arXiv subjects

Zura Kakushadze

Publications and source records attributed to Zura Kakushadze.

At least 55 records · Page 3Linked to original sources

arXiv2016

FX Options in Target Zone

In this note we discuss - in what is intended to be a pedagogical fashion - FX option pricing in target zones with attainable boundaries. The boundaries must be reflecting. The no-arbitrage requirement implies that the differential (foreign minus domestic) short-rate is not deterministic. When the band is narrow, we can pick the functional form of the FX rate process based on computational convenience. With a thoughtful choice, the FX option pricing problem can be solved analytically. The European option prices are expressed via (fast converging) series of elementary functions. We discuss the general approach to solving the pricing PDE and explicit examples, including analytically tractable models with (non-Ornstein-Uhlenbeck) mean-reversion.

q-fin.PR↗

arXiv2016

How to Combine a Billion Alphas

We give an explicit algorithm and source code for computing optimal weights for combining a large number N of alphas. This algorithm does not cost O(N^3) or even O(N^2) operations but is much cheaper, in fact, the number of required operations scales linearly with N. We discuss how in the absence of binary or quasi-binary clustering of alphas, which is not observed in practice, the optimization problem simplifies when N is large. Our algorithm does not require computing principal components or inverting large matrices, nor does it require iterations. The number of risk factors it employs, which typically is limited by the number of historical observations, can be sizably enlarged via using position data for the underlying tradables.

q-fin.PM↗

arXiv2016

Performance v. Turnover: A Story by 4,000 Alphas

We analyze empirical data for 4,000 real-life trading portfolios (U.S. equities) with holding periods of about 0.7-19 trading days. We find a simple scaling C ~ 1/T, where C is cents-per-share, and T is the portfolio turnover. Thus, the portfolio return R has no statistically significant dependence on the turnover T. We also find a scaling R ~ V^X, where V is the portfolio volatility, and the power X is around 0.8-0.85 for holding periods up to 10 days or so. To our knowledge, this is the only publicly available empirical study on such a large number of real-life trading portfolios/alphas.

q-fin.PM↗

arXiv2016

101 Formulaic Alphas

We present explicit formulas - that are also computer code - for 101 real-life quantitative trading alphas. Their average holding period approximately ranges 0.6-6.4 days. The average pair-wise correlation of these alphas is low, 15.9%. The returns are strongly correlated with volatility, but have no significant dependence on turnover, directly confirming an earlier result based on a more indirect empirical analysis. We further find empirically that turnover has poor explanatory power for alpha correlations.

q-fin.PM↗

arXiv2016

Multifactor Risk Models and Heterotic CAPM

We give a complete algorithm and source code for constructing general multifactor risk models (for equities) via any combination of style factors, principal components (betas) and/or industry factors. For short horizons we employ the Russian-doll risk model construction to obtain a nonsingular factor covariance matrix. This generalizes the heterotic risk model construction to include arbitrary non-industry risk factors as well as industry risk factors with generic "weights". The aim of sharing our proprietary know-how with the investment community is to encourage organic risk model building. The presentation is intended to be essentially self-contained and pedagogical. So, stop wasting money and complaining, start building risk models and enjoy!

q-fin.PM↗

arXiv2016

Mean-Reversion and Optimization

The purpose of these notes is to provide a systematic quantitative framework - in what is intended to be a "pedagogical" fashion - for discussing mean-reversion and optimization. We start with pair trading and add complexity by following the sequence "mean-reversion via demeaning -> regression -> weighted regression -> (constrained) optimization -> factor models". We discuss in detail how to do mean-reversion based on this approach, including common pitfalls encountered in practical applications, such as the difference between maximizing the Sharpe ratio and minimizing an objective function when trading costs are included. We also discuss explicit algorithms for optimization with linear costs, constraints and bounds.

q-fin.PM↗

arXiv2016

Heterotic Risk Models

We give a complete algorithm and source code for constructing what we refer to as heterotic risk models (for equities), which combine: i) granularity of an industry classification; ii) diagonality of the principal component factor covariance matrix for any sub-cluster of stocks; and iii) dramatic reduction of the factor covariance matrix size in the Russian-doll risk model construction. This appears to prove a powerful approach for constructing out-of-sample stable short-lookback risk models. Thus, for intraday mean-reversion alphas based on overnight returns, Sharpe ratio optimization using our heterotic risk models sizably improves the performance characteristics compared to weighted regressions based on principal components or industry classification. We also give source code for: a) building statistical risk models; and ii) Sharpe ratio optimization with homogeneous linear constraints and position bounds.

q-fin.PM↗

arXiv2015

Shrinkage = Factor Model

Shrunk sample covariance matrix is a factor model of a special form combining some (typically, style) risk factor(s) and principal components with a (block-)diagonal factor covariance matrix. As such, shrinkage, which essentially inherits out-of-sample instabilities of the sample covariance matrix, is not an alternative to multifactor risk models but one out of myriad possible regularization schemes. We give an example of a scheme designed to be less prone to said instabilities. We contextualize this within multifactor models.

q-fin.PM↗

arXiv2015

An Index for SSRN Downloads

We propose a new index to quantify SSRN downloads. Unlike the SSRN downloads rank, which is based on the total number of an author's SSRN downloads, our index also reflects the author's productivity by taking into account the download numbers for the papers. Our index is inspired by - but is not the same as - Hirsch's h-index for citations, which cannot be directly applied to SSRN downloads. We analyze data for about 30,000 authors and 367,000 papers. We find a simple empirical formula for the SSRN author rank via a Gaussian function of the log of the number of downloads.

cs.DL↗

arXiv2015

Quantization Rules for Dynamical Systems

We discuss a manifestly covariant way of arriving at the quantization rules based on causality, with no reference to Poisson or Peierls brackets of any kind.

hep-th↗

arXiv2015

A Spectral Model of Turnover Reduction

We give a simple explicit formula for turnover reduction when a large number of alphas are traded on the same execution platform and trades are crossed internally. We model turnover reduction via alpha correlations. Then, for a large number of alphas, turnover reduction is related to the largest eigenvalue and the corresponding eigenvector of the alpha correlation matrix.

q-fin.GN↗

arXiv2015

On Origins of Alpha

We argue that an important contributing factor into market inefficiency is the lack of a robust mechanism for the stock price to rise if a company has good earnings, e.g., via buybacks/dividends. Instead, the stock price is prone to volatility due to rather random perception/interpretation of earnings announcements (among other data) by market participants. We present empirical evidence indicating that dividend paying stocks on average are less volatile, even factoring out market cap. We further ponder possible ways of increasing market efficiency via 1) instituting such a mechanism, 2) a taxation scheme that would depend on holding periods, and 3) a universal crossing engine/exchange for mutual and pension funds (and similar long holding horizon vehicles) with no dark pools, 100% transparency, and no advantage for timing orders.

q-fin.PM↗

arXiv2015

Combining Alphas via Bounded Regression

We give an explicit algorithm and source code for combining alpha streams via bounded regression. In practical applications typically there is insufficient history to compute a sample covariance matrix (SCM) for a large number of alphas. To compute alpha allocation weights, one then resorts to (weighted) regression over SCM principal components. Regression often produces alpha weights with insufficient diversification and/or skewed distribution against, e.g., turnover. This can be rectified by imposing bounds on alpha weights within the regression procedure. Bounded regression can also be applied to stock and other asset portfolio construction. We discuss illustrative examples.

q-fin.PM↗

arXiv2015

Coping with Negative Short-Rates

We discuss a simple extension of the Ho and Lee model with generic time-dependent drift in which: 1) we compute bond prices analytically; 2) the yield curve is sensible and the asymptotic yield is positive; and 3) our analytical solution provides a clean and simple way of separating volatility from the drift in the short-rate process. Our extension amounts to introducing one or two reflecting barriers for the underlying Brownian motion (as opposed to the short-rate), which allows to have more realistic time-dependent drift (as opposed to constant drift). In our model the spectrum -- or, roughly, the set of short-rate values contributing to bond and other claim prices -- is discrete and positive. We discuss how to calibrate our model using empirical yield data by fitting three parameters and then read off the time-dependent drift.

q-fin.MF↗

arXiv2015

4-Factor Model for Overnight Returns

We propose a 4-factor model for overnight returns and give explicit definitions of our 4 factors. Long horizon fundamental factors such as value and growth lack predictive power for overnight (or similar short horizon) returns and are not included. All 4 factors are constructed based on intraday price and volume data and are analogous to size (price), volatility, momentum and liquidity (volume). Historical regressions a la Fama and MacBeth (1973) suggest that our 4 factors have sizable serial t-statistic and appear to be relevant predictors for overnight returns. We check this by using our 4-factor model in an explicit intraday mean-reversion alpha.

q-fin.PM↗

arXiv2015

Custom v. Standardized Risk Models

We discuss when and why custom multi-factor risk models are warranted and give source code for computing some risk factors. Pension/mutual funds do not require customization but standardization. However, using standardized risk models in quant trading with much shorter holding horizons is suboptimal: 1) longer horizon risk factors (value, growth, etc.) increase noise trades and trading costs; 2) arbitrary risk factors can neutralize alpha; 3) "standardized" industries are artificial and insufficiently granular; 4) normalization of style risk factors is lost for the trading universe; 5) diversifying risk models lowers P&L correlations, reduces turnover and market impact, and increases capacity. We discuss various aspects of custom risk model building.

q-fin.PM↗

arXiv2015

Notes on Alpha Stream Optimization

In these notes we discuss investment allocation to multiple alpha streams traded on the same execution platform, including when trades are crossed internally resulting in turnover reduction. We discuss approaches to alpha weight optimization where one maximizes P&L subject to bounds on volatility (or Sharpe ratio). The presence of negative alpha weights, which are allowed when alpha streams are traded on the same execution platform, complicates the optimization problem. By using factor model approach to alpha covariance matrix, the original optimization problem can be viewed as a 1-dimensional root searching problem plus an optimization problem that requires a finite number of iterations. We discuss this approach without costs and with linear costs, and also with nonlinear costs in a certain approximation, which makes the allocation problem tractable without forgoing nonlinear portfolio capacity bound effects.

q-fin.PM↗

arXiv2015

Phynance

These are the lecture notes for an advanced Ph.D. level course I taught in Spring'02 at the C.N. Yang Institute for Theoretical Physics at Stony Brook. The course primarily focused on an introduction to stochastic calculus and derivative pricing with various stochastic computations recast in the language of path integral, which is used in theoretical physics, hence "Phynance". I also included several "quiz" problems (with solutions) comprised of (pre-)interview questions quantitative finance job candidates were sometimes asked back in those days. The course to a certain extent follows an excellent book "Financial Calculus: An Introduction to Derivative Pricing" by M. Baxter and A. Rennie.

q-fin.MF↗