Algorithmic Finance

The underlying stochastic nature of the requirements for the Solvency II regulations has introduced significant challenges if the required calculations are to be performed correctly, without resorting to excessive approximations, within practical timescales. It is generally acknowledged by practising actuaries within UK life offices that it is currently impossible to correctly fulfil the requirements imposed by Solvency II using existing computational techniques based on commercially available valuation packages.

Our work has already shown that it is possible to perform profitability calculations at a far higher rate than is achievable using commercial packages. One of the key factors in achieving these gains is to calculate reserves using recurrence relations that scale linearly with the number of time steps.

Here, we present a general vector recurrence relation which can be used for a wide range of non-unit linked policies that are covered by Solvency II; such contracts include annuities, term assurances, and endowments. Our results suggest that by using an optimised parallel implementation of this algorithm, on an affordable hardware platform, it is possible to perform the ‘brute force’ approach to demonstrating solvency in a realistic timescale (of the order of a few hours).

This paper presents an intuitively simple asset pricing model designed to predict stock returns and volatilities, when stock prices may follow a fractal walk rather than a random walk. The model utilizes similarity ratio of the return fractals as the basis for forecasting. We argue that a collection of past returns such as moving average statistics can be “expanded” to generate future returns through the similarity ratios by trading time multiples. We also argue that stock returns with fractal dimensions in excess of 1.5, the norm for market efficiency, may be prone to frequent jumps and discontinuity. The paper builds an econometrically testable model to estimate fractal dimensions to offer an alternative way to forecast return volatilities without having to estimate separately the bipower variation and the jumps in existing studies. We argue that fractional Brownian motion may be a more suitable description than the standard Wiener process when describing stock return behaviors. The paper also demonstrates the application aspect of our asset pricing model to high frequency algorithmic trading.

Institutional equity portfolios are typically constructed via taking expected stock returns and then applying the computationally expensive processes of covariance matrix estimation and mean-variance optimization. Unfortunately, these computational costs make it prohibitive to comprehensively backtest and tune higher frequency strategies over long histories. In this paper, we introduce a recursive algorithm which significantly lowers the computational cost of calculating the covariance matrix and its inverse as well as an iterative heuristic which provides a very fast approximation to mean-variance optimization. Together, these techniques cut backtesting time to a fraction of that of standard techniques. Where possible, the additional step of caching pre-calculated covariance matrices, can result in overall backtesting speeds up to orders of magnitude faster than the standard methods. We demonstrate the efficacy of our approach by selecting a prediction strategy in a fraction of the time taken by standard methods.

The stock market index is one of the main tools used by investors and financial managers to describe the market and compare the returns on specific investments. Common approaches to index calculation rely on a company's market value generating a weighted average as the index. This work presents new methods of computing adaptive stock market indices based on dynamical properties of the underlying index constituents, and introduces measures to evaluate their performance. The premise behind this work is that the influence of each stock on other stocks should be a major factor in determining the weight given to each stock in the index composition. The methodologies presented here provide the means to construct a dynamic adaptive index, which can be used as a benchmark for the underlying dynamics of the market. We investigate the components of the S&P500 index, and the components of the TA25 index, representing a large (NYSE) and a small (TASE) developed market, respectively. We focus our study on periods before and during the 2008 Sub-prime mortgage crisis. Our results provide evidence that the adaptive-indices provide an effective tool for policy and decision makers to monitor the stability and dynamics of the markets, and identify bubble formation and their ensuing collapse.

The maximum drawdown control strategy dynamically allocates wealth between cash and a risky portfolio, keeping losses below a chosen pre-defined level. This paper introduces variations of the strategy, namely the excess drawdown and the relative drawdown control strategies. The excess drawdown control is a more flexible strategy that can cope with common (re)allocation restrictions such as lock-up periods, cash bans or liquidity constraints through an implementation with a hedging overlay. The relative drawdown control strategy is adapted to contexts in which investors seek to limit benchmark underperformance instead of absolute losses. A formal proof that the loss-control objectives introduced can be insured using dynamic allocation is provided and the potential benefits and implementation aspects of the strategies are illustrated with examples.

This paper is a further extension of the method proposed in Itkin (2014) as applied to another set of jump-diffusion models: Inverse Normal Gaussian, Hyperbolic and Meixner. To solve the corresponding PIDEs we accomplish few steps. First, a second-order operator splitting on financial processes (diffusion and jumps) is applied to these PIDEs. To solve the diffusion equation we use standard finite-difference methods. For the jump part, we transform the jump integral into a pseudo-differential operator and construct its second order approximation on a grid which supersets the grid used for the diffusion part. The proposed schemes are unconditionally stable in time and preserve positivity of the solution which is computed either via a matrix exponential, or via its Páde approximation. Various numerical experiments are provided to justify these results.