Finite Difference Schemes for Option Pricing under Stochastic Volatility and Lévy Processes: Numerical Analysis and Computing

[EN] In the stock markets, the process of estimating a fair price for a stock, option or commodity is consider the corner stone for this trade. There are several attempts to obtain a suitable mathematical model in order to enhance the estimation process for evaluating the options for short or long periods. The Black-Scholes partial differential equation (PDE) and its analytical solution, 1973, are considered a breakthrough in the mathematical modeling for the stock markets. Because of the ideal assumptions of Black-Scholes several alternatives have been developed to adequate the models to the real markets. Two strategies have been done to capture these behaviors; the first modification is to add jumps into the asset following Lévy processes, leading to a partial integro-differential equation (PIDE); the second is to allow the volatility to evolve stochastically leading to a PDE with two spatial variables. Here in this work, we solve numerically PIDEs for a wide class of Lévy processes using finite difference schemes for European options and also, the associated linear complementarity problem (LCP) for American option. Moreover, the models for options under stochastic volatility incorporated with jump-diffusion are considered. Numerical analysis for the proposed schemes is studied since it is the efficient and practical way to guarantee the convergence and accuracy of numerical solutions. In fact, without numerical analysis, careless computations may waste good mathematical models. This thesis consists of four chapters; the first chapter is an introduction containing historically review for stochastic processes, Black-Scholes equation and preliminaries on numerical analysis. Chapter two is devoted to solve the PIDE for European option under CGMY process. The PIDE for this model is solved numerically using two distinct discretization approximations; the first approximation guarantees unconditionally consistency while the second approximation provides unconditional positivity and stability. In the first approximation, the differential part is approximated using the explicit scheme and the integral part is approximated using the trapezoidal rule. In the second approximation, the differential part is approximated using the Patankar-scheme and the integral part is approximated using the four-point open type formula. Chapter three provides a unified treatment for European and American options under a wide class of Lévy processes as CGMY, Meixner and Generalized Hyperbolic. First, the reaction and convection terms of the differential part of the PIDE are removed using appropriate mathematical transformation. The differential part for European case is explicitly discretized , while the integral part is approximated using Laguerre-Gauss quadrature formula. Numerical properties such as positivity, stability and consistency for this scheme are studied. For the American case, the differential part of the LCP is discretized using a three-time level approximation with the same integration technique. Next, the Projected successive over relaxation and multigrid techniques have been implemented to obtain the numerical solution. Several numerical examples are given including discussion of the errors and computational cost. Finally in Chapter four, the PIDE for European option under Bates model is considered. Bates model combines both stochastic volatility and jump diffusion approaches resulting in a PIDE with a mixed derivative term. Since the presence of cross derivative terms involves the existence of negative coefficient terms in the numerical scheme deteriorating the quality of the numerical solution, the mixed derivative is eliminated using suitable mathematical transformation. The new PIDE is solved numerically and the numerical analysis is provided. Moreover, the LCP for American option under Bates model is studied.[ES] El proceso de estimación del precio de una acción, opción u otro derivado en los mercados de valores es objeto clave de estudio de las matemáticas financieras. Se pueden encontrar diversas técnicas para obtener un modelo matemático adecuado con el fin de mejorar el proceso de valoración de las opciones para periodos cortos o largos. Históricamente, la ecuación de Black-Scholes (1973) fue un gran avance en la elaboración de modelos matemáticos para los mercados de valores. Es un modelo práctico para estimar el valor razonable de una opción. Sobre unos supuestos determinados, F. Black y M. Scholes obtuvieron una ecuación diferencial parcial lineal y su solución analítica. Desde entonces se han desarrollado modelos más complejos para adecuarse a la realidad de los mercados. Un tipo son los modelos con volatilidad estocástica que vienen descritos por una ecuación en derivadas parciales con dos variables espaciales. Otro enfoque consiste en añadir saltos en el precio del subyacente por medio de modelos de Lévy lo que lleva a resolver una ecuación integro-diferencial parcial (EIDP). En esta memoria se aborda la resolución numérica de una amplia clase de modelos con procesos de Lévy. Se desarrollan esquemas en diferencias finitas para opciones europeas y también para opciones americanas con su problema de complementariedad lineal (PCL) asociado. Además se tratan modelos con volatilidad estocástica incorporando difusión con saltos. Se plantea el análisis numérico ya que es el camino eficiente y práctico para garantizar la convergencia y precisión de las soluciones numéricas. De hecho, la ausencia de análisis numérico debilita un buen modelo matemático. Esta memoria está organizada en cuatro capítulos. El primero es una introducción con un breve repaso de los procesos estocásticos, el modelo de Black-Scholes así como nociones preliminares de análisis numérico. En el segundo capítulo se trata la EIDP para las opciones europeas según el modelo CGMY. Se proponen dos esquemas en diferencias finitas; el primero garantiza consistencia incondicional de la solución mientras que el segundo proporciona estabilidad y positividad incondicionales. Con el primer enfoque, la parte diferencial se discretiza por medio de un esquema explícito y para la parte integral se usa la regla del trapecio. En la segunda aproximación, para la parte diferencial se usa un esquema tipo Patankar y la parte integral se aproxima por medio de la fórmula de tipo abierto con cuatro puntos. En el capítulo tercero se propone un tratamiento unificado para una amplia clase de modelos de opciones en procesos de Lévy como CGMY, Meixner e hiperbólico generalizado. Se eliminan los términos de reacción y convección por medio de un apropiado cambio de variables. Después la parte diferencial se aproxima por un esquema explícito mientras que para la parte integral se usa la fórmula de cuadratura de Laguerre-Gauss. Se analizan positividad, estabilidad y consistencia. Para las opciones americanas, la parte diferencial del LCP se discretiza con tres niveles temporales mediante cuadratura de Laguerre-Gauss para la integración numérica. Finalmente se implementan métodos iterativos de proyección y relajación sucesiva y la técnica de multimalla. Se muestran varios ejemplos incluyendo estudio de errores y coste computacional. El capítulo 4 está dedicado al modelo de Bates que combina los enfoques de volatilidad estocástica y de difusión con saltos derivando en una EIDP con un término con derivadas cruzadas. Ya que la discretización de una derivada cruzada comporta la existencia de coeficientes negativos en el esquema que deterioran la calidad de la solución numérica, se propone un cambio de variables que elimina dicha derivada cruzada. La EIDP transformada se resuelve numéricamente y se muestra el análisis numérico. Por otra parte se estudia el LCP para opciones americanas con el modelo de Bates.[CA] El procés d'estimació del preu d'una acció, opció o un altre derivat en els mercats de valors és objecte clau d'estudi de les matemàtiques financeres . Es poden trobar diverses tècniques per a obtindre un model matemàtic adequat a fi de millorar el procés de valoració de les opcions per a períodes curts o llargs. Històricament, l'equació Black-Scholes (1973) va ser un gran avanç en l'elaboració de models matemàtics per als mercats de valors. És un model matemàtic pràctic per a estimar un valor raonable per a una opció. Sobre uns suposats F. Black i M. Scholes van obtindre una equació diferencial parcial lineal amb solució analítica. Des de llavors s'han desenrotllat models més complexos per a adequar-se a la realitat dels mercats. Un tipus és els models amb volatilitat estocástica que ve descrits per una equació en derivades parcials amb dos variables espacials. Un altre enfocament consistix a afegir bots en el preu del subjacent per mitjà de models de Lévy el que porta a resoldre una equació integre-diferencial parcial (EIDP) . En esta memòria s'aborda la resolució numèrica d'una àmplia classe de models baix processos de Lévy. Es desenrotllen esquemes en diferències finites per a opcions europees i també per a opcions americanes amb el seu problema de complementarietat lineal (PCL) associat. A més es tracten models amb volatilitat estocástica incorporant difusió amb bots. Es planteja l'anàlisi numèrica ja que és el camí eficient i pràctic per a garantir la convergència i precisió de les solucions numèriques. De fet, l'absència d'anàlisi numèrica debilita un bon model matemàtic. Esta memòria està organitzada en quatre capítols. El primer és una introducció amb un breu repàs dels processos estocásticos, el model de Black-Scholes així com nocions preliminars d'anàlisi numèrica. En el segon capítol es tracta l'EIDP per a les opcions europees segons el model CGMY. Es proposen dos esquemes en diferències finites; el primer garantix consistència incondicional de la solució mentres que el segon proporciona estabilitat i positivitat incondicionals. Amb el primer enfocament, la part diferencial es discretiza per mitjà d'un esquema explícit i per a la part integral s'empra la regla del trapezi. En la segona aproximació, per a la part diferencial s'usa l'esquema tipus Patankar i la part integral s'aproxima per mitjà de la fórmula de tipus obert amb quatre punts. En el capítol tercer es proposa un tractament unificat per a una àmplia classe de models d'opcions en processos de Lévy com ara CGMY, Meixner i hiperbòlic generalitzat. S'eliminen els termes de reacció i convecció per mitjà d'un apropiat canvi de variables. Després la part diferencial s'aproxima per un esquema explícit mentres que per a la part integral s'usa la fórmula de quadratura de Laguerre-Gauss. S'analitzen positivitat, estabilitat i consistència. Per a les opcions americanes, la part diferencial del LCP es discretiza amb tres nivells temporals amb quadratura de Laguerre-Gauss per a la integració numèrica. Finalment s'implementen mètodes iteratius de projecció i relaxació successiva i la tècnica de multimalla. Es mostren diversos exemples incloent estudi d'errors i cost computacional. El capítol 4 està dedicat al model de Bates que combina els enfocaments de volatilitat estocástica i de difusió amb bots derivant en una EIDP amb un terme amb derivades croades. Ja que la discretización d'una derivada croada comporta l'existència de coeficients negatius en l'esquema que deterioren la qualitat de la solució numèrica, es proposa un canvi de variables que elimina dita derivada croada. La EIDP transformada es resol numèricament i es mostra l'anàlisi numèrica. D'altra banda s'estudia el LCP per a opcions americanes en el model de Bates.El-Fakharany, MMR. (2015). Finite Difference Schemes for Option Pricing under Stochastic Volatility and Lévy Processes: Numerical Analysis and Computing [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/53917TESI

Target Detection in a Known Number of Intervals Based on Cooperative Search Technique

Finding hidden/lost targets in a broad region costs strenuous effort and takes a long time. From a practical view, it is convenient to analyze the available data to exclude some parts of the search region. This paper discusses the coordinated search technique of a one-dimensional problem with a search region consisting of several mutual intervals. In other words, if the lost target has a probability of existing in a bounded interval, then the successive bounded interval has a far-fetched probability. Moreover, the search domain is swept by two searchers moving in opposite directions, leading to three categories of target distribution truncations: commensurate, uneven, and symmetric. The truncated probability distributions are defined and applied based on the proposed classification to calculate the expected value of the elapsed time to find the hidden object. Furthermore, the optimization of the associated expected time values of various cases is investigated based on Newton's method. Several examples are presented to discuss the behavior of various distributions under each case of truncation. Also, the associated expected time values are calculated as their minimum values.Comment: 32 pages, 11 figure

Solving partial integro-differential option pricing problems for a wide class of infinite activity Lévy processes

[EN] In this paper, numerical analysis of finite difference schemes for partial integro-differential models related to European and American option pricing problems under a wide class of Lévy models is studied. Apart from computational and accuracy issues, qualitative properties such as positivity are treated. Consistency of the proposed numerical scheme and stability in the von Neumann sense are included. Gauss Laguerre quadrature formula is used for the discretization of the integral part. Numerical examples illustrating the potential advantages of the presented results are included.This work has been partially supported by the European Union in the FP7-PEOPLE-2012-ITN program under Grant Agreement Number 304617 (FP7 Marie Curie Action, Project Multi-ITN STRIKE-Novel Methods in Computational Finance) and the Ministerio de Economia y Competitividad Spanish grant MTM2013-41765-P.El-Fakharany, M.; Company Rossi, R.; Jódar Sánchez, LA. (2016). Solving partial integro-differential option pricing problems for a wide class of infinite activity Lévy processes. Journal of Computational and Applied Mathematics. 296:739-752. https://doi.org/10.1016/j.cam.2015.10.027S73975229

Positive Solutions of European Option Pricing with CGMYProcess Models Using Double Discretization Difference Schemes

[EN] This paper deals with the numerical analysis of PIDE option pricing models with CGMY process using double discretization schemes. This approach assumes weaker hypotheses of the solution on the numerical boundary domain than other relevant papers. Positivity, stability, and consistency are studied. An explicit scheme is proposed after a suitable change of variables. Advantages of the proposed schemes are illustrated with appropriate examples.This work has been partially supported by the European Union in the FP7-PEOPLE-2012-ITN program under Grant Agreement no. 304617 (FP7 Marie Curie Action, Project Multi-ITN STRIKE-Novel Methods in Computational Finance) and by the Spanish M.E.Y.C. Grant DPI2010-20891-C02-01.Company Rossi, R.; Jódar Sánchez, LA.; El-Fakharany, M. (2013). Positive Solutions of European Option Pricing with CGMYProcess Models Using Double Discretization Difference Schemes. Abstract and Applied Analysis. 2013:1-12. https://doi.org/10.1155/2013/517480S1122013Kou, S. G. (2002). A Jump-Diffusion Model for Option Pricing. Management Science, 48(8), 1086-1101. doi:10.1287/mnsc.48.8.1086.166Merton, R. C. (1976). Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics, 3(1-2), 125-144. doi:10.1016/0304-405x(76)90022-2Barndorff-Nielsen, O. E. (1997). Processes of normal inverse Gaussian type. Finance and Stochastics, 2(1), 41-68. doi:10.1007/s007800050032Koponen, I. (1995). Analytic approach to the problem of convergence of truncated Lévy flights towards the Gaussian stochastic process. Physical Review E, 52(1), 1197-1199. doi:10.1103/physreve.52.1197Madan, D. B., & Milne, F. (1991). Option Pricing With V. G. Martingale Components. Mathematical Finance, 1(4), 39-55. doi:10.1111/j.1467-9965.1991.tb00018.xCarr, P., Geman, H., Madan, D. B., & Yor, M. (2002). The Fine Structure of Asset Returns: An Empirical Investigation. The Journal of Business, 75(2), 305-333. doi:10.1086/338705BOYARCHENKO, S. I., & LEVENDORSKIǏ, S. Z. (2000). OPTION PRICING FOR TRUNCATED LÉVY PROCESSES. International Journal of Theoretical and Applied Finance, 03(03), 549-552. doi:10.1142/s0219024900000541Matache *, A.-M., Nitsche, P.-A., & Schwab, C. (2005). Wavelet Galerkin pricing of American options on Lévy driven assets. Quantitative Finance, 5(4), 403-424. doi:10.1080/14697680500244478Poirot, J., & Tankov, P. (2007). Monte Carlo Option Pricing for Tempered Stable (CGMY) Processes. Asia-Pacific Financial Markets, 13(4), 327-344. doi:10.1007/s10690-007-9048-7Fang, F., & Oosterlee, C. W. (2009). A Novel Pricing Method for European Options Based on Fourier-Cosine Series Expansions. SIAM Journal on Scientific Computing, 31(2), 826-848. doi:10.1137/080718061Benhamou, E., Gobet, E., & Miri, M. (2009). Smart expansion and fast calibration for jump diffusions. Finance and Stochastics, 13(4), 563-589. doi:10.1007/s00780-009-0102-3Cont, R., & Voltchkova, E. (2005). A Finite Difference Scheme for Option Pricing in Jump Diffusion and Exponential Lévy Models. SIAM Journal on Numerical Analysis, 43(4), 1596-1626. doi:10.1137/s0036142903436186Wang, I., Wan, J., & Forsyth, P. (2007). Robust numerical valuation of European and American options under the CGMY process. The Journal of Computational Finance, 10(4), 31-69. doi:10.21314/jcf.2007.169Casabán, M.-C., Company, R., Jódar, L., & Romero, J.-V. (2012). Double Discretization Difference Schemes for Partial Integrodifferential Option Pricing Jump Diffusion Models. Abstract and Applied Analysis, 2012, 1-20. doi:10.1155/2012/120358Andersen, L., & Andreasen, J. (2000). Review of Derivatives Research, 4(3), 231-262. doi:10.1023/a:1011354913068Almendral, A., & Oosterlee, C. W. (2007). Accurate Evaluation of European and American Options Under the CGMY Process. SIAM Journal on Scientific Computing, 29(1), 93-117. doi:10.1137/050637613Sachs, E. W., & Strauss, A. K. (2008). Efficient solution of a partial integro-differential equation in finance. Applied Numerical Mathematics, 58(11), 1687-1703. doi:10.1016/j.apnum.2007.11.002Salmi, S., & Toivanen, J. (2011). An iterative method for pricing American options under jump-diffusion models. Applied Numerical Mathematics, 61(7), 821-831. doi:10.1016/j.apnum.2011.02.002Toivanen, J. (2008). Numerical Valuation of European and American Options under Kou’s Jump-Diffusion Model. SIAM Journal on Scientific Computing, 30(4), 1949-1970. doi:10.1137/060674697Almendral, A., & Oosterlee, C. W. (2005). Numerical valuation of options with jumps in the underlying. Applied Numerical Mathematics, 53(1), 1-18. doi:10.1016/j.apnum.2004.08.037Lee, J., & Lee, Y. (2013). Tridiagonal implicit method to evaluate European and American options under infinite activity Lévy models. Journal of Computational and Applied Mathematics, 237(1), 234-243. doi:10.1016/j.cam.2012.07.028Madan, D. B., & Seneta, E. (1990). The Variance Gamma (V.G.) Model for Share Market Returns. The Journal of Business, 63(4), 511. doi:10.1086/296519Milgram, M. S. (1985). The generalized integro-exponential function. Mathematics of Computation, 44(170), 443-443. doi:10.1090/s0025-5718-1985-0777276-4Matache, A.-M., von Petersdorff, T., & Schwab, C. (2004). Fast deterministic pricing of options on Lévy driven assets. ESAIM: Mathematical Modelling and Numerical Analysis, 38(1), 37-71. doi:10.1051/m2an:2004003Kangro, R., & Nicolaides, R. (2000). Far Field Boundary Conditions for Black--Scholes Equations. SIAM Journal on Numerical Analysis, 38(4), 1357-1368. doi:10.1137/s0036142999355921Madan, D. B., Carr, P. P., & Chang, E. C. (1998). The Variance Gamma Process and Option Pricing. Review of Finance, 2(1), 79-105. doi:10.1023/a:100970343153

Unconditional Positive Stable Numerical Solution of Partial Integrodifferential Option Pricing Problems

This paper is concerned with the numerical solution of partial integrodifferential equation for option pricing models under a tempered stable process known as CGMY model. A double discretization finite difference scheme is used for the treatment of the unbounded nonlocal integral term. We also introduce in the scheme the Patankar-trick to guarantee unconditional nonnegative numerical solutions. Integration formula of open type is used in order to improve the accuracy of the approximation of the integral part. Stability and consistency are also studied. Illustrative examples are included.This work has been partially supported by the European Union in the FP7-PEOPLE-2012-ITN Program under Grant Agreement no. 304617 (FP7 Marie Curie Action, Project Multi-ITN STRIKE-Novel Methods in Computational Finance).Fakharany, MMRE.; Company Rossi, R.; Jódar Sánchez, LA. (2015). Unconditional Positive Stable Numerical Solution of Partial Integrodifferential Option Pricing Problems. Journal of Applied Mathematics. 2015:1-10. https://doi.org/10.1155/2015/960728S110201

Removing the Correlation Term in Option Pricing HestonModel: Numerical Analysis and Computing

[EN] This paper deals with the numerical solution of option pricing stochastic volatility model described by a time-dependent, twodimensional convection-diffusion reaction equation. Firstly, the mixed spatial derivative of the partial differential equation (PDE) is removed bymeans of the classical technique for reduction of second-order linear partial differential equations to canonical form. An explicit difference scheme with positive coefficients and only five-point computational stencil is constructed. The boundary conditions are adapted to the boundaries of the rhomboid transformed numerical domain. Consistency of the scheme with the PDE is shown and stepsize discretization conditions in order to guarantee stability are established. Illustrative numerical examples are included.This work has been partially supported by the European Union in the FP7-PEOPLE-2012-ITN Program under Grant Agreement no. 304617 (FP7 Marie Curie Action, Project Multi-ITN STRIKE-Novel Methods in Computational Finance) and by the Spanish MEYC Grant DPI2010-20891-C02-01.Company Rossi, R.; Jódar Sánchez, LA.; El-Fakharany, M.; Casabán Bartual, MC. (2013). Removing the Correlation Term in Option Pricing HestonModel: Numerical Analysis and Computing. Abstract and Applied Analysis. 2013:1-11. https://doi.org/10.1155/2013/246724S1112013HULL, J., & WHITE, A. (1987). The Pricing of Options on Assets with Stochastic Volatilities. The Journal of Finance, 42(2), 281-300. doi:10.1111/j.1540-6261.1987.tb02568.xHeston, S. L. (1993). A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options. Review of Financial Studies, 6(2), 327-343. doi:10.1093/rfs/6.2.327Pascucci, A. (2011). PDE and Martingale Methods in Option Pricing. Bocconi & Springer Series. doi:10.1007/978-88-470-1781-8Benhamou, E., Gobet, E., & Miri, M. (2010). Time Dependent Heston Model. SIAM Journal on Financial Mathematics, 1(1), 289-325. doi:10.1137/090753814Hilber, N., Matache, A.-M., & Schwab, C. (2005). Sparse wavelet methods for option pricing under stochastic volatility. The Journal of Computational Finance, 8(4), 1-42. doi:10.21314/jcf.2005.131Zhu, W., & Kopriva, D. A. (2009). A Spectral Element Approximation to Price European Options with One Asset and Stochastic Volatility. Journal of Scientific Computing, 42(3), 426-446. doi:10.1007/s10915-009-9333-xClarke, N., & Parrott, K. (1999). Multigrid for American option pricing with stochastic volatility. Applied Mathematical Finance, 6(3), 177-195. doi:10.1080/135048699334528Düring, B., & Fournié, M. (2012). High-order compact finite difference scheme for option pricing in stochastic volatility models. Journal of Computational and Applied Mathematics, 236(17), 4462-4473. doi:10.1016/j.cam.2012.04.017Zvan, R., Forsyth, P., & Vetzal, K. (2003). Negative coefficients in two-factor option pricing models. The Journal of Computational Finance, 7(1), 37-73. doi:10.21314/jcf.2003.096Company, R., Jódar, L., & Pintos, J.-R. (2009). Consistent stable difference schemes for nonlinear Black-Scholes equations modelling option pricing with transaction costs. ESAIM: Mathematical Modelling and Numerical Analysis, 43(6), 1045-1061. doi:10.1051/m2an/2009014Company, R., Jódar, L., & Pintos, J.-R. (2010). Numerical analysis and computing for option pricing models in illiquid markets. Mathematical and Computer Modelling, 52(7-8), 1066-1073. doi:10.1016/j.mcm.2010.02.037Casabán, M.-C., Company, R., Jódar, L., & Pintos, J.-R. (2011). Numerical analysis and computing of a non-arbitrage liquidity model with observable parameters for derivatives. Computers & Mathematics with Applications, 61(8), 1951-1956. doi:10.1016/j.camwa.2010.08.009Kangro, R., & Nicolaides, R. (2000). Far Field Boundary Conditions for Black--Scholes Equations. SIAM Journal on Numerical Analysis, 38(4), 1357-1368. doi:10.1137/s0036142999355921EHRHARDT, M., & MICKENS, R. E. (2008). A FAST, STABLE AND ACCURATE NUMERICAL METHOD FOR THE BLACK–SCHOLES EQUATION OF AMERICAN OPTIONS. International Journal of Theoretical and Applied Finance, 11(05), 471-501. doi:10.1142/s021902490800489

Azides in the Synthesis of Various Heterocycles

In this review, we focus on some interesting and recent examples of various applications of organic azides such as their intermolecular or intramolecular, under thermal, catalyzed, or noncatalyzed reaction conditions. The aforementioned reactions in the aim to prepare basic five-, six-, organometallic heterocyclic-membered systems and/or their fused analogs. This review article also provides a report on the developed methods describing the synthesis of various heterocycles from organic azides, especially those reported in recent papers (till 2020). At the outset, this review groups the synthetic methods of organic azides into different categories. Secondly, the review deals with the functionality of the azido group in chemical reactions. This is followed by a major section on the following: (1) the synthetic tools of various heterocycles from the corresponding organic azides by one-pot domino reaction; (2) the utility of the chosen catalysts in the chemoselectivity favoring C−H and C-N bonds; (3) one-pot procedures (i.e., Ugi four-component reaction); (4) nucleophilic addition, such as Aza-Michael addition; (5) cycloaddition reactions, such as [3+2] cycloaddition; (6) mixed addition/cyclization/oxygen; and (7) insertion reaction of C-H amination. The review also includes the synthetic procedures of fused heterocycles, such as quinazoline derivatives and organometal heterocycles (i.e., phosphorus-, boron- and aluminum-containing heterocycles). Due to many references that have dealt with the reactions of azides in heterocyclic synthesis (currently more than 32,000), we selected according to generality and timeliness. This is considered a recent review that focuses on selected interesting examples of various heterocycles from the mechanistic aspects of organic azides

Effectiveness of Shock Wave Therapy versus Intra-Articular Corticosteroid Injection in Diabetic Frozen Shoulder Patients’ Management: Randomized Controlled Trial

Frozen shoulder is a major musculoskeletal illness in diabetic patients. This study aimed to compare the effectiveness of shock wave and corticosteroid injection in the management of diabetic frozen shoulder patients. Fifty subjects with diabetic frozen shoulder were divided randomly into group A (the intra-articular corticosteroid injection group) and group B that received 12 sessions of shock wave therapy, while each patient in both groups received the traditional physiotherapy program. The level of pain and disability, the range of motion, as well as the glucose triad were evaluated before patient assignment to each group, during the study and at the end of the study. Compared to the pretreatment evaluations there were significant improvements of shoulder pain and disability and in shoulder flexion and abduction range of motion in both groups (p < 0.05). The shock wave group revealed a more significant improvement the intra-articular corticosteroid injection group, where p was 0.001 for shoulder pain and disability and shoulder flexion and abduction. Regarding the effect of both interventions on the glucose triad, there were significant improvements in glucose control with group B, where p was 0.001. Shock waves provide a more effective and safer treatment modality for diabetic frozen shoulder treatment than corticosteroid intra-articular injection

Novel Pyridinium Based Ionic Liquid Promoter for Aqueous Knoevenagel Condensation: Green and Efficient Synthesis of New Derivatives with Their Anticancer Evaluation

Herein, a distinctive dihydroxy ionic liquid ([Py-2OH]OAc) was straightforwardly assembled from the sonication of pyridine with 2-chloropropane-1,3-diol by employing sodium acetate as an ion exchanger. The efficiency of the ([Py-2OH]OAc as a promoter for the sono-synthesis of a novel library of condensed products through DABCO-catalyzed Knoevenagel condensation process of adequate active cyclic methylenes and ninhydrin was next investigated using ultimate greener conditions. All of the reactions studied went cleanly and smoothly, and the resulting Knoevenagel condensation compounds were recovered in high yields without detecting the aldol intermediates in the end products. Compared to traditional strategies, the suggested approach has numerous advantages including mild reaction conditions with no by-products, eco-friendly solvent, outstanding performance in many green metrics, and usability in gram-scale synthesis. The reusability of the ionic liquid was also studied, with an overall retrieved yield of around 97% for seven consecutive runs without any substantial reduction in the performance. The novel obtained compounds were further assessed for their in vitro antitumor potential toward three human tumor cell lines: Colo-205 (colon cancer), MCF-7 (breast cancer), and A549 (lung cancer) by employing the MTT assay, and the findings were evaluated with the reference Doxorubicin. The results demonstrated that the majority of the developed products had potent activities at very low doses. Compounds comprising rhodanine (5) or chromane (12) moieties exhibited the most promising cytotoxic effects toward three cell lines, particularly rhodanine carboxylic acid derivative (5c), showing superior cytotoxic effects against the investigated cell lines compared to the reference drug. Furthermore, automated docking simulation studies were also performed to support the results obtained