sec><sec>In this paper, we construct a Feynman path integral pricing model of stock index options with stochastic volatility by taking Hang Seng index option as the research object and Heston model as the control group. It is found that the Feynman path integral pricing model is significantly superior to the Heston model either at different strike prices on the same expiration date or at different expiration dates for the same strike price. The stock index option pricing model constructed in this paper can give the numerical solution of Feynman's pricing kernel, and directly realizes the forecast of stock index option price. The pricing accuracy is significantly improved compared with the pricing accuracy given by the Heston model through using the characteristic function method.</sec><sec>The remarkable advantages of Feynman path integral stock index option pricing model are as follows. Firstly, the path integral has advantages in solving multivariate problems: the Feynman pricing kernel represents all the information about pricing and can be easily expanded from one-dimensional to multidimensional case, so the change of closing price of stock index and volatility of underlying index can be taken into account simultaneously. Secondly, based on the relationship between the Feynman path generation principle and the law of large number, the mean values of pricing kernel obtained by MATLAB not only optimizes the calculation process, but also significantly improves the pricing accuracy. Feynman path integral is the main method in quantum finance, and the research in this paper will provide reference for its further application in the pricing of financial derivatives.</sec>

Option Writing: Using VIX to Improve Returns

Buy-write and put-write strategies have been shown to match market returns with lower volatility, resulting in higher risk-adjusted performance. The strategies benefit from the fact that the implied volatility of options is generally higher than actual realized volatility. In this article, we show that this premium is higher at elevated levels of implied volatility (as represented by the VIX index level). Based on this finding, we propose a simple conditional strategy in which one sells options at elevated levels of the VIX. Using data from 1990 through 2018, we find that this conditional strategy outperforms both the market and continuous option-selling strategies on an absolute and risk-adjusted basis.

Numerical Contour Integral Methods for Free-Boundary Partial Differential Equations Arising in American Volatility Options Pricing

The aim of this paper is to study the numerical contour integral methods (NCIMs) for solving free-boundary partial differential equations (PDEs) from American volatility options pricing. Firstly, the governing free-boundary PDEs are modified as a unified form of PDEs on the fixed space region; then performing Laplace-Carson transform (LCT) leads to ordinary differential equations (ODEs) which involve the unknown inverse functions of free boundaries. Secondly, the inverse free-boundary functions are approximated and optimized by solving of the free-boundary values of the perpetual American volatility options. Finally, the ODEs are solved by the finite difference methods (FDMs), and the results are restored via the numerical Laplace inversion. Numerical results confirm that the NCIMs outperform the FDMs for solving free-boundary PDEs in regard to the accuracy and computational time.

Trading Events: Smiles, Frowns and Moustaches – the Many Faces of the Options Market

Financial markets are becoming increasingly event driven. These can take the form of economic releases, speeches and elections. The arrival of this new information often causes the underlying asset to jump. Consequently, the expectation of these known announcements can produce significant distortions in option volatility surfaces. This presents a challenge, or opportunity, for pricing and risk management. We therefore undertake an examination of large events. Firstly, we perform an empirical analysis of the UK referendum in 2016. This is in order to give market practitioners intuition for these extreme scenarios and to guide us towards a quantitative approach. As a result, we introduce a new bimodal jump model. By varying the two jump parameters, we show how to parsimoniously produce the aforementioned smile distortions – specifically: skews, frowns and W-shapes. Furthermore, we demonstrate the model’s calibration to the observed volatility surface leading up to the referendum, and the French election in 2017. If jump parameters are frozen with time, the model can be used in the temporal volatility interpolation. Finally, we discuss how the jump probabilities are becoming increasingly observable via alternative data, such as betting markets, election polls and research reports.

Approaches to forecasing option volatility

The article investigates a new approach to the idea of volatility. In spite of the well-known assumption that option volatility in future will be exactly the same as today, the author puts forward a method, which links the change in volatility to change of only one parameter, i.e. the price of basic asset. The idea that the price of basic asset is a ‘guide’ for option volatility does not need any proof, as like terminal contracts options are estimated proceeding from their basic asset. This method can help estimate future volatility for one (or even more steps) ahead. Like any other forecast method it builds up the error as the number of steps in the future increases, however the simplicity of its use and low resource-intensiveness make it a worthy alternative to the method accepted now, which shows volatility while presenting prospects of the current option position. To forecast volatility for one step ahead we used the following basic statistic methods and economic models: the method of linear regression, Newton-Rafson method for finding option strikes for the set deltas, the method of spline-interpolation, the model of calculating ‘option smile’ Vanna-Volga.

Modeling Airport Business Risks, Enplanement Volatility, and Valuation of Flexibility Options in Airport Expansion Projects

Airport capital improvement programs involve considerable flexibility in investment timing and engineering design. Even though the valuation of flexibility options may depend on several factors, the volatility of future airport activity levels, which largely defines the business risk for airport operators, makes up the focus of the present paper. As such, the paper proposes a model that can be used to value two types of flexibility options common in airport expansion projects. The first type of option—flexibility in investment timing decisions—creates value by conditioning capacity expansion decisions on trends in airport activity levels. The second, flexibility in engineering design, permits operators to influence their demand composition and to reconfigure airport facilities if their business environment changes unfavorably. A Monte Carlo simulation example also demonstrates the application of the proposed valuation model. The results show that flexibility options add economic value by reducing downside exposures and by providing the ability to increase capacity if enplanements stay on a rising trajectory. Moreover, the paper provides a comparison of enplanement growth rates by airport size for the largest 140 U.S. airports from 1990 to 2016. The analysis shows that medium airports may be uniquely positioned to benefit the most from flexible design approaches. For these airports, results imply higher exposures to excess capacity risks because of the increased persistence of losses, despite the higher year-on-year volatility of small airports.

A switching self-exciting jump diffusion process for stock prices

This study proposes a new Markov switching process with clustering effects. In this approach, a hidden Markov chain with a finite number of states modulates the parameters of a self-excited jump process combined to a geometric Brownian motion. Each regime corresponds to a particular economic cycle determining the expected return, the diffusion coefficient and the long-run frequency of clustered jumps. We study first the theoretical properties of this process and we propose a sequential Monte-Carlo method to filter the hidden state variables. We next develop a Markov Chain Monte-Carlo procedure to fit the model to the S&P 500. We find that self-exciting jumps occur mainly during economic recession and nearly disappear in periods of economic growth. Finally, we analyse the impact of such a jump clustering on implied volatilities of European options.

A self-exciting switching jump diffusion: properties, calibration and hitting time

A way to model the clustering of jumps in asset prices consists in combining a diffusion process with a jump Hawkes process in the dynamics of the asset prices. This article proposes a new alternative model based on regime switching processes, referred to as a self-exciting switching jump diffusion (SESJD) model. In this model, jumps in the asset prices are synchronized with changes of states of a hidden Markov chain. The matrix of transition probabilities of this chain is designed in order to approximate the dynamics of a Hawkes process. This model presents several advantages compared to other jump clustering models. Firstly, the SESJD model is easy to fit to time series since estimation can be performed with an enhanced Hamilton filter. Secondly, the model explains various forms of option volatility smiles. Thirdly, several properties about the hitting times of the SESJD model can be inferred by using a fluid embedding technique, which leads to closed form expressions for some financial derivatives, like perpetual binary options.

The implied volatility of Forward-Start options: ATM short-time level, skew and curvature

ABSTRACTUsing Malliavin Calculus techniques, we derive closed-form expressions for the at-the-money behaviour of the forward implied volatility, its skew and its curvature, in general Markovian stochastic volatility models with continuous paths.

Real Option Identification Framework for Mine Operational Decision-Making

Identification of opportunities for applying real options (RO) in mining operations is a major challenge to decision-makers. In order to make optimal decisions in uncertain times, managers require a full understanding of the relationships between risk, uncertainty and flexibility. RO analysis, which captures the value of any managerial flexibility that may exist in a project, provides a proactive management of uncertainty. Thus, it enhances optimal decision-making. However, it is important that a structured framework is created to identify project uncertainties and areas available to cultivate flexibility. In this paper, uncertainty identification framework in a mining operation is proposed, and areas for managerial flexibility and their application domains within the mining cycle are mapped as well. To avoid complex mathematical models, which hinder the adoption of RO analysis in mining operations, a relationship between risk measure (beta) and flexibility (flexibility index) is derived and applied. This implies that if a project beta is known, then the expected option values and volatility of future cash flows can be precisely estimated. Once the option value is calculated using the derived equation, a modified smooth pasting condition with the mean value theorem is subsequently applied to estimate the optimal value. This combination of beta, flexibility index and mean value theorem can be used as a decision criterion for screening various options within a mining project.

Firm opacity and informed trading around spinoffs

PurposeThis paper aims to examine the prevalence of informed trading around corporate spinoffs and the relation between firm opacity and informed trading using option market data.Design/methodology/approachThe author investigates the prevalence of informed trading by examining the relationship between abnormal stock returns associated with spinoffs and the volatility spread/volatility skewness of options prior to the spinoffs. Furthermore, the author examines how opacity and organizational complexity prior to the spinoffs affect informed trading.FindingsThe study shows that option volatility spread and volatility skewness for the five days prior to the spinoffs can predict the abnormal stock returns on the spinoff announcement days, suggesting that there is informed trading in the options market prior to spinoffs. The study shows that informed trading is more prevalent for firms that are more opaque prior to the spinoff. Furthermore, informed trading decreases after spinoffs.Originality/valueTo the best of knowledge, this is the first empirical research that examines the prevalence of informed trading around spinoffs by using options volatility spread/skewness and the relation between firm opacity and informed options trading.

Downside and Upside Volatility, Value-at-Risk, Expected Shortfall and Pricing of Options Based on a Skewed Generalized Logistic Distribution

This paper derives the moment functions of the truncated skewed type III generalized logistic (SGL). These are then applied in finance for the development of value-at-risk, expected shortfall and downside risk measures for investment returns and values. The SGL distribution provides and good fit to the empirical distribution of a representative set of long series of financial data. Moreover, the SGL generates accurate value at risk measures.

Examining the Equilibrium Relationship Between the Shanghai 50 Stock Index Futures and the Shanghai 50 ETF Options Markets

ABSTRACTBased on the put-call-futures parity model, this article studies the equilibrium relationship between the Shanghai 50 stock index futures and the Shanghai 50 Exchange-Traded Fund (ETF) options markets by analyzing the arbitrage opportunities and profits between these two derivative markets. This article reveals that the cost spread, option volatility, days from the expiration date, moneyness of options, trading strategy, and policy factors all have a great impact on the arbitrage profits and opportunities. In addition, significant arbitrage profits and opportunities indicate violations of put-call-futures parity. Although no equilibrium relationship exists between the Shanghai 50 stock index futures and the Shanghai 50 ETF options markets, efficiency in these markets has gradually improved.

Debt rollover-induced local volatility model

This paper introduces a structural scenario-based model with debt rollover risk and a higher-fidelity treatment of the bankruptcy procedure. The emerging stock price process is a generalized Brownian motion with state-dependent local volatility, and the resultant implied volatility smile is due exclusively to structural features (debt rollover and credit risks). Therefore, the model reinforces structural foundations of local volatility option pricing models. The paper advocates a joint modeling and calibration framework for multiple classes of derivatives on the firm’s asset value. In particular, an empirical application to Solar City equity and stock option valuation demonstrates the versatility and efficiency gains of the suggested model.

On the Seasonality in the Implied Volatility of Electricity Options

Seasonality is an important topic in electricity markets, as both supply and demand are dependent on the time of the year. Clearly the level of prices shows a seasonal behaviour, but not only. Also the price fluctuations are typically seasonal. In this paper, we study empirically the implied volatility of options on electricity futures, investigate whether seasonality is present and we aim to quantify its structure. Although typically futures prices can be described well with multi-factor models including exponentially decreasing components, we do not find evidence of exponential behaviour in our data set. Generally, a simple linear shape reflects the squared volatilities very well as a curve depending on the time to maturity. Moreover, we find clear seasonal patterns in the level of the volatility depending on the delivery month of the futures, and compare the performance of several implementations of seasonality in the one factor framework.

Taylor expansion for derivative securities pricing as a precondition for strategic market decisions

The strategy of managing the pricing processes, in particular managing the dynamics of the price of the underlying asset and its volatility, the prices of indices, shares, options, the magnitude of financial flows, in the method of calculating the company’s rating based on the available quotations of securities, is developed. The article deals with the study of pricing and calculating the volatility of European options with general local-stochastic volatility, applying Taylor series methods for degenerate diffusion processes. The application of this idea requires new approaches caused by degradation difficulties. Price approximation is obtained by solving the Cauchy problem of partial differential equations diffusion with inertia, and the volatility approximation is completely explicit, that is, it does not require special functions. If the payoff of options is a function of only x, then the Taylor series expansion does not depend on t and an analytical expression of the fundamental solution is considerably simplified. Applied an approach to the pricing of derivative securities on the basis of classical Taylor series expansion, when the stochastic process is described by the diffusion equation with inertia (degenerate parabolic equation). Thus, the approximate value of options can be calculated as effectively as the Black-Scholes pricing of derivative securities. On the basis of the solved problem, it is possible to calculate their turns step-by-step. This enables to predict the dynamics of the pricing of derivatives and to create a strategy of behavior at options according to the passage of the process. For each approximation, price volatility is calculated, which makes it possible to take into account all changes in the market and to calculate possible situations. The step-by-step finding of the change in yield and volatility in the relevant analysis enables us to make informed strategic decisions by traders in the financial markets.

Global Volatility Spillover, Transaction Cost and CNY Exchange Rate Parities

AbstractThe present study examines the intertemporal association between CBOE market volatility indices (VIX), foreign exchange rates and respective bid-ask spread for four CNY exchange rate parities. For this purpose, the study utilizes the stylized EGARCH (1, 1) model for the period of 2011 to 2016. Results report that negative slopes of EUVIX, BPVIX, and JYVIX imply a higher level of volatility, hence improves the underlying exchange rate through appreciation, while positive slopes of VXFXI deteriorates exchange rates during the sample period. Similarly, high volatility widens bid-ask spread which, in turn, deteriorates respective exchange rate and vice versa. The market-oriented policies of China increased the forecasting capability of options volatility indexes to anticipate exchange rate dynamics from 2% to 5%. This indicates that flexible exchange rate regimes lead to increase the predicting power of micro structural components. Assessments of Post-reforms in CNY exchange rate evidence the rise in volatility in financial markets of China, which may discourage investor confidence and seeks for ‘flight to safety’ effect. While, low volatility reduces bid-ask spread which improves underlying exchange rate. The level and variance estimates of exchange rates and spreads reveal that there exists a significant relationship with VIX indices which implies that GARCH forecasts outperform in anticipating future volatility. The volatility estimates of variances show the persistence of volatility and absence of leverage effect. Overall, this article suggests that VIX index can act as ‘fear gauge’ indicator and its potential direction may guide investors in anticipating the movements of CNY exchange rate parities. Moreover, outcomes provide imperative implications to monetary and financial institutions for policy framing.

Are there gains from using information over the surface of implied volatilities?

We investigate the out‐of‐sample predictability of implied volatility using the information over the implied volatility surface. We show that implied volatility surface is useful for the out‐of‐sample forecast of implied volatility up to 1 week ahead. Trading strategies based on the predictability of implied volatility could generate significant risk‐adjusted gains after controlling for transaction costs. Significant results also depend on the way of modeling implied volatility surface. We then calibrate a two‐factor stochastic volatility option pricing model to implied volatility data. Results show that implied volatility is better explained by both long‐ and short‐term variance factors.

Volatility Options in Rough Volatility Models

We discuss the pricing and hedging of volatility options in some rough volatility models. First, we develop efficient Monte Carlo methods and asymptotic approximations for computing option prices and hedge ratios in models where log-volatility follows a Gaussian Volterra process. While providing a good fit for European options, these models are unable to reproduce the VIX option smile observed in the market, and are thus not suitable for VIX products. To accommodate these, we introduce the class of modulated Volterra processes, and show that they successfully capture the VIX smile.

Effects of IFRS-13 on the relevance of fair value adjusted by credit risk: Evidence from Europe

Accounting harmonization in Europe by International Financial Reporting Standard adoption is a recurrent object of study in the accounting literature. In this paper the consequences of the adoption of Standard-13 are analyzed. In particular, this research analyzes the effects on the implied volatility option (risk) for non-financial companies of three variables: financial leverage, own probability default (Debt Value Adjusted) and financial institutions credit risk (Credit Value Adjusted), before and after the adoption of the accounting standard on fair value. The empirical study focuses on member companies of the European Monetary Union zone to avoid other risk factors different to market risk (such as exchange rate or different risk free rate) and at the same time, easily identify the market portfolio (EUROSTOXX-50). To overcome the problems of endogeneity in the panel data, we use the technique System Generalized Method of Moments with instrumental variables to estimate the parameters. The results show that the leverage effect on excess risk does not change after adopting the Standard, however, its own and the financial institutions default probabilities become statistically significant. Furthermore, this novel methodology allows estimate industry asset betas and, in all cases the asset betas were lower than equity betas and, found an average debt beta of 0.4 for the sample period.