مدل های بازار مالی با استفاده از فرآیندهای لوی و نوسانات متغیر با زمان

Asset management and pricing models require the proper modeling of the return distribution of financial assets. While the return distribution used in the traditional theories of asset pricing and portfolio selection is the normal distribution, numerous studies that have investigated the empirical behavior of asset returns in financial markets throughout the world reject the hypothesis that asset return distributions are normally distribution. Alternative models for describing return distributions have been proposed since the 1960s, with the strongest empirical and theoretical support being provided for the family of stable distributions (with the normal distribution being a special case of this distribution). Since the turn of the century, specific forms of the stable distribution have been proposed and tested that better fit the observed behavior of historical return distributions. More specifically, subclasses of the tempered stable distribution have been proposed. In this paper, we propose one such subclass of the tempered stable distribution which we refer to as the “KR distribution”. We empirically test this distribution as well as two other recently proposed subclasses of the tempered stable distribution: the Carr–Geman–Madan–Yor (CGMY) distribution and the modified tempered stable (MTS) distribution. The advantage of the KR distribution over the other two distributions is that it has more flexible tail parameters. For these three subclasses of the tempered stable distribution, which are infinitely divisible and have exponential moments for some neighborhood of zero, we generate the exponential Lévy market models induced from them. We then construct a new GARCH model with the infinitely divisible distributed innovation and three subclasses of that GARCH model that incorporates three observed properties of asset returns: volatility clustering, fat tails, and skewness. We formulate the algorithm to find the risk-neutral return processes for those GARCH models using the “change of measure” for the tempered stable distributions. To compare the performance of those exponential Lévy models and the GARCH models, we report the results of the parameters estimated for the S&P 500 index and investigate the out-of-sample forecasting performance for those GARCH models for the S&P 500 option prices.

Since Mandelbrot (1963) introduced the Lévy stable (or αα-stable) distribution to model the empirical distribution of asset prices, the αα-stable distribution became the most popular alternative to the normal distribution, the latter distribution being rejected by numerous empirical studies that have found financial return series to be heavy-tailed and possibly skewed. Rachev and Mitnik, 2000 and Rachev et al., 2005 have developed financial models with αα-stable distributions and applied them to market and credit risk management, option pricing, and portfolio selection as will as discussing the major attacks on the αα-stable models. A fair conclusion of the literature is that while the empirical evidence does not support the normal distribution, it is also not consistent with an αα-stable distribution. The distribution of returns for assets has heavier tails relative to the normal distribution and thinner tails than the αα-stable distribution. Partly in response to those empirical inconsistencies, various alternatives to the αα-stable distribution were proposed in the literature. The “classical tempered stable” (CTS) distribution (Koponen, 1995, Boyarchenko and Levendorskii˘, 2000 and Carr et al., 2002) and the “modified tempered stable” (MTS) distribution (Kim et al. (2006)) are two examples; an extension of the CTS distribution named the “KR” distribution (Kim et al. (in press)) is another. These distributions, sometimes called the tempered stable distributions, have not only heavier tails than the normal distribution and thinner than the αα-stable distribution, but also have finite moments for all orders. The tempered stable distributions are used for constructing the exponential Lévy model. If the driving process is the CTS process, then the exponential Lévy model is called the CGMY model, and if the driving process is the MTS process or KR process, then we refer to the exponential Lévy models as the MTS model or the KR model, respectively. The main problem with the exponential Lévy models is that they generate an incomplete market; that is, the equivalent martingale measure (EMM) of a given market measure is not unique in general. For this reason, we need a method to select one reasonable EMM in the incomplete market generated by an exponential Lévy model. One classical method in selecting an EMM is the Esscher transform presented by Gerber and Shiu, 1994 and Gerber and Shiu, 1996; another reasonable method is finding the “minimal entropy martingale measure” presented by Fujiwara and Miyahara (2003). While these methods are mathematically elegant and have a financial interpretation within the context of a utility maximization problem, empirically the model prices obtained from the EMM have not matched the market prices observed for options. The other method for handling the problem is to estimate the risk-neutral measure by using current option price data independent of the historical underlying distribution. This method can fit model prices to market prices directly, but it has a problem: the historical market measure and the risk-neutral measure need not to be equivalent and it conflicts with the no-arbitrage property for option prices. To overcome these drawbacks, one must estimate the market measure and the risk-neutral measure simultaneously, and preserve the equivalent property between two measures. One method for doing so is “the least-squares calibration with a prior” proffered by Cont and Tankov (2004). Basically, their method finds an EMM of the market measure that minimizes the least squares error of the model option prices relative to the market option prices. In spite of the skewness and the fat-tail property of the driving process, the exponential Lévy model has been rejected by empirical evidence (e.g., the finding that there is volatility clustering). The Markov property of the exponential Lévy model is one reason for the rejection. GARCH option pricing models have been developed to price options under the assumption of a non-Markovian property, more precisely, the assumption of volatility clustering. GARCH models of Duan, 1995 and Heston and Nandi, 2000 are important works on the non-Markovian structure of asset returns with the normal innovation process, but the normal innovation process disregards the empirical innovation process of asset returns. Duan et al. (2004) enhanced the classical GARCH model by adding jumps to the innovation processes. Subsequently, Menn and Rachev, 2005a and Menn and Rachev, 2005b introduced an enhanced GARCH model with innovations which follow the smoothly truncated stable (STS) distribution. In this paper, we present market models based on the tempered stable distributions and provide empirical tests of these distributions. First, we consider the CGMY, MTS, and KR models. Then we find their EMM using the method of least-squares calibration with a prior and verify empirically the advantages of the KR model. We can find the parameters of the EMM such that the least squares error of the KR model prices are less than the error of the CGMY and MTS model prices. The change of measure between two KR processes has more freedom than that of the CGMY and MTS, and this freedom provides some empirical benefit which will be discussed. We then construct a new GARCH model that combines the volatility clustering property of Duan’s GARCH model and the skewness and fat-tail property of the infinitely divisible distribution which induces the Lévy process. This combination approach was first attempted by Menn and Rachev, 2005a and Menn and Rachev, 2005b using the αα-stable distribution. In this paper, we improve and extend their approach. We consider the GARCH model and apply infinitely divisible distribution for modeling the residual distribution of the GARCH model. Technically, we apply the three tempered stable distributions (the CTS, the MTS, and the KR distributions) to modeling the residual distribution. The market parameters of the GARCH models are estimated for S&P 500 index and the prices of three individual stocks. The out-of-sample forecasting performance of those GARCH models for the S&P 500 call option prices is investigated and compared to the performance of the Black–Scholes model with the historical volatility, the CGMY model with Esscher transform, and the Duan’s normal-GARCH model. The implied volatility curves of the call option prices computed using the three tempered stable GARCH option pricing models are also compared to the curves calculated using the market prices and the model prices of the CGMY model and the normal-GARCH model. The contribution of this paper is threefold. First, we present a new market model which is an extension of the CGMY model that allows for flexible modeling of the tail behavior of the return distribution. We verify the theoretical and empirical benefit of the market model generated by the distribution. Second, we construct new GARCH models with non-normally distributed innovation with flexible tail characteristics. Finally, we formulate a method to ascertain the risk-neutral stock price process corresponding to the new GARCH model and obtain good empirical performance when applying this model to option prices. The remainder of this paper is organized as follows. Section 2 presents the continuous-time market model and reviews the CGMY model. The MTS distribution and the MTS model are given in Section 3, while the KR distribution and the KR model are given in Section 4. The empirical tests are provided in Section 5 where we show the estimation results for the market parameters and test the calibration performance of the risk-neutral parameters for the CGMY, MTS, and KR models. The GARCH model with the innovations with the infinitely divisible law and its tempered stable subclasses are discussed in Section 6, and its empirical results are reported in Section 7. Section 8 summarizes the principal conclusions of the paper.

In this paper, we investigate two types of market models: the continuous market model and the discrete market model. The former model presented in this paper is the extended tempered stable market model that we have named the KR model. This model is an extension of the CGMY model that allows for more flexible parameters than both the CGMY model and the MTS model. This greater flexibility allows for a more efficient means for calibrating the risk-neutral measure. With respect to the discrete market model, we present a new non-normal GARCH model with infinitely divisible distributed innovation. More specifically, we consider three subclasses of the GARCH model: the CTS-GARCH model, MTS-GARCH model, and the KR-GARCH model. Not only do these models describe volatility clustering, but other tail characteristics that have been observed for asset returns. Hence, these models are more realistic GARCH models that can be employed in option pricing. The three tempered stable GARCH models investigated in this paper are not rejected in statistical hypothesis testing, while the normal-GARCH model is rejected. The reason for the good statistical results for tempered stable GARCH models relative to the normal-GARCH model is that skewness and fat-tail property of their innovation are taken into account. Moreover, the tempered stable GARCH models have better forecasting performance than the normal-GARCH model and the CGMY model with Esser transform, in addition to generating a more realistic implied volatility curve. Consequently, the clustering volatility with the tempered stable innovation explains the behavior of stock and option markets better than the Markovian model of the exponential CGMY distribution and the clustering volatility with the standard normal residual distribution.