What Does Today's Smile Imply About Future Volatilities (original) (raw)

Arbitrage-Free Prediction of the Implied Volatility Smile

SSRN Electronic Journal, 2000

This paper gives an arbitrage-free prediction for future prices of an arbitrary co-terminal set of options with a given maturity, based on the observed time series of these option prices. The statistical analysis of such a multi-dimensional time series of option prices corresponding to n strikes (with n large, e.g. n ≥ 40) and the same maturity, is a difficult task due to the fact that option prices at any moment in time satisfy non-linear and non-explicit no-arbitrage restrictions. Hence any n-dimensional time series model also has to satisfy these implicit restrictions at each time step, a condition that is impossible to meet since the model innovations can take arbitrary values. We solve this problem for any n ∈ N in the context of Foreign Exchange (FX) by first encoding the option prices at each time step in terms of the parameters of the corresponding risk-neutral measure and then performing the time series analysis in the parameter space. The option price predictions are obtained from the predicted riskneutral measure by effectively integrating it against the corresponding option payoffs.

Alternative asset-price dynamics and volatility smile

Quantitative Finance, 2003

We propose a general class of analytically tractable models for the dynamics of an asset price leading to smiles or skews in the implied volatility structure. The considered asset can be an exchange rate, a stock index, or even a forward Libor rate. The class is based on an explicit SDE under a given forward measure. The models we propose feature i) explicit asset-price dynamics, ii) virtually unlimited number of parameters, iii) analytical formulas for European options.

Smoothing the volatility smile using the Corrado-Su model

International Journal of Financial Markets and Derivatives, 2015

The expansion of the derivatives market both globally and particularly in Brazil has driven users to enhance and develop tools for more efficient pricing. However, despite this expansion of the derivatives market, certain characteristics of the Brazilian options market still impose limits on its analysis and thus the studies performed. Despite the most commonly used pricing model in Brazil being that of the Black-Scholes, its assumption of constant asset's volatility up to maturity and log-normal distribution price are criticized by a number of authors, because those are not sustained in real-life situations and consequently leads to inconsistencies. The objective of the model suggested by Corrado and Su, and improved by Brown and Robinson, is to adapt the BS model to the distribution of the asset intended for pricing by introducing the returns' kurtosis and skewness. For this reason, it is expected that the model will be more apt for calculating implied volatility in order to reduce the volatility smile. The purpose of this paper is to show which window of observation generates the kurtosis and skewness which smoothes the volatility smile most, using the Corrado-Su model. Therefore, the companies chosen for the study were Petrobrás PN and Vale PNA, because their stocks and options are the most liquid on the Brazilian market. The data analysis indicated a smoother volatility smile using short term observation windows rather than long term windows and a performance of earlier windows equivalent to those of the Black-Scholes model.

Why do we smile? On the determinants of the implied volatility function

Journal of Banking & Finance, 1999

We report simple regressions and Granger causality tests in order to understand the pattern of implied volatilities across exercise prices. We employ all calls and puts transacted between 16:00 and 16:45 on the Spanish IBEX-35 index from January 1994 to April 1996. Transaction costs, proxied by the bid±ask spread, seem to be a key determinant of the curvature of the volatility smile. Moreover, time to expiration, the uncertainty associated with the market and the relative market momentum are also important variables in explaining the smile.

Option Bounds and the Pricing of the Volatility Smile

Review of Derivatives Research, 2000

This paper presents a new approach forthe estimation of the risk-neutral probability distribution impliedby observed option prices in the presence of a non-horizontalvolatility smile. This approach is based on theoretical considerationsderived from option pricing in incomplete markets. Instead ofa single distribution, a pair of risk-neutral distributions areestimated, that bracket the option prices defined by the volatilitybid/ask midpoint. These distributions define upper and lowerbounds on option prices that are consistent with the observableoption parameters and are the tightest ones possible, in thesense of minimizing the distance between the option upper andlower bounds. The application of the new approach to a sampleof observations on the S&P 500 option market showsthat the bounds produces are quite tight, and also that theirderivation is robust to the presence of violations of arbitragerelations in option quotes, which cause many other methods tofail.

Dynamics of implied volatility surfaces

Quantitative Finance, 2002

The prices of index options at a given date are usually represented via the corresponding implied volatility surface, presenting skew/smile features and term structure which several models have attempted to reproduce. However, the implied volatility surface also changes dynamically over time in a way that is not taken into account by current modelling approaches, giving rise to 'Vega' risk in option portfolios. Using time series of option prices on the SP500 and FTSE indices, we study the deformation of this surface and show that it may be represented as a randomly fluctuating surface driven by a small number of orthogonal random factors. We identify and interpret the shape of each of these factors, study their dynamics and their correlation with the underlying index. Our approach is based on a Karhunen-Loève decomposition of the daily variations of implied volatilities obtained from market data. A simple factor model compatible with the empirical observations is proposed. We illustrate how this approach models and improves the well known 'sticky moneyness' rule used by option traders for updating implied volatilities. Our approach gives a justification for use of 'Vega's for measuring volatility risk and provides a decomposition of volatility risk as a sum of contributions from empirically identifiable factors.

New Insights on the Implied and Realized Volatility Relation

Review of Pacific Basin Financial Markets and Policies, 2012

Existing studies on the informational content of at-the-money implied volatility (ATMIV) and past realized volatility (PRV) and the relation between the two have mainly focused on a single short forecast horizon and conclude that ATMIV outperforms PRV. We examine the relation between implied and realized volatility over both short and longer forecasting horizons to provide a forecasting competition. We analytically demonstrate the option maturity effect on the sensitivity of the implied volatility (IV) estimation. As time to maturity increases, vega increases but at a decreasing rate, up to [Formula: see text]. At shorter (longer) maturities, small pricing changes should have greater (smaller) corrective impact on IVs. We find that IV outperforms PRV over a one month forecast horizon. However, as the forecast horizon increases, PRV outperforms IV and subsumes the information contained in it. These mixed results may be attributed to the reduced efficiency of longer dated sections of ...