The Variance Gamma Process and Option Pricing

A three parameter stochastic process, termed the variance gammaprocess, that generalizes Brownian motion is developed as amodel for the dynamics of log stock prices. Theprocess is obtainedby evaluating Brownian motion with drift at a random time givenby a gamma process. The two additional parameters are the driftof the Brownian motion and the volatility of the time change.These additional parameters provide control over the skewnessand kurtosis of the return distribution. Closed forms are obtainedfor the return density and the prices of European options.Thestatistical and risk neutral densities are estimated for dataon the S&P500 Index and the prices of options on this Index.It is observed that the statistical density is symmetric withsome kurtosis, while the risk neutral density is negativelyskewed with a larger kurtosis. The additional parameters alsocorrect for pricing biases of the Black Scholes model that isa parametric special case of the option pricing model developedhere.     

The Variance Gamma Process and Option Pricing

A three parameter stochastic process, termed the variance gamma process, that generalizes Brownian motion is developed as a model for the dynamics of log stock prices. The process is obtained by evaluating Brownian motion with drift at a random time given by a gamma process. The two additional parameters are the drift of the Brownian motion and the volatility of the time change. These additional parameters provide control over the skewness and kurtosis of the return distribution. Closed forms are obtained for the return density and the prices of European options. The statistical and risk neutral densities are estimated for data on the S & P500 Index and the prices of options on this Index. It is observed that the statistical density is symmetric with some kurtosis, while the risk neutral density is negatively skewed with a larger kurtosis. The additional parameters also correct for pricing biases of the Black Scholes model that is a parametric special case of the option pricing model developed here.     

Estimation and pricing under long-memory stochastic volatility

We treat the problem of option pricing under a stochastic volatility model that exhibits long-range dependence. We model the price process as a Geometric Brownian Motion with volatility evolving as a fractional Ornstein–Uhlenbeck process. We assume that the model has long-memory, thus the memory parameter H in the volatility is greater than 0.5. Although the price process evolves in continuous time, the reality is that observations can only be collected in discrete time. Using historical stock price information we adapt an interacting particle stochastic filtering algorithm to estimate the stochastic volatility empirical distribution. In order to deal with the pricing problem we construct a multinomial recombining tree using sampled values of the volatility from the stochastic volatility empirical measure. Moreover, we describe how to estimate the parameters of our model, including the long-memory parameter of the fractional Brownian motion that drives the volatility process using an implied method. Finally, we compute option prices on the S&P 500 index and we compare our estimated prices with the market option prices.     

Short-time at-the-money skew and rough fractional volatility

The Black–Scholes implied volatility skew at the money of SPX options is known to obey a power law with respect to the time to maturity. We construct a model of the underlying asset price process which is dynamically consistent to the power law. The volatility process of the model is driven by a fractional Brownian motion with Hurst parameter less than half. The fractional Brownian motion is correlated with a Brownian motion which drives the asset price process. We derive an asymptotic expansion of the implied volatility as the time to maturity tends to zero. For this purpose, we introduce a new approach to validate such an expansion, which enables us to treat more general models than in the literature. The local-stochastic volatility model is treated as well under an essentially minimal regularity condition in order to show such a standard model cannot be dynamically consistent to the power law.     

Implied integrated variance and hedging

If the volatility is stochastic, stock price returns and European option prices depend on the time average of the variance, i.e. the integrated variance, not on the path of the volatility. Applying a Bayesian statistical approach, we compute a forward-looking estimate of this variance, an option-implied integrated variance. Simultaneously, we obtain estimates of the correlation coefficient between stock price and volatility shocks, and of the parameters of the volatility process. Due to the convexity of the Black–Scholes formula with respect to the volatility, pricing and hedging with Black–Scholes-type formulas and the implied volatility often lead to inaccuracies if the volatility is stochastic. Theoretically, this problem can be avoided by using Hull–White-type option pricing and hedging formulas and the integrated variance. We use the implied integrated variance and Hull–White-type formulas to hedge European options and certain volatility derivatives.     

Solvable local and stochastic volatility models: supersymmetric methods in option pricing

In this paper we provide an extensive classification of one- and two-dimensional diffusion processes which admit an exact solution to the Kolmogorov (and hence Black–Scholes) equation (in terms of hypergeometric functions). By identifying the one-dimensional solvable processes with the class of integrable superpotentials introduced recently in supersymmetric quantum mechanics, we obtain new analytical solutions. In particular, by applying supersymmetric transformations on a known solvable diffusion process (such as the Natanzon process for which the solution is given by a hypergeometric function), we obtain a hierarchy of new solutions. These solutions are given by a sum of hypergeometric functions, generalizing the results obtained in a paper by Albanese et al. (Albanese, C., Campolieti, G., Carr, P. and Lipton, A., Black–Scholes goes hypergeometric. Risk Mag., 2001, 14, 99–103). For two-dimensional processes, more precisely stochastic volatility models, the classification is achieved for a specific class called gauge-free models including the Heston model, the 3?/?2-model and the geometric Brownian model. We then present a new exact stochastic volatility model belonging to this class.     

A new closed-form formula for pricing European options under a skew Brownian motion

In this paper, we present a new pricing formula based on a modified Black–Scholes (B-S) model with the standard Brownian motion being replaced by a particular process constructed with a special type of skew Brownian motions. Although Corns and Satchell [2007. “Skew Brownian Motion and Pricing European Options.” The European Journal of Finance 13 (6): 523–544] have worked on this model, the results they obtained are incorrect. In this paper, not only do we identify precisely where the errors in Although Corns and Satchell [2007. “Skew Brownian Motion and Pricing European Options”. The European Journal of Finance 13 (6): 523–544] are, we also present a new closed-form pricing formula based on a newly proposed equivalent martingale measure, called ‘endogenous risk neutral measure’, by which only endogenous risks should and can be fully hedged. The newly derived option pricing formula takes the B-S formula as a special case and it does not induce any significant additional burden in terms of numerically computing option values, compared with the effort involved in computing the B-S formula.     

Learning minimum variance discrete hedging directly from the market

Option hedging is a critical risk management problem in finance. In the Black–Scholes model, it has been recognized that computing a hedging position from the sensitivity of the calibrated model option value function is inadequate in minimizing variance of the option hedge risk, as it fails to capture the model parameter dependence on the underlying price (see e.g. Coleman et al., J. Risk, 2001, 5(6), 63–89; Hull and White, J. Bank. Finance, 2017, 82, 180–190). In this paper, we demonstrate that this issue can exist generally when determining hedging position from the sensitivity of the option function, either calibrated from a parametric model from current option prices or estimated nonparametricaly from historical option prices. Consequently, the sensitivity of the estimated model option function typically does not minimize variance of the hedge risk, even instantaneously. We propose a data-driven approach to directly learn a hedging function from the market data by minimizing variance of the local hedge risk. Using the S&P 500 index daily option data for more than a decade ending in August 2015, we show that the proposed method outperforms the parametric minimum variance hedging method proposed in Hull and White [J. Bank. Finance, 2017, 82, 180–190], as well as minimum variance hedging corrective techniques based on stochastic volatility or local volatility models. Furthermore, we show that the proposed approach achieves significant gain over the implied BS delta hedging for weekly and monthly hedging.     

Saddlepoint approximations to option price in a regime-switching model

In this paper we consider the saddlepoint approximation for the valuation of a European-style call option in a Markovian, regime-switching, Black–Scholes–Merton economy, where the price process of an underlying risky asset is assumed to follow a Markov-modulated geometric Brownian motion. The standard option pricing procedure under this model becomes problematic as the occupation time of chains for a given state cannot be evaluated easily. In the case of two-state Markov chains, we present an explicit analytic formula of the cumulant generating function (CGF). When the process has more than two states, an approximate formula of the CGF is provided. We adopt a splitting method to reduce the complexity of computing the matrix exponential function. Then we use these CGFs to develop the use of the saddlepoint approximations. The numerical results show that the saddlepoint approximation is an efficient and reliable approach for option pricing under a multi-state regime-switching model.     

The multivariate Variance Gamma model: basket option pricing and calibration

In this paper, we propose a methodology for pricing basket options in the multivariate Variance Gamma model introduced in Luciano and Schoutens [Quant. Finance 6(5), 385–402]. The stock prices composing the basket are modelled by time-changed geometric Brownian motions with a common Gamma subordinator. Using the additivity property of comonotonic stop-loss premiums together with Gauss-Laguerre polynomials, we express the basket option price as a linear combination of Black & Scholes prices. Furthermore, our new basket option pricing formula enables us to calibrate the multivariate VG model in a fast way. As an illustration, we show that even in the constrained situation where the pairwise correlations between the Brownian motions are assumed to be equal, the multivariate VG model can closely match the observed Dow Jones index options.     

The Instantaneous Volatility and the Implied Volatility Surface for a Generalized Black–Scholes Model

Takaoka (Asia–Pacific Financial Markets 11:431–444, 2004) proposed a generalization of the Black–Scholes stock price model by taking a weighted average of geometric Brownian motions of different variance parameters. The model can be classified as a local volatility model, though its local volatility function is not explicitly given. In the present paper, we prove some properties concerning the instantaneous volatility process, the implied volatility curve, and the local volatility function of the generalized model. Some numerical computations are also carried out to confirm our results.     

Alternative methods to derive option pricing models: review and comparison

The main purposes of this paper are: (1) to review three alternative methods for deriving option pricing models (OPMs), (2) to discuss the relationship between binomial OPM and Black–Scholes OPM, (3) to compare Cox et al. method and Rendleman and Bartter method for deriving Black–Scholes OPM, (4) to discuss lognormal distribution method to derive Black–Scholes OPM, and (5) to show how the Black–Scholes model can be derived by stochastic calculus. This paper shows that the main methodologies used to derive the Black–Scholes model are: binomial distribution, lognormal distribution, and differential and integral calculus. If we assume risk neutrality, then we don’t need stochastic calculus to derive the Black–Scholes model. However, the stochastic calculus approach for deriving the Black–Scholes model is still presented in Sect. 6. In sum, this paper can help statisticians and mathematicians understand how alternative methods can be used to derive the Black–Scholes option model.     

A new closed-form solution as an extension of the Black–Scholes formula allowing smile curve plotting

In this paper, as a generalization of the Black–Scholes (BS) model, we elaborate a new closed-form solution for a uni-dimensional European option pricing model called the J-model. This closed-form solution is based on a new stochastic process, called the J-process, which is an extension of the Wiener process satisfying the martingale property. The J-process is based on a new statistical law called the J-law, which is an extension of the normal law. The J-law relies on four parameters in its general form. It has interesting asymmetry and tail properties, allowing it to fit the reality of financial markets with good accuracy, which is not the case for the normal law. Despite the use of one state variable, we find results similar to those of Heston dealing with the bi-dimensional stochastic volatility problem for pricing European calls. Inverting the BS formula, we plot the smile curve related to this closed-form solution. The J-model can also serve to determine the implied volatility by inverting the J-formula and can be used to price other kinds of options such as American options.     

On the qualitative effect of volatility and duration on prices of Asian options

We show that under the Black–Scholes assumption the price of an arithmetic average Asian call option with fixed strike increases with the level of volatility. This statement is not trivial to prove and for other models in general wrong. In fact we demonstrate that in a simple binomial model no such relationship holds. Under the Black–Scholes assumption however, we give a proof based on the maximum principle for parabolic partial differential equations. Furthermore we show that an increase in the length of duration over which the average is sampled also increases the price of an arithmetic average Asian call option, if the discounting effect is taken out. To show this, we use the result on volatility and the fact that a reparametrization in time corresponds to a change in volatility in the Black–Scholes model. Both results are extremely important for the risk management and risk assessment of portfolios that include Asian options.     

An Empirical Comparison of Two Stochastic Volatility Models using Indian Market Data

We conduct an empirical comparison of hedging strategies for two different stochastic volatility models proposed in the literature. One is an asymptotic expansion approach and the other is the risk-minimizing approach applied to a Markov-switched geometric Brownian motion. We also compare these with the Black–Scholes delta hedging strategies using historical and implied volatilities. The derivatives we consider are European call options on the NIFTY index of the Indian National Stock Exchange. We compare a few cases with profit and loss data from a trading desk. We find that for the cases that we analyzed, by far the better results are obtained for the Markov-switched geometric Brownian motion.     

When are Options Overpriced? The Black--Scholes Model and Alternative Characterisations of the Pricing Kernel

An important determinant of option prices is the elasticityof the pricing kernel used to price all claims in the economy.In this paper, we first show that for a given forward priceof the underlying asset, option prices are higher when the elasticityof the pricing kernel is declining than when it is constant.We then investigate the implications of the elasticity of thepricing kernel for the stochastic process followed by the underlyingasset. Given that the underlying information process followsa geometric Brownian motion, we demonstrate that constant elasticityof the pricing kernel is equivalent to a Brownian motion forthe forward price of the underlying asset, so that the Black–Scholesformula correctly prices options on the asset. In contrast,declining elasticity implies that the forward price processis no longer a Brownian motion: it has higher volatility andexhibits autocorrelation. In this case, the Black–Scholesformula underprices all options.     

Calibration of a nonlinear feedback option pricing model

We consider the option pricing model proposed by Mancino and Ogawa, where the implementation of dynamic hedging strategies has a feedback impact on the price process of the underlying asset. We present numerical results showing that the smile and skewness patterns of implied volatility can actually be reproduced as a consequence of dynamical hedging. The simulations are performed using a suitable semi-implicit finite difference method. Moreover, we perform a calibration of the nonlinear model to market data and we compare it with more popular models, such as the Black–Scholes formula, the Jump-Diffusion model and Heston's model. In judging the alternative models, we consider the following issues: (i) the consistency of the implied structural parameters with the times-series data; (ii) out-of-sample pricing; and (iii) parameter uniformity across different moneyness and maturity classes. Overall, nonlinear feedback due to hedging strategies can, at least in part, contribute to the explanation from a theoretical and quantitative point of view of the strong pricing biases of the Black–Scholes formula, although stochastic volatility effects are more important in this regard.     

Stochastic volatility options pricing with wavelets and artificial neural networks

Abstract

The paper describes an alternative options pricing method which uses a binomial tree linked to an innovative stochastic volatility model. The volatility model is based on wavelets and artificial neural networks. Wavelets provide a convenient signal/noise decomposition of the volatility in the nonlinear feature space. Neural networks are used to infer future volatility from the wavelets feature space in an iterative manner. The bootstrap method provides the 95% confidence intervals for the options prices. Market options prices as quoted on the Chicago Board Options Exchange are used for performance comparison between the Black‐Scholes model and a new options pricing scheme. The proposed dynamic volatility model produces as good as and often better options prices than the conventional Black‐Scholes formulae.     

Option pricing with Weyl–Titchmarsh theory

In the Black–Merton–Scholes framework, the price of an underlying asset is assumed to follow a pure diffusion process. No-arbitrage theory shows that the price of an option contract written on the asset can be determined by solving a linear diffusion equation with variable coefficients. Applying the separating variable method, the problem of option pricing under state-dependent deterministic volatility can be transformed into a Schrödinger spectral problem, which has been well studied in quantum mechanics. With Weyl–Titchmarsh theory, we are able to determine the boundary condition and the nature of the eigenvalues and eigenfunctions. The solution can be written analytically in a Stieltjes integral. A few case studies demonstrate that a new analytical option pricing formula can be produced with our method.