Computational Finance

We discuss two numerical techniques, based on the path integral approach described in a previous paper, for solving the stochastic equations underlying the financial markets: the path integral Monte Carlo, and the path integral deterministic evaluation. In particular, we apply the latter to some specific financial problems: the pricing of a European option, a zero-coupon bond, a caplet, an American option, and a Bermudan swaption.

With the latest developments in the area of advanced computer architectures, we are already seeing large scale machines at petascale level and we are faced with the exascale computing challenge. All these require scalability at system, algorithmic and mathematical model level. In particular, efficient scalable algorithms are required to bridge the performance gap. In this paper, examples of various approaches of designing scalable algorithms for such advanced architectures will be given. We will briefly present our approach to Monte Carlo scalable algorithms for Linear Algebra and explain how these approaches are extended to the field of Computational Finance. Implementation examples will be presented using Linear Algebra Problems and problems from Computational Finance. Furthermore, the corresponding properties of these algorithms will be outlined and discussed.

The Impact of dividend decision on the share price volatility in the Indian Stock Market is the subject matter of this study. Using the correlation analysis and the least square multiple regression methods, the study found only 32% of the changes in the share prices are explained by Dividend Yield, Dividend Payout Ratio, Investment growth, Size of the Firm, Leverage, and Earnings Volatility.

High Level Synthesis (HLS) tools for Field Programmable Gate Arrays (FPGAs) have made considerable progress, and are now sufficiently mature that a novice developer could create functionally correct implementation with limited understanding of the target hardware. In this case study, a novice developer considers a benchmark of financial problems for implementation upon FPGA via HLS. This novice starts by extending an existing implementation for a CPU or GPU using tools such as Xilinx's Vivado HLS, the Altera OpenCL SDK or Maxeler's MaxCompiler. When their direct source code translation inevitably didn't meet performance expectations, this developer then applies optimisations such as exploiting task or pipeline parallelism as well as C-slowing. When a combination of these optimisations are considered for a range of devices and process technologies, an acceleration of up to 220 times is achieved using these tools, the sort of acceleration expected of custom architectures. Compared to the 31 times improvement shown by an optimised Multicore CPU implementation, the 60 times improvement by a GPU and 207 times by a Xeon Phi, these results suggest that HLS is indeed ready for industrial adoption.

Dalam perhitungan harga opsi eropa, sampai saat ini rumus Black-Scholes masih merupakan satu-satunya cara ekplisit untuk melakukan perhitungan. Namun dengan menggunakan rumus ini, hanya akan diperoleh satu harga opsi saja. Dengan mengaproksimasi PDP Black-Scholes dengan menggunakan Skema Forward in Time and Central in Space (FTCN), Backward in Time and Central in Space (BTCN), dan Crank-Nicolson, memungkinkan kita memperoleh berbagai nilai opsi untuk berbagai nilai saham sadal satu kali iterasi. Tugas ini bertujuan untuk membahas aplikasi metode Forward in Time and Central in Space (FTCN), Backward in Time and Central in Space (BTCN), dan Crank-Nicolson untuk menyelesaikan persamaan diferensial parsial Black-Scholes untuk opsi call Eropa. Selain itu juga akan dianalisis error yang dihasilkan ketiga metode tersebut, terhadap hasil rumus Black-Scholes. Diketahui persamaan di¤erensial di¤erensial parsial Black-Scholes seperti berikut @f @t + 1 2 2 s 2 @ 2 f @s 2 + rs @f @s rf = 0 dimana f menyatakan harga opsi yang dapat berubah terhadap waktu (t) dan dipengaruhi juga oleh harga saham (s). : volatility harga saham r : interest rate 2 Teori Dasar 2.1 Transformasi PDP Black-Scholes

We propose a framework for studying optimal market making policies in a limit order book (LOB). The bid-ask spread of the LOB is modelled by a Markov chain with finite values, multiple of the tick size, and subordinated by the Poisson process of the tick-time clock. We consider a small agent who continuously submits limit buy/sell orders at best bid/ask quotes, and may also set limit orders at best bid (resp. ask) plus (resp. minus) a tick for getting the execution order priority, which is a crucial issue in high frequency trading. By trading with limit orders, the agent faces an execution risk since her orders are executed only when they meet counterpart market orders, which are modelled by Cox processes with intensities depending on the spread and on her limit prices. By holding non-zero positions on the risky asset, the agent is also subject to the inventory risk related to price volatility. Then the agent can also choose to trade with market orders, and therefore get immediate execution, but at a least favorable price because she has to cross the bid-ask spread.

This thesis demonstrates the use of deep learning for automating hourly price forecasts in continuous intraday electricity markets, using various types of neural networks on comprehensive sequential market data and cutting-edge image processing networks on spatio-temporal weather forecast maps. Deep learning is a subfield of artificial intelligence that excels on problems such as these with multifarious input data and manifold interacting factors. It has seen tremendous success on a range of problems across industries, and while it is important to have realistic expectations, there is little reason to believe that intraday electricity markets are different. Focusing on Nord Pool’s intraday market Elbas, we predict Nordic buyers’ volume-weighted average price over the last six hours of trading prior to each delivery hour. Aggregating this window gives buyers flexibility from many trades and sufficient time in which to act on the predictions, and solves issues with data sparsity while keeping sufficient resolution for predictions to be informative.

We develop various neural networks via extensive experimentation, with inspiration from other research and problem domains. To make the findings relevant in practice, we impose constraints on the input data based on what would be available to Elbas market participants six trading hours ahead of delivery. The neural networks are benchmarked against a set of simple domain- based heuristics and traditional methods from econometrics and machine learning. We conclude with a holistic evaluation of the efficacy of deep learning on our problem, whether it is economically justifiable in light of its value-add, what the salient hurdles are to implementing it in practice, and what the implications are for broader applications of AI in intraday markets.

The deep learning models are relatively accurate and reliable under normal market conditions. The average price across all delivery hours in the held-out data is 30.95 EUR/MWh, where our best network is on average off by 2.72 EUR/MWh. It beats the best simple heuristic by 21–25%, and the best benchmark model by 12–16%. The network also anticipates major fluctuations in prices relatively consistently, and generally outperforms all alternative methods when prices are especially volatile or trading activity particularly high. In contrast to the benchmarks, there are also ample avenues for improving the network further. Beyond being promising in its own right, we also argue that the network demonstrates the wider potential of deep learning in a range of applications in intraday markets, and that these are worthy of serious consideration — though one should be aware of the practical hurdles to implementing them operationally.

In this project, we have implemented an exact pricer using a continuous analytic formula and a Partial Differential Equation (PDE) pricer for barrier options in C++ and compared the results obtained from both functions. For the PDE pricer, we have also added a setAlignment method that allows for the grid to be aligned to the barrier level. Furthermore, we have investigated the convergence properties of the PDE pricer. We find that the PDE pricer exhibits a linear order of convergence, and that the fully implicit scheme has smoother convergence (fewer fluctuations) compared to the Crank-Nicholson scheme. Moreover, we also find that the Broadie-Glasserman-Kou adjustment formula for the continuous analytic formula provides a close approximation to the discrete PDE pricer.

The Markowitz mean-variance portfolio theory posits that the optimal portfolio weights can be chosen based off an efficient tradeoff between profit modeled as the mean and risk measured as the variance-covariance matrix. These values must be historically estimated, occasionally leading to poor portfolio performance and asset misallocation. Moreover, as Brodie et. al. point out, the Markowitz optimization scheme, “is empirically unstable,” making, “the classic Markowitz portfolio optimization an ill-posed (or ill-conditioned) inverse problem.”  Consequently, this paper formulates the global minimum variance portfolio as a regression problem with added penalties. This analysis thus utilizes penalty regularization on training data to more efficiently estimate global minimum variance portfolio weights and resolve many of the classic scheme’s issues.

The widespread use and proven profitability of technical trading rules in financial markets has long been a puzzle in academic finance. In this paper we show, using an agent-based model of an evolving stock market, that widespread technical trading can arise due to a multi-person prisoners' dilemma in which the inclusion of techinical trading rules to a single agent's repertoire of rules is a dominant strategy. The use of this dominant strategy by all traders in the market creates a symmetric Nash equilibrium in which wealth earned is lower and the volatility of prices is higher than in the hypothetical case in which all agents rely only on fundamental rules. Our explanation of this lower wealth and higher volatility is that the use of technical trading rules worsens the accuracy of the predictions of all agents' market forecasts by contributing to the reinforcement of price trends, augmenting volatility, and increasing the amount of noise in the market. Submitted to Con...

Asian options belong to the so-called path-dependent derivatives. They are among the most difficult to price and hedge, both analytically and numerically. Basket options are even harder to price and hedge because of the large number of state variables. Several approaches have been proposed in the literature, including, among others, Monte Carlo simulations, tree-based methods, partial differential equations, and analytical approximations. The last category is the most appealing because most of the other methods are very complex and slow. Our method belongs to the analytical approximation class. It is based on the observation that, though the weighted average of lognormal variables is no longer lognormal, it can be approximated by a lognormal random variable if the first two moments match the true first two moments. To have a better approximation, we consider the Taylor expansion of the ratio of the characteristic function of the average to that of the approximating lognormal random variable around zero volatility. We include terms up to σ 6 in the expansion. The resulting option formulas are in closed form. We treat discrete Asian options as a special case of basket options. Formulas for continuous Asian options are obtained from their discrete counterparts. Numerical tests indicate that the formulas are very accurate. Comparisons with all other leading analytical approximations show that our method has performed the best overall in terms of accuracy for both short-and long-maturity options. Furthermore, unlike some other methods, our approximation treats basket (portfolio) and Asian options in a unified way. Lastly, in an Appendix we point out a serious mathematical error in a popular method of pricing Asian options in the literature.

Shout option is a modified contract of European type option. This contract can only be exercised at the expiration date, and holder has a right to reset the exercise price into the current level of underlying asset in along time to expiry date. This work try to price the option by solving a linear complementary problem, and consider to use Discrete Morse Flow (DMF) to approximate the solution. DMF is a minimizing scheme to solve differential equation depend on time, and concerning space variable. It solves differential equation in divergence form, and it can ensure gradient continuity of solution.

A stock market, equity market or share market is the aggregation of buyers and sellers (a loose network of economic transactions, not a physical facility or discrete entity) of stocks (also called shares); these may include securities listed on a stock exchange as well as those only traded privately.

Bitcoin, the first cryptocurrency, was created in 2009 and ever since it has rattled the financial market. Soaring from $1,000 to just under $20,000 in 2017, Bitcoin was just the start of the cryptocurrency financial sphere piquing the interest from common man to Nobel laureates alike. With such a short history, and only gaining mainstream attention in 2017, the cryptocurrency assets have been little studied. Thus, this paper hopes to gain a better insight into the field’s risks, modeling, dependencies, and causalities by comparing a cryptocurrency to a financial equity portfolio and a portfolio of 10 cryptocurrencies.

The paper is broken into three key scopes of analysis to be explored and referenced throughout as Portfolio Statistics and Risk Profile, Time Scaling, and Cryptocurrency Dependency and Causality. The first area of exploration analyzes the statistics of two financial data portfolios, presenting the opportunity to gain a preliminary comparison of FX/equity portfolios and cryptocurrencies’ risk profiles. It aims via parametric and non-parametric calculations of Value-at-Risk and Conditional Value at Risk to gain the best overall insight of each respective portfolios’ risks. The second investigation analyzes the portfolios as random walks, signals, and time series to detect deviations from these analysis assumptions and evaluate financial returns at different time scales to more accurately model each’s intrinsic statistics and create better forecasts. The last consideration then breaks from these two portfolio comparisons, analyzing various cryptocurrency dependencies and causalities to better understand the returns of these “hot” financial products.

We survey some new progress on the pricing models driven by fractional Brownian motion \cb{or} mixed fractional Brownian motion. In particular, we give results on arbitrage opportunities, hedging, and option pricing in these models. We summarize some recent results on fractional Black & Scholes pricing model with transaction costs. We end the paper by giving some approximation results and indicating

We propose two main applications of Gy\"{o}ngy (1986)'s construction of inhomogeneous Markovian stochastic differential equations that mimick the one-dimensional marginals of continuous It\^{o} processes. Firstly, we prove Dupire (1994) and Derman and Kani (1994)'s result. We then present Bessel-based stochastic volatility models in which this relation is used to compute analytical formulas for the local volatility. Secondly, we use these

The objective of this study is to review genetic algorithms, which are evolution and natural genetic process based stochastic search and optimization techniques, and their financial applications. First, the concepts of evolutionary algorithms and genetic algortihms are explained. Then, literature on the applications of genetic algorithms on finance is reviewed. Financial applications of genetic algoritms are analyzed under two main headings: applications of genetic algorithms in the field of corporate finance and applications on financial market systems and financial economics. Although, there is an enormous literature on financial applications of genetic algorithms, applications on Turkish markets are quite limited. With the review of the financial application of this algoithm it is intented to provide directions for Turkish researchers in this field.
Keywords: Evolutionary Algorithms, Genetic Algorithms, Financial Applications.

We present a new numerical method to price vanilla options quickly in time-changed Brownian motion models. The method is based on rational function approximations of the Black-Scholes formula. Detailed numerical results are given for a number of widely used models. In particular, we use the variance-gamma model, the CGMY model and the Heston model without correlation to illustrate our results. Comparison to the standard fast Fourier transform method with respect to accuracy and speed appears to favour the newly developed method in the cases considered. We present error estimates for the option prices. Additionally, we use this method to derive a procedure to compute, for a given set of arbitrage-free European call option prices, the corresponding Black-Scholes implied volatility surface. To achieve this, rational function approximations of the inverse of the Black-Scholes formula are used. We are thus able to work out implied volatilities more efficiently than one can by the use of other common methods. Error estimates are presented for a wide range of parameters.

This paper provides an introduction to Modern Portfolio Theory (MPT) and compares two fundamental models: Markowitz's Mean-Variance model and Konno's Mean-Absolute Deviation model. We implemented the models using Python as an array of software packages for solving complex programming problems was readily available. Using historical data from the S&P 500, we compareded the computed optimal portfolios and find that Mean Absolute Deviation model outperforms the Mean-Variance model.

Find the code here: https://github.com/kgeoffrey/Portfolio-Optimization

Dalam paper ini membahas perilaku harga saham perusahaan PT. Unilever Indonesia Tbk menggunakan data close price selama setahun dari 2013 – 2014. Dengan mengunakan return, rata – rata return, variansi. Paper ini bertujuan untuk mengetahui perilaku harga saham PT. Unilever Indonesia Tbk, sehingga didapatkan gambaran jika kita ingin membeli sebagian saham dari perusahaan tersebut. Hasil rata – rata return sebesar 0,11%, variansi sebesar 0,07%. Disimpulkan bahwa perilaku harga saham perusahaan PT. Unilever Indonesia Tbk Fluktuatif dan Acak.

The purpose of this study was to examine the relationship between dividend policy and share price volatility with a focus on consumer product companies listed in Malaysian stock market. For this purpose, a sample of 84 companies from 142 consumer product companies listed in main market of Bursa Malaysia were selected and the relationship between share price volatility with two main measurements of dividend policy, dividend yield and payout ,were examined by applying multiple regression for a period of six years from 2005 to 2010. The primarily regression model was expanded by adding control variables including size, earning volatility, leverage, debt and growth. The empirical results of this study showed significant negative relationship between share price volatility with two main measurements of dividend policy which are dividend yield and dividend payout. Moreover, a significant negative relationship between share price volatility and size is found. Based on findings of this stud...

It is shown that time series about financial market variables are highly nonlinearly dependent on time. Fluctuations or volatility of returns on assets is one of them. Portfolio managers, option traders and market makers are all interested in volatility forecasting in order to get higher profits and less risky positions. The nonlinear dependence on time is very complex and parametric approaches, and linear models fail. Therefore as nonparametric tools artificial neural networks (ANNs) are candidates to deal with the volatility and/or return forecasting problems. On the other hand, based on the fact that volatility is time varying and that periods of high volatility tend to cluster, the most popular models in modeling volatility are GARCH type models because they can account excess kurtosis and asymmetric effects of financial time series. A standard GARCH(1,1) model usually indicates high persistence in the conditional variance, which may originate from structural changes. Hence it is natural that artificial neural networks (ANN) will be constructed to capture the nonlinear relationship between past return innovations and conditional variance which may be missed by linear regression models. First a usual feedforward, back propagation network is used. The structure of the return data makes FFANN difficult to converge. To overcome this difficulty a neural network with appropriate recurrent connections in the context of nonlinear ARMA models are used. These are the Jordan neural networks (JNN). Then Elman recurrent networks (ENN) and a mixture of the two (EJNN) are also used. The data set consists of returns of the S&P100  index daily closing prices obtained from the S&P100  website. The results indicate that the selected JNN(1,1,1) model has superior performances compared to the standard GARCH(1,1) model. The contribution of this paper can be seen in determining the appropriate NN that is comparable to the standard GARCH(1,1) model and its application in forecasting conditional variance of stock returns. Moreover, from the econometric perspective, NN models are used as a semi-parametric method that combines flexibility of nonparametric methods and the interpretability of parameters of parametric methods.

We study Atlas-type models of equity markets with local characteristics that depend on both name and rank, and in ways that induce a stable capital distribution. Ergodic properties and rankings of processes are examined with reference to the theory of reflected Brownian motions in polyhedral domains. In the context of such models we discuss properties of various investment strategies, including the so-called growth-optimal and universal portfolios.

The volume Computational Finance 1999 contains a selection of the papers presented at Computational Finance '99 at the Stern School of Business, New York Univ. in January 1999. This conference is an annual refereed meeting, which was previosly called "Neural Networks in the Capital Markets."

This chapter surveys research on agent-based models used in finance. It will concentrate on models where the use of computational tools is critical for the process of crafting models which give insights into the importance and dynamics of investor heterogeneity in many financial settings.

Taking advantage of the recent litterature on exact simulation algorithms (Beskos, Papaspiliopoulos and Roberts [1]) and unbiased estimation of the expectation of certain fonctional integrals (Wagner [27], Beskos et al. [2] and Fearnhead et al. [6]), we apply an exact simulation based technique for pricing continuous arithmetic average Asian options in the Black & Scholes framework. Unlike existing Monte Carlo methods, we are no longer prone to the discretization bias resulting from the approximation of continuous time processes through discrete sampling. Numerical results of simulation studies are presented and variance reduction problems are considered.

The purpose of this paper is introducing rigorous methods and formulas for bilateral counterparty risk credit valuation adjustments (CVA's) on interest-rate portfolios. In doing so, we summarize the general arbitrage-free valuation framework for counterparty risk adjustments in presence of bilateral default risk, as developed more in detail in , including the default of the investor. We illustrate the symmetry in the valuation and show that the adjustment involves a long position in a put option plus a short position in a call option, both with zero strike and written on the residual net present value of the contract at the relevant default times. We allow for correlation between the default times of the investor and counterparty, and for correlation of each with the underlying risk factor, namely interest rates. We also analyze the often neglected impact of credit spread volatility. We include Netting in our examples, although other agreements such as Margining and Collateral are left for future work.

In this paper we describe the gpusvcalibration R package for accelerating stochastic volatility model calibration on GPUs. The package is designed for use with existing CRAN packages for optimization such as DEOptim and nloptr. Stochastic volatility models are used extensively across the capital markets for pricing and risk management of exchange traded financial options. However, there are many challenges to calibration, including comparative assessment of the ro-bustness of different models and optimization routines. For example, we observe that when fitted to sub-minute level mid-market quotes, models require frequent calibration every few minutes and the quality of the fit is routine sensitive. The R statistical software environment is popular with quantitative analysts in the financial industry partly because it facilitates application design space exploration. However, a typical R based implementation of a stochastic volatility model calibration on a CPU does not meet the performance requirements for sub-minute level trading, i.e. mid to high frequency trading. We identified the most computationally intensive part of the calibration process in R and off-loaded that to the GPU. We created a map-reduce interface to the compu-tationally intensive kernel so that it can be easily integrated in a variety of R based calibration codes using our package. We demonstrate that the new R based implementation using our package is comparable in performance to a C/C ++ GPU based calibration code.

In this paper we present a new multi-asset pricing model, which is built upon newly developed families of solvable multi-parameter single-asset diffusions with a nonlinear smile-shaped volatility and an affine drift. Our multi-asset pricing model arises by employing copula methods. In particular, all discounted single-asset price processes are modeled as martingale diffusions under a risk-neutral measure. The price processes are so-called UOU diffusions and they are each generated by combining a variable (Ito) transformation with a measure change performed on an underlying Ornstein-Uhlenbeck (Gaussian) process. Consequently, we exploit the use of a normal bridge copula for coupling the single-asset dynamics while reducing the distribution of the multi-asset price process to a multivariate normal distribution. Such an approach allows us to simulate multidimensional price paths in a precise and fast manner and hence to price path-dependent financial derivatives such as Asian-style and...

This structured product analysis and rating model (the Model) is intended for analyzing structured notes that are to be issued as asset-backed securities (ABS) along with their indicative credit-rating consequences. The main driver of these notes is a cash flow pattern of a loan portfolio that is in turn driven by other variables and parameters. Principal to the analysis in this Model is how to construct the loan portfolio profile and its forecast until maturity. Along the timeline, many scheduled and assumed events could take place that will affect the cash flow pattern of the loan portfolio as well as that of the structured note portfolio. The Model also considers the effect of credit risk and default risk on the quantitative rating of the notes in its Rating Module. Such a rating module can also be used on a standalone basis with its own historical data inputs and simple cash flow projection.

In this note we describe the HJM(LLM) model for pricing Mid-Curve Money Market Future Options. The model is based on assuming a lognormal process for the relevant forward money market (“LIBOR”) rates and imbedding it into the general no-arbitrage HJM interest rate modeling framework (understood in the most general sense as a manifestly self-consistent framework for formulating interest rate models in terms of any type of interest rate, including the observable market rates [2, 3, 4]). The relevant convexity adjustment factor is calculated by leveraging on the known functional form for the discount bond price volatility in any no-arbitrage lognormal “LIBOR” rate model, and approximating it with a leading order deterministic approximation. This makes it possible to derive an analytic expression for the HJM(LLM) model Discount Propagator using the path integral techniques and leads to the explicit analytic pricing formulas for the Mid-Curve futures options.

In this paper, we model dependence between operational risks by allowing risk profiles to evolve stochastically in time and to be dependent. This allows for a flexible correlation structure where the dependence between frequencies of different risk categories and between severities of different risk categories as well as within risk categories can be modeled. The model is estimated using Bayesian inference methodology, allowing for combination of internal data, external data and expert opinion in the estimation procedure. We use a specialized Markov chain Monte Carlo simulation methodology known as Slice sampling to obtain samples from the resulting posterior distribution and estimate the model parameters.

We obtain new closed-form pricing formulas for contingent claims when the asset follows a Dupire-type local volatility model. To obtain the formulas we use the Dyson-Taylor commutator method that we have recently developed in [5, 6, 8] for short-time asymptotic expansions of heat kernels, and obtain a family of general closed-form approximate solutions for both the pricing kernel and derivative price. A bootstrap scheme allows us to extend our method to large time. We also perform analytic as well as a numerical error analysis, and compare our results to other known methods.

The purpose of this study was to examine the relationship between dividend policy and share price volatility with a focus on companies represent four sectors listed in Jordanian stock market. For this purpose, a sample of 53 companies listed in main market of Bursa Amman were selected and the relationship between share price volatility with two main measurements of dividend policy, dividend yield and payout,were examined by applying multiple regression for a period of 13 years from 2001 to 2013. The primarily regression model was expanded by adding control variables including size, stock repurchase, and stock dividend. The empirical results of this study showed significant negative relationship between share price volatility with dividend payout and a very weak positive relationship between dividend yield and share price volatility. Moreover, a significant positive relationship between share price volatility and size is found. Based on findings of this study, dividend payout and sto...

This paper discusses properties of a Doubly Stochastic Poisson Process (DSPP) where the intensity process belongs to a class of affine diffusions. For any intensity process from this class we derive an analytical expression for probability distribution functions of the corresponding DSPP. A specification of our results is provided in a particular case where the intensity is given by one-dimensional Feller process and its parameters are estimated by Kalman filtering for high frequency transaction data.