Some exotic options under symmetric and asymmetric conditional volatility of returns

For a plain vanilla call and three of the more popular exotic (path-dependent) types of options, this study examines the impact of symmetric and asymmetric GARCH processes in returns. The price, delta and gamma of European call options, Black–Scholes implied volatilities and convergence of these factors are all studied, through a simulation of price paths. For comparison, we ensure that the unconditional volatility of each process is identical. The impact of a standard symmetric GARCH volatility structure on the option parameters is usually to bias price and delta downwards, but to bias gamma upwards, sometimes quite considerably. Asymmetric GARCH effects exacerbate this effect, and it varies across the different options. GARCH effects appear not to induce a smile. Finally, as time to maturity shortens, at-the-money call prices and deltas converge slowly but gammas can change wildly when GARCH effects are added.     

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.     

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.     

The intraday behaviors and relationships with its underlying assets: evidence on option market in Taiwan

In this paper, we use daily data to investigate the information asymmetric effects and the relationships between the trading volume of options and their underlying spot trading volume. Our results reveal that options with higher liquidity are near-the-money and expiration periods with 2 to 4 weeks have higher trading activity. We classify them into two parts with the ARIMA model: the expected trading activity impact and the unexpected trading activity impact. Using the bivariate generalized autoregressive conditional heteroscedasticity (GARCH) model, we investigate the trading activity effect and information asymmetric effect. In conclusion, the trading volume volatility of the spot and options markets move together, and a greater expected and unexpected trading volume volatility of the spot (options) market is associated with greater volatility in the options (spot) market. However, both markets generate higher trading volume volatility when people expect such an impact rather than when they do not. We also find that there are feedback effects within these two markets. Furthermore, when the spot (options) market has negative innovations, it generates a greater impact on the options (spot) market than do positive innovations. Finally, the conditional correlation coefficient between the spot and the option markets changes over time based on the bivariate GARCH model.     

The GARCH Option Pricing Model: A Modification of Lattice Approach

Ritchken and Trevor (1999) proposed a lattice approach for pricing American options under discrete time-varying volatility GARCH frameworks. Even though the lattice approach worked well for the pricing of the GARCH options, it was inappropriate when the option price was computed on the lattice using standard backward recursive procedures, even if the concepts of Cakici and Topyan (2000) were incorporated. This paper shows how to correct the deficiency and that with our adjustment, the lattice method performs properly for option pricing under the GARCH process. JEL Classification: C10, C32, C51, F37, G12     

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.     

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.     

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.     

BIVARIATE BINOMIAL OPTIONS PRICING WITH GENERALIZED INTEREST RATE PROCESSES

We extend existing pricing models and develop a bivariate binomial option pricing technique that accommodates correlated state variables. This technique offers the ability to price American-style options, thereby accommodating early exercise, despite the existence of two correlated underlying state variables. Our technique is computationally efficient and can be further generalized for multiple-state variables, albeit with an accompanying rise in computational expense.     

Evaluation of Black-Scholes and GARCH Models Using Currency Call Options Data

This paper empirically examines the performance of Black-Scholes and Garch-M call option pricing models using call options data for British Pounds, Swiss Francs and Japanese Yen. The daily exchange rates exhibit an overwhelming presence of volatility clustering, suggesting that a richer model with ARCH/GARCH effects might have a better fit with actual prices. We perform dominant tests and calculate average percent mean squared errors of model prices. Our findings indicate that the Black-Scholes model outperforms the GARCH models. An implication of this result is that participants in the currency call options market do not seem to price volatility clusters in the underlying process.     

Dynamic copula models for the spark spread

We propose a non-symmetric copula to model the evolution of electricity and gas prices by a bivariate non-Gaussian autoregressive process. We identify the marginal dynamics as driven by normal inverse Gaussian processes, estimating them from a series of observed UK electricity and gas spot data. We estimate the copula by modeling the difference between the empirical copula and the independent copula. We then simulate the joint process and price options written on the spark spread. We find that option prices are significantly influenced by the copula and the marginal distributions, along with the seasonality of the underlying prices.     

Monte Carlo Option Pricing for Tempered Stable (CGMY) Processes

Lévy processes are popular models for stock price behavior since they allow to take into account jump risk and reproduce the implied volatility smile. In this paper, we focus on the tempered stable (also known as CGMY) processes, which form a flexible 6-parameter family of Lévy processes with infinite jump intensity. It is shown that under an appropriate equivalent probability measure a tempered stable process becomes a stable process whose increments can be simulated exactly. This provides a fast Monte Carlo algorithm for computing the expectation of any functional of tempered stable process. We use our method to price European options and compare the results to a recent approximate simulation method for tempered stable process by Madan and Yor (CGMY and Meixner Subordinators are absolutely continuous with respect to one sided stable subordinators, 2005).     

Adaptive placement method on pricing arithmetic average options

Since there is no analytic solution for arithmetic average options until present, developing an efficient numerical algorithm becomes a promising alternative. One of the most famous numerical algorithms is introduced by Hull and White (J Deriv 1:21–31, 1993). Motivated by the common idea of reducing the nonlinearity error in the adaptive mesh model in Figlewski and Gao (J Financ Econ 53:313–351, 1999) and the adaptive quadrature method, we propose an adaptive placement method to replace the logarithmically equally-spaced placement rule in the Hull and White’s model by placing more representative average prices in the highly nonlinear area of the option value as the function of the arithmetic average stock price. The basic idea of this method is to design a recursive algorithm to limit the error of the linear interpolation between each pair of adjacent representative average prices. Numerical experiments verify the superior performance of this method for reducing the interpolation error and hence improving the convergence rate. To show that the adaptive placement method can improve any numerical algorithm with the techniques of augmented state variables and the piece-wise linear interpolation approximation, we also demonstrate how to integrate the adaptive placement method into the GARCH option pricing algorithm in Ritchken and Trevor (J Finance 54:377–402, 1999). Similarly great improvement of the convergence rate suggests the potential applications of this novel method to a broad class of numerical pricing algorithms for exotic options and complex underlying processes.     

GARCH modelling of banking integration in the Eurozone

We investigate the progress of integration in the European banking industry and its effects on the price of the common stock of banks listed on European stock exchanges. We estimate the overall effect of progress by comparing the changes in the stock price volatility of listed banks over the period from January 1990 to December 2005. Using univariate and bivariate GARCH models, we document that the introduction of the Euro and the enlargement of the European Union in May 2004 have contributed to the integration process of the banking industry in Europe. We also find evidence of negative volatility spillovers among bank stock returns for different groups of countries that have been involved in various recent stages of the European economic and political integration.     

Valuing fade-in options with default risk in Heston–Nandi GARCH models

Review of Derivatives Research - In this paper, we present a pricing model to value fade-in options with default risk, where the underlying asset price is driven by the Heston–Nandi GARCH...     

Bivariate FIGARCH and fractional cointegration

We consider the modelling of volatility on closely related markets. Univariate fractional volatility (FIGARCH) models are now standard, as are multivariate GARCH models. In this paper, we adopt a combination of the two methodologies. There is as yet little consensus on the methodology for testing for fractional cointegration. The contribution of this paper is to demonstrate the feasibility of estimating and testing cointegrated bivariate FIGARCH models. We apply these methods to volatility on the NYMEX and IPE crude oil markets. We find a common order of fractional integration for the two volatility processes and confirm that they are fractionally cointegrated. An estimated error correction FIGARCH model indicates that the preponderant adjustment is of the IPE towards NYMEX.     

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.     

International stock markets interactions and conditional correlations

This paper investigates the transmission of price and volatility spillovers across the US and European stock markets in bivariate combinations. The framework used encompasses the most popular multivariate GARCH models, with News Impact Surfaces employed for interpretation. By using synchronous data the dynamic conditional correlation model (Engle, R., 2002. Dynamic conditional correlation: a simple class of multivariate GARCH models. Journal of Business and Economic Statistics 20, 339–350) is found to best capture the relationships for over half of the bivariate combinations of markets. Other findings include volatility spillovers from the US to European markets, and a reverse spillover. In addition, the magnitude of the correlation between markets is higher not only for negative shocks in both markets, but also when a combination of shocks of opposite signs occurs.     

Long‐Run Risk and Its Implications for the Equity Premium Puzzle: New Evidence from a Multivariate Framework

This paper investigates the empirical evidence of long‐run risk and its implications for the equity premium puzzle. We find that the long‐run risk model is generally weakly identified and that standard inferences tend to underestimate the uncertainty of long‐run risk. We extend the LM‐type test of Ma and Nelson (2010) that remains valid under weak identification to the bivariate VARMA‐GARCH model of consumption and dividend growth. The results cast doubt on the validity of long‐run risk as an explanation for the equity premium puzzle. We also evaluate the approach of Bansal, Kiku, and Yaron (2007a), which extracts long‐run risk by regressing consumption growth and its volatility on predictive variables. The results using the Bonferroni Q‐test of Campbell and Yogo (2006) suggest that consumption and dividend growth are generally unpredictable by the price‐dividend ratio and risk‐free rate. This casts doubt on the validity of the BKY approach.