Nonlinear Transaction Research Articles

A meshless method for numerical solutions of linear and nonlinear time-fractional Black-Scholes models

<abstract><p>The numerical solution of the time-fractional Black-Scholes model for European and American options is presented using a local meshless collocation approach based on hybrid Gaussian-cubic radial basis functions with polynomials is presented. The approach is then expanded to a nonlinear time-fractional model for an option with transaction costs in a market with low liquidity. The spatial derivatives of the models are discretized using the proposed meshless technique. Numerical experiments are carried out for the American option, European option, and nonlinear transaction cost option models. In order to evaluate the effectiveness and precision of the suggested meshless approach, $ L_{\infty} $ and $ L_{rel} $ error norms are utilized. Both call and put option volatility is explored. A non-uniform grid customized around the strike price region is also used to determine the prices of European call and American put options. The methods described in literature are compared with the numerical results.</p></abstract>

Quantitative Trading through Random Perturbation Q-Network with Nonlinear Transaction Costs

In recent years, reinforcement learning (RL) has seen increasing applications in the financial industry, especially in quantitative trading and portfolio optimization when the focus is on the long-term reward rather than short-term profit. Sequential decision making and Markov decision processes are rather suited for this type of application. Through trial and error based on historical data, an agent can learn the characteristics of the market and evolve an algorithm to maximize the cumulative returns. In this work, we propose a novel RL trading algorithm utilizing random perturbation of the Q-network and account for the more realistic nonlinear transaction costs. In summary, we first design a new near-quadratic transaction cost function considering the slippage. Next, we develop a convolutional deep Q-learning network (CDQN) with multiple price input based on this cost functions. We further propose a random perturbation (rp) method to modify the learning network to solve the instability issue intrinsic to the deep Q-learning network. Finally, we use this newly developed CDQN-rp algorithm to make trading decisions based on the daily stock prices of Apple (AAPL), Meta (FB), and Bitcoin (BTC) and demonstrate its strengths over other quantitative trading methods.

Optimal Portfolio Insurance under Nonlinear Transaction Costs

The minimization of the costs related to portfolio insurance is a very important investment strategy. In this article, by adding the transaction costs to the classical minimum cost portfolio insurance (MCPI) problem, we define and study the MCPI under transaction costs (MCPITC) problem as a nonlinear programming (NLP) problem. In this way, the MCPI problem becomes more realistic. Since such NLP problems are commonly solved by heuristics, we use the Beetle Antennae Search (BAS) algorithm to provide a solution to the MCPITC problem. Numerical experiments and computer simulations in real-world data sets confirm that our approach is an excellent alternative to other evolutionary computation algorithms.

Deep Reinforcement Learning for Option Replication and Hedging

The authors propose models for the solution of the fundamental problem of option replication subject to discrete trading, round lotting, and nonlinear transaction costs using state-of-the-art methods in deep reinforcement learning (DRL), including deep Q-learning, deep Q-learning with Pop-Art, and proximal policy optimization (PPO). Each DRL model is trained to hedge a whole range of strikes, and no retraining is needed when the user changes to another strike within the range. The models are general, allowing the user to plug in any option pricing and simulation library and then train them with no further modifications to hedge arbitrary option portfolios. Through a series of simulations, the authors show that the DRL models learn similar or better strategies as compared to delta hedging. Out of all models, PPO performs the best in terms of profit and loss, training time, and amount of data needed for training. TOPICS:Big data/machine learning, options, risk management, simulations Key Findings • The authors propose models for the replication of options over a whole range of strikes subject to discrete trading, round lotting, and nonlinear transaction costs based on state-of-the-art methods in deep reinforcement learning including deep Q-learning and proximal policy optimization. • The models allow the user to plug in any option pricing and simulation library and then train them with no further modifications to hedge arbitrary option portfolios. • A series of simulations demonstrates that the deep reinforcement learning models learn similar or better strategies as compared to delta hedging. • Proximal policy optimization outperforms the other models in terms of profit and loss, training time, and amount of data needed for training.

Visualizing and Optimizing Portfolio with Nonlinear Transaction Costs and Specific Constraints

In this paper, we consider the portfolio selection problem, with nonlinear transaction costs, basic constraints and probabilistic constraints. Such a problem cannot be handled by the usual quadratic or convex optimization methods.  We develop a heuristic method which yields to computation of efficient (suboptimal) solution of the problem. We describe our heuristic method for finding optimal portfolio based on solving many small optimization problems over large generation number, thus we obtain a good suboptimal solution.  Experimental results are demonstrated and visualized with various widely used indexes: stocks in US market: U.S. three – months treasury bills, U.S. long – term government bonds, S&P 500, Wilshire 5000, NASDAQ, Lehman Brothers corporate bond index, EAFE foreign stock index, and Gold recorded from (Jan 2000 – Jan 2008) with enhanced performance.  Finally, the results suggest that nonlinear transactions costs improve considerably the value of optimal portfolio over investment period, especially for portfolio with smaller among all assets.

Visualizing and Optimizing Portfolio with Nonlinear Transaction Costs and Specific Constraints

Visualizing and Optimizing Portfolio with Nonlinear Transaction Costs and Specific Constraints

Contemporary approaches and models of socialization

The article presents concepts and models of socialization/upbringing present in contemporary psychology. Currently, upbringing is recognized as a non-linear transaction process. It emphasizes the qualitative nature of changes, which are the result of the clash of natural contradictions having their origins in: internal factors related to the educator and the charge, external ones in the dyad educator-charge, and ones related to the nature and specificity of the educational relationship. The presentation of the latest approach to socialization is preceded by a concise discussion of previous approaches, including historical ones. Considerations are also made about the role of time and dialectical understanding of the educational relationship in explaining the upbringing process. Furthermore, meaningful questions leading to deepening the knowledge about socialization and allowing a more effective application have been included. Finally, the comparison of selected models of upbringing is drawn, as well as the author’s definition of this process.

Robust Portfolio Optimization in an Illiquid Market in Discrete-Time

We present a robust dynamic programming approach to the general portfolio selection problem in the presence of transaction costs and trading limits. We formulate the problem as a dynamic infinite game against nature and obtain the corresponding Bellman-Isaacs equation. Under several additional assumptions, we get an alternative form of the equation, which is more feasible for a numerical solution. The framework covers a wide range of control problems, such as the estimation of the portfolio liquidation value, or portfolio selection in an adverse market. The results can be used in the presence of model errors, non-linear transaction costs and a price impact.

Novel Financial Capital Flow Forecast Framework Using Time Series Theory and Deep Learning: A Case Study Analysis of Yu’e Bao Transaction Data

Appropriate monetary liquidity is important for financial institutions. When institutions lack adequate cash flow for customer redemption, their income will decrease, their reputation will be affected, and they may even go bankrupt. However, the opposite extreme in which more cash is reserved than needed may result in lost opportunities to make successful investments. This study uses Yu'e Bao transaction data to investigate a method for forecasting financial capital flow. Yu'e Bao, which is a financial product launched by Alibaba, faces the core challenge of maximizing commercial profits to reduce investment risks. Liquidity risk is considered the main factor in Yu'e Bao's investment strategy. First, a linear model called YEB_ARIMA is proposed by determining the autocorrelation (ACF) and partial autocorrelation (PACF) parameters, which are optimized by the grid search method. Second, a deep learning model called YEB_LSTM is introduced to strengthen the expressiveness of the model that yields nonlinear transaction features. Then, a hybrid learning method called YEB_Hybrid is applied to improve the original weak classifiers. This model includes both a linear combination and logistic regression learning. Third, a set of experiments and analyses are conducted based on subscription and redemption datasets to demonstrate that the hybrid model achieves an accuracy of 84.39% and 84.36%, respectively, under a variety of evaluation indexes. Finally, various proposed fund reserve ratios are provided based on capital forecasts.

A smoothing SAA algorithm for a portfolio choice model based on second-order stochastic dominance measures

In this paper, we provide a smoothing sample average approximation (SAA) method to solve a portfolio choice model based on second-order stochastic dominance (SSD) measure. Introducing a second-order stochastic dominance constraint in portfolio choice is theoretically attractive since all risk-averse investors would prefer a dominating portfolio. However, how to get the best choice among SSD efficient portfolios which is based on a stochastic optimization model is a challenge. We use the sample average to approximate the expected return rate function in the model and get a linear/nonlinear programming when the benchmark has discrete distribution. Then we propose a smoothing penalty algorithm to solve this problem. Meanwhile, we investigate the convergence of the optimal value of the transformed model and show that the optimal value converges to its counterpart with probability approaching to one at exponential rate as the sample size increases. By comparing the numerical results of the smoothing SAA algorithm with the common linear programming (LP) algorithm, we find that the smoothing algorithm has better performance than the LP algorithm in three aspects: (ⅰ)the smoothing SAA method can avoid the infinite constraints in the transformed models and the size of the smoothing algorithm model will not increase as the sample grows; (ⅱ)the smoothing SAA algorithm can deal with the nonlinear portfolio models with nonlinear transaction cost function; (ⅲ) the smoothing algorithm can get the global optimal solution because the smoothing function maintains the original convexity.

A high-order finite difference method for option valuation

In this paper, we propose the use of an efficient high-order finite difference algorithm to price options under several pricing models including the Black–Scholes model, the Merton’s jump–diffusion model, the Heston’s stochastic volatility model and the nonlinear transaction costs or illiquidity models. We apply a local mesh refinement strategy at the points of singularity usually found in the payoff of most financial derivatives to improve the accuracy and restore the rate of convergence of a non-uniform high-order five-point stencil finite difference scheme. For linear models, the time-stepping is dealt with by using an exponential time integration scheme with Carathéodory–Fejér approximations to efficiently evaluate the product of a matrix exponential with a vector of option prices. Nonlinear Black–Scholes equations are solved using an efficient iterative scheme coupled with a Richardson extrapolation. Our numerical experiments clearly demonstrate the high-order accuracy of the proposed finite difference method, making the latter a method of choice for solving both linear and nonlinear partial differential equations in financial engineering problems.

Super-replication with nonlinear transaction costs and volatility uncertainty

We study super-replication of contingent claims in an illiquid market with model uncertainty. Illiquidity is captured by nonlinear transaction costs in discrete time and model uncertainty arises as our only assumption on stock price returns is that they are in a range specified by fixed volatility bounds. We provide a dual characterization of super-replication prices as a supremum of penalized expectations for the contingent claim’s payoff. We also describe the scaling limit of this dual representation when the number of trading periods increases to infinity. Hence, this paper complements the results in [Finance Stoch. 17 (2013) 447–475] and [Ann. Appl. Probab. 5 (1995) 198–221] for the case of model uncertainty.

Optimal liquidation in a finite time regime switching model with permanent and temporary pricing impact

In this paper we discuss the optimal liquidation over a finite time horizon until the exit time. The drift and diffusion terms of the asset price are general functions depending on all variables including control andmarket regime. There is also a local nonlinear transaction cost associated to the liquidation. The model deals with both the permanent impact and the temporary impact in a regime switching framework. The problem can be solved with the dynamic programming principle. The optimal value function is the unique continuous viscosity solution to the HJB equation and can be computed with the finite difference method.

Approximate hedging for nonlinear transaction costs on the volume of traded assets

This paper is dedicated to the replication of a convex contingent claim h(S 1) in a financial market with frictions, due to deterministic order books or regulatory constraints. The corresponding transaction costs can be rewritten as a nonlinear function G of the volume of traded assets, with G′(0)>0. For a stock with Black–Scholes midprice dynamics, we exhibit an asymptotically convergent replicating portfolio, defined on a regular time grid with n trading dates. Up to a well-chosen regularization h n of the payoff function, we first introduce the frictionless replicating portfolio of \(h^{n}(S^{n}_{1})\), where S n is a fictitious stock with enlarged local volatility dynamics. In the market with frictions, a suitable modification of this portfolio strategy provides a terminal wealth that converges in \(\mathbb{L}^{2}\) to the claim h(S 1) as n goes to infinity. In terms of order book shapes, the exhibited replicating strategy only depends on the size 2G′(0) of the bid–ask spread. The main innovation of the paper is the introduction of a “Leland-type” strategy for nonvanishing (nonlinear) transaction costs on the volume of traded shares, instead of the commonly considered traded amount of money. This induces lots of technicalities, which we overcome by using an innovative approach based on the Malliavin calculus representation of the Greeks.

The Capacity of Trading Strategies

Due to non-linear transaction costs, the financial performance of a trading strategy decreases with portfolio size. Using a dynamic trading model a la Garleanu and Pedersen (2013), we derive closed-form formulas for the performance-to-scale frontier reached by a trader endowed with a signal predicting stock returns. The decay with scale of the realized Sharpe ratio is slower for strategies that (1) trade more liquid stocks (2) are based on signals that do not fade away quickly and (3) have strong frictionless performance. For an investor ready to accept a Sharpe reduction by 30%, portfolio scale (measured in dollar volatility) is given by a simple formula that is a function of the frictionless Sharpe, a measure of price impact, and a measure of the speed at which the signal fades away. We apply the framework to four well-known strategies. Because stocks have become more liquid, the capacity of strategies has increased in the 2000s compared to the 1990s. Due to high signal persistence, the capacity of a quality strategy is an order of magnitude larger than the others and is the only one highly scalable in the mid-cap range.

Arbitrage Pricing Theory Under Transaction Costs and Application to the Tobin Tax

This paper studies arbitrage pricing theory in financial markets with transaction costs. We extend the existing theory to include the more realistic possibility that the price at which the investors trade is dependent on the traded volume. The investors in the market always buy at the ask and sell at the bid price. Transaction costs are composed of two terms, one is able to capture the implicit transaction costs, and the second the price impact. Moreover, a new definition of a self-financing portfolio is obtained. The self-financing condition suggests that continuous trading is possible, but is restricted to predictable trading strategies having caglad (right-continuous with left limits) and caglad (left-continuous with right limits) paths of bounded quadratic variation and of finitely many jumps. That is, caglad and caglad predictable trading strategies of infinite variation, with finitely many jumps and of finite quadratic variation are allowed in our setting. Restricting ourselves to caglad predictable trading strategies, we show that the existence of an equivalent probability measure is equivalent to the absence of arbitrage opportunities, so that the first fundamental theorem of asset pricing (FFTAP) holds. It is also shown that the use of continuous and bounded variation trading strategies can improve the efficiency of hedging in a market with transaction costs. To better understand how to apply the theory proposed we provide an example with linear and non-linear transaction costs.

Hedging without sweat: a genetic programming approach

Hedging in the presence of transaction costs leads to complex optimization problems. These problems typically lack closed-form solutions, and their implementation relies on numerical methods that provide hedging strategies for specific parameter values. In this paper, we use a genetic programming algorithm to derive explicit formulas for near-optimal hedging strategies under nonlinear transaction costs. The strategies are valid over a large range of parameter values and require no information about the structure of the optimal hedging strategy.

Transaction Costs and Tradable Permits: Empirical Evidence from the EU Emissions Trading Scheme

In this paper, transaction costs in the EU emissions trading scheme (EU ETS) are examined empirically based on survey data from German companies. Transaction costs from measurement, reporting and verification (MRV) of emissions, permit trading and general informational costs are considered. Transaction costs from MRV and permit trading are of non-linear form and dependent on annual emission and trading volumes based on OLS and nonparametric estimation. As a consequence of non-linear transaction costs, welfare losses occur compared to a first-best setup under zero transaction costs. In practice, smaller emitters will emit less (abate more) compared to larger emitters and vice versa due to economies of scale in the management of emissions trading. The results further imply that optimal firm size might be influenced by transaction costs in the EU ETS because of relatively high average transaction costs amongst smaller emitters. Overall annual transaction costs in the EU ETS in Germany are estimated to be EUR 8.7 million.

Constant Rebalanced Portfolio Optimization Under Nonlinear Transaction Costs

We study the constant rebalancing strategy for multi-period portfolio optimization via conditional value-at-risk (CVaR) when there are nonlinear transaction costs. This problem is difficult to solve because of its nonconvexity. The nonlinear transaction costs and CVaR constraints make things worse; state-of-the-art nonlinear programming (NLP) solvers have trouble in reaching even locally optimal solutions. As a practical solution, we develop a local search algorithm in which linear approximation problems and nonlinear equations are iteratively solved. Computational results are presented, showing that the algorithm attains a good solution in a practical time. It is better than the revised version of an existing global optimization. We also assess the performance of the constant rebalancing strategy in comparison with the buy-and-hold strategy.

THE COST OF ILLIQUIDITY AND ITS EFFECTS ON HEDGING

Though liquidity is commonly believed to be a major effect in financial markets, there appears to be no consensus definition of what it is or how it is to be measured. In this paper, we understand liquidity as a nonlinear transaction cost incurred as a function of rate of change of portfolio. Using this definition, we obtain the optimal hedging policy for the hedging of a call option in a Black-Scholes model. This is a more challenging question than the more common studies of optimal strategy for liquidating an initial position, because our goal requires us to match a random final value. The solution we obtain reduces in the case of quadratic loss to the solution of three partial differential equations of Black-Scholes type, one of them nonlinear.