A Study on Modeling Financial Mathematics by Computational Program and Its Applications

A Keynesian Approach to Modeling the Long-Term Interest Rate

The space of probability distributions on a given sample space possesses natural geometric properties. For example, in the case of a smooth parametric family of probability distributions on the real line, the parameter space has a Riemannian structure induced by the embedding of the family into the Hilbert space of square-integrable functions, and is characterised by the Fisher-Rao metric. In the nonparametric case the relevant geometry is determined by the spherical distance function of Bhattacharyya. In the context of term structure modelling, we show that minus the derivative of the discount function with respect to the maturity date gives rise to a probability density. This follows as a consequence of the positivity of interest rates. Therefore, by mapping the density functions associated with a given family of term structures to Hilbert space, the resulting metrical geometry can be used to analyse the relationship of yield curves to one another. We show that the general arbitrage-free yield curve dynamics can be represented as a process taking values in the convex space of smooth density functions on the positive real line. It follows that the theory of interest rate dynamics can be represented by a class of processes in Hilbert space. We also derive the dynamics for the central moments associated with the distribution determined by the yield curve.

A Hybrid Model for Equity Indices and Stochastic Interest Rates

EMPIRICAL STUDIES ON INTEREST RATE DERIVATIVES by Xudong Sun Dr. Hongtao Yang, Examination Committee Chair Associate Professor of Mathematics University of Nevada, Las Vegas, USA Interest rate models are the building blocks of nancial market and the interest rate derivatives market is the largest derivatives market in the world. In this dissertation, we shall focus on numerical pricing of interest rate derivatives, estimating model parameters by Kalman lter, and studying various models empirically. We shall propose a frontxing nite element method to price the American put option under the quadratic term structure framework and compare it with a trinomial tree method and common nite element method. Numerical test results show the superiority of our frontxing nite element method in the aspects of computing the option and free boundary simultaneously with high accuracy. We shall also employ the Kalman lter and its variant techniques to estimate parameters of the a ne term structure models as well as quadratic term structure models. Various comparisons of di erent Kalman lter performance and both the in-sample t and out-sample t for Monte Carlo simulations as well as real treasury yield data are presented. In addition, we shall propose a general one-factor interest rate model and apply a homotopy perturbation method to valuate bond prices. One of the attractive qualities of the approximated solution of homotopy perturbation method is its fast speed of achieving the same accuracy compared to the tree method.

Quantitative analysis of energy markets

Out-of-Sample Performance of Spot Interest Rate Models

The 2024 3rd International Conference on Economics, Mathematical Finance and Risk Management (EMFRM 2024) was successfully held during December 07-08, 2024 in Dubai, United Arab Emirates. This conference aims to bring together innovative scholars and experts in economic system, business statistics, mathematical finance, corporate governance, and risk management in a common forum. The main objective of EMFRM 2024 is to promote economic development research, another objective is to facilitate the exchange of scientific information among researchers, developers, engineers, students, and practitioners around the world. We are confident that the participants benefited from the theoretical insights, practical knowledge, and personal contacts that will help them to build a long-term and sustainable communication among researchers and practitioners in related scientific areas. EMFRM 2024 invited scholars and experts from various universities to participate in the review and guidance of this conference. The conference focuses on the latest research fields, aims to provide an international cooperation and exchange platform for experts, scholars, and business managers to share their research achievements. This collection of proceedings compiles oral and paper presentations submitted by the authors and scrutinized by the special committee. EMFRM Organizing Committees Dubai, United Arab Emirates

Option pricing is essential for market participants as it facilitates risk management and price discovery. This study provides a comprehensive elucidation of diverse option pricing models. The author delineates the development of option pricing models from the fundamental Black-Scholes-Merton model (BSM) to more intricate methodologies. The importance of comprehending option pricing for hedging strategies and risk management is crucial in financial mathematics. The study analyses several models, including the BS model, Jump Diffusion model (JDM), Binomial Tree model (BTM), Monte Carlo simulation (MCS), and stochastic volatility models (SVM), while addressing their applications, limitations, advantages, and disadvantages. This work delineates the history and distinctions between other models and the Black-Scholes-Merton model. The Black-Scholes-Merton model exerts the most substantial influence on financial markets. The alternative models exhibit greater flexibility and enhanced accuracy when confronting complicated and uncertain market trends. The trend of future development is bound to be the continuous updating of models to adapt to more complex situations.

Preface. 1. Simulations and the Monte Carlo method. 1.1 Description of the method. 1.2 Convergence theorems. 1.3 Simulation of random variables. 1.4 Variance reduction techniques. 1.5 Exercises. 2. Markov chains. 2.1 Definitions and elementary properties. 2.2 Examples. 2.3 Strong Markov property. 2.4 Recurrent and transient states. 2.5 The irreducible and recurrent case. 2.6 The aperiodic case. 2.7 Reversible Markov chain. 2.8 Rate of convergence to equilibrium. 2.9 Statistics of Markov chains. 2.10 Exercises. 3. Stochastic algorithms. 3.1 Markov chain Monte Carlo. 3.2 Simulation of the invariant probability. 3.3 Rate of convergence towards the invariant probability. 3.4 Simulated annealing. 3.5 Exercises. 4. Markov chains and the genome. 4.1 Reading DNA. 4.2 The i.i.d. model. 4.3 The Markov model. 4.4 Hidden Markov models. 4.5 Hidden semi-Markov model. 4.6 Alignment of two sequences. 4.7 A multiple alignment algorithm. 4.8 Exercises. 5. Control and filtering of Markov chains. 5.1 Deterministic optimal control. 5.2 Control of Markov chains. 5.3 Linear quadratic optimal control. 5.4 Filtering of Markov chains. 5.5 The Kalman-Bucy filter. 5.6 Linear-quadratic control with partial observation. 5.7 Exercises. 6. The Poisson process. 6.1 Point processes and counting processes. 6.2 The Poisson process. 6.3 The Markov property. 6.4 Large time behaviour. 6.5 Exercises. 7. Jump Markov processes. 7.1 General facts. 7.2 Infinitesimal generator. 7.3 The strong Markov property. 7.4 Embedded Markov chain. 7.5 Recurrent and transient states. 7.6 The irreducible recurrent case. 7.7 Reversibility. 7.8 Markov models of evolution and phylogeny. 7.9 Application to discretized partial differential equations. 7.10 Simulated annealing. 7.11 Exercises. 8. Queues and networks. 8.1 M/M/1 queue. 8.2 M/M/1/K queue. 8.3 M/M/s queue. 8.4 M/M/s/s queue. 8.5 Repair shop. 8.6 Queues in series. 8.7 M/G/ queue. 8.8 M/G/1 queue. 8.9 Open Jackson network. 8.10 Closed Jackson network. 8.11 Telephone network. 8.12 Kelly networks. 8.13 Exercises. 9. Introduction to mathematical finance. 9.1 Fundamental concepts. 9.2 European options in the discrete model. 9.3 The Black-Scholes model and formula. 9.4 American options in the discrete model. 9.5 American options in the Black-Scholes model. 9.6 Interest rate and bonds. 9.7 Exercises. 10. Solutions to selected exercises. 10.1 Chapter 1. 10.2 Chapter 2. 10.3 Chapter 3. 10.4 Chapter 4. 10.5 Chapter 5. 10.6 Chapter 6. 10.7 Chapter 7. 10.8 Chapter 8. 10.9 Chapter 9. References Index.

Spike Modeling for Interest Rate Derivatives with an Application to SOFR Caplets

This study examines the integration of financial mathematics and artificial intelligence (AI), focusing on credit risk analysis within the financial industry. Traditionally, financial mathematics has played a central role in modeling and assessing credit risk. However, recent technological advancements have enabled a more synergistic relationship between financial mathematics and AI, particularly through the use of machine learning. This integration enhances risk assessment by leveraging classification algorithms to uncover complex patterns in historical data patterns that are often difficult to detect using conventional methods. Through a case study in the financial sector, this research explores the application of classification algorithms to better understand customer behavior, evaluate credit history, and predict the likelihood of future loan repayment. In the realm of machine learning, classification techniques are essential for processing large volumes of data and generating accurate risk predictions. By combining the analytical rigor of financial mathematics with the predictive power of machine learning, this study aims to develop a robust and adaptive model for credit risk analysis one capable of responding to dynamic market and economic conditions, and improving the overall decision-making process in credit assessment.

This paper presents an analysis of models for Japanese short-term inter- est rate. The models are constructed based on mean reverting model using Bayesian method to capture the dynamics of short-term interest rate. The parameters of our models are estimated by marginal likelihood and posterior expectation and we shall make model selection using the information criterion EIC(extended information cri- terion). An application of the models will be implemented using weekly Japanese average interest rates on certificates of deposit (new issues) less than 30 days in the period from January 2001 to December 2008. 1 Introduction Correct modeling of the short-term interest rate is essential in finace, as it is this rate that is fundamental to the pricing of securities and important for risk man- agement. In the study of the short-term interest rate dynamics, various models have been suggested. There are examples of these models, such as Vasicek model by Vasicek(1977), CIR model by Cox, Ingersoll and Ross(1985) and so on. An empirical comparison of these models was made by Chan, Karolyi, Longstaff and Sanders(1992). In the paper, the param- eters are estimated by generalized method of moments(GMM) and they implemented the hypothesis testing methods developed by Newey and West for evaluation of the models. In the parameter estimation of mean reverting model using GMM, it is known that the results are easy to be influenced by an initial values, and it becomes often unstable. Recently, Ahangarani(2005) employed maximum likelihood method for parameter esti- mation and log likelihood ratio test for model selection, and Kawada(2007) implemented GMM and made model selection using the information criterion EIC(extended information criterion) for various short-term interest rate models. Another articles of the interest rate model, using Bayesian framework, Jones(2003) published study about nonlinear drift of the interest rate model in detail, and Gray(2005) studied continuous time short rate models and the parameters estimated by Markov chain Monte Carlo(MCMC) method. Sanford and Martin(2006) made estimation using MCMC algorithm and model selection is made by Bayes factors for each model calculated using Savage-Dickey density ratio. In addition, a recent study by Hong and Lin(2006) examined a variety of short-term interest rate mod- els including the single-factor diffusion models, GARCH models, Markov regime-switching models and jump-diffusion models for Chinese rates. However, more useful models which capture the dynamics of the short-term interest rate are needed. In this paper, we suggest two hierarchical Bayes models based on mean reverting model. The parameters of the models are treated as random variables so that we have to consider appropriate prior distributions for them. By regarding parameters as random variables, we can consider that model construction and estimation involve necessarily some uncertainty and utilize the prior information of interest rate. In the first hierarchical Bayes model, we assume that volatility is constant, and in the second one, it depends on the

This paper analyzes an interest rate model with self-exciting jumps, in which a jump in the interest rate model increases the intensity of jumps in the same model. This self-exciting property leads to clustering effects in the interest rate model. We obtain a closed-form expression for the conditional moment-generating function when the model coefficients have affine structures. Based on the Girsanov-type measure transformation for general jump-diffusion processes, we derive the evolution of the interest rate under the equivalent martingale measure and an explicit expression of the zero-coupon bond pricing formula. Furthermore, we give a pricing formula for the European call option written on zero-coupon bonds. Finally, we provide an interpretation for the clustering effects in the interest rate model within a simple framework of general equilibrium. Indeed, we construct an interest rate model, the equilibrium state of which coincides with the interest rate model with clustering effects proposed in this paper.

The Indonesian stock market, especially PT Bank Central Asia (BCA) stock experiences high volatility. Therefore, this research is very important to provide a deeper understanding of the pricing of put and call options as a risk management instrument. This study aims to analyze the pricing of put and call options on PT Bank Central Asia (BCA) shares using the Black-Scholes model and identify factors that affect stock price fluctuations on option prices. This study aims to understand the changes in put and call option prices on PT Bank Central Asia (BCA) shares caused by fluctuations in stock prices. This study uses historical data of PT Bank Central Asia (BCA) shares, volatility, and current interest rates as the basis for analysis. The Black-Scholes method is used as a framework to analyze the pricing of put and call options by considering volatility and interest rates. The results of this study analyze the price changes of put and call options in the face of stock price fluctuations and the factors that affect option prices so as to provide insight into risk management and investment decision making. This research is expected to provide practical guidance for investors and risk managers in making decisions related to put and call options on PT Bank Central Asia (BCA) shares.

This volume in the Mastering Mathematical Finance series strikes just the right balance between mathematical rigour and practical application. Existing books on the challenging subject of stochastic interest rate models are often too advanced for Master's students or fail to include practical examples. Stochastic Interest Rates covers practical topics such as calibration, numerical implementation and model limitations in detail. The authors provide numerous exercises and carefully chosen examples to help students acquire the necessary skills to deal with interest rate modelling in a real-world setting. In addition, the book's webpage at www.cambridge.org/9781107002579 provides solutions to all of the exercises as well as the computer code (and associated spreadsheets) for all numerical work, which allows students to verify the results.