Superconvergence of the finite element solutions of the Black–Scholes equation

We investigate the performances of the finite element method in solving the Black–Scholes option pricing model. Such an analysis highlights that, if the finite element method is carried out properly, then the solutions obtained are superconvergent at the boundaries of the finite elements. In particular, this is shown to happen for quadratic and cubic finite elements, and for the pricing of European vanilla and barrier options. To the best of our knowledge, lattice-based approximations of the Black–Scholes model that exhibit nodal superconvergence have never been observed so far, and are somehow unexpected, as the solutions of the associated partial differential problems have various kinds of irregularities.     

Statistical arbitrage in the Black–Scholes framework

In this study, we prove the existence of statistical arbitrage opportunities in the Black–Scholes framework by considering trading strategies that consist of borrowing at the risk-free rate and taking a long position in the stock until it hits a deterministic barrier level. We derive analytical formulas for the expected value, variance and probability of loss for the discounted cumulative trading profits. The statistical arbitrage condition is derived in the Black–Scholes framework, which imposes a constraint on the Sharpe ratio of the stock. Furthermore, we verify our theoretical results via extensive Monte Carlo simulations.     

Comparison of Black–Scholes Formula with Fractional Black–Scholes Formula in the Foreign Exchange Option Market with Changing Volatility

In this paper, the authors discuss the fractional option pricing with Black–Scholes formula, deduce the Fractional Black–Scholes formula, show the empirical results by using China merchants bank foreign exchange call option price, and find when the volatility is smaller, the asymptotic mean squared error of Fractional Black–Scholes is bigger than the Traditional Black–Scholes’, while the volatility is bigger—the market mechanism has a full play, the result is reverse. Namely when the market mechanism is given a full scope, the estimating effect of Fractional Black–Scholes is better than Traditional Black–Scholes’.     

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.     

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.     

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.     

Revisiting the ICAPM under the distortion of risk–return tradeoff in short-horizon stock returns

We suggest that the distortion of the positive risk–return relation in the ICAPM is a consequence of trading by informed investors to exploit mispricing. We hypothesize and demonstrate that a non-positive (strongly positive) risk–return relation following positive (negative) market returns is attributed to short-selling (purchasing) of overpriced (underpriced) stocks along with optimistic (pessimistic) expectations conditional on good (bad) market news. We verify this asymmetry in the risk–return relation through the indirect risk–return relation conditional on good (bad) market news. We also find that the attenuation (reinforcement) of the positive risk–return relation is more profound in high- (low-) sentiment periods.     

New development: Corporatization of local authorities in England in the wake of austerity 2010–2016

A key institutional driver of current reforms within English local government is ‘alternative service delivery’. Our review of councils’ annual accounts between 2010/11 and 2016/17 suggests ‘corporatization’—the creation of local authority companies—is a growing phenomenon across the whole of English local government. This represents such a significant and far-reaching development in the governance, performance and efficiency of local public services that it constitutes a major field-level change at the interstices of the institutions of state, market, corporation and community. In this article, the authors briefly sketch ways corporatization could be regarded as a field-level change, before presenting findings and reflecting on their implications.     

Market uncertainty and disclosure of internal control deficiencies under the Sarbanes–Oxley Act

This study examines cross-sectional differences in stock market reactions to the disclosure of internal control deficiencies under Section 302 of the Sarbanes–Oxley Act. We hypothesize that the market punishment for internal control problems will be less severe for internal control disclosure that helps reduce market uncertainty around the disclosure. We also predict that such a relation is dependent on the types of disclosure and the market’s prior knowledge of the credibility of firms’ financial reporting. Consistent with our hypothesis, we find that when firms disclose their internal control deficiencies, their abnormal stock returns are negatively associated with changes in market uncertainty (e.g., changes in the standard deviations of daily stock returns) around the disclosure. We also find that the impact of the uncertainty reduction is greater for voluntary disclosures of non-material weakness, especially those made in the context of previous suspicious events. The negative impact of changes in market uncertainty on the abnormal stock returns remains intact even after controlling for possible simultaneity. An analysis using financial analysts’ earnings forecasts dispersion as an alternative proxy for uncertainty confirms the results.     

New Development: Scotland's Concordat—An assessment of the new financial framework in central–local relations

The Scottish Concordat was presented as a new model of central–local relations, based on mutual respect and joint accountability. In practice, it delivered an uncosted package with inadequate measures of outcomes. The result is service reductions, job losses, and blurred accountability.     

Optimal time to sell a stock in the Black–Scholes model: comment on ‘Thou shalt buy and hold’, by A. Shiryaev,Z. Xu and X.Y. Zhou

We reconsider the problem of the optimal time to sell a stock studied by Shiryaev et al. (2008 Shiryaev, A, Xu, Z and Zhou, XY. 2008. Thou shalt buy and hold. Quant. Finan., 8: 765–776. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) (following in this issue of Quantitative Finance) using path integral methods. These methods allow us to confirm the results obtained by these authors and extend them to the entire parameter region. We also obtain the full distribution of the time t_m at which the maximum of the price is reached for arbitrary values of the drift.