Pricing Of Financial Derivatives Research Articles

Modelling and Forecasting of Exchange Rate Pairs Using the Kalman Filter

ABSTRACTDeveloping and employing practically useful and easy to calibrate models for prediction of exchange rates remains a challenging task, especially for highly volatile emerging market currencies. In this paper, we propose a novel approach for joint prediction of correlated exchange rates for two different currencies with respect to the same base currency. For this purpose, we reformulate a generalized version of a bivariate ARMA model into a state space model and use the Kalman filter for estimation and forecasting of the underlying exchange rates as latent variables. With extensive numerical experiments spanning 18 different exchange rates (across both emerging markets, developing and developed economies), we demonstrate that our approach consistently outperforms univariate ARMA models as well as the random walk model in short term out‐of‐sample prediction for various exchange rate pairs. Our study fills a gap in the empirical finance literature in terms of robust, explainable, accurate, and easy to calibrate models for forecasting correlated exchange rates. The proposed methodology has applications in exchange rate risk management as well as pricing of financial derivatives based on two exchange rates.

Series form solutions of time–space fractional Black–Scholes model via extended He-Aboodh algorithm

The objective of the current study is analyze linear and nonlinear time–space fractional Black–Scholes models via modified homotopy perturbation method (m-HPM). In current investigation, memory effects in financial markets are explored through fractional derivative in Caputo sense. The effectiveness of proposed methodology is checked numerically by finding residual errors and presented in tables. These tables also provide a benchmark for the comparison with already existing results in literature. Furthermore, solutions are graphically analyzed via 3D and contour plots across a range of parameters under varying market conditions. Analysis confirms the efficiency of m-HPM for predicting solutions of time–space fractional Black–Scholes models. The current study can contribute in understanding the applications of fractional calculus in finance, and can be a valuable computational tool for pricing financial derivatives in fractional environments.

Random Processes and Their Applications in Financial Mathematics

Random processes are fundamental tools in modern financial mathematics, widely applied in fields such as financial derivatives pricing, risk management, and interest rate models. This paper systematically introduces the basic theory of random processes, including definitions and classifications, Brownian motion and geometric Brownian motion, and stochastic differential equations. Furthermore, the paper explores specific applications of random processes in financial mathematics, focusing on financial derivatives pricing, risk management and portfolio optimization, and applications in interest rate models and credit risk. Through theoretical and empirical research, this paper reveals the significant role and development prospects of random processes in financial mathematics, providing new ideas and methods for risk management and pricing models in financial markets.

Neural network empowered liquidity pricing in a two-price economy under conic finance settings

In the article at hand neural networks are used to model liquidity in financial markets, under conic finance settings, in two different contexts. That is, on the one hand this paper illustrates how the use of neural networks within a two-price economy allows to obtain accurate pricing and Greeks of financial derivatives, enhancing computational performances compared to classical approaches such as (conic) Monte Carlo. The methodology proposed for this purpose is agnostic of the underlying valuation model, and it easily adapts to all models suitable for pricing in conic financial markets. On the other hand, this article also investigates the possibility of valuing contingent claims under conic assumptions, using local stochastic volatility models, where the local volatility is approximated by means of a (combination of) neural network(s). Moreover, we also show how it is possible to generate hybrid families of distortion functions to better fit the implied liquidity of the market, as well as we introduce a conic version of the SABR model, based on the Wang transform, that still allows for analytical bid and ask pricing formulae.

MODELLING STOCK PRICES OF A BANK WITH EXTREME VALUE DISTRIBUTIONS

The study investigates the application of Extreme Value Theory in modelling stock prices, aiming to capture the tail behaviour and extreme movements that conventional distributions often fail to represent accurately. The use of Extreme Value Theory has gained considerable attention in the field of finance due to its ability to model rare events, such as financial crises or market crashes. By incorporating Extreme Value Theory, researchers aim to improve risk management, portfolio optimization, and pricing of financial derivatives. In this study, the Log-normal, Weibull, Gamma, and Normal distributions were used to model the stock price closing data, with a specific focus on extreme value distributions. Both graphical explorations and goodness-of-fit criteria were considered together to evaluate the suitability of these distributions. When assessing the data, it was observed that the Weibull distribution provided the best fit for the given stock price closing data.

Derivative Pricing using Quantum Signal Processing

Pricing financial derivatives on quantum computers typically includes quantum arithmetic components which contribute heavily to the quantum resources required by the corresponding circuits. In this manuscript, we introduce a method based on Quantum Signal Processing (QSP) to encode financial derivative payoffs directly into quantum amplitudes, alleviating the quantum circuits from the burden of costly quantum arithmetic. Compared to current state-of-the-art approaches in the literature, we find that for derivative contracts of practical interest, the application of QSP significantly reduces the required resources across all metrics considered, most notably the total number of T-gates by ∼16x and the number of logical qubits by ∼4x. Additionally, we estimate that the logical clock rate needed for quantum advantage is also reduced by a factor of ∼5x. Overall, we find that quantum advantage will require 4.7k logical qubits, and quantum devices that can execute 109 T-gates at a rate of 45MHz. While in this work we focus specifically on the payoff component of the derivative pricing process where the method we present is most readily applicable, similar techniques can be employed to further reduce the resources in other applications, such as state preparation.

Foundations and trends in option pricing models: a 45 years global examination based on bibliometric analysis

PurposeThe purpose of this study is to analyse and compile the literature on various option pricing models (OPM) or methodologies. The report highlights the gaps in the existing literature review and builds recommendations for potential scholars interested in the subject area.Design/methodology/approachIn this study, the researchers used a systematic literature review procedure to collect data from Scopus. Bibliometric and structured network analyses were used to examine the bibliometric properties of 864 research documents.FindingsAs per the findings of the study, publication in the field has been increasing at a rate of 6% on average. This study also includes a list of the most influential and productive researchers, frequently used keywords and primary publications in this subject area. In particular, Thematic map and Sankey’s diagram for conceptual structure and for intellectual structure co-citation analysis and bibliographic coupling were used.Research limitations/implicationsBased on the conclusion presented in this paper, there are several potential implications for research, practice and society.Practical implicationsThis study provides useful insights for future research in the area of OPM in financial derivatives. Researchers can focus on impactful authors, significant work and productive countries and identify potential collaborators. The study also highlights the commonly used OPMs and emerging themes like machine learning and deep neural network models, which can inform practitioners about new developments in the field and guide the development of new models to address existing limitations.Social implicationsThe accurate pricing of financial derivatives has significant implications for society, as it can impact the stability of financial markets and the wider economy. The findings of this study, which identify the most commonly used OPMs and emerging themes, can help improve the accuracy of pricing and risk management in the financial derivatives sector, which can ultimately benefit society as a whole.Originality/valueIt is possibly the initial effort to consolidate the literature on calibration on option price by evaluating and analysing alternative OPM applied by researchers to guide future research in the right direction.

Retraction: The value of Monte Carlo model-based variance reduction technology in the pricing of financial derivatives.

Retraction: The value of Monte Carlo model-based variance reduction technology in the pricing of financial derivatives.

Research on the Pricing Model of DerivativesBased on the Black-Scholes Model and Binomial Model

The pricing of financial derivatives plays a significant role in risk management and financial transactions. The models of pricing are mainly divided into Black-Scholes Model and Binomial Model. The two models have different characteristics and distinguishing the models is helpful for price makers. The paper aims to analyse the distinction between the two models on the Huaxia Stock Exchange 50ETF by using Python. This paper selected the stock price, the volatility and the option price from the Eastmoney database. First, edit the formulas of two models and then introduce the parameters into the formula by Python and get the model calculated price. Comparison between the calculated prices helps to compare the two pricing models. The result is that Black-Scholes Model is more accurate and Binomial Model is more simplified which is suitable for new officers.

An Empirical Study on the Logarithmic Return of Securities

Objective: Previous studies have shown that the distribution of security returns has the characteristics of peak and fat tail, and it is not realistic to use logarithmic normal distribution to describe security returns. This paper mainly studies the specific form of security returns distribution and conducts empirical research. Methods: We first assume that the security returns follow a logarithmic t-distribution. Then, the maximum likelihood estimation and moment estimation are used to explore the degree of freedom of the distribution. Finally, the goodness of fit test is used to conduct an empirical analysis about Shanghai Securities Composite Index. Results: The specific expression formulas for the estimator of the degree of freedom are obtained. Empirical testing shows that the security returns follow a logarithmic t-distribution. Conclusion: The probability distribution of security returns is unrealistic to assume that the security returns follow the logarithmic normal distribution. It is more practical to use the logarithmic t-distribution to characterize the distribution of security returns. Our results provide theoretical assistance for exploring the pricing of financial derivatives.

Learning the random variables in Monte Carlo simulations with stochastic gradient descent: Machine learning for parametric PDEs and financial derivative pricing

AbstractIn financial engineering, prices of financial products are computed approximately many times each trading day with (slightly) different parameters in each calculation. In many financial models such prices can be approximated by means of Monte Carlo (MC) simulations. To obtain a good approximation the MC sample size usually needs to be considerably large resulting in a long computing time to obtain a single approximation. A natural deep learning approach to reduce the computation time when new prices need to be calculated as quickly as possible would be to train an artificial neural network (ANN) to learn the function which maps parameters of the model and of the financial product to the price of the financial product. However, empirically it turns out that this approach leads to approximations with unacceptably high errors, in particular when the error is measured in the ‐norm, and it seems that ANNs are not capable to closely approximate prices of financial products in dependence on the model and product parameters in real life applications. This is not entirely surprising given the high‐dimensional nature of the problem and the fact that it has recently been proved for a large class of algorithms, including the deep learning approach outlined above, that such methods are in general not capable to overcome the curse of dimensionality for such approximation problems in the ‐norm. In this article we introduce a new numerical approximation strategy for parametric approximation problems including the parametric financial pricing problems described above and we illustrate by means of several numerical experiments that the introduced approximation strategy achieves a very high accuracy for a variety of high‐dimensional parametric approximation problems, even in the ‐norm. A central aspect of the approximation strategy proposed in this article is to combine MC algorithms with machine learning techniques to, roughly speaking, learn the random variables (LRV) in MC simulations. In other words, we employ stochastic gradient descent (SGD) optimization methods not to train parameters of standard ANNs but instead to learn random variables appearing in MC approximations. In that sense, the proposed LRV strategy has strong links to Quasi‐Monte Carlo (QMC) methods as well as to the field of algorithm learning. Our numerical simulations strongly indicate that the LRV strategy might indeed be capable to overcome the curse of dimensionality in the ‐norm in several cases where the standard deep learning approach has been proven not to be able to do so. This is not a contradiction to the established lower bounds mentioned above because this new LRV strategy is outside of the class of algorithms for which lower bounds have been established in the scientific literature. The proposed LRV strategy is of general nature and not only restricted to the parametric financial pricing problems described above, but applicable to a large class of approximation problems. In this article we numerically test the LRV strategy in the case of the pricing of European call options in the Black‐Scholes model with one underlying asset, in the case of the pricing of European worst‐of basket put options in the Black‐Scholes model with three underlying assets, in the case of the pricing of European average put options in the Black‐Scholes model with three underlying assets and knock‐in barriers, as well as in the case of stochastic Lorentz equations. For these examples the LRV strategy produces highly convincing numerical results when compared with standard MC simulations, QMC simulations using Sobol sequences, SGD‐trained shallow ANNs, and SGD‐trained deep ANNs.

Deep Equal Risk Pricing of Financial Derivatives with Non-Translation Invariant Risk Measures

The objective is to study the use of non-translation invariant risk measures within the equal risk pricing (ERP) methodology for the valuation of financial derivatives. The ability to move beyond the class of convex risk measures considered in several prior studies provides more flexibility within the pricing scheme. In particular, suitable choices for the risk measure embedded in the ERP framework, such as the semi-mean-square-error (SMSE), are shown herein to alleviate the price inflation phenomenon observed under the tail value at risk-based ERP as documented in previous work. The numerical implementation of non-translation invariant ERP is performed through deep reinforcement learning, where a slight modification is applied to the conventional deep hedging training algorithm so as to enable obtaining a price through a single training run for the two neural networks associated with the respective long and short hedging strategies. The accuracy of the neural network training procedure is shown in simulation experiments not to be materially impacted by such modification of the training algorithm.

Study of Pricing of High-Dimensional Financial Derivatives Based on Deep Learning

Many problems in the fields of finance and actuarial science can be transformed into the problem of solving backward stochastic differential equations (BSDE) and partial differential equations (PDEs) with jumps, which are often difficult to solve in high-dimensional cases. To solve this problem, this paper applies the deep learning algorithm to solve a class of high-dimensional nonlinear partial differential equations with jump terms and their corresponding backward stochastic differential equations (BSDEs) with jump terms. Using the nonlinear Feynman-Kac formula, the problem of solving this kind of PDE is transformed into the problem of solving the corresponding backward stochastic differential equations with jump terms, and the numerical solution problem is turned into a stochastic control problem. At the same time, the gradient and jump process of the unknown solution are separately regarded as the strategy function, and they are approximated, respectively, by using two multilayer neural networks as function approximators. Thus, the deep learning-based method is used to overcome the “curse of dimensionality” caused by high-dimensional PDE with jump, and the numerical solution is obtained. In addition, this paper proposes a new optimization algorithm based on the existing neural network random optimization algorithm, and compares the results with the traditional optimization algorithm, and achieves good results. Finally, the proposed method is applied to three practical high-dimensional problems: Hamilton-Jacobi-Bellman equation, bond pricing under the jump Vasicek model and option pricing under the jump diffusion model. The proposed numerical method has obtained satisfactory accuracy and efficiency. The method has important application value and practical significance in investment decision-making, option pricing, insurance and other fields.

Credit risk pricing in a consumption‐based equilibrium framework with incomplete accounting information

AbstractWe present a consumption‐based equilibrium framework for credit risk pricing based on the Epstein–Zin (EZ) preferences where the default time is modeled as the first hitting time of a default boundary and bond investors have imperfect/partial information about the firm value. The imperfect information is generated by the underlying observed state variables and a noisy observation process of the firm value. In addition, the consumption, the volatility, and the firm value process are modeled to follow affine diffusion processes. Using the EZ equilibrium solution as the pricing kernel, we provide an equivalent pricing measure to compute the prices of financial derivatives as discounted values of the future payoffs given the incomplete information. The price of a zero‐coupon bond is represented in terms of the solutions of a stochastic partial differential equation (SPDE) and a deterministic PDE; the self‐contained proofs are provided for both this representation and the well‐posedness of the involved SPDE. Furthermore, this SPDE is numerically solved, which yields some insights into the relationship between the structure of the yield spreads and the model parameters.

Parameters identification for an inverse problem arising from a binary option using a Bayesian inference approach

No-arbitrage property provides a simple method for pricing financial derivatives. However, arbitrage opportunities exist in various fields, even for a very short time. By knowing that an arbitrage property exists, we can adopt a financial trading strategy. This paper investigates the inverse option problems (IOP) in the backward parabolic equation with a suitable initial condition in financial markets. We identify the coefficients of this problem from the measured data and attempt to find arbitrage opportunities in financial markets using a Bayesian inference approach, which is presented as an IOP solution. The posterior probability density function of the parameters is computed from the measured data. The statistics of the unknown parameters are estimated by a Markov Chain Monte Carlo (MCMC) algorithm, which exploits the posterior state space. The efficient sampling strategy of the MCMC algorithm enables us to solve inverse problems by the Bayesian inference technique. Our numerical results indicate that the Bayesian inference approach can simultaneously estimate the unknown trend and volatility coefficients from the artificial measured data and the real financial market data.

Local Hölder continuity of inverse variation-inequality problem constructed by non-Newtonian polytropic operators in finance

<abstract><p>This paper aims to explore the inverse variation-inequality problems of a specific type of degenerate parabolic operators in a non-divergence form. These problems have significant implications in financial derivative pricing. The study focuses on analyzing the Hölder continuity of weak solutions by employing cut-off factors.</p></abstract>

Numerical methods for backward stochastic differential equations: A survey

Backward Stochastic Differential Equations (BSDEs) have been widely employed in various areas of social and natural sciences, such as the pricing and hedging of financial derivatives, stochastic optimal control problems, optimal stopping problems and gene expression. Most BSDEs cannot be solved analytically and thus numerical methods must be applied to approximate their solutions. There have been a variety of numerical methods proposed over the past few decades as well as many more currently being developed. For the most part, they exist in a complex and scattered manner with each requiring a variety of assumptions and conditions. The aim of the present work is thus to systematically survey various numerical methods for BSDEs, and in particular, compare and categorize them, for further developments and improvements. To achieve this goal, we focus primarily on the core features of each method based on an extensive collection of 333 references: the main assumptions, the numerical algorithm itself, key convergence properties and advantages and disadvantages, to provide an up-to-date coverage of numerical methods for BSDEs, with insightful summaries of each and a useful comparison and categorization.

Saddlepoint Method for Pricing European Options under Markov-Switching Heston’s Stochastic Volatility Model

This paper evaluates the prices of European-style options when dynamics of the underlying asset is assumed to follow a Markov-switching Heston’s stochastic volatility model. Under this framework, the expected return and the long-term mean of the variance of the underlying asset rely on states of the economy modeled by a continuous-time Markov chain. There is evidence that the Markov-switching Heston’s stochastic volatility model performs well in capturing major events affecting price dynamics. However, due to the nature of the model, analytic solutions for the prices of options or other financial derivatives do not exist. By means of the saddlepoint method, an analytic approximation for European-style option price is presented. The saddlepoint method gives an effective approximation to option prices under the Markov-switching Heston’s stochastic volatility model.

General Restrictions on Prices of Financial Derivatives Written on Underlying Diffusions

We apply probabilistic solutions of parabolic PDEs with terminal and boundary conditions to obtain restrictions on contingent claims written on one-dimensional diffusions. For term structure derivatives, we obtain monotonicity and convexity results with respect to the short-term interest rate. We apply them to bonds, calls on bonds, and puts on interest rates and we find a condition for the price of these derivatives to be convex in that rate. We find that yield curves corresponding to higher short-term rates lie uniformly above curves with lower rates. Regarding options on assets with local volatility, we obtain probabilistic representations, bounds, and asymptotic results for delta, rho, and theta. Similar results are obtained for Asian options.

Quantum algorithms for numerical differentiation of expected values with respect to parameters

The quantum algorithms for Monte Carlo integration (QMCI), which are based on quantum amplitude estimation (QAE), speed up expected value calculation compared with classical counterparts and have been widely investigated along with their applications to industrial problems such as financial derivative pricing. In this paper, we consider an expected value of a function of a stochastic variable and a real-valued parameter and how to calculate derivatives of the expectation with respect to the parameter. This problem is related to calculating sensitivities of financial derivatives and so of industrial importance. Based on QMCI and the general-order central difference formula for numerical differentiation, we propose two quantum methods for this problem and evaluate their complexities. The first one, which we call the naive iteration method, simply calculates the formula by iterative computations and additions of the terms in it and then estimates its expected value by QAE. The second one, which we name the sum-in-QAE method, performs the summation of the terms at the same time as the sum over the possible values of the stochastic variable in a single QAE. We see that, depending on the smoothness of the function and the number of qubits available, either of two methods is better than the other. In particular, when the function is nonsmooth or we want to save the qubit number, the sum-in-QAE method can be advantageous.