An empirical model of volatility of returns and option pricing

This paper reports several entirely new results on financial market dynamics and option pricing We observe that empirical distributions of returns are much better approximated by an exponential distribution than by a Gaussian. This exponential distribution of asset prices can be used to develop a new pricing model for options (in closed algebraic form) that is shown to provide valuations that agree very well with those used by traders. We show how the Fokker-Planck formulation of fluctuations can be used with a local volatility (diffusion coeffficient) to generate an exponential distribution for asset returns, and also how fat tails for extreme returns are generated dynamically by a simple generalization of our new volatility model. Nonuniqueness in deducing dynamics from empirical data is discussed and is shown to have no practical effect over time scales much less than one hundred years. We derive an option pricing pde and explain why it‘s superfluous, because all information required to price options in agreement with the delta-hedge is already included in the Green function of the Fokker-Planck equation for a special choice of parameters. Finally, we also show how to calculate put and call prices for a stretched exponential returns density.Market instability; market dynamics; finance; option pricing

Pricing Options by Simulation Using Realized Volatility

A growing literature advocates the use of high-frequency data for the purpose of volatility estimation. However, despite the successes in modeling the conditional mean of realized volatility empirical evaluations of this class of models outside the realm of short run forecasting is limited. How can realized volatility be used for pricing options? What are the modeling qualities introduced by realized volatility models for pricing derivatives? In this short paper, we propose an option pricing framework based on a new realized volatility model that captures all the relevant empirical regularities of the realized volatility series of the S&P 500 index. We emphasize two main empirical regularities for our volatility model and that are potentially very relevant for option pricing purposes. First, realized variation measures constructed from high-frequency returns reveal a large degree of time series unpredictability in the volatility of asset returns. Even though returns standardized by (ex-post) quadratic variation measures are nearly gaussian, this unpredictability brings substantially more uncertainty to the empirically relevant (ex-ante) distribution of returns. In this setting carefully modeling the stochastic structure of the time series disturbances of realized volatility is fundamental. Second, there is evidence of very large leverage effects; large falls (rises) in prices being associated with persistent regimes of high (low) variance in the index returns. We propose a model for the conditional volatility, skewness and kurtosis of daily index and stocks returns. The main new feature of this model is to recognize that volatility is itself more volatile and more persistent in high volatility periods. Contrary to "peso problem" considerations, we show that when volatility is (nearly) observable it is not necessary to rely on rare realizations on past return data to learn about the tails of the return distribution, an unexplored and large modeling gain enabled by high frequency data. We conduct a brief empirical illustration analysis of the pricing performance of this approach against some benchmark models using data from the S&P 500 options in the 2001-2004 period. The results indicate that as expected the superior forecasting accuracy of realized volatility translates into significantly smaller pricing errors when compared to models of the GARCH family. More significantly, our results indicate that modeling leverage effects and the volatility of volatility are paramount reducing common pricing anomalies.Volatility, Option pricing, Volatility of volatility, Forecasting

A Stochastic Volatility Model With Realized Measures for Option Pricing

Based on the fact that realized measures of volatility are affected by measurement errors, we introduce a new family of discrete-time stochastic volatility models having two measurement equations relating both observed returns and realized measures to the latent conditional variance. A semi-analytical option pricing framework is developed for this class of models. In addition, we provide analytical filtering and smoothing recursions for the basic specification of the model, and an effective MCMC algorithm for its richer variants. The empirical analysis shows the effectiveness of filtering and smoothing realized measures in inflating the latent volatility persistence—the crucial parameter in pricing Standard and Poor’s 500 Index options

Econometric Analysis of Financial Derivatives

__Abstract__ One of the fastest growing areas in empirical finance, and also one of the least rigorously analyzed, especially from a financial econometrics perspective, is the econometric analysis of financial derivatives, which are typically complicated and difficult to analyze. The purpose of this special issue of the journal on “Econometric Analysis of Financial Derivatives” is to highlight several areas of research by leading academics in which novel econometric, financial econometric, mathematical finance and empirical finance methods have contributed significantly to the econometric analysis of financial derivatives, including market-based estimation of stochastic volatility models, the fine structure of equity-index option dynamics, leverage and feedback effects in multifactor Wishart stochastic volatility for option pricing, option pricing with non-Gaussian scaling and infinite-state switching volatility, stock return and cash flow predictability: the role of volatility risk, the long and the short of the risk-return trade-off, What’s beneath the surface? option pricing with multifrequency latent states, bootstrap score tests for fractional integration in heteroskedastic ARFIMA models, with an application to price dynamics in commodity spot and futures markets, a stochastic dominance approach to financial risk management strategies, empirical evidence on the importance of aggregation, asymmetry, and jumps for volatility prediction, non-linear dynamic model of the variance risk premium, pricing with finite dimensional dependence, quanto option pricing in the presence of fat tails and asymmetric dependence, smile from the past: a general option pricing framework with multiple volatility and leverage components, COMFORT: A common market factor non-Gaussian returns model, divided governments and futures prices, and model-based pricing for financial derivative

Econometric Analysis of Financial Derivatives: An Overview

One of the fastest growing areas in empirical finance, and also one of the least rigorously analyzed, especially from a financial econometrics perspective, is the econometric analysis of financial derivatives, which are typically complicated and difficult to analyze. The purpose of this special issue of the journal on “Econometric Analysis of Financial Derivatives” is to highlight several areas of research by leading academics in which novel econometric, financial econometric, mathematical finance and empirical finance methods have contributed significantly to the econometric analysis of financial derivatives, including market-based estimation of stochastic volatility models, the fine structure of equity-index option dynamics, leverage and feedback effects in multifactor Wishart stochastic volatility for option pricing, option pricing with non-Gaussian scaling and infinite-state switching volatility, stock return and cash flow predictability: the role of volatility risk, the long and the short of the risk-return trade-off, What’s beneath the surface? option pricing with multifrequency latent states, bootstrap score tests for fractional integration in heteroskedastic ARFIMA models, with an application to price dynamics in commodity spot and futures markets, a stochastic dominance approach to financial risk management strategies, empirical evidence on the importance of aggregation, asymmetry, and jumps for volatility prediction, non-linear dynamic model of the variance risk premium, pricing with finite dimensional dependence, quanto option pricing in the presence of fat tails and asymmetric dependence, smile from the past: a general option pricing framework with multiple volatility and leverage components, COMFORT: A common market factor non-Gaussian returns model, divided governments and futures prices, and model-based pricing for financial derivative

Pricing options by simulation using realized volatility

A growing literature advocates the use of high-frequency data for the purpose of volatility estimation. However, despite the successes in modeling the conditional mean of realized volatility empirical evaluations of this class of models outside the realm of short run forecasting is limited. How can realized volatility be used for pricing options? What are the modeling qualities introduced by realized volatility models for pricing derivatives? In this short paper, we propose an options pricing framework based on a new realized volatility model that captures all the relevant empirical regularities of the realized volatility series of the S&P 500 index. We emphasize two main empirical regularities for our volatility model and that are potentially very relevant for option pricing purposes. Fist, realized variation measures constructed from high-frequency returns reveal a large degree of time series unpredictability in the volatility of asset returns. Even though returns standardized by (ex-post) quadratic variation measures are nearly gaussian, this unpredictability brings substantially more uncertainty to the empirically relevant (ex-ante) distribution of returns. In this setting carefully modeling the stochastic structure of the time series disturbances of realized volatility is fundamental. Second, there is evidence of very large leverage effects; large falls (rises) in prices being associated with persistent regimes of high (low) variance in the index returns. We propose a model for the conditional volatility, skewness and kurtosis of daily index and stocks returns. The main new feature of this model is to recognize that volatility is itself more volatile and more persistent in high volatility periods. Contrary to “peso problem” considerations, we show that when volatility is (nearly) observable it is not necessary to rely on rare realizations on past return data to learn about the tails of the return distribution, an unexplored and large modeling gain enabled by high frequency data. We conduct a brief empirical illustration analysis of the pricing performance of this approach against some benchmark models using data from the S&P 500 options in the 2001-2004 period. The results indicate thatas expected the superior forecasting accuracy of realized volatility translates into significantly smaller pricing errors when compared to models of the GARCH family. More significantly, our results indicate that modeling leverage effects and the volatility of volatility are paramount reducing common pricing anomalies

Assessing the Impact of Market Microstructure Noise and Random Jumps on the Relative Forecasting Performance of Option-Implied and Returns-Based Volatility

This paper presents a comprehensive empirical evaluation of option-implied and returns-based forecasts of volatility, in which new developments related to the impact on measured volatility of market microstructure noise and random jumps are explicitly taken into account. The option-based component of the analysis also accommodates the concept of model-free implied volatility, such that the forecasting performance of the options market is separated from the issue of misspecification of the option pricing model. The forecasting assessment is conducted using an extensive set of observations on equity and option trades for News Corporation for the 1992 to 2001 period, yielding certain clear results. According to several different criteria, the model-free implied volatility is the best performing forecast, overall, of future volatility, with this result being robust to the way in which alternative measures of future volatility accommodate microstructure noise and jumps. Of the volatility measures considered, the one which is, in turn, best forecast by the option-implied volatility is that measure which adjusts for microstructure noise, but which retains some information about random jumps.Volatility Forecasts; Quadratic Variation; Intraday Volatility Measures; Model-free Implied Volatility.

Option pricing under the double exponential jump‐diffusion model with stochastic volatility and interest rate

This paper proposes an efficient option pricing model that incorporates stochastic interest rate (SIR), stochastic volatility (SV), and double exponential jump into the jump‐diffusion settings. The model comprehensively considers the leptokurtosis and heteroscedasticity of the underlying asset’s returns, rare events, and an SIR. Using the model, we deduce the pricing characteristic function and pricing formula of a European option. Then, we develop the Markov chain Monte Carlo method with latent variable to solve the problem of parameter estimation under the double exponential jump‐diffusion model with SIR and SV. For verification purposes, we conduct time efficiency analysis, goodness of fit analysis, and jump/drift term analysis of the proposed model. In addition, we compare the pricing accuracy of the proposed model with those of the Black–Scholes and the Kou (2002) models. The empirical results show that the proposed option pricing model has high time efficiency, and the goodness of fit and pricing accuracy are significantly higher than those of the other two models