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Journal ArticleDOI

Dynamic Programming Approach for Valuing Options in the GARCH Model

01 Feb 2009-Management Science (INFORMS)-Vol. 55, Iss: 2, pp 252-266
TL;DR: An efficient algorithm based on dynamic programming coupled with piecewise polynomial approximation to compute the value of a given option, at all observation dates and levels of the state vector is developed.
Abstract: In this paper, we develop an efficient algorithm to value options under discrete-time GARCH processes. We propose a procedure based on dynamic programming coupled with piecewise polynomial approximation to compute the value of a given option, at all observation dates and levels of the state vector. The method can be used for the large GARCH family of models based on Gaussian innovations and may accommodate all low-dimensional European as well as American derivatives. Numerical implementations show that this method competes very advantageously with other available valuation methods.
Citations
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Journal ArticleDOI
TL;DR: In this article, a comprehensive review of option valuation under generalized autoregressive conditional heteroskadasticity (GARCH) is presented, covering both theory and empirical estimation.
Abstract: Black and Scholes assumed stock volatility was a constant and known parameter, but the evidence for stochastically time-varying volatility is overwhelming. Most stochastic volatility models introduce a second stochastic process for variance, driven by shocks that are (negatively) correlated with the return process. But this makes volatility a latent variable and the volatility risk factors cannot be hedged, so they should logically carry risk premia. Estimating the variance equation requires sophisticated econometric methods whose small sample properties may be questioned. Generalized autoregressive conditional heteroskadasticity (GARCH) also offers a specification in which volatility is random, but both volatility and returns are driven by the same shocks, so the variance equation is fitted directly on the observed returns. This article presents a comprehensive review of option valuation under GARCH, covering both theory and empirical estimation. The basic GARCH specification captures time-varying volatility, and the authors review modifications that incorporate known characteristics of real-world return processes, including asymmetric response to up and down shocks, reversion toward a slowly time-varying long-run mean, non-affine and/or non-Gaussian shocks, and return processes subject to multiple shocks, including jumps. GARCH models are typically fitted to series of stock returns, but the authors strongly recommend using both stock returns and option data in model fitting. This helps tie the statistical variance estimates to the markets for variance-dependent securities and exploits information from many additional prices available in the options market. They describe practical suggestions for optimizing and review methods for pricing options under GARCH, including early exercise of American options. The article ends with a series of Appendices giving Matlab code for option pricing under alternative GARCH specifications.

49 citations

Journal ArticleDOI
TL;DR: This paper develops a machine learning approach, called ARMA-GARCH-NN, to capture intra-day patterns for stock market shock forecasting, which integrates classical financial pricing models with artificial neural networks, with explicitly designed feature selection and cross-validation methods.
Abstract: Discovering hidden patterns under unexpected market shocks is a significant and challenging problem, which continually attracts attention from research communities of mathematics, economics, and data science. Classic financial pricing models present unsatisfactory prediction accuracy when applied to real-world data due to limited capacity in depicting complex market movements. In this paper, we develop a machine learning approach, called ARMA-GARCH-NN, to capture intra-day patterns for stock market shock forecasting. Specifically, we integrate classical financial pricing models with artificial neural networks, with explicitly designed feature selection and cross-validation methods. We conduct empirical studies on high-frequency data of the U.S. stock market for evaluation. Our results provide initial evidence of the predictability of market shocks. Additionally, we confirm the effectiveness of ARMA-GARCH-NN by recognizing patterns in massive stock data without strong assumptions on distribution. Our method can serve as a portable methodology that integrates the advantages of traditional financial models and data-driven methods to reveal hidden patterns in large-scale financial data.

31 citations

Journal ArticleDOI
TL;DR: In this article, a method of approximation for American options which can preserve both convexity and monotonicity properties is proposed, which can then be used to define exercise times and can also be used in combination with primal-dual methods to get sharper bounds.
Abstract: It can be shown that when the payoff function is convex and decreasing (respectively increasing) with respect to the underlying (multidimensional) assets, then the same is true for the value of the associated American option, provided some conditions are satisfied. In such a case, all Monte Carlo methods proposed so far in the literature do not preserve the convexity or monotonicity properties. In this paper, we propose a method of approximation for American options which can preserve both convexity and monotonicity. The resulting values can then be used to define exercise times and can also be used in combination with primal-dual methods to get sharper bounds. Other application of the algorithm include finding optimal hedging strategies.

19 citations

Journal ArticleDOI
TL;DR: A partial differential equation formulation for the value of an option when the underlying asset price is described by a discrete-time GARCH process is proposed, achieving a high level of precision in a few seconds of computing time.
Abstract: In this paper, we propose a partial differential equation formulation for the value of an option when the underlying asset price is described by a discrete-time GARCH process. Our numerical approach involves a spectral Fourier-Chebyshev interpolation. Numerical illustrations are provided, and the results are compared with other available valuation methods. Our numerical procedure converges exponentially fast and allows for the efficient computation of option prices, achieving a high level of precision in a few seconds of computing time.

15 citations


Cites methods from "Dynamic Programming Approach for Va..."

  • ...The reference solution computed in Ben-Ameur et al. (2009) is 1.0991, and the 95% confidence interval obtained by a least squares Monte Carlo method in Stentoft (2005) is 1 0765 1 1177 ....

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  • ...The reference solution computed in Ben-Ameur et al. (2009) is 1....

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  • ...Lastly, Ben-Ameur et al. (2009) use dynamic programming combined with finite-element interpolation and show that their method supports Markov-chain approximation as a special case....

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  • ...Lastly, Ben-Ameur et al. (2009) use dynamic programming combined with finite-element interpolation and show that their method supports Markov-chain approximation as a special case. The empirical literature on option prices under GARCH includes Duan (1995), Bollerslev and Mikkelsen (1996, 1999), Heston and Nandi (2000), Hsieh and Ritchken (2005), Duan and Zhang (2001), Myers and Hanson (1993), and Christoffersen and Jacobs (2004) for European options; and Stentoft (2005) for American options....

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  • ...As a reference solution, we use the price computed in Ben-Ameur et al. (2009) using a dynamic programming approach with two state variables (price and variance)....

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References
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Journal ArticleDOI
TL;DR: In this paper, a theoretical valuation formula for options is derived, based on the assumption that options are correctly priced in the market and it should not be possible to make sure profits by creating portfolios of long and short positions in options and their underlying stocks.
Abstract: If options are correctly priced in the market, it should not be possible to make sure profits by creating portfolios of long and short positions in options and their underlying stocks. Using this principle, a theoretical valuation formula for options is derived. Since almost all corporate liabilities can be viewed as combinations of options, the formula and the analysis that led to it are also applicable to corporate liabilities such as common stock, corporate bonds, and warrants. In particular, the formula can be used to derive the discount that should be applied to a corporate bond because of the possibility of default.

28,434 citations

Journal ArticleDOI
TL;DR: In this article, a new class of stochastic processes called autoregressive conditional heteroscedastic (ARCH) processes are introduced, which are mean zero, serially uncorrelated processes with nonconstant variances conditional on the past, but constant unconditional variances.
Abstract: Traditional econometric models assume a constant one-period forecast variance. To generalize this implausible assumption, a new class of stochastic processes called autoregressive conditional heteroscedastic (ARCH) processes are introduced in this paper. These are mean zero, serially uncorrelated processes with nonconstant variances conditional on the past, but constant unconditional variances. For such processes, the recent past gives information about the one-period forecast variance. A regression model is then introduced with disturbances following an ARCH process. Maximum likelihood estimators are described and a simple scoring iteration formulated. Ordinary least squares maintains its optimality properties in this set-up, but maximum likelihood is more efficient. The relative efficiency is calculated and can be infinite. To test whether the disturbances follow an ARCH process, the Lagrange multiplier procedure is employed. The test is based simply on the autocorrelation of the squared OLS residuals. This model is used to estimate the means and variances of inflation in the U.K. The ARCH effect is found to be significant and the estimated variances increase substantially during the chaotic seventies.

20,728 citations

Journal ArticleDOI
TL;DR: In this paper, a natural generalization of the ARCH (Autoregressive Conditional Heteroskedastic) process introduced in 1982 to allow for past conditional variances in the current conditional variance equation is proposed.

17,555 citations

Journal ArticleDOI
TL;DR: An overview of some of the developments in the formulation of ARCH models and a survey of the numerous empirical applications using financial data can be found in this paper, where several suggestions for future research, including the implementation and tests of competing asset pricing theories, market microstructure models, information transmission mechanisms, dynamic hedging strategies, and pricing of derivative assets, are also discussed.

4,206 citations


"Dynamic Programming Approach for Va..." refers methods in this paper

  • ...The GARCH and MGARCH (multivariate GARCH) models were discussed by Bollerslev et al. (1992) and Bera and Higgins (1993), among others....

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Journal ArticleDOI
TL;DR: This paper defined the news impact curve which measures how new information is incorporated into volatility estimates and compared various ARCH models including a partially nonparametric one with daily Japanese stock return data.
Abstract: This paper defines the news impact curve which measures how new information is incorporated into volatility estimates. Various new and existing ARCH models including a partially nonparametric one are compared and estimated with daily Japanese stock return data. New diagnostic tests are presented which emphasize the asymmetry of the volatility response to news. Our results suggest that the model by Glosten, Jagannathan, and Runkle is the best parametric model. The EGARCH also can capture most of the asymmetry; however, there is evidence that the variability of the conditional variance implied by the EGARCH is too high.

3,151 citations