Post trade allocation: how much are bunched orders costing your performance?

Individual trade orders are often bunched into a block order for processing efficiency, where in post execution, they are allocated into individual accounts. Since Regulators have not mandated any specific post trade allocation practice or methodology, entities try to rigorously follow internal policies and procedures to meet the minimum Regulatory ask of being procedurally fair and equitable. However, as many have found over the years, there is no simple solution for post trade allocation between accounts that results in a uniform distribution of returns. Furthermore, in many instances, the divergences between returns do not dissipate with more transactions, and tend to increase in some cases. This paper is the first systematic treatment of trade allocation risk. We shed light on the reasons for return divergence among accounts, and we present a solution that supports uniform allocation of return irrespective of number of accounts and trade sizes.Comment: 16 pages, 2 figures, 12 table

A fast method for pricing American options under the variance gamma model

We investigate methods for pricing American options under the variance gamma model. The variance gamma process is a pure jump process which is constructed by replacing the calendar time by the gamma time in a Brownian motion with drift, which makes it a time-changed Brownian motion. In general, the finite difference method and the simulation method can be used for pricing under this model, but their speed is not satisfactory. So there is a need for fast but accurate approximation methods. In the case of Black-Merton-Scholes model, there are fast approximation methods, but they cannot be utilized for the variance gamma model. We develop a new fast method inspired by the quadratic approximation method, while reducing the error by making use of a machine learning technique on pre-calculated quantities. We compare the performance of our proposed method with those of the existing methods and show that this method is efficient and accurate for practical use.Comment: 16 pages, 1 Figure, 4 Table

An unsupervised deep learning approach in solving partial integro-differential equations

We investigate solving partial integro-differential equations (PIDEs) using unsupervised deep learning in this paper. To price options, assuming underlying processes follow Levy processes, we require to solve PIDEs. In supervised deep learning, pre-calculated labels are used to train neural networks to fit the solution of the PIDE. In an unsupervised deep learning, neural networks are employed as the solution, and the derivatives and the integrals in the PIDE are calculated based on the neural network. By matching the PIDE and its boundary conditions, the neural network gives an accurate solution of the PIDE. Once trained, it would be fast for calculating options values as well as option Greeks.Comment: 22 pages, 4 figure

The VIX index under scrutiny of machine learning techniques and neural networks

The CBOE Volatility Index, known by its ticker symbol VIX, is a popular measure of the market's expected volatility on the SP 500 Index, calculated and published by the Chicago Board Options Exchange (CBOE). It is also often referred to as the fear index or the fear gauge. The current VIX index value quotes the expected annualized change in the SP 500 index over the following 30 days, based on options-based theory and current options-market data. Despite its theoretical foundation in option price theory, CBOE's Volatility Index is prone to inadvertent and deliberate errors because it is weighted average of out-of-the-money calls and puts which could be illiquid. Many claims of market manipulation have been brought up against VIX in recent years. This paper discusses several approaches to replicate the VIX index as well as VIX futures by using a subset of relevant options as well as neural networks that are trained to automatically learn the underlying formula. Using subset selection approaches on top of the original CBOE methodology, as well as building machine learning and neural network models including Random Forests, Support Vector Machines, feed-forward neural networks, and long short-term memory (LSTM) models, we will show that a small number of options is sufficient to replicate the VIX index. Once we are able to actually replicate the VIX using a small number of SP options we will be able to exploit potential arbitrage opportunities between the VIX index and its underlying derivatives. The results are supposed to help investors to better understand the options market, and more importantly, to give guidance to the US regulators and CBOE that have been investigating those manipulation claims for several years

Deep reinforcement learning on a multi-asset environment for trading

Financial trading has been widely analyzed for decades with market participants and academics always looking for advanced methods to improve trading performance. Deep reinforcement learning (DRL), a recently reinvigorated method with significant success in multiple domains, still has to show its benefit in the financial markets. We use a deep Q-network (DQN) to design long-short trading strategies for futures contracts. The state space consists of volatility-normalized daily returns, with buying or selling being the reinforcement learning action and the total reward defined as the cumulative profits from our actions. Our trading strategy is trained and tested both on real and simulated price series and we compare the results with an index benchmark. We analyze how training based on a combination of artificial data and actual price series can be successfully deployed in real markets. The trained reinforcement learning agent is applied to trading the E-mini S&P 500 continuous futures contract. Our results in this study are preliminary and need further improvement