Option pricing in a lognormal securities market with discrete trading

This paper derives a call option valuation equation assuming discrete trading in securities markets where the underlying asset and market returns are bivariate lognormally distributed and investors have increasing, concave utility functions exhibiting skewness preference. Since the valuation does not require the continouus time riskfree hedging of Black and Scholes, nor the discrete time riskfree hedging of Cox, Ross and Rubinstein, market effects are introduced into the option valuation relation. The new option valuation seems to correct for the systematic mispricing of well-in and well-out of the money options by the Black and Scholes option pricing formula.     

Faster Implied Volatilities via the Implicit Function Theorem

We present a faster, more accurate technique for estimating implied volatility using the standard partial derivatives of the Black‐Scholes option‐pricing formula. Beside Newton‐Raphson and slower approximation methods, this technique is the first to provide an error tolerance, which is essential for practical application. All existing noniterative approximation methods do not provide error tolerances and have the potential for large errors.     

A revised option pricing formula with the underlying being banned from short selling

An important issue in derivative pricing that hasn't been explored much until very recently is the impact of short selling to the price of an option. This paper extends a recent publication in this area to the case in which a ban of short selling of the underlying alone is somewhat less ‘effective’ than the extreme case discussed by Guo and Zhu [Equal risk pricing under convex trading constraints. J. Econ. Dyn. Control, 2017, 76, 136–151]. The case presented here is closer to reality, in which the effect of a ban on the underlying of an option alone may quite often be ‘diluted’ due to market interactions of the underlying asset with other correlated assets. Under a new assumption that there exists at least a correlated asset in the market, which is allowed to be short sold and thus can be used by traders for hedging purposes even though short selling of the underlying itself is banned, a new closed-form equal-risk pricing formula for European options is successfully derived. The new formula contains two distinguishable advantages; (a) it does not induce any significantly extra burden in terms of numerically computing option values, compared with the effort involved in using the Black–Scholes formula, which is still popularly used in finance industry today; (b) it remains simple and elegant as only one additional parameter beyond the Black–Scholes formula is introduced, to reflect the dilution effect to the ban as a result of market interactions.     

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.     

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.     

Pricing Perpetual Put Options by the Black–Scholes Equation with a Nonlinear Volatility Function

We investigate qualitative and quantitative behavior of a solution of the mathematical model for pricing American style of perpetual put options. We assume the option price is a solution to the stationary generalized Black–Scholes equation in which the volatility function may depend on the second derivative of the option price itself. We prove existence and uniqueness of a solution to the free boundary problem. We derive a single implicit equation for the free boundary position and the closed form formula for the option price. It is a generalization of the well-known explicit closed form solution derived by Merton for the case of a constant volatility. We also present results of numerical computations of the free boundary position, option price and their dependence on model parameters.     

Model uncertainty,recalibration, and the emergence of delta–vega hedging

We study option pricing and hedging with uncertainty about a Black–Scholes reference model which is dynamically recalibrated to the market price of a liquidly traded vanilla option. For dynamic trading in the underlying asset and this vanilla option, delta–vega hedging is asymptotically optimal in the limit for small uncertainty aversion. The corresponding indifference price corrections are determined by the disparity between the vegas, gammas, vannas and volgas of the non-traded and the liquidly traded options.     

Dynamic, nonparametric hedging of European style contingent claims using canonical valuation

The canonical valuation, proposed by Stutzer [1996. Journal of Finance 51, 1633–1652], is a nonparametric option pricing approach for valuing European-style contingent claims. This paper derives risk-neutral dynamic hedge formulae for European call and put options under canonical valuation that obey put–call parity. Further, the paper documents the error-metrics of the canonical hedge ratio and analyzes the effectiveness of discrete dynamic hedging in a stochastic volatility environment. The results suggest that the nonparametric hedge formula generates hedges that are substantially unbiased and is capable of producing hedging outcomes that are superior to those produced by Black and Scholes [1973. Journal of Political Economy 81, 637–654] delta hedging.     

Discrete time hedging with liquidity risk

We study a discrete time hedging and pricing problem in a market with liquidity costs. Using Leland’s discrete time replication scheme [Leland, H.E., 1985. Journal of Finance, 1283–1301], we consider a discrete time version of the Black–Scholes model and a delta hedging strategy. We derive a partial differential equation for the option price in the presence of liquidity costs and develop a modified option hedging strategy which depends on the size of the parameter for liquidity risk. We also discuss an analytic method of solving the pricing equation using a series solution.     

The QLBS Q-Learner goes NuQLear: fitted Q iteration,inverse RL,and option portfolios

The QLBS model is a discrete-time option hedging and pricing model that is based on Dynamic Programming (DP) and Reinforcement Learning (RL). It combines the famous Q-Learning method for RL with the Black–Scholes (–Merton) (BSM) model's idea of reducing the problem of option pricing and hedging to the problem of optimal rebalancing of a dynamic replicating portfolio for the option, which is made of a stock and cash. Here we expand on several NuQLear (Numerical Q-Learning) topics with the QLBS model. First, we investigate the performance of Fitted Q Iteration for an RL (data-driven) solution to the model, and benchmark it versus a DP (model-based) solution, as well as versus the BSM model. Second, we develop an Inverse Reinforcement Learning (IRL) setting for the model, where we only observe prices and actions (re-hedges) taken by a trader, but not rewards. Third, we outline how the QLBS model can be used for pricing portfolios of options, rather than a single option in isolation, thus providing its own, data-driven and model-independent solution to the (in)famous volatility smile problem of the Black–Scholes model.     

GARCH vs. stochastic volatility: Option pricing and risk management

In this paper we compare the out-of-sample performance of two common extensions of the Black–Scholes option pricing model, namely GARCH and stochastic volatility (SV). We calibrate the three models to intraday FTSE 100 option prices and apply two sets of performance criteria, namely out-of-sample valuation errors and Value-at-Risk (VaR) oriented measures. When we analyze the fit to observed prices, GARCH clearly dominates both SV and the benchmark Black–Scholes model. However, the predictions of the market risk from hypothetical derivative positions show sizable errors. The fit to the realized profits and losses is poor and there are no notable differences between the models. Overall, we therefore observe that the more complex option pricing models can improve on the Black–Scholes methodology only for the purpose of pricing, but not for the VaR forecasts.     

Using the short-lived arbitrage model to compute minimum variance hedge ratios: application to indices,stocks and commodities

The short-lived arbitrage model has been shown to significantly improve in-sample option pricing fit relative to the Black–Scholes model. Motivated by this model, we imply both volatility and virtual interest rates to adjust minimum variance hedge ratios. Using several error metrics, we find that the hedging model significantly outperforms the traditional delta hedge and a current benchmark hedge based on the practitioner Black–Scholes model. Our applications include hedges of index options, individual stock options and commodity futures options. Hedges on gold and silver are especially sensitive to virtual interest rates.     

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.     

Hedging jump risk,expected returns and risk premia in jump-diffusion economies

This article presents a pure exchange economy that extends Rubinstein [Bell J. Econ. Manage. Sci., 1976, 7, 407–425] to show how the jump-diffusion option pricing model of Black and Scholes [J. Political Econ., 1973, 81, 637–654] and Merton [J. Financ. Econ., 1976, 4, 125–144] evolves in gamma jumping economies. From empirical analysis and theoretical study, both the aggregate consumption and the stock price are unknown in determining jumping times. By using the pricing kernel, we determine both the aggregate consumption jump time and the stock price jump time from the equilibrium interest rate and CCAPM (Consumption Capital Asset Pricing Model). Our general jump-diffusion option pricing model gives an explicit formula for how the jump process and the jump times alter the pricing. This innovation with predictable jump times enhances our analysis of the expected stock return in equilibrium and of hedging jump risks for jump-diffusion economies.     

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.     

A general closed-form spread option pricing formula

We propose a new accurate method for pricing European spread options by extending the lower bound approximation of Bjerksund and Stensland (2011) beyond the classical Black–Scholes framework. This is possible via a procedure requiring a univariate Fourier inversion. In addition, we are also able to obtain a new tight upper bound. Our method provides also an exact closed form solution via Fourier inversion of the exchange option price, generalizing the Margrabe (1978) formula. The method is applicable to models in which the joint characteristic function of the underlying assets forming the spread is known analytically. We test the performance of these new pricing algorithms performing numerical experiments on different stochastic dynamic models.     

EVIDENCE ON DELTA HEDGING AND IMPLIED VOLATILITIES FOR THE BLACK‐SCHOLES,GAMMA, AND WEIBULL OPTION PRICING MODELS

Modifying the distributional assumptions of the Black‐Scholes model is one way to accommodate the skewness of underlying asset returns. Simple models based on the compensated gamma and Weibull distributions of asset prices are shown to produce some improvements in option pricing. To evaluate these assertions, I construct and compare delta hedges of all S&P 500 options traded on the Chicago Board Options Exchange between September 2001 and October 2003 for the Weibull, Black‐Scholes, and gamma models. I also compare implied volatilities and their smiles (i.e., nonlinearities) among the three models. None of the three models improves over the others as far as delta hedging is concerned. Volatilities implied by all three models exhibit statistically significant smiles.     

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.     

Hedging derivative securities with genetic programming

One of the most recent applications of GP to finance is to use genetic programming to derive option pricing formulas. Earlier studies take the Black–Scholes model as the true model and use the artificial data generated by it to train and to test GP. The aim of this paper is to provide some initial evidence of the empirical relevance of GP to option pricing. By using the real data from S&P 500 index options, we train and test our GP by distinguishing the case in-the-money from the case out-of-the-money. Unlike most empirical studies, we do not evaluate the performance of GP in terms of its pricing accuracy. Instead, the derived GP tree is compared with the Black–Scholes model in its capability to hedge. To do so, a notion of tracking error is taken as the performance measure. Based on the post-sample performance, it is found that in approximately 20% of the 97 test paths GP has a lower tracking error than the Black–Scholes formula. We further compare our result with the ones obtained by radial basis functions and multilayer perceptrons and one-stage GP. Copyright © 1999 John Wiley & Sons, Ltd.     

A test of the beta model on Eurodollar futures options

The paper reports empirical tests of the beta model for pricing fixed-income options. The beta model resembles the Black–Scholes model with the lognormal probability distribution replaced by a beta probability distribution. The test is based on 32 817 daily prices of Eurodollar futures options and concludes that the beta model is more accurate than alternative option pricing models.