Efficient Markets and Contingent Claims Valuation: An Information Theoretic Approach

This research article shows how the pricing of derivative securities can be seen from the context of stochastic optimal control theory and information theory. The financial market is seen as an information processing system, which optimizes an information functional. An optimization problem is constructed, for which the linearized Hamilton–Jacobi–Bellman equation is the Black–Scholes pricing equation for financial derivatives. The model suggests that one can define a reasonable Hamiltonian for the financial market, which results in an optimal transport equation for the market drift. It is shown that in such a framework, which supports Black–Scholes pricing, the market drift obeys a backwards Burgers equation and that the market reaches a thermodynamical equilibrium, which minimizes the free energy and maximizes entropy.     

Generalised Geometric Brownian Motion: Theory and Applications to Option Pricing

Classical option pricing schemes assume that the value of a financial asset follows a geometric Brownian motion (GBM). However, a growing body of studies suggest that a simple GBM trajectory is not an adequate representation for asset dynamics, due to irregularities found when comparing its properties with empirical distributions. As a solution, we investigate a generalisation of GBM where the introduction of a memory kernel critically determines the behaviour of the stochastic process. We find the general expressions for the moments, log-moments, and the expectation of the periodic log returns, and then obtain the corresponding probability density functions using the subordination approach. Particularly, we consider subdiffusive GBM (sGBM), tempered sGBM, a mix of GBM and sGBM, and a mix of sGBMs. We utilise the resulting generalised GBM (gGBM) in order to examine the empirical performance of a selected group of kernels in the pricing of European call options. Our results indicate that the performance of a kernel ultimately depends on the maturity of the option and its moneyness.     

A Quantitative Comparison between Shannon and Tsallis–Havrda–Charvat Entropies Applied to Cancer Outcome Prediction

In this paper, we propose to quantitatively compare loss functions based on parameterized Tsallis–Havrda–Charvat entropy and classical Shannon entropy for the training of a deep network in the case of small datasets which are usually encountered in medical applications. Shannon cross-entropy is widely used as a loss function for most neural networks applied to the segmentation, classification and detection of images. Shannon entropy is a particular case of Tsallis–Havrda–Charvat entropy. In this work, we compare these two entropies through a medical application for predicting recurrence in patients with head–neck and lung cancers after treatment. Based on both CT images and patient information, a multitask deep neural network is proposed to perform a recurrence prediction task using cross-entropy as a loss function and an image reconstruction task. Tsallis–Havrda–Charvat cross-entropy is a parameterized cross-entropy with the parameter α . Shannon entropy is a particular case of Tsallis–Havrda–Charvat entropy for α=1 . The influence of this parameter on the final prediction results is studied. In this paper, the experiments are conducted on two datasets including in total 580 patients, of whom 434 suffered from head–neck cancers and 146 from lung cancers. The results show that Tsallis–Havrda–Charvat entropy can achieve better performance in terms of prediction accuracy with some values of α .     

A data-centric approach to understanding the pricing of financial options

We investigate what can be learned from a purely phenomenological study of options prices without modelling assumptions. We fitted neural net (NN) models to LIFFE “ESX” European style FTSE 100 index options using daily data from 1992 to 1997. These non-parametric models reproduce the Black-Scholes (BS) analytic model in terms of fit and performance measures using just the usual five inputs (S, X, t, r, IV). We found that adding transaction costs (bid-ask spread) to these standard five parameters gives a comparable fit and performance. Tests show that the bid-ask spread can be a statistically significant explanatory variable for option prices. The difference in option prices between the models with transaction costs and those without ranges from about -3.0 to +1.5 index points, varying with maturity date. However, the difference depends on the moneyness (S/X), being greatest in-the-money. This suggests that use of a five-factor model can result in a pricing difference of up to ￡10 to ￡30 per call option contract compared with modelling under transaction costs. We found that the influence of transaction costs varied between different yearly subsets of the data. Open interest is also a significant explanatory variable, but volume is not. Received 31 December 2001     

Scaling and long-range dependence in option pricing I: Pricing European option with transaction costs under the fractional Black–Scholes model

This paper deals with the problem of discrete time option pricing by the fractional Black–Scholes model with transaction costs. By a mean self-financing delta-hedging argument in a discrete time setting, a European call option pricing formula is obtained. The minimal price of an option under transaction costs is obtained as timestep , which can be used as the actual price of an option. In fact, is an adjustment to the volatility in the Black–Scholes formula by using the modified volatility to replace the volatility σ, where is the Hurst exponent, and k is a proportional transaction cost parameter. In addition, we also show that timestep and long-range dependence have a significant impact on option pricing.     

The Cross-Sectional Intrinsic Entropy—A Comprehensive Stock Market Volatility Estimator

To take into account the temporal dimension of uncertainty in stock markets, this paper introduces a cross-sectional estimation of stock market volatility based on the intrinsic entropy model. The proposed cross-sectional intrinsic entropy (CSIE) is defined and computed as a daily volatility estimate for the entire market, grounded on the daily traded prices—open, high, low, and close prices (OHLC)—along with the daily traded volume for all symbols listed on The New York Stock Exchange (NYSE) and The National Association of Securities Dealers Automated Quotations (NASDAQ). We perform a comparative analysis between the time series obtained from the CSIE and the historical volatility as provided by the estimators: close-to-close, Parkinson, Garman–Klass, Rogers–Satchell, Yang–Zhang, and intrinsic entropy (IE), defined and computed from historical OHLC daily prices of the Standard & Poor’s 500 index (S&P500), Dow Jones Industrial Average (DJIA), and the NASDAQ Composite index, respectively, for various time intervals. Our study uses an approximate 6000-day reference point, starting 1 January 2001, until 23 January 2022, for both the NYSE and the NASDAQ. We found that the CSIE market volatility estimator is consistently at least 10 times more sensitive to market changes, compared to the volatility estimate captured through the market indices. Furthermore, beta values confirm a consistently lower volatility risk for market indices overall, between 50% and 90% lower, compared to the volatility risk of the entire market in various time intervals and rolling windows.     

The Relationship between Crude Oil Futures Market and Chinese/US Stock Index Futures Market Based on Breakpoint Test

Combined with the B-P (breakpoint) test and VAR–DCC–GARCH model, the relationship between WTI crude oil futures and S&P 500 index futures or CSI 300 index futures was investigated and compared. The results show that breakpoints exist in the relationship in the mean between WTI crude oil futures market and Chinese stock index futures market or US stock index futures market. The relationship in mean between WTI crude oil futures prices and S&P 500 stock index futures, or CSI 300 stock index futures is weakening. Meanwhile, there is a decreasing dynamic conditional correlation between the WTI crude oil futures market and Chinese stock index futures market or US stock index futures market after the breakpoint in the price series. The Chinese stock index futures are less affected by short-term fluctuations in crude oil futures returns than US stock index futures.     

Stock price dynamics and option valuations under volatility feedback effect

According to the volatility feedback effect, an unexpected increase in squared volatility leads to an immediate decline in the price–dividend ratio. In this paper, we consider the properties of stock price dynamics and option valuations under the volatility feedback effect by modeling the joint dynamics of stock price, dividends, and volatility in continuous time. Most importantly, our model predicts the negative effect of an increase in squared return volatility on the value of deep-in-the-money call options and, furthermore, attempts to explain the volatility puzzle. We theoretically demonstrate a mechanism by which the market price of diffusion return risk, or an equity risk-premium, affects option prices and empirically illustrate how to identify that mechanism using forward-looking information on option contracts. Our theoretical and empirical results support the relevance of the volatility feedback effect. Overall, the results indicate that the prevailing practice of ignoring the time-varying dividend yield in option pricing can lead to oversimplification of the stock market dynamics.     

Non-parametric extraction of implied asset price distributions

We present a fully non-parametric method for extracting risk neutral densities (RNDs) from observed option prices. The aim is to obtain a continuous, smooth, monotonic, and convex pricing function that is twice differentiable. Thus, irregularities such as negative probabilities that afflict many existing RND estimation techniques are reduced. Our method employs neural networks to obtain a smoothed pricing function, and a central finite difference approximation to the second derivative to extract the required gradients.This novel technique was successfully applied to a large set of FTSE 100 daily European exercise (ESX) put options data and as an Ansatz to the corresponding set of American exercise (SEI) put options. The results of paired t-tests showed significant differences between RNDs extracted from ESX and SEI option data, reflecting the distorting impact of early exercise possibility for the latter. In particular, the results for skewness and kurtosis suggested different shapes for the RNDs implied by the two types of put options. However, both ESX and SEI data gave an unbiased estimate of the realised FTSE 100 closing prices on the options’ expiration date. We confirmed that estimates of volatility from the RNDs of both types of option were biased estimates of the realised volatility at expiration, but less so than the LIFFE tabulated at-the-money implied volatility.     

A nonparametric approach for European option valuation

A nonparametric approach for European option valuation is proposed in this paper, which adopts a purely jump model to describe the price dynamics of the underlying asset, and the minimal entropy martingale measure for those jumps is used as the pricing measure of this market. A simple Monte Carlo simulation method is proposed to calculate the price of derivatives under this risk neural measure. And the volatility of the spot market can be renewed automatically without particular specification in the proposed method. The performances of the proposed method are compared to that of the Black-Scholes formula in an artificial world and the real world. The results of our investigations suggest that the proposed method is a valuable method.     

Simulation of nonlinear interest rates in quantum finance: Libor Market Model

The simulation of the Libor Market Model (LMM) is extensively studied in the framework of quantum finance. The imperfectly correlated Libor rates are simulated based on a Gaussian quantum field and a recursion equation of nontrivial stochastic drift. The Libor options are studied using both the simulation method and the analytical formula. The caplet price of simulation is compared with Black’s caplet formula which can be exactly derived from the LMM. The invariance of caplet price for different forward bond numeraire is verified by using the simulation. The simulation results for coupon bond options and swaptions are compared with the approximate price, which are limited for the reason that the approximate price is derived using the small volatility expansion. The simulation method is shown to have great potential in the application of pricing interest rate instruments.     

How to Promote the Performance of Parametric Volatility Forecasts in the Stock Market? A Neural Networks Approach

This study uses the fourteen stock indices as the sample and then utilizes eight parametric volatility forecasting models and eight composed volatility forecasting models to explore whether the neural network approach and the settings of leverage effect and non-normal return distribution can promote the performance of volatility forecasting, and which one of the sixteen models possesses the best volatility forecasting performance. The eight parametric volatility forecasts models are composed of the generalized autoregressive conditional heteroskedasticity (GARCH) or GJR-GARCH volatility specification combining with the normal, Student’s t, skewed Student’s t, and generalized skewed Student’s t distributions. Empirical results show that, the performance for the composed volatility forecasting approach is significantly superior to that for the parametric volatility forecasting approach. Furthermore, the GJR-GARCH volatility specification has better performance than the GARCH one. In addition, the non-normal distribution does not have better forecasting performance than the normal distribution. In addition, the GJR-GARCH model combined with both the normal distribution and a neural network approach has the best performance of volatility forecasting among sixteen models. Thus, a neural network approach significantly promotes the performance of volatility forecasting. On the other hand, the setting of leverage effect can encourage the performance of volatility forecasting whereas the setting of non-normal distribution cannot.     

Inequalities for Jensen–Sharma–Mittal and Jeffreys–Sharma–Mittal Type f–Divergences

In this paper, we introduce new divergences called Jensen–Sharma–Mittal and Jeffreys–Sharma–Mittal in relation to convex functions. Some theorems, which give the lower and upper bounds for two new introduced divergences, are provided. The obtained results imply some new inequalities corresponding to known divergences. Some examples, which show that these are the generalizations of Rényi, Tsallis, and Kullback–Leibler types of divergences, are provided in order to show a few applications of new divergences.     

Analysis on International Competitiveness of Service Trade in the Guangdong–Hong Kong–Macao Greater Bay Area Based on Using the Entropy and Gray Correlation Methods

Based on the analysis and measurement of the overall situation, import and export structure and international competitiveness of the various sectors of service trade in the Guangdong–Hong Kong–Macao Greater Bay Area, with the help of MATLAB and Gray System Modeling software, the synergy degree model was established to quantitatively analyze the synergy level of service trade in the Greater Bay Area with the help of grey correlation analysis method and entropy weight method. The results show that the overall development trend of service trade in the Guangdong–Hong Kong–Macao Greater Bay Area is good. The service trade industries in different regions are highly complementary and have a high degree of correlation. The potential for the coordinated development of internal service trade is excellent, and the overall situation of service trade in the Greater Bay Area is in a stage of transition from a moderate level of synergy to a high level of synergy. The Greater Bay Area can achieve industrial synergy by accelerating industrial integration and green transformation, establishing a coordinated development mechanism, sharing market platform, strengthening personnel security, and further enhancing the international competitiveness of service trade. The established model better reflects the current coordination of service trade in the Guangdong–Hong Kong–Macao Greater Bay Area and has good applicability. In the future, more economic, technological, geographic, and policy data and information can be comprehensively used to study the spatial pattern, evolution rules, and mechanisms of coordinated development in the broader area.     

Estimating Distributions of Parameters in Nonlinear State Space Models with Replica Exchange Particle Marginal Metropolis–Hastings Method

Extracting latent nonlinear dynamics from observed time-series data is important for understanding a dynamic system against the background of the observed data. A state space model is a probabilistic graphical model for time-series data, which describes the probabilistic dependence between latent variables at subsequent times and between latent variables and observations. Since, in many situations, the values of the parameters in the state space model are unknown, estimating the parameters from observations is an important task. The particle marginal Metropolis–Hastings (PMMH) method is a method for estimating the marginal posterior distribution of parameters obtained by marginalization over the distribution of latent variables in the state space model. Although, in principle, we can estimate the marginal posterior distribution of parameters by iterating this method infinitely, the estimated result depends on the initial values for a finite number of times in practice. In this paper, we propose a replica exchange particle marginal Metropolis–Hastings (REPMMH) method as a method to improve this problem by combining the PMMH method with the replica exchange method. By using the proposed method, we simultaneously realize a global search at a high temperature and a local fine search at a low temperature. We evaluate the proposed method using simulated data obtained from the Izhikevich neuron model and Lévy-driven stochastic volatility model, and we show that the proposed REPMMH method improves the problem of the initial value dependence in the PMMH method, and realizes efficient sampling of parameters in the state space models compared with existing methods.     

Study of Nonlinear Models of Oscillatory Systems by Applying an Intelligent Computational Technique

In this paper, we have analyzed the mathematical model of various nonlinear oscillators arising in different fields of engineering. Further, approximate solutions for different variations in oscillators are studied by using feedforward neural networks (NNs) based on the backpropagated Levenberg–Marquardt algorithm (BLMA). A data set for different problem scenarios for the supervised learning of BLMA has been generated by the Runge–Kutta method of order 4 (RK-4) with the “NDSolve” package in Mathematica. The worth of the approximate solution by NN-BLMA is attained by employing the processing of testing, training, and validation of the reference data set. For each model, convergence analysis, error histograms, regression analysis, and curve fitting are considered to study the robustness and accuracy of the design scheme.     

Time-changed geometric fractional Brownian motion and option pricing with transaction costs

This paper deals with the problem of discrete time option pricing by a fractional subdiffusive Black–Scholes model. The price of the underlying stock follows a time-changed geometric fractional Brownian motion. By a mean self-financing delta-hedging argument, the pricing formula for the European call option in discrete time setting is obtained.     

Refined Young Inequality and Its Application to Divergences

We give bounds on the difference between the weighted arithmetic mean and the weighted geometric mean. These imply refined Young inequalities and the reverses of the Young inequality. We also studied some properties on the difference between the weighted arithmetic mean and the weighted geometric mean. Applying the newly obtained inequalities, we show some results on the Tsallis divergence, the Rényi divergence, the Jeffreys–Tsallis divergence and the Jensen–Shannon–Tsallis divergence.     

Comparative Analysis of Different Univariate Forecasting Methods in Modelling and Predicting the Romanian Unemployment Rate for the Period 2021–2022

Unemployment has risen as the economy has shrunk. The coronavirus crisis has affected many sectors in Romania, some companies diminishing or even ceasing their activity. Making forecasts of the unemployment rate has a fundamental impact and importance on future social policy strategies. The aim of the paper is to comparatively analyze the forecast performances of different univariate time series methods with the purpose of providing future predictions of unemployment rate. In order to do that, several forecasting models (seasonal model autoregressive integrated moving average (SARIMA), self-exciting threshold autoregressive (SETAR), Holt–Winters, ETS (error, trend, seasonal), and NNAR (neural network autoregression)) have been applied, and their forecast performances have been evaluated on both the in-sample data covering the period January 2000–December 2017 used for the model identification and estimation and the out-of-sample data covering the last three years, 2018–2020. The forecast of unemployment rate relies on the next two years, 2021–2022. Based on the in-sample forecast assessment of different methods, the forecast measures root mean squared error (RMSE), mean absolute error (MAE), and mean absolute percent error (MAPE) suggested that the multiplicative Holt–Winters model outperforms the other models. For the out-of-sample forecasting performance of models, RMSE and MAE values revealed that the NNAR model has better forecasting performance, while according to MAPE, the SARIMA model registers higher forecast accuracy. The empirical results of the Diebold–Mariano test at one forecast horizon for out-of-sample methods revealed differences in the forecasting performance between SARIMA and NNAR, of which the best model of modeling and forecasting unemployment rate was considered to be the NNAR model.     

Decision Rules Construction: Algorithm Based on EAV Model

In the paper, an approach for decision rules construction is proposed. It is studied from the point of view of the supervised machine learning task, i.e., classification, and from the point of view of knowledge representation. Generated rules provide comparable classification results to the dynamic programming approach for optimization of decision rules relative to length or support. However, the proposed algorithm is based on transformation of decision table into entity–attribute–value (EAV) format. Additionally, standard deviation function for computation of averages’ values of attributes in particular decision classes was introduced. It allows to select from the whole set of attributes only these which provide the highest degree of information about the decision. Construction of decision rules is performed based on idea of partitioning of a decision table into corresponding subtables. In opposite to dynamic programming approach, not all attributes need to be taken into account but only these with the highest values of standard deviation per decision classes. Consequently, the proposed solution is more time efficient because of lower computational complexity. In the framework of experimental results, support and length of decision rules were computed and compared with the values of optimal rules. The classification error for data sets from UCI Machine Learning Repository was also obtained and compared with the ones for dynamic programming approach. Performed experiments show that constructed rules are not far from the optimal ones and classification results are comparable to these obtained in the framework of the dynamic programming extension.