Pricing options under stochastic volatility

Research in International Business and …, 2004

1998

This paper examines the behaviour of European option price (Duan (1995)) and the Black-Scholes model bias when stock returns follow a GARCH (1,1) process. The GARCH option price is not preferenceneutral and depends on the unit risk premium (λ) as well as the two GARCH (1,1) process parameters (α1 , β1). In general, the GARCH option price does not seem overly sensitive to these parameters. Deep-out-ofthe-money and short maturity options are an exception. The variance persistence parameter, γ = α1 + β1, has a material bearing on the magnitude of the Black-Scholes model bias. The risk preference parameter, l, on the other hand, determines the so called “leverage effect” and can be important in determining the direction of the Black-Scholes model bias. Consequently, a time varying risk premium (l) may help explain a general underpricing or overpricing of traded options (Black (1975)). Consistent with "volatility smile" and similar to the bias noted by Merton (1976), deep-out-o...

Options are instruments which have the special property of limiting the downside risk, while not limiting the upside potential, thus their use in hedging. The share of the options market in the Indian capital market has increased to 64% in just over a decade. The trading turnover of options in the FY11 was Rs. 193,95,710 crore, and the trading volume generated by options market was almost two times that of the volume generated in the cash market and futures market put together. So trading and pricing of stock option have occupied an important place in the Indian derivatives market. Volatility is a critical factor influencing the option pricing; however, it is an extremely difficult factor to forecast. Hence the crucial problem lies with the accurate estimation of volatility. The estimated volatility can be used to determine future prices of the stock or the stock option. Empirical research has shown that using historical volatility in different option pricing models leads to pricing biases. The GARCH (1, 1) model can be a solution for this

Review of Derivatives Research, 2006

Heston and Nandi (2000) provide considerable empirical support for their GARCH option pricing model. Their model has the advantage that analytical solutions are available for pricing European options. This article takes a closer look at this model and compares its performance with the NGARCH option model of Duan (1995). We con¯rm Heston and Nandi's¯ndings, namely that their model can explain a signi¯cant portion of the volatility smile. However, we show that the NGARCH model is superior in removing biases from pricing residuals for all moneyness and maturity categories especially for out-the-money contracts. The out-of-sample performance of both GARCH models is closely examined, and the NGARCH model is shown to have very attractive properties. The NGARCH model continues to perform well, even when the parameters of the model are not re-estimated for long periods of time. Given the existence of relatively e±cient algorithms for pricing American claims and exotics under NGARCH processes, we recommend that traders and risk managers consider the NGARCH model.

2012

This study develops a GARCH-type model, i.e., the variance-gamma GARCH (VG GARCH) model, based on the two major strands of option pricing literature. The first strand of the literature uses the variance-gamma process, a time-changed Brownian motion, to model the underlying asset price process such that the possible skewness and excess kurtosis on the distributions of asset returns are considered. The second strand of the literature considers the propagation of the previously arrived news by including the feedback and leverage effects on price movement volatility in a GARCH framework. The proposed VG GARCH model is shown to obey a locally risk-neutral valuation relationship (LRNVR) under the sufficient conditions postulated by . This new model provides a unified framework for estimating the historical and risk-neutral distributions, and thus facilitates option pricing calibration using historical underlying asset prices. An empirical study is performed comparing the proposed VG GARCH model with four competing pricing models: benchmark Black-Scholes, ad hoc Black-Scholes, normal NGARCH, and stochastic volatility VG. The performance of the VG GARCH model versus these four competing models is then demonstrated.

SSRN Electronic Journal, 2000

Derivatives have become widely accepted as tools for hedging and risk-management, and also to some extent for speculation. A more recent trend has been gaining some ground, that of arbitrage in derivatives.

The Journal of Finance, 1999

This paper discusses the pricing of arithmetic Asian options when the underlying stock follows the constant elasticity of variance (CEV) process. We build a binomial tree method to estimate the CEV process and use it to price arithmetic Asian options. We find that the binomial tree method for the lognormal case can effectively solve the computational problems arising from the inherent complexities of arithmetic Asian options when the stock price follows CEV process. We present numerical results to demonstrate the validity and the convergence of the approach for the different parameter values set in CEV process.

Physica A: Statistical Mechanics and its Applications, 2003

In a seminal paper in 1973, Black and Scholes argued how expected distributions of stock prices can be used to price options. Their model assumed a directed random motion for the returns and consequently a lognormal distribution of asset prices after a finite time. We point out two problems with their formulation. First, we show that the option valuation is not uniquely determined; in particular ,strategies based on the delta-hedge and CAPM (the Capital Asset Pricing Model) are shown to provide different valuations of an option. Second, asset returns are known not to be Gaussian distributed. Empirically, distributions of returns are seen to be much better approximated by an exponential distribution. This exponential distribution of asset prices can be used to develop a new pricing model for options that is shown to provide valuations that agree very well with those used by traders. We show how the Fokker-Planck formulation of fluctuations (i.e., the dynamics of the distribution) can be modified to provide an exponential distribution for returns. We also show how a singular volatility can be used to go smoothly from exponential to Gaussian returns and thereby illustrate why exponential returns cannot be reached perturbatively starting from Gaussian ones, and explain how the theory of 'stochastic volatility' can be obtained from our model by making a bad approximation. Finally, we show how to calculate put and call prices for a stretched exponential density.

Journal of CENTRUM Cathedra: The Business and Economics Research Journal, 2012

Derivatives have become widely accepted as tools for hedging and risk-management, as well as speculation to some extent. A more recent trend has been gaining ground, namely, arbitrage in derivatives.

This paper analyses the Black-Scholes' and Heston Option Pricing Model. We discuss the concept of historical volatility in the two Models. We compare the two models for the parameter-'Volatility'. A mathematical tool, UMBRAE (Unscaled Mean Bounded Relative Absolute Error) is used to compare the two models for historical volatility while pricing European call options. Real data from NSE (National Stock Exchange) is considered for three different sectors like-Banking, Automobiles and, Pharmaceuticals for comparison through Moneyness (which is defined as the percentage difference of stock price and strike price) and Time-To-Maturity. Mathematical software-Matlab is used for all mathematical calculations.