Implied Volatility Research Articles

Modeling options markets by focusing on active tradersr

In this work, we study the complex behavior of options markets characterized by the volatility smile phenomenon, through microsimulation (MS). We adopt two types of active traders in our MS model: speculators and arbitrageurs, and call and put options on one underlying asset. Speculators make decisions based on their expectations of the asset price at the option expiration time. Arbitrageurs trade at dierent arbitrage opportunities such as violation of put-call parity. Dierence in liquidity among options is also included. Notwithstanding its simplicity, our model can generate implied volatility (IV) curves similar to empirical observations. Our results suggest that the volatility smile is related to the competing eect of heterogeneous trading behavior and the impact of dierential liquidity.

Coarse Thinking, Implied Volatility, and the Price of Call and Put Options

Coarse Thinking, Implied Volatility, and the Price of Call and Put Options

Common Factors and Causality in the Dynamics of Implied Volatility Surfaces: Evidence from the FX OTC Market

In this paper we explore the dynamics of implied volatility surfaces (IVS) both in a single-currency framework and in the context of a global integrated market. We construct a parametric function of “moneyness” and “time-to-maturity” factors that correspond to common shapes of IVS with evident financial intuition. In the first part of the paper, we analyze the time series properties of the estimated factor loadings coefficients, and at the same time the intuition behind the derived associations between them. The second part of the paper explores the interaction between IVS of different currency pairs in two stages. In the first stage, shape transmission between IVS is examined, using simple linear causality tests. We answer the question of whether certain shapes in one currency are transmitted to another. The second stage involves an exploratory factor analysis to uncover latent common factors, across all currencies, which explain part of IVS variability. Our results uncover large idiosyncratic components in the emerging markets’ IVS and dominant common factors that explain most of the variability in the main European currencies.

Shape-Preserving Interpolation and Smoothing for Options Market Implied Volatility

The interpolation of the market implied volatility function from several observations of option prices is often required in financial practice and empirical study. However, the results from existing interpolation methods may not satisfy the property that the European call option price function is monotonically decreasing and convex with respect to the strike price. In this paper, a modified convex interpolation method (with and without smoothing) is developed to approximate the option price function while explicitly incorporating the shape restrictions. The method is optimal for minimizing the distance between the implied risk-neutral density function and a prior density function, which allows us to benefit from nonparametric methodology and empirical experience. Numerical performance shows that the method is accurate and robust. Whether or not the sample satisfies the convexity and decreasing constraints, the method always works.

Modeling the Dynamics of Implied Volatility Surfaces

The objective of this study is to model implied volatility surfaces and identify risk factors that account for most of the randomness in the volatility surface. The approach is similar to the Dumas, Fleming and Whaley (DFW) (1998) study; we use moneyness (e.g., in forward price) and out-of-the-money (OTM) put-call options on FTSE100 index. After these adjustments, the nonlinear parametric optimization technique is employed to estimate different models of DFW in order to characterize the implied volatility surfaces and produce smooth implied volatility surfaces. Next, principal component analysis (PCA) is applied to the implied volatility surfaces to extract principal components that account for most of the dynamics in the shape of the surface. Hence, we estimate and obtain smooth implied volatility surfaces with the parametric models that account for both smile and time to maturity, and, therefore, the constant volatility model fails to explain the variations in the surfaces. Finally, we find that the first three principal components can explain about 69-82% of the variances in the implied volatility surfaces. The applications of our study are hedging of derivatives positions, trading, risk management, and policy making.

Joint modeling of call and put implied volatility

This paper exploits the fact that implied volatilities calculated from identical call and put options have often been empirically found to differ, although they should be equal in theory. We propose a new bivariate mixture multiplicative error model and show that it is a good fit to Nikkei 225 index call and put option implied volatility (IV). A good model fit requires two mixture components in the model, allowing for different mean equations and error distributions for calmer and more volatile days. Forecast evaluation indicates that, in addition to jointly modeling the time series of call and put IV, cross effects should be added to the model: put-side implied volatility helps forecast call-side IV, and vice versa. Impulse response functions show that the IV derived from put options recovers faster from shocks, and the effect of shocks lasts for up to six weeks.

Common functional principal components

Functional principal component analysis (FPCA) based on the Karhunen–Loève decomposition has been successfully applied in many applications, mainly for one sample problems. In this paper we consider common functional principal components for two sample problems. Our research is motivated not only by the theoretical challenge of this data situation, but also by the actual question of dynamics of implied volatility (IV) functions. For different maturities the log-returns of IVs are samples of (smooth) random functions and the methods proposed here study the similarities of their stochastic behavior. First we present a new method for estimation of functional principal components from discrete noisy data. Next we present the two sample inference for FPCA and develop the two sample theory. We propose bootstrap tests for testing the equality of eigenvalues, eigenfunctions, and mean functions of two functional samples, illustrate the test-properties by simulation study and apply the method to the IV analysis.

An Investor Sentiment Barometer - Greek Implied Volatility Index

In this paper a new measure of Greek stock market volatility based on the prices of FTSE/ATHEX-20 index options is proposed. Greek Implied Volatility Index is calculated using the model-free methodology that involves option prices summations and is independent from the Black and Scholes pricing formula. The specific method is applied for the first time in a peripheral and illiquid market as the Athens Exchange. The empirical findings of this paper show that the proposed volatility index includes information about future realized volatility beyond that contained in past volatility and in addition, show that there is a statistically significant negative and asymmetric contemporaneous relationship between the returns of the implied volatility index and the underlying equity index. Finally, the volatility transmission effects on the Greek stock exchange from two leading markets, namely the New York Stock Exchange and the Deutsche Borse, are tested and documented.

SWARCH e Volatilidade Implícita no Câmbio do Real/USD

Este artigo avalia empiricamente a previsão de volatilidade e o conteúdo informacional das variações do câmbio US$/R$, usando opções negociadas na Bolsa de Futuros brasileira. A comparação usa dois modelos de previsão alternativos. O primeiro modelo supõe mudanças markovianas de regime na variância condicional – SWARCH (Hamilton, 1994). No segundo, a volatilidade é implicitamente calculada usando o modelo de Garman e Kohlhagen (1983). Os resultados mostram que o desempenho do SWARCH é superior em termos de previsão e conteúdo informacional, contrastando com a literatura corrente em dois aspectos: primeiro porque utiliza um modelo com mudança de regimes ao invés de um sem mudanças, segundo porque o melhor modelo ´e o que usa dados passados em vez da volatilidade implícita.

The Predictive Power of Interest Rate Volatility on Economic Sentiment: Evidence for Germany and the U.K.

In this paper we examine the explanatory and predictive power of interest rate volatility on the economic agent expectations measured by the Economic Sentiment Indicator (ESI), which is elaborated by the European Commission as a leading indicator of business cycle. In particular, we use implicit volatilities from the UK and German cap markets as explanatory variables, together with the yield spread and the short rate, obtaining that it is possible to explain nearly an overwhelming 70% of the variability of the ESI in contrast with historical or conditional volatility estimates which can hardly add any additional explanatory power.

Risiko-Renditeprofil des neuen Covered-Call-Index der Deutschen Börse

Zusammenfassung Risiko-Renditeprofil des neuen Covered-Call-Index der Deutschen Borse Wir analysieren in dieser Arbeit das Risiko-Renditeprofil des neuen DAXplus Covered-Call-Index der Deutschen Borse AG im Zeitraum 01/1993 bis 06/2005. Es zeigt sich, dass der DAXplus Covered Call im Betrachtungszeitraum auf risikoadjustierter Basis einer Direktanlage in den DAX vorzuziehen war. Dies lasst sich unter anderem damit erklaren, dass die impliziten Volatilitaten der Call-Optionen auf den DAX im Vergleich zu den historischen Volatilitaten zu hoch waren und dadurch zu hohe Optionspramien nach sich zogen. Zusatzlich untersuchen wir die Auswirkungen verschiedener Moneynesses und Restlaufzeiten auf die Risiko-Renditeprofile von Covered-Call-Strategien auf den DAX. Der DAXplus Covered Call weist hierbei gegenuber anderen Covered-Call-Strategien die vorteilhafteste Entwicklung im Untersuchungszeitraum auf. Abschliesend vergleichen wir den DAXplus Covered Call mit drei Covered-Call-Strategien auf internationale Aktien...

Using the Black and Scholes Formula (in Spanish)

I show: a) how to use the Black and Scholes formula to value options; b) what is the arbitrage portfolio; c) how to calculate the volatility; d) how the early exercise affects the value of the option; and e) how the dividends affect the value of the option; and f) what is the implicit volatility.

Joint Modeling of Call and Put Implied Volatility

This paper exploits the fact that implied volatilities calculated from identical call and put options have often been empirically found to differ, although they should be equal in theory. We propose a new bivariate mixture multiplicative error model and show that it is a good fit to Nikkei 225 index call and put option implied volatility (IV). A good model fit requires two mixture components in the model, allowing for different mean equations and error distributions for calmer and more volatile days. Forecast evaluation indicates that in addition to jointly modeling the time series of call and put IV, cross effects should be added to the model: put-side implied volatility helps forecast call-side IV, and vice versa. Impulse response functions show that the IV derived from put options recovers faster from shocks, and the effect of shocks lasts for up to six weeks.

On extracting information implied in options

Options are financial instruments with a payoff depending on future states of the underlying asset. Therefore option markets contain information about expectations of the market participants about market conditions, e.g. current uncertainty on the market and corresponding risk. A standard measure of risk calculated from plain vanilla options is the implied volatility (IV). IV can be understood as an estimate of the volatility of returns in future period. Another concept based on the option markets is the state-price density (SPD) that is a density of the future states of the underlying asset. From raw data we can recover the IV function by nonparametric smoothing methods. Smoothed IV estimated by standard techniques may lead to a non-positive SPD which violates no arbitrage criteria. In this paper, we combine the IV smoothing with SPD estimation in order to correct these problems. We propose to use the local polynomial smoothing technique. The elegance of this approach is that it yields all quantities needed to calculate the corresponding SPD. Our approach operates only on the IVs--a major improvement comparing to the earlier multi-step approaches moving through the Black---Scholes formula from the prices to IVs and vice-versa.

Can the VIX Signal Market's Direction? An Asymmetric Dynamic Strategy

The article shows statistically that the VIX Implied Volatility Index is an important driver of the S&P 500 future returns. The statistical analysis is performed by means of a regression based on dummy variables in order to circumvent the difficulties posed by the lack of linearity between the variables. The results obtained are then used to construct an automated procedure that signals daily whether it is convenient to invest in the S&P 500 or to stay put. Finally, we test the quality of the signal by implementing an asymmetrical buy-and-hold strategy with 3-months horizon on the S&P 500. Our results show that the strategy outperforms the long-only strategy on the same index, thus confirming a widespread belief among traders.

EXACT SOLUTION OF A MARTINGALE STOCHASTIC VOLATILITY OPTION PROBLEM AND ITS EMPIRICAL EVALUATION

Exact explicit solution of the log‐normal stochastic volatility (SV) option model has remained an open problem for two decades. In this paper, I consider the case where the risk‐neutral measure induces a martingale volatility process, and derive an exact explicit solution to this unsolved problem which is also free from any inverse transforms. A representation of the asset price shows that its distribution depends on that of two random variables, the terminal SV as well as the time average of future stochastic variances. Probabilistic methods, using the author's previous results on stochastic time changes, and a Laplace–Girsanov Transform technique are applied to produce exact explicit probability distributions and option price formula. The formulae reveal interesting interplay of forces between the two random variables through the correlation coefficient. When the correlation is set to zero, the first random variable is eliminated and the option formula gives the exact formula for the limit of the Taylor series in Hull and White's (1987) approximation. The SV futures option model, comparative statics, price comparisons, the Greeks and practical and empirical implementation and evaluation results are also presented. A PC application was developed to fit the SV models to current market prices, and calculate other option prices, and their Greeks and implied volatilities (IVs) based on the results of this paper. This paper also provides a solution to the option implied volatility problem, as the empirical studies show that, the SV model can reproduce market prices, better than Black–Scholes and Black‐76 by up to 2918%, and its IV curve can reproduce that of market prices very closely, by up to within its 0.37%.

Commonality in the time-variation of stock–stock and stock–bond return comovements

We jointly investigate time-varying comovements between stock returns across countries and between long-term government bond and stock returns within countries. Our focus is on how daily return comovements vary with stock uncertainty, as measured by the implied volatility (IV) from equity index options. Cross-country stock return comovements tend to be stronger (weaker) following high (low) IV days and on days with large (small) changes in IV. Stock-bond return comovements tend to be substantially positive (negative) following low (high) IV days and on days with small (large) changes in IV. A regime-switching analysis also indicates a striking temporal commonality in the stock?stock and stock-bond comovement variations. Our findings bear on understanding the influence of time-varying uncertainty on price formation and the diversification benefits of stock-bond and cross-country stock holdings.

Extracting Model-Free Volatility from Option Prices

The CBOE9s VIX index is a measure of the implied volatility (IV) in 30-day stock index options. Originally constructed as a weighted average of Black-Scholes IVs from 8 at the money calls and puts, the VIX was redesigned in 2003. The new VIX uses a nonparametric procedure to extract an IV from out of the money calls and puts over the full range of strikes. Implementation of the theoretical procedure, however, requires several approximations, for example to deal with the fact that only a discrete set of strikes are traded in the market, rather than a continuum over the full range from zero to infinity, as required by the theory. In this article, Jiang and Tian look carefully at the new VIX algorithm to assess the impact of these approximations on its accuracy. They find that some of them may produce substantial errors, even in simply recovering the volatility input from a set of options in a pure Black-Scholes world. They then propose a modified calculation technique using a smoothing algorithm, that can almost entirely eliminate the errors. <b>TOPICS:</b>Options, statistical methods, risk management

Commonality in the time-variation of stock–stock and stock–bond return comovements

We jointly investigate time-varying comovements between stock returns across countries and between long-term government bond and stock returns within countries. Our focus is on how daily return comovements vary with stock uncertainty, as measured by the implied volatility (IV) from equity index options. Cross-country stock return comovements tend to be stronger (weaker) following high (low) IV days and on days with large (small) changes in IV. Stock–bond return comovements tend to be substantially positive (negative) following low (high) IV days and on days with small (large) changes in IV. A regime-switching analysis also indicates a striking temporal commonality in the stock–stock and stock–bond comovement variations. Our findings bear on understanding the influence of time-varying uncertainty on price formation and the diversification benefits of stock–bond and cross-country stock holdings.

You Don't Have to Bother Newton for Implied Volatility

The Black-Scholes formula is often used in the backward direction to invert the implied volatility, usually with some solver method. Solver methods, being aesthetically unappealing, are also slower than closed-form approximations. However, closed-form approximations in previous works lack accuracy, often providing option pricing errors well exceeding the bid-ask spreads. We develop a new closed-form method based on the rational approximation. By exploiting the homogeneity in the Black-Scholes formula, we are able to show explicitly our domain of approximation and investigate thoroughly the accuracy of our method. The rational approximation is much faster than typical solver methods and very accurate for both at-the-money and away-from-the-money options. Its accuracy can be further improved by one or two steps of Newton-Raphson iterations.