On accurate and provably efficient GARCH option pricing algorithms

The GARCH model has been very successful in capturing the serial correlation of asset return volatilities. As a result, applying the model to options pricing attracts a lot of attention. However, previous tree-based GARCH option pricing algorithms suffer from exponential running time, a cut-off maturity, inaccuracy, or some combination thereof. Specifically, this paper proves that the popular trinomial-tree option pricing algorithms of Ritchken and Trevor (Ritchken, P. and Trevor, R., Pricing options under generalized GARCH and stochastic volatility processes. J. Finance, , 54(1), 377–402.) and Cakici and Topyan (Cakici, N. and Topyan, K., The GARCH option pricing model: a lattice approach. J. Comput. Finance, , 3(4), 71–85.) explode exponentially when the number of partitions per day, n, exceeds a threshold determined by the GARCH parameters. Furthermore, when explosion happens, the tree cannot grow beyond a certain maturity date, making it unable to price derivatives with a longer maturity. As a result, the algorithms must be limited to using small n, which may have accuracy problems. The paper presents an alternative trinomial-tree GARCH option pricing algorithm. This algorithm provably does not have the short-maturity problem. Furthermore, the tree-size growth is guaranteed to be quadratic if n is less than a threshold easily determined by the model parameters. This level of efficiency makes the proposed algorithm practical. The surprising finding for the first time places a tree-based GARCH option pricing algorithm in the same complexity class as binomial trees under the Black–Scholes model. Extensive numerical evaluation is conducted to confirm the analytical results and the numerical accuracy of the proposed algorithm. Of independent interest is a simple and efficient technique to calculate the transition probabilities of a multinomial tree using generating functions.     

Model risk of the implied GARCH-normal model

Model risk causes significant losses in financial derivative pricing and hedging. Investors may undertake relatively risky investments due to insufficient hedging or overpaying implied by flawed models. The GARCH model with normal innovations (GARCH-normal) has been adopted to depict the dynamics of the returns in many applications. The implied GARCH-normal model is the one minimizing the mean square error between the market option values and the GARCH-normal option prices. In this study, we investigate the model risk of the implied GARCH-normal model fitted to conditional leptokurtic returns, an important feature of financial data. The risk-neutral GARCH model with conditional leptokurtic innovations is derived by the extended Girsanov principle. The option prices and hedging positions of the conditional leptokurtic GARCH models are obtained by extending the dynamic semiparametric approach of Huang and Guo [Statist. Sin., 2009, 19, 1037–1054]. In the simulation study we find significant model risk of the implied GARCH-normal model in pricing and hedging barrier and lookback options when the underlying dynamics follow a GARCH-t model.     

An analysis of option pricing under systematic consumption risk using GARCH

An issue in the pricing of contingent claims is whether to account for consumption risk. This is relevant for contingent claims on stock indices, such as the FTSE 100 share price index, as investor’s desire for smooth consumption is often used to explain risk premiums on stock market portfolios, but is not used to explain risk premiums on contingent claims themselves. This paper addresses this fundamental question by allowing for consumption in an economy to be correlated with returns. Daily data on the FTSE 100 share price index are used to compare three option pricing models: the Black–Scholes option pricing model, a GARCH (1, 1) model priced under a risk-neutral framework, and a GARCH (1, 1) model priced under systematic consumption risk. The findings are that accounting for systematic consumption risk only provides improved accuracy for in-the-money call options. When the correlation between consumption and returns increases, the model that accounts for consumption risk will produce lower call option prices than observed prices for in-the-money call options. These results combined imply that the potential consumption-related premium in the market for contingent claims is constant in the case of FTSE 100 index options.     

Option pricing under regime switching

Abstract

This paper develops a family of option pricing models when the underlying stock price dynamic is modelled by a regime switching process in which prices remain in one volatility regime for a random amount of time before switching over into a new regime. Our family includes the regime switching models of Hamilton (Hamilton J 1989 Econometrica 57 357–84), in which volatility influences returns. In addition, our models allow for feedback effects from returns to volatilities. Our family also includes GARCH option models as a special limiting case. Our models are more general than GARCH models in that our variance updating schemes do not only depend on levels of volatility and asset innovations, but also allow for a second factor that is orthogonal to asset innovations. The underlying processes in our family capture the asymmetric response of volatility to good and bad news and thus permit negative (or positive) correlation between returns and volatility. We provide the theory for pricing options under such processes, present an analytical solution for the special case where returns provide no feedback to volatility levels, and develop an efficient algorithm for the computation of American option prices for the general case.     

An empirical comparison of GARCH option pricing models

Recent empirical studies have shown that GARCH models can be successfully used to describe option prices. Pricing such contracts requires knowledge of the risk neutral cumulative return distribution. Since the analytical forms of these distributions are generally unknown, computationally intensive numerical schemes are required for pricing to proceed. Heston and Nandi (2000) consider a particular GARCH structure that permits analytical solutions for pricing European options and they provide empirical support for their model. The analytical tractability comes at a potential cost of realism in the underlying GARCH dynamics. In particular, their model falls in the affine family, whereas most GARCH models that have been examined fall in the non-affine family. This article takes a closer look at this model with the objective of establishing whether there is a cost to restricting focus to models in the affine family. We confirm Heston and Nandi's findings, namely that their model can explain a significant portion of the volatility smile. However, we show that a simple non affine NGARCH option model is superior in removing biases from pricing residuals for all moneyness and maturity categories especially for out-the-money contracts. The implications of this finding are examined. JEL Classification G13     

Empirical Study of Nikkei 225 Options with the Markov Switching GARCH Model

This paper investigates the pricing of Nikkei 225 Options using the Markov Switching GARCH (MSGARCH) model, and examines its practical usefulness in option markets. We assume that investors are risk-neutral and then compute option prices by using Monte Carlo simulation. The results reveal that, for call options, the MSGARCH model with Student’s t-distribution gives more accurate pricing results than GARCH models and the Black–Scholes model. However, this model does not have good performance for put options.     

Fast accurate binomial pricing

We discuss here an alternative interpretation of the familiar binomial lattice approach to option pricing, illustrating it with reference to pricing of barrier options, one- and two-sided, with fixed, moving or partial barriers, and also the pricing of American put options. It has often been observed that if one tries to price a barrier option using a binomial lattice, then one can find slow convergence to the true price unless care is taken over the placing of the grid points in the lattice; see, for example, the work of Boyle & Lau [2]. The placing of grid points is critical whether one uses a dynamic programming approach, or a Monte Carlo approach, and this can make it difficult to compute hedge ratios, for example. The problems arise from translating a crossing of the barrier for the continuous diffusion process into an event for the binomial approximation. In this article, we show that it is not necessary to make clever choices of the grid positioning, and by interpreting the nature of the binomial approximation appropriately, we are able to derive very quick and accurate pricings of barrier options. The interpretation we give here is applicable much more widely, and helps to smooth out the ‘odd-even’ ripples in the option price as a function of time-to-go which are a common feature of binomial lattice pricing.     

A lattice model for option pricing under GARCH-jump processes

This study extends the GARCH pricing tree in Ritchken and Trevor (J Financ 54:366–402, 1999) by incorporating an additional jump process to develop a lattice model to value options. The GARCH-jump model can capture the behavior of asset prices more appropriately given its consistency with abundant empirical findings that discontinuities in the sample path of financial asset prices still being found even allowing for autoregressive conditional heteroskedasticity. With our lattice model, it shows that both the GARCH and jump effects in the GARCH-jump model are negative for near-the-money options, while positive for in-the-money and out-of-the-money options. In addition, even when the GARCH model is considered, the jump process impedes the early exercise and thus reduces the percentage of the early exercise premium of American options, particularly for shorter-term horizons. Moreover, the interaction between the GARCH and jump processes can raise the percentage proportions of the early exercise premiums for shorter-term horizons, whereas this effect weakens when the time to maturity increases.     

American option pricing with discrete and continuous time models: An empirical comparison

This paper considers discrete time GARCH and continuous time SV models and uses these for American option pricing. We first of all show that with a particular choice of framework the parameters of the SV models can be estimated using simple maximum likelihood techniques. We then perform a Monte Carlo study to examine their differences in terms of option pricing, and we study the convergence of the discrete time option prices to their implied continuous time values. Finally, a large scale empirical analysis using individual stock options and options on an index is performed comparing the estimated prices from discrete time models to the corresponding continuous time model prices. The results show that, while the overall differences in performance are small, for the in the money put options on individual stocks the continuous time SV models do generally perform better than the discrete time GARCH specifications.     

An FFT-network for Lévy option pricing

This paper develops a simple network approach to American exotic option valuation under Lévy processes using the fast Fourier transform (FFT). The forward shooting grid (FSG) technique of the lattice approach is then generalized to expand the FFT-network to accommodate path-dependent variables. This network pricing approach is applicable to all Lévy processes for which the characteristic function is readily available. In other words, the log-value of the underlying asset can follow finite-activity or infinite-activity Lévy processes. With the powerful computation of FFT, the proposed network has a negligible additional computational burden compared to the binomial tree approach. The early exercise policy and option values in the continuation region are determined in a way very similar to that of the lattice approach. Numerical examples using American-style barrier, lookback, and Asian options demonstrate that the FFT-network is accurate and efficient.     

Stock market momentum, business conditions, and GARCH option pricing models

This paper examines the forecasting performance of GARCH option pricing models from a market momentum perspective, and the possible impacts of financial crises and business conditions are also examined. The empirical results demonstrate that market momentum impacts the forecasting performance of GARCH option pricing models. The EGARCH model performs better under downward market momentum, while the standard GARCH performs better under upward market momentum. In addition, parsimonious models generally outperform richly parameterized ones. The above findings are robust to financial crises, and the results further demonstrate that business conditions influence the forecasting performance of GARCH option pricing models.     

An Empirical Comparison of Option‐Pricing Models in Hedging Exotic Options

This paper examines the empirical performance of various option‐pricing models in hedging exotic options, such as barrier options and compound options. A practical and relevant testing approach is adopted to capture the essence of model risk in option pricing and hedging. Our results indicate that the exotic feature of the option under consideration has a great impact on the relative performance of different option‐pricing models. In addition, for any given model, the more “exotic” the option, the poorer the hedging effectiveness.     

Modeling time series information into option prices: An empirical evaluation of statistical projection and GARCH option pricing model

This paper compares the empirical performances of statistical projection models with those of the Black–Scholes (adapted to account for skew) and the GARCH option pricing models. Empirical analysis on S&P500 index options shows that the out-of-sample pricing and projected trading performances of the semi-parametric and nonparametric projection models are substantially better than more traditional models. Results further indicate that econometric models based on nonlinear projections of observable inputs perform better than models based on OLS projections, consistent with the notion that the true unobservable option pricing model is inherently a nonlinear function of its inputs. The econometric option models presented in this paper should prove useful and complement mainstream mathematical modeling methods in both research and practice.     

Estimating and using GARCH models with VIX data for option valuation

This paper uses information on VIX to improve the empirical performance of GARCH models for pricing options on the S&P 500. In pricing multiple cross-sections of options, the models’ performance can clearly be improved by extracting daily spot volatilities from the series of VIX rather than by linking spot volatility with different dates by using the series of the underlying’s returns. Moreover, in contrast to traditional returns-based Maximum Likelihood Estimation (MLE), a joint MLE with returns and VIX improves option pricing performance, and for NGARCH, joint MLE can yield empirically almost the same out-of-sample option pricing performance as direct calibration does to in-sample options, but without costly computations. Finally, consistently with the existing research, this paper finds that non-affine models clearly outperform affine models.     

Covariance dependent kernels,a Q-affine GARCH for multi-asset option pricing

This paper introduces a class of multivariate GARCH models that extends the existing literature by explicitly modeling correlation dependent pricing kernels. A large subclass admits closed-form recursive solutions for the moment generating function under the risk-neutral measure, which permits efficient pricing of multi-asset options. We perform a full calibration to three bivariate series of index returns and their corresponding volatility indexes in a joint maximum likelihood estimation. The results empirically confirm the presence of correlation dependance in addition to the well known variance dependance in the pricing kernel. The model improves both the overall likelihood and the VIX-implied likelihoods, with a better fitting of marginal distributions, e.g., 15% less error on one-asset option prices. The new degree of freedom is also shown to significantly impact the shape of marginal and joint pricing kernels, and leads to up to 53% differences for out-of-the-money two-asset correlation option prices.     

Is the realized volatility good for option pricing during the recent financial crisis?

The contributions of this paper are threefold. The first contribution is the proposed logarithmic HAR (log-HAR) option-pricing model, which is more convenient compared with other option pricing models associated with realized volatility in terms of simpler estimation procedure. The second contribution is the test of the empirical implications of heterogeneous autoregressive model of the realized volatility (HAR)-type models in the S&P 500 index options market with comparison of the non-linear asymmetric GARCH option-pricing model, which is the best model in pricing options among generalized autoregressive conditional heteroskedastic-type models. The third contribution is the empirical analysis based on options traded from July 3, 2007 to December 31, 2008, a period covering a recent financial crisis. Overall, the HAR-type models successfully predict out-of-sample option prices because they are based on realized volatilities, which are closer to the expected volatility in financial markets. However, mixed results exist between the log-HAR and the heterogeneous auto-regressive gamma models in pricing options because the former is better than the latter in times of turmoil, whereas it is worse during the rather stable periods.     

An empirical test of the variance gamma option pricing model

In this paper, we test the three-parameter symmetric variance gamma (SVG) option pricing model and the four-parameter asymmetric variance gamma (AVG) option pricing model empirically. Prices of the Hang Seng Index call options, which are of European style, are used as the data for the empirical test. Since the variance gamma option pricing model is developed for the pricing of European options, the empirical test gives a more conclusive answer than previous papers, which used American option data to the applicability of the VG models. The present study uses a large number of intraday option data, which span over a period of 3 years. Synchronous option and futures data are used throughout the study. Pairwise comparisons between the accuracy of model prices are carried out using both parametric and nonparametric methods.The conclusion is that the VG option pricing model performs marginally better than the Black–Scholes (BS) model. Under the historical approach, the VG models can moderately iron out some of the systematic biases inherent in the BS model. However, under the implied approach, the VG models continue to exhibit predictable biases and its overall performance in pricing and hedging is still far less than desirable.     

Pricing Hang Seng Index options around the Asian financial crisis – A GARCH approach

This paper investigates how well the Hang Seng Index options, the most important class of option contracts traded in Hong Kong, are priced using the GARCH approach. We calibrated the GARCH parameters using the call and put option data and used them to price options in the subsequent weeks. We found the GARCH model performs very well in comparison with the Black–Scholes model even after allowing for a smile/smirk adjustment. Its superior performance was also evident both before and during the recent Asian financial turmoil.     

Pricing European-type,early-exercise and discrete barrier options using an algorithm for the convolution of Legendre series

This paper applies an algorithm for the convolution of compactly supported Legendre series (the CONLeg method) (cf. Hale and Townsend, An algorithm for the convolution of Legendre series. SIAM J. Sci. Comput., 2014, 36, A1207–A1220), to pricing European-type, early-exercise and discrete-monitored barrier options under a Lévy process. The paper employs Chebfun (cf. Trefethen et al., Chebfun Guide, 2014 (Pafnuty Publications: Oxford), Available online at: http://www.chebfun.org/) in computational finance and provides a quadrature-free approach by applying the Chebyshev series in financial modelling. A significant advantage of using the CONLeg method is to formulate option pricing and option Greek curves rather than individual prices/values. Moreover, the CONLeg method can yield high accuracy in option pricing when the risk-free smooth probability density function (PDF) is smooth/non-smooth. Finally, we show that our method can accurately price options deep in/out of the money and with very long/short maturities. Compared with existing techniques, the CONLeg method performs either favourably or comparably in numerical experiments.