Research

27 pages, 660 KiB

Open AccessArticle

Discounted Optimal Stopping of a Brownian Bridge, with Application to American Options under Pinning

by Bernardo D’Auria, Eduardo García-Portugués and Abel Guada

Mathematics 2020, 8(7), 1159; https://doi.org/10.3390/math8071159 - 15 Jul 2020

Cited by 3 | Viewed by 2687

Mathematically, the execution of an American-style financial derivative is commonly reduced to solving an optimal stopping problem. Breaking the general assumption that the knowledge of the holder is restricted to the price history of the underlying asset, we allow for the disclosure of [...] Read more.

Mathematically, the execution of an American-style financial derivative is commonly reduced to solving an optimal stopping problem. Breaking the general assumption that the knowledge of the holder is restricted to the price history of the underlying asset, we allow for the disclosure of future information about the terminal price of the asset by modeling it as a Brownian bridge. This model may be used under special market conditions, in particular we focus on what in the literature is known as the “pinning effect”, that is, when the price of the asset approaches the strike price of a highly-traded option close to its expiration date. Our main mathematical contribution is in characterizing the solution to the optimal stopping problem when the gain function includes the discount factor. We show how to numerically compute the solution and we analyze the effect of the volatility estimation on the strategy by computing the confidence curves around the optimal stopping boundary. Finally, we compare our method with the optimal exercise time based on a geometric Brownian motion by using real data exhibiting pinning. Full article

(This article belongs to the Special Issue Financial Mathematics)

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22 pages, 2100 KiB

Open AccessArticle

Markov-Switching Stochastic Processes in an Active Trading Algorithm in the Main Latin-American Stock Markets

by Oscar V. De la Torre-Torres, Evaristo Galeana-Figueroa and José Álvarez-García

Mathematics 2020, 8(6), 942; https://doi.org/10.3390/math8060942 - 08 Jun 2020

Cited by 3 | Viewed by 2282

Abstract

In the present paper, we review the use of two-state, Generalized Auto Regressive Conditionally Heteroskedastic Markovian stochastic processes (MS-GARCH). These show the quantitative model of an active stock trading algorithm in the three main Latin-American stock markets (Brazil, Chile, and Mexico). By backtesting [...] Read more.

In the present paper, we review the use of two-state, Generalized Auto Regressive Conditionally Heteroskedastic Markovian stochastic processes (MS-GARCH). These show the quantitative model of an active stock trading algorithm in the three main Latin-American stock markets (Brazil, Chile, and Mexico). By backtesting the performance of a U.S. dollar based investor, we found that the use of the Gaussian MS-GARCH leads, in the Brazilian market, to a better performance against a buy and hold strategy (BH). In addition, we found that the use of t-Student MS-ARCH models is preferable in the Chilean market. Lastly, in the Mexican case, we found that is better to use Gaussian time-fixed variance MS models. Their use leads to the best overall performance than the BH portfolio. Our results are of use for practitioners by the fact that MS-GARCH models could be part of quantitative and computer algorithms for active trading in these three stock markets. Full article

(This article belongs to the Special Issue Financial Mathematics)

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32 pages, 3229 KiB

Open AccessEditor’s ChoiceArticle

Global Research Trends in Financial Transactions

by Emilio Abad-Segura and Mariana-Daniela González-Zamar

Mathematics 2020, 8(4), 614; https://doi.org/10.3390/math8040614 - 16 Apr 2020

Cited by 28 | Viewed by 6938

Abstract

Traditionally, financial mathematics has been used to solve financial problems. With globalization, financial transactions require new analysis based on tools of probability, statistics, and economic theory. Global research trends in this topic during the period 1935–2019 have been analyzed. With this objective, a [...] Read more.

Traditionally, financial mathematics has been used to solve financial problems. With globalization, financial transactions require new analysis based on tools of probability, statistics, and economic theory. Global research trends in this topic during the period 1935–2019 have been analyzed. With this objective, a bibliometric methodology of 1486 articles from the Scopus database was applied. The obtained results offer data on the scientific activity of countries, institutions, authors, and institutions that promote this research topic. The results reveal an increasing trend, mainly in the last decade. The main subjects of knowledge are social sciences and economics, econometrics, and finance. The author with the most articles is Khare from the Indian Institute of Management Rohtak. The most prolific affiliation is the British University of Oxford. The country with the most academic publications and international collaborations is the United States. In addition, the most used keywords in articles are “financial management”, “financial transaction tax”, “banking”, “financial service”, “blockchain”, “decision making”, and “financial market”. The increase in publications in recent years at the international level confirms the growing trend in research on financial transactions. Full article

(This article belongs to the Special Issue Financial Mathematics)

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16 pages, 714 KiB

Open AccessArticle

Efficiency of China’s Listed Securities Companies: Estimation through a DEA-Based Method

by Tao Xu, Jianxin You and Yilei Shao

Mathematics 2020, 8(4), 589; https://doi.org/10.3390/math8040589 - 15 Apr 2020

Cited by 4 | Viewed by 2556

Abstract

Accurate assessment of the efficiency of securities companies is of great significance to improve the competitiveness of companies, due to their increasingly important role in supporting economic development. As the main contribution, this paper proposes a novel efficiency estimation framework for securities companies [...] Read more.

Accurate assessment of the efficiency of securities companies is of great significance to improve the competitiveness of companies, due to their increasingly important role in supporting economic development. As the main contribution, this paper proposes a novel efficiency estimation framework for securities companies based on data envelopment analysis (DEA), which takes into account operational risks and technical heterogeneity. First, the risk variable is incorporated in the evaluation system as an undesirable output through the setting of weak disposability. Subsequently, the meta-frontier model is introduced to consider the impact of the technical heterogeneity of different companies to improve the accuracy of the assessment. Furthermore, this article also provides the meta-frontier Malmquist model, which can be utilized to analyze in detail technological progress. Finally, the securities companies listed in the Chinese stock market were selected as samples for empirical analysis. The efficiency evaluation model for securities companies proposed in this paper will provide a reference for related evaluation issues. Full article

(This article belongs to the Special Issue Financial Mathematics)

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7 pages, 252 KiB

Open AccessArticle

Fractional Partial Differential Equations Associated with Lêvy Stable Process

by Reem Abdullah Aljedhi and Adem Kılıçman

Mathematics 2020, 8(4), 508; https://doi.org/10.3390/math8040508 - 02 Apr 2020

Cited by 3 | Viewed by 2042

Abstract

In this study, we first present a time-fractional L

\hat{e}

vy diffusion equation of the exponential option pricing models of European option pricing and the risk-neutral parameter. Then, we modify a particular L

\hat{e}

vy-time fractional diffusion equation of European-style options. [...] Read more.

In this study, we first present a time-fractional L

\hat{e}

vy diffusion equation of the exponential option pricing models of European option pricing and the risk-neutral parameter. Then, we modify a particular L

\hat{e}

vy-time fractional diffusion equation of European-style options. Further, we introduce a more general model based on the L

\hat{e}

vy-time fractional diffusion equation and review some recent findings associated with risk-neutral free European option pricing. Full article

(This article belongs to the Special Issue Financial Mathematics)

13 pages, 536 KiB

Open AccessFeature PaperArticle

Super-Fast Computation for the Three-Asset Equity-Linked Securities Using the Finite Difference Method

by Chaeyoung Lee, Jisang Lyu, Eunchae Park, Wonjin Lee, Sangkwon Kim, Darae Jeong and Junseok Kim

Mathematics 2020, 8(3), 307; https://doi.org/10.3390/math8030307 - 26 Feb 2020

Cited by 1 | Viewed by 2677

Abstract

In this article, we propose a super-fast computational algorithm for three-asset equity-linked securities (ELS) using the finite difference method (FDM). ELS is a very popular investment product in South Korea. There are one-, two-, and three-asset ELS. The three-asset ELS is the most [...] Read more.

In this article, we propose a super-fast computational algorithm for three-asset equity-linked securities (ELS) using the finite difference method (FDM). ELS is a very popular investment product in South Korea. There are one-, two-, and three-asset ELS. The three-asset ELS is the most popular financial product among them. FDM has been used for pricing the one- and two-asset ELS because it is accurate. However, the three-asset ELS is still priced using the Monte Carlo simulation (MCS) due to the curse of dimensionality for FDM. To overcome the limitation of dimension for FDM, we propose a systematic non-uniform grid with an explicit Euler scheme and an optimal implementation of the algorithm. The computational time is less than 6 s. We perform standard ELS option pricing and compare the results from the fast FDM with the ones from MCS. The computational results confirm the superiority and practicality of the proposed algorithm. Full article

(This article belongs to the Special Issue Financial Mathematics)

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14 pages, 490 KiB

Open AccessArticle

A New Approach for the Black–Scholes Model with Linear and Nonlinear Volatilities

by Seda Gulen, Catalin Popescu and Murat Sari

Mathematics 2019, 7(8), 760; https://doi.org/10.3390/math7080760 - 19 Aug 2019

Cited by 16 | Viewed by 6849

Abstract

Since financial engineering problems are of great importance in the academic community, effective methods are still needed to analyze these models. Therefore, this article focuses mainly on capturing the discrete behavior of linear and nonlinear Black–Scholes European option pricing models. To achieve this, [...] Read more.

Since financial engineering problems are of great importance in the academic community, effective methods are still needed to analyze these models. Therefore, this article focuses mainly on capturing the discrete behavior of linear and nonlinear Black–Scholes European option pricing models. To achieve this, this article presents a combined method; a sixth order finite difference (FD6) scheme in space and a third–order strong stability preserving Runge–Kutta (SSPRK3) over time. The computed results are compared with available literature and the exact solution. The computed results revealed that the current method seems to be quite strong both quantitatively and qualitatively with minimal computational effort. Therefore, this method appears to be a very reliable alternative and flexible to implement in solving the problem while preserving the physical properties of such realistic processes. Full article

(This article belongs to the Special Issue Financial Mathematics)

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24 pages, 1014 KiB

Open AccessArticle

Introducing Weights Restrictions in Data Envelopment Analysis Models for Mutual Funds

by Antonella Basso and Stefania Funari

Mathematics 2018, 6(9), 164; https://doi.org/10.3390/math6090164 - 10 Sep 2018

Cited by 5 | Viewed by 4656

Abstract

Data envelopment analysis has been applied in a number of papers to measure the performance of mutual funds, besides a great many applications on the more diverse fields of performance evaluation. The data envelopment analysis models proposed in the mutual funds literature do [...] Read more.

Data envelopment analysis has been applied in a number of papers to measure the performance of mutual funds, besides a great many applications on the more diverse fields of performance evaluation. The data envelopment analysis models proposed in the mutual funds literature do not generally set restrictions on the weights assigned to the input and output variables. In this paper, we study the effects of the introduction of different weight restrictions on the results of the performance evaluation of mutual funds. In addition, we provide a unified matrix representation for three widely used approaches on weight restrictions: virtual weight restrictions with constraints on all decision-making units (DMUs) (on all funds); virtual weight restrictions with constraints only on the target unit; assurance regions. Using the unified matrix representation of the weights constraints, we formulate the data envelopment analysis (DEA ) efficiency model and express the efficient frontier in a unified way for the different weight restrictions considered. We investigate the effects of the different weight restrictions on the performance evaluation by means of an empirical application on a set of European mutual funds. Moreover, we study the behaviour of the fund performance scores as the restrictions on the weights become increasingly strict. Full article

(This article belongs to the Special Issue Financial Mathematics)

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19 pages, 1095 KiB

Open AccessArticle

Advanced Expected Tail Loss Measurement and Quantification for the Moroccan All Shares Index Portfolio

by Marouane Airouss, Mohamed Tahiri, Amale Lahlou and Abdelhak Hassouni

Mathematics 2018, 6(3), 38; https://doi.org/10.3390/math6030038 - 07 Mar 2018

Cited by 4 | Viewed by 3833

Abstract

In this paper, we have analyzed and tested the Expected Tail Loss (ETL) approach for the Value at Risk (VaR) on the Moroccan stock market portfolio. We have compared the results with the general approaches for the standard VaR, which has been the [...] Read more.

In this paper, we have analyzed and tested the Expected Tail Loss (ETL) approach for the Value at Risk (VaR) on the Moroccan stock market portfolio. We have compared the results with the general approaches for the standard VaR, which has been the most suitable method for Moroccan stock investors up to now. These methods calculate the maximum loss that a portfolio is likely to experience over a given time span. Our work advances those modeling methods with supplementation by inputs from the ETL approach for application to the Moroccan stock market portfolio—the Moroccan All Shares Index (MASI). We calculate these indicators using several methods, according to an easy and fast implementation with a high-level probability and with accommodation for extreme risks; this is in order to numerically simulate and study their behavior to better understand investment opportunities and, thus, form a clear view of the Moroccan financial landscape. Full article

(This article belongs to the Special Issue Financial Mathematics)

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242 KiB

Open AccessArticle

On the Inception of Financial Representative Bubbles

by Massimiliano Ferrara, Bruno A. Pansera and Francesco Strati

Mathematics 2017, 5(4), 64; https://doi.org/10.3390/math5040064 - 17 Nov 2017

Cited by 5 | Viewed by 3452

Abstract

In this work, we aim to formalize the inception of representative bubbles giving the condition under which they may arise. We will find that representative bubbles may start at any time, depending on the definition of a behavioral component. This result is at [...] Read more.

In this work, we aim to formalize the inception of representative bubbles giving the condition under which they may arise. We will find that representative bubbles may start at any time, depending on the definition of a behavioral component. This result is at odds with the theory of classic rational bubbles, which are those models that rely on the fulfillment of the transversality condition by which a bubble in a financial asset can arise just at its first trade. This means that a classic rational bubble (differently from our model) cannot follow a cycle since if a bubble exists, it will burst by definition and never arise again. Full article

(This article belongs to the Special Issue Financial Mathematics)

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