In this paper, finite element method is applied to Leland's model for numerical simulation of option pricing with transaction costs. Spatial finite element models based on P1 and/or P2 elements are formulated in combination with a Crank-Nicolson-type temporal scheme. The temporal scheme is implemented using the Rannacher approach. Examples with several sets of parameter values are presented and compared with finite difference results in the literature. Spatial-temporal mesh-size ratios are observed for controlling the stability of our method. Our results compare favorably with the finite difference results in the literature for the model.

Many empirical studies have discussed market liquidity, which is regarded as a measure of a booming financial market. Further, various indicators for objectively evaluating market liquidity have also been proposed and their merits have been discussed. In recent years, the impact of high-frequency traders (HFTs) on financial markets has been a focal concern, but no studies have systematically discussed their relationship with major market liquidity indicators, including volume, tightness, resiliency, and depth. In this study, we used agent-based simulations to compare the major liquidity indicators in an artificial market where an HFT participated was compared to one where no HFT participated. The results showed that all liquidity indicators in the market where an HFT participated improved more than those in the market where no HFT participated. Furthermore, as a result of investigating the correlations between the major liquidity indicators in our simulations and the extant empirical literature, we found that market liquidity can be measured not only by the major liquidity indicators but also by execution rate. Therefore, it is suggested that it could be appropriate to employ execution rate as a novel liquidity indicator in future studies.

This paper introduces forward-looking measures of the network connectedness of fears in the financial system, arising due to the good and bad beliefs of market participants about uncertainty that spreads unequally across a network of banks. We argue that this asymmetric network structure extracted from call and put traded option prices of the main U.S. banks contains valuable information for predicting macroeconomic conditions and economic uncertainty, and it can serve as a tool for forward-looking systemic risk monitoring.

In this paper, we study a semi-martingale optimal transport problem and its application to the calibration of Local-Stochastic Volatility (LSV) models. Rather than considering the classical constraints on marginal distributions at initial and final time, we optimise our cost function given the prices of a finite number of European options. We formulate the problem as a convex optimisation problem, for which we provide a PDE formulation along with its dual counterpart. Then we solve numerically the dual problem, which involves a fully non-linear Hamilton-Jacobi-Bellman equation. The method is tested by calibrating a Heston-like LSV model with simulated data and foreign exchange market data.

This research is to assess cryptocurrencies with the conditional beta, compared with prior studies based on unconditional beta or fixed beta. It is a new approach to building a pricing model for cryptocurrencies. Therefore, we expect that the use of conditional beta will increase the explanatory ability of factors in previous pricing models. Besides, this research is also a pioneer in placing the uncertainty factor in the cryptocurrency pricing model. Earlier studies on cryptocurrency pricing have ignored this factor. However, it is a significant factor in the valuation of cryptocurrencies because uncertainty leads to investor sentiment and affects prices.

We present a neural network (NN) approach to fit and predict implied volatility surfaces (IVSs). Atypically to standard NN applications, financial industry practitioners use such models equally to replicate market prices and to value other financial instruments. In other words, low training losses are as important as generalization capabilities. Importantly, IVS models need to generate realistic arbitrage-free option prices, meaning that no portfolio can lead to risk-free profits. We propose an approach guaranteeing the absence of arbitrage opportunities by penalizing the loss using soft constraints. Furthermore, our method can be combined with standard IVS models in quantitative finance, thus providing a NN-based correction when such models fail at replicating observed market prices. This lets practitioners use our approach as a plug-in on top of classical methods. Empirical results show that this approach is particularly useful when only sparse or erroneous data are available. We also quantify the uncertainty of the model predictions in regions with few or no observations. We further explore how deeper NNs improve over shallower ones, as well as other properties of the network architecture. We benchmark our method against standard IVS models. By evaluating our method on both training sets, and testing sets, namely, we highlight both their capacity to reproduce observed prices and predict new ones.

Discrete-choice life cycle models can be used to, e.g., estimate how social security reforms change employment rate. Optimal employment choices during the life course of an individual can be solved in the framework of life cycle models. This enables estimating how a social security reform influences employment rate. Mostly, life cycle models have been solved with dynamic programming, which is not feasible when the state space is large, as often is the case in a realistic life cycle model. Solving such life cycle models requires the use of approximate methods, such as reinforced learning algorithms. We compare how well a deep reinforced learning algorithm ACKTR and dynamic programming solve a relatively simple life cycle model. We find that the average utility is almost the same in both algorithms, however, the details of the best policies found with different algorithms differ to a degree. In the baseline model representing the current Finnish social security scheme, we find that reinforced learning yields essentially as good results as dynamics programming. We then analyze a straight-forward social security reform and find that the employment changes due to the reform are almost the same. Our results suggest that reinforced learning algorithms are of significant value in analyzing complex life cycle models.

The general method is proposed for constructing a family of martingale measures for a wide class of evolution of risky assets. The sufficient conditions are formulated for the evolution of risky assets under which the family of equivalent martingale measures to the original measure is a non-empty set. The set of martingale measures is constructed from a set of strictly nonneg ative random variables, satisfying certain conditions. The inequalities are obtained for the non-negative random variables satisfying certain conditions. Using these inequalities, a new simple proof of optional decomposition theorem for the nonnegative super-martingale is proposed. The family of spot measures is introduced and the representation is found for them. The conditions are found under which each martingale measure is an integral over the set of spot measures. On the basis of nonlinear processes such as ARCH and GARCH, the parametric family of random processes is introduced for which the interval of non-arbitrage prices are found. The formula is obtained for the fair price of the contract with option of European type for the considered parametric processes. The parameters of the introduced random processes are estimated and the estimate is found at which the fair price of contract with option is the least.

A significant number of the non-financial firms listed at Nairobi Securities Exchange (NSE) have been experiencing declining financial performance which deter investors from investing in such firms. The lenders are also not willing to lend to such firms. As such, the firms struggle to raise funds for their operations. Prudent financing decisions can lead to financial growth of the firm. The purpose of this study is to assess the effect of Long-term debt on the financial growth of Non-financial firms listed at Nairobi Securities Exchange. Financial firms were excluded because of their specific sector characteristics and stringent regulatory framework. The study is guided by Trade-Off Theory and Theory of Growth of the Firm. Explanatory research design was adopted. The population of the study comprised of 45 non-financial firms listed at the NSE for a period of ten years from 2008 to 2017. The study conducted both descriptive statistics analysis and panel data analysis. The result indicates that Long term debt explains 21.6% and 5.16% of variation in financial growth as measured by growth in earnings per share and growth in market capitalization respectively. Long term debt positively and significantly influences financial growth measured using both growth in earnings per share and growth in market capitalization. The study recommends that, the management of non-financial firms listed at Nairobi Securities Exchange to employ financing means that can improve the earnings per share, market capitalization and enhance the value of the firm for the benefit of its stakeholders.

Factor modeling of asset returns has been a dominant practice in investment science since the introduction of the Capital Asset Pricing Model (CAPM) and the Arbitrage Pricing Theory (APT). The factors, which account for the systematic risk, are either specified or interpreted to be exogenous. They explain a significant portion of the risk in large portfolios. We propose a framework that asks how much of the risk, that we see in equity markets, may be explained by the asset returns themselves. To answer this question, we decompose the asset returns into an endogenous component and the remainder, and analyze the properties of the resulting risk decomposition. Statistical methods to estimate this decomposition from data are provided along with empirical tests. Our results point to the possibility that most of the risk in equity markets may be explained by a sparse network of interacting assets (or their issuing firms). This sparse network can give the appearance of a set exogenous factors where, in fact, there may be none. We illustrate our results with several case studies.

We provide a survey of the Kolkata index of social inequality, focusing in particular on income inequality. Based on the observation that inequality functions (such as the Lorenz function), giving the measures of income or wealth against that of the population, to be generally nonlinear, we show that the fixed point (like Kolkata index k) of such a nonlinear function (or related, like the complementary Lorenz function) offer better measure of inequality than the average quantities (like Gini index). Indeed the Kolkata index can be viewed as a generalized Hirsch index for a normalized inequality function and gives the fraction k of the total wealth possessed by the rich (1-k) fraction of the population. We analyze the structures of the inequality indices for both continuous and discrete income distributions. We also compare the Kolkata index to some other measures like the Gini coefficient and the Pietra index. Lastly, we provide some empirical studies which illustrate the differences between the Kolkata index and the Gini coefficient.

We investigate a machine learning approach to option Greeks approximation based on Gaussian process (GP) surrogates. The method takes in noisily observed option prices, fits a nonparametric input-output map and then analytically differentiates the latter to obtain the various price sensitivities. Our motivation is to compute Greeks in cases where direct computation is expensive, such as in local volatility models, or can only ever be done approximately. We provide a detailed analysis of numerous aspects of GP surrogates, including choice of kernel family, simulation design, choice of trend function and impact of noise.

We further discuss the application to Delta hedging, including a new Lemma that relates quality of the Delta approximation to discrete-time hedging loss. Results are illustrated with two extensive case studies that consider estimation of Delta, Theta and Gamma and benchmark approximation quality and uncertainty quantification using a variety of statistical metrics. Among our key take-aways are the recommendation to use Matern kernels, the benefit of including virtual training points to capture boundary conditions, and the significant loss of fidelity when training on stock-path-based datasets.

This paper aims to make a new contribution to the study of lifetime ruin problem by considering investment in two hedge funds with high-watermark fees and drift uncertainty. Due to multi-dimensional performance fees that are charged whenever each fund profit exceeds its historical maximum, the value function is expected to be multi-dimensional. New mathematical challenges arise as the standard dimension reduction cannot be applied, and the convexity of the value function and Isaacs condition may not hold in our ruin probability minimization problem with drift uncertainty. We propose to employ the stochastic Perron's method to characterize the value function as the unique viscosity solution to the associated Hamilton Jacobi Bellman (HJB) equation without resorting to the proof of dynamic programming principle. The required comparison principle is also established in our setting to close the loop of stochastic Perron's method.

This paper concerns portfolio selection with multiple assets under rough covariance matrix. We investigate the continuous-time Markowitz mean-variance problem for a multivariate class of affine and quadratic Volterra models. In this incomplete non-Markovian and non-semimartingale market framework with unbounded random coefficients, the optimal portfolio strategy is expressed by means of a Riccati backward stochastic differential equation (BSDE). In the case of affine Volterra models, we derive explicit solutions to this BSDE in terms of multi-dimensional Riccati-Volterra equations. This framework includes multivariate rough Heston models and extends the results of \cite{han2019mean}. In the quadratic case, we obtain new analytic formulae for the the Riccati BSDE and we establish their link with infinite dimensional Riccati equations. This covers rough Stein-Stein and Wishart type covariance models. Numerical results on a two dimensional rough Stein-Stein model illustrate the impact of rough volatilities and stochastic correlations on the optimal Markowitz strategy. In particular for positively correlated assets, we find that the optimal strategy in our model is a `buy rough sell smooth' one.

In this paper, we study the mean-variance portfolio selection problem under partial information with drift uncertainty. First we show that the market model is complete even in this case while the information is not complete and the drift is uncertain. Then, the optimal strategy based on partial information is derived, which reduces to solving a related backward stochastic differential equation (BSDE). Finally, we propose an efficient numerical scheme to approximate the optimal portfolio that is the solution of the BSDE mentioned above. Malliavin calculus and the particle representation play important roles in this scheme.

Stock price movement prediction is commonly accepted as a very challenging task due to the volatile nature of financial markets. Previous works typically predict the stock price mainly based on its own information, neglecting the cross effect among involved stocks. However, it is well known that an individual stock price is correlated with prices of other stocks in complex ways. To take the cross effect into consideration, we propose a deep learning framework, called Multi-GCGRU, which comprises graph convolutional network (GCN) and gated recurrent unit (GRU) to predict stock movement. Specifically, we first encode multiple relationships among stocks into graphs based on financial domain knowledge and utilize GCN to extract the cross effect based on these pre-defined graphs. To further get rid of prior knowledge, we explore an adaptive relationship learned by data automatically. The cross-correlation features produced by GCN are concatenated with historical records and then fed into GRU to model the temporal dependency of stock prices. Experiments on two stock indexes in China market show that our model outperforms other baselines. Note that our model is rather feasible to incorporate more effective stock relationships containing expert knowledge, as well as learn data-driven relationship.

This paper studies the multilevel Monte-Carlo estimator for the expectation of a maximum of conditional expectations. This problem arises naturally when considering many stress tests and appears in the calculation of the interest rate module of the standard formula for the SCR. We obtain theoretical convergence results that complements the recent work of Giles and Goda and gives some additional tractability through a parameter that somehow describes regularity properties around the maximum. We then apply the MLMC estimator to the calculation of the SCR at future dates with the standard formula for an ALM savings business on life insurance. We compare it with estimators obtained with Least Square Monte-Carlo or Neural Networks. We find that the MLMC estimator is computationally more efficient and has the main advantage to avoid regression issues, which is particularly significant in the context of projection of a balance sheet by an insurer due to the path dependency. Last, we discuss the potentiality of this numerical method and analyze in particular the effect of the portfolio allocation on the SCR at future~dates.

We propose a robust risk measurement approach that minimizes the expectation of overestimation plus underestimation costs. We consider uncertainty by taking the supremum over a collection of probability measures, relating our approach to dual sets in the representation of coherent risk measures. We provide results that guarantee the existence of a solution and explore the properties of minimizer and minimum as risk and deviation measures, respectively. An empirical illustration is carried out to demonstrate the use of our approach in capital determination.

We consider a semilinear equation linked to the finite horizon consumption - investment problem under the stochastic factor framework and we prove it admits a classical solution and provide all obligatory estimates to successfully apply a verification reasoning. The paper covers the standard time additive utility, as well as the recursive utility framework. We extend existing results by considering more general factor dynamics including a non-trivial diffusion part and a stochastic correlation between assets and factors. In addition, this is the first paper which compromises many other optimization problems in finance, for example those related to the indifference pricing or the quadratic hedging problem. The extension of the result to the stochastic differential utility and robust portfolio optimization is provided as well. The essence of our paper lays in using improved stochastic methods to prove gradient estimates for suitable HJB equations with restricted control space.

Mining blocks on a blockchain equipped with a proof of work consensus protocol is well-known to be resource-consuming. A miner bears the operational cost, mainly electricity consumption and IT gear, of mining, and is compensated by a capital gain when a block is discovered. This paper aims at quantifying the profitability of mining when the possible event of ruin is also considered. This is done by formulating a tractable stochastic model and using tools from applied probability and analysis, including the explicit solution of a certain type of advanced functional differential equation. The expected profit at a future time point is determined for the situation when the miner follows the protocol as well as when he/she withholds blocks. The obtained explicit expressions allow to analyze the sensitivity with respect to the different model ingredients and to identify conditions under which selfish mining is a strategic advantage.

\noindent We address the issue of market making on electronic markets when taking into account the clustering and long memory properties of market order flows. We consider a market model with one market maker and order flows driven by general Hawkes processes. We formulate the market maker's objective as a stochastic control problem. We characterize an optimal control by proving existence and uniqueness of a viscosity solution to the associated Hamilton-Jacobi-Bellman equation. Finally we propose a fully consistent numerical method allowing to implement this optimal strategy in practice.

The BS equations with fractional order two asset price models give a better prediction of options pricing in the monetary market. In this paper, the changed form of BS-condition with two asset price models dependent on the Liovelle-Caputo derivative for good predictions of options prices are utilized. The analytical solution is demonstrated in form of convergent infinite series and obtained by the properties of Samudu Transform.

Carpooling is a system in which drivers accept to add some limited detours to their habitual journeys to pick-up and drop-off other riders. Most research and operating platforms present carpooling as an alternative to fixed schedule transit and only very little work has attempted to integrate it with fixed-schedule mass transit. The aim of this paper is to showcase the benefits of such integration, under the philosophy of Mobility as a Service (MaaS), in a daily commuting scenario. We present an integrated mass transit plus carpooling system that, by design, constructs multimodal trips, including transit and carpooling legs. To this aim, the system generates vehicle detours in order to serve transit stations. We evaluate the performance of this system via simulation. We compare the ``Current'' System, where carpooling is an alternative to transit, to our ``Integrated'' System, where carpooling and transit are integrated in a single system. We show that, by doing this, the transportation accessibility greatly increases: about 40\% less users remain without feasible travel options and the overall travel time decreases by about 10\%. We achieve this by requiring relatively small driver detours, thanks to a better utilization vehicle routes, with drivers' vehicles driving on average with more riders on board. The simulation code is available open source.

The field of portfolio selection is an active research topic, which combines elements and methodologies from various fields, such as optimization, decision analysis, risk management, data science, forecasting, etc. The modeling and treatment of deep uncertainties for future asset returns is a major issue for the success of analytical portfolio selection models. Recently, robust optimization (RO) models have attracted a lot of interest in this area. RO provides a computationally tractable framework for portfolio optimization based on relatively general assumptions on the probability distributions of the uncertain risk parameters. Thus, RO extends the framework of traditional linear and non-linear models (e.g., the well-known mean-variance model), incorporating uncertainty through a formal and analytical approach into the modeling process. Robust counterparts of existing models can be considered as worst-case re-formulations as far as deviations of the uncertain parameters from their nominal values are concerned. Although several RO models have been proposed in the literature focusing on various risk measures and different types of uncertainty sets about asset returns, analytical empirical assessments of their performance have not been performed in a comprehensive manner. The objective of this study is to fill in this gap in the literature. More specifically, we consider different types of RO models based on popular risk measures and conduct an extensive comparative analysis of their performance using data from the US market during the period 2005-2016.

A leveraged ETF is a fund aimed at achieving a rate of return several times greater than that of the underlying asset such as Nikkei 225 futures. Recently, it has been suggested that rebalancing trades of a leveraged ETF may destabilize the financial markets. An empirical study using an agent-based simulation indicated that a rebalancing trade strategy could affect the price formation of an underlying asset market. However, no leveraged ETF trading method for suppressing the increase in volatility as much as possible has yet been proposed. In this paper, we compare different strategies of trading for a proposed trading model and report the results of our investigation regarding how best to suppress an increase in market volatility. As a result, it was found that as the minimum number of orders in a rebalancing trade increases, the impact on the market price formation decreases.

The art of systematic financial trading evolved with an array of approaches, ranging from simple strategies to complex algorithms all relying, primary, on aspects of time-series analysis. Recently, after visiting the trading floor of a leading financial institution, we noticed that traders always execute their trade orders while observing images of financial time-series on their screens. In this work, we built upon the success in image recognition and examine the value in transforming the traditional time-series analysis to that of image classification. We create a large sample of financial time-series images encoded as candlestick (Box and Whisker) charts and label the samples following three algebraically-defined binary trade strategies. Using the images, we train over a dozen machine-learning classification models and find that the algorithms are very efficient in recovering the complicated, multiscale label-generating rules when the data is represented visually. We suggest that the transformation of continuous numeric time-series classification problem to a vision problem is useful for recovering signals typical of technical analysis.

Companies survey their customers to measure their satisfaction levels with the company and its services. The received responses are crucial as they allow companies to assess their respective performances and find ways to make needed improvements. This study focuses on the non-systematic bias that arises when customers assign numerical values in ordinal surveys. Using real customer satisfaction survey data of a large retail bank, we show that the common practice of segmenting ordinal survey responses into uneven segments limit the value that can be extracted from the data. We then show that it is possible to assess the magnitude of the irreducible error under simple assumptions, even in real surveys, and place the achievable modeling goal in perspective. We finish the study by suggesting that a thoughtful survey design, which uses either a careful binning strategy or proper calibration, can reduce the compounding non-systematic error even in elaborated ordinal surveys. A possible application of the calibration method we propose is efficiently conducting targeted surveys using active learning.