Challenges in approximating the Black and Scholes call formula with hyperbolic tangents

In this paper, we introduce the concept of standardized call function and we obtain a new approximating formula for the Black and Scholes call function through the hyperbolic tangent. Differently from other solutions proposed in the literature, this formula is invertible; hence, it is useful for pricing and risk management as well as for extracting the implied volatility from quoted options. The latter is of particular importance since it indicates the risk of the underlying and it is the main component of the option’s price. That is what trading desks focus on. Further we estimate numerically the approximating error of the suggested solution and, by comparing our results in computing the implied volatility with the most common methods available in the literature, we discuss the challenges of this approach

Forecasting interest rates through Vasicek and CIR models: a partitioning approach

The aim of this paper is to propose a new methodology that allows forecasting, through Vasicek and CIR models, of future expected interest rates (for each maturity) based on rolling windows from observed financial market data. The novelty, apart from the use of those models not for pricing but for forecasting the expected rates at a given maturity, consists in an appropriate partitioning of the data sample. This allows capturing all the statistically significant time changes in volatility of interest rates, thus giving an account of jumps in market dynamics. The performance of the new approach is carried out for different term structures and is tested for both models. It is shown how the proposed methodology overcomes both the usual challenges (e.g. simulating regime switching, volatility clustering, skewed tails, etc.) as well as the new ones added by the current market environment characterized by low to negative interest rates.Comment: Research artcile, 23 pages, 8 figures, 7 table

Non-universal behaviour of helical two-dimensional three-component turbulence

The dynamics of two-dimensional three-component (2D3C) flows is relevant to describe the long-time evolution of strongly rotating flows and/or of conducting fluids with a strong mean magnetic field. We show that in the presence of a strong helical forcing, the out-of-plane component ceases to behave as a passive advected quantity and develops a nontrivial dynamics which deeply changes its large-scale properties. We show that a small-scale helicity injection correlates the input on the 2D component with the one on the out-of-plane component. As a result, the third component develops a nontrivial energy transfer. The latter is mediated by homochiral triads, confirming the strong 3D nature of the leading dynamical interactions. In conclusion, we show that the out-of-plane component in a 2D3C flow enjoys strong nonuniversal properties as a function of the degree of mirror symmetry of the small-scale forcing

A Semigroup Approach to generalized Black-Scholes Type Equations in Incomplete Markets.

In this paper we will study an option pricing problem in incomplete markets by an analytic point of view. The incompleteness is generated by the presence of a non-traded asset. The aim of this paper is to use the semigroup theory in order to prove existence and uniqueness of solutions to generalized Black-Scholes type equations that are non-linearly associated with the price of European claims written exclusively on non-traded assets. Then, we derive analytic expressions of the solutions. An approximate representation in terms of a generalized Feynman-Kac type formula is derived in cases where an explicit closed form solution is not available. Numerical examples are also given (see Appendix E) where theoretical approximations and numerical tests reveal a remarkable agreement

Interest rates calibration with a CIR model

The purpose of this paper is to model interest rates from observed financial market data through a new approach to the Cox–Ingersoll–Ross (CIR) model. This model is popular among financial institutions mainly because it is a rather simple (uni-factorial) and better model than the former Vasicek framework. However, there are a number of issues in describing interest rate dynamics within the CIR framework on which focus should be placed. Therefore, a new methodology has been proposed that allows forecasting future expected interest rates from observed financial market data by preserving the structure of the original CIR model, even with negative interest rates. The performance of the new approach, tested on monthly-recorded interest rates data, provides a good fit to current data for different term structures. Design/methodology/approach To ensure a fitting close to current interest rates, the innovative step in the proposed procedure consists in partitioning the entire available market data sample, usually showing a mixture of probability distributions of the same type, in a suitable number of sub-sample having a normal/gamma distribution. An appropriate translation of market interest rates to positive values has been introduced to overcome the issue of negative/near-to-zero values. Then, the CIR model parameters have been calibrated to the shifted market interest rates and simulated the expected values of interest rates by a Monte Carlo discretization scheme. We have analysed the empirical performance of the proposed methodology for two different monthly-recorded EUR data samples in a money market and a long-term data set, respectively. Findings Better results are shown in terms of the root mean square error when a segmentation of the data sample in normally distributed sub-samples is considered. After assessing the accuracy of the proposed procedure, the implemented algorithm was applied to forecast next-month expected interest rates over a historical period of 12 months (fixed window). Through an error analysis, it was observed that our algorithm provides a better fitting of the predicted expected interest rates to market data than the exponentially weighted moving average model. A further confirmation of the efficiency of the proposed algorithm and of the quality of the calibration of the CIR parameters to the observed market interest rates is given by applying the proposed forecasting technique. Originality/value This paper has the objective of modelling interest rates from observed financial market data through a new approach to the CIR model. This model is popular among financial institutions mainly because it is a rather simple (uni-factorial) and better model than the former Vasicek model (Section 2). However, there are a number of issues in describing short-term interest rate dynamics within the CIR framework on which focus should be placed. A new methodology has been proposed that allows us to forecast future expected short-term interest rates from observed financial market data by preserving the structure of the original CIR model. The performance of the new approach, tested on monthly data, provides a good fit for different term structures. It is shown how the proposed methodology overcomes both the usual challenges (e.g. simulating regime switching, clustered volatility and skewed tails), as well as the new ones added by the current market environment (particularly the need to model a downward trend to negative interest rates

What it is like to be \u201cex\u201d? Psycho-discursive analysis of a dangling identity

The goal of this article is to observe how communicative processes intervene in constructing and maintaining given self-narratives. Proceeding from the idea that the Self is, rst of all, narration, we shall scrutinise the topic of identity permanence and change in our linguistic system by exploring the con gurations of reality produced by the use of a speci c morpheme: the \u201cex\u201d. After considering some differences about the intricate relationship between language, mind and reality, we tried to enter the etym and the meanings linked to the use of this linguistic particle. Making use of Wittgenstein\u2019s proposed method called \u201cperspicuous representation\u201d we examined the implicit meanings it assumes in different usage contexts such as daily language and self-narratives provided by prisoners asked about their image of the \u201cex-prisoner\u201d. The analysis revealed a substantial ambiguity between the identity change and its nega- tion since the use of the pre x \u201cex\u201d seemed to con rm and at the same time deny the meaning of the noun accompanying it. The use of the pre x can also prevent the re-positioning of the persons with respect to their past. These ndings may contribute to the understanding of the need for a change in linguistic direction. Our conceptu- alisations of identity cannot be divorced from the linguistic devices used to express them. Major implications for daily language and for the study of clinical and deviant phenomena have been considered

A generalized derivation of the Black-Scholes implied volatility through hyperbolic tangents

This article extends the previous research on the notion of a standardized call function and how to obtain an approximate model of the Black-Scholes formula via the hyperbolic tangent. Although the Black-Scholes approach is outdated and suffers from many limitations, it is still widely used to derive the implied volatility of options. This is particularly important for traders because it represents the risk of the underlying, and is the main factor in the option price. The approximation error of the suggested solution was estimated and the results compared with the most common methods available in the literature. A new formula was provided to correct some cases of underestimation of implied volatility. Graphic evidence, stress tests and Monte Carlo analysis confirm the quality of the results obtained. Finally, further literature is provided as to why implied volatility is used in decision making