Monte Carlo option pricing with asymmetric realized volatility dynamics
Introduction
The advent of high frequency stock market data and the subsequent introduction of realized volatility measures represented a substantial step forward in the accuracy with which econometric models of volatility could be evaluated and allowed for the development of new and more precise parametric models of time varying volatility. Several researchers have looked into the properties of ex post volatility measures derived from high frequency data and developed time series models that invariably outperform latent variable models of the GARCH (Generalized Autoregressive Conditional Heteroskedasticity) or stochastic volatility family of models [3] in forecasting future volatility, to the point that the comparison has been dropped altogether in recent papers.
Recent contributions to the realized volatility modeling and forecasting literature are exemplified by Andersen et al. [3], the HAR (heterogeneous autoregressive) model of Corsi [10], the MIDAS (mixed data sample) approach of Ghysels et al. [15] and the unobserved ARMA component model of Refs. [18], [23]. Martens et al. [20] develop a nonlinear (ARFIMA) model to accommodate level shifts, day of the week, leverage and volatility level effects. Andersen et al. [2] and Tauchen and Zhou [26] argue that the inclusion of jump components significantly improves forecasting performance. McAleer and Medeiros [21] extend the HAR model to account for nonlinearities. Hillebrand and Medeiros [17] also consider nonlinear models and evaluate the benefits of bootstrap aggregation (bagging) for volatility forecasting. Ghysels et al. [15] argue that realized absolute values outperform square return-based volatility measures in predicting future increments in quadratic variation. Scharth and Medeiros [22] introduce multiple regime models linked to asymmetric effects. Liu and Maheu [19] derive a bayesian averaging approach for forecasting realized volatility. Bollerslev et al. [8] propose a full system for returns, jumps and continuous time for components of price movements using realized variation measures.
Despite these 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. Fleming et al. [14] examine the economic value of volatility timing using realized volatility. Bandi et al. [5] evaluate and compare the quality of several recently proposed realized volatility estimators in the context of option pricing and trading of short term options on a stylized setting. Stentoft [25] derives an appropriate return and volatility dynamics to be used for option pricing purposes in the context of realized volatility and perform an empirical analysis using stock options for three large American companies.
We emphasize two main empirical regularities 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.
In this 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, the dually asymmetric realized volatility (DARV) model of Allen et al. [1]. In this setting returns display conditional volatility, skewness and kurtosis. 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 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. This exercise supplements the extensive empirical analysis of this model conducted in that paper. The results indicate that as expected the superior forecasting accuracy of the proposed realized volatility model 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 for reducing common pricing anomalies.
Section snippets
Realized volatility and data
Suppose that at day t the logarithmic prices of a given asset follow a continuous time diffusion:where p(t + τ) is the logarithmic price at time t + τ, is the drift component, σ(t + τ) is the instantaneous volatility (or standard deviation), and dW(t + τ) is a standard Brownian motion. Andersen et al. [3] (and others) showed that the daily compound returns, defined as rt = p(t) − p(t + 1), are Gaussian conditionally on , the σ-algebra (information
The dually asymmetric realized volatility model
The dually asymmetric realized volatility (DARV) model introduced by Allen et al. [1] is a first step in analyzing and incorporating the modeling qualities of a more realistic setting for the volatility risk within a standard realized volatility model. The dual asymmetry in the model comes from leverage effects (as seen in the last section) and the positive relation between the level of volatility and the degree of volatility risk. The fundamental issue that arises in specifying the model is
Empirical illustration
In this section we perform a brief empirical analysis of our option pricing model. For conciseness we focus on put options with 9–60 calendar days to expiration. Defining moneyness by M = St/X, where St is the underlying index price at the time when the option is observed and X is the strike price, we divide the options into the following groups: at-the-money (0.98 < M < 1.02), out-the money (1.02 < M < 1.05), in-the-money (0.95 < M < 0.98), deep out-of-the-money (1.05 < M < 1.1) and the deep in-the-money (0.9 < M <
Conclusion
As exemplified by the long memory property case, the volatility literature grows in the middle of a theoretical gap. There currently exists no fully compelling theoretically parametric model of asset returns and the theoretical underpinnings of the patterns observed in volatility processes are not fully understood. Nevertheless, the importance of empirical findings of the volatility literature for option pricing and other applications should not be understated. In this paper we have analyzed an
Acknowledgments
This article is a revised version of the paper presented at MODSIM09 conference. The authors wish to acknowledge the helpful comments and suggestions of the conference participants and two anonymous referees. The financial support of the Australian Research Council is gratefully acknowledged. Michael McAleer wishes to thank the financial support of the National Science Council, Taiwan. Data supplied by Securities Industry Research Centre of Asia-Pacific (SIRCA).
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