Cardinality versus q-norm constraints for index tracking

Index tracking aims at replicating a given benchmark with a smaller number of its constituents. Different quantitative models can be set up to determine the optimal index replicating portfolio. In this paper, we propose an alternative based on imposing a constraint on the q-norm (0?<?q?<?1) of the replicating portfolios’ asset weights: the q-norm constraint regularises the problem and identifies a sparse model. Both approaches are challenging from an optimization viewpoint due to either the presence of the cardinality constraint or a non-convex constraint on the q-norm. The problem can become even more complex when non-convex distance measures or other real-world constraints are considered. We employ a hybrid heuristic as a flexible tool to tackle both optimization problems. The empirical analysis of real-world financial data allows us to compare the two index tracking approaches. Moreover, we propose a strategy to determine the optimal number of constituents and the corresponding optimal portfolio asset weights.     

Volatility as an Asset Class: European Evidence

Abstract

Volatility movements are known to be negatively correlated with stock index returns. Hence, investing in volatility appears to be attractive for investors seeking risk diversification. The most common instruments for investing in pure volatility are variance swaps, which now enjoy an active over-the-counter (OTC) market. This paper investigates the risk-return tradeoff of variance swaps on the Deutscher Aktienindex and Euro STOXX 50 index over the time period from 1995 to 2004. We synthetically derive variance swap rates from the smile in option prices. Using quotes from two large investment banks over two months, we validate that the synthetic values are close to OTC market prices. We find that variance swap returns exhibit an option-like profile compared to returns of the underlying index. Given this pattern, it is crucial to account for the non-normality of returns in measuring the performance of variance swap investments. As in the US, the average returns of selling variance swaps are found to be strongly positive and too large to be compatible with standard equilibrium models. The magnitude of the estimated risk premium is related to variance uncertainty and past index returns. This indicates that the variance swap rate does not seem to incorporate all past information relevant for forecasting future realized variance.     

High-dimensional index tracking based on the adaptive elastic net

When a portfolio consists of a large number of assets, it generally incorporates too many small and illiquid positions and needs a large amount of rebalancing, which can involve large transaction costs. For financial index tracking, it is desirable to avoid such atomized, unstable portfolios, which are difficult to realize and manage. A natural way of achieving this goal is to build a tracking portfolio that is sparse with only a small number of assets in practice. The cardinality constraint approach, by directly restricting the number of assets held in the tracking portfolio, is a natural idea. However, it requires the pre-specification of the maximum number of assets selected, which is rarely practicable. Moreover, the cardinality constrained optimization problem is shown to be NP-hard. Solving such a problem will be computationally expensive, especially in high-dimensional settings. Motivated by this, this paper employs a regularization approach based on the adaptive elastic-net (Aenet) model for high-dimensional index tracking. The proposed method represents a family of convex regularization methods, which nests the traditional Lasso, adaptive Lasso (Alasso), and elastic-net (Enet) as special cases. To make the formulation more practical and general, we also take the full investment condition and turnover restrictions (or transaction costs) into account. An efficient algorithm based on coordinate descent with closed-form updates is derived to tackle the resulting optimization problem. Empirical results show that the proposed method is computationally efficient and has competitive out-of-sample performance, especially in high-dimensional settings.     

Index tracking through deep latent representation learning

We consider the problem of index tracking whose goal is to construct a portfolio that minimizes the tracking error between the returns of a benchmark index and the tracking portfolio. This problem carries significant importance in financial economics as the tracking portfolio represents a parsimonious index that facilitates a practical means to trade the benchmark index. For this reason, extensive studies from various optimization and machine learning-based approaches have ensued. In this paper, we solve this problem through the latest developments from deep learning. Specifically, we associate a deep latent representation of asset returns, obtained through a stacked autoencoder, with the benchmark index's return to identify the assets for inclusion in the tracking portfolio. Empirical results indicate that to improve the performance of previously proposed deep learning-based index tracking, the deep latent representation needs to be learned in a strictly hierarchical manner and the relationship between the returns of the index and the assets should be quantified by statistical measures. Various deep learning-based strategies have been tested for the stock market indices of the S&P 500, FTSE 100 and HSI, and it is shown that our proposed methodology generates the best index tracking performance.     

Index tracking with constrained portfolios

Passive portfolio management strategies, such as index tracking, are popular in the industry, but so far little research has been done on the cardinality of such a portfolio, i.e. on how many different assets ought to be included in it. One reason for this is the computational complexity of the associated optimization problems. Traditional optimization techniques cannot deal appropriately with the discontinuities and the many local optima emerging from the introduction of explicit cardinality constraints. More recent approaches, such as heuristic methods, on the other hand, can overcome these hurdles. This paper demonstrates how one of these methods, differential evolution, can be used to solve the constrained index-tracking problem. We analyse the financial implication of cardinality constraints for a tracking portfolio using an empirical study of the Down Jones Industrial Average. We find that the index can be tracked satisfactorily with a subset of its components and, more important, that the deviation between computed actual tracking error and the theoretically achievable tracking error out of sample is negligibly affected by the portfolio's cardinality. Copyright © 2007 John Wiley & Sons, Ltd.     

The term structure of S&P 100 model-free volatilities

We develop an improved method to obtain the model-free volatility more accurately despite the limitations of a finite number of options and large strike price intervals. Our method computes the model-free volatility from European-style S&P 100 index options over a horizon of up to 450 days, the first time that this has been attempted, as far as we are aware. With the estimated daily term structure over the long horizon, we find that (i) changes in model-free volatilities are asymmetrically more positively impacted by a decrease in the index level than negatively impacted by an increase in the index level; (ii) the negative relationship between the daily change in model-free volatility and the daily change in index level is stronger in the near term than in the far term; and (iii) the slope of the term structure is positively associated with the index level, having a tendency to display a negative slope during bear markets and a positive slope during bull markets. These significant results have important implications for pricing and hedging index derivatives and portfolios.     

Mean-reversion properties of implied volatilities

In this paper, we present a new stylized fact for options whose underlying asset is a stock index. Extracting implied volatility time series from call and put options on the Deutscher Aktien index (DAX) and financial times stock exchange index (FTSE), we show that the persistence of these volatilities depends on the moneyness of the options used for its computation. Using a functional autoregressive model, we show that this effect is statistically significant. Surprisingly, we show that the diffusion-based stochastic volatility models are not consistent with this stylized fact. Finally, we argue that adding jumps to a diffusion-based volatility model help recovering this volatility pattern. This suggests that the persistence of implied volatilities can be related to the tails of the underlying volatility process: this corroborates the intuition that the liquidity of the options across moneynesses introduces an additional risk factor to the one usually considered.     

Fast Ninomiya–Victoir calibration of the double-mean-reverting model

Abstract

We consider the three-factor double mean reverting (DMR) option pricing model of Gatheral [Consistent Modelling of SPX and VIX Options, 2008], a model which can be successfully calibrated to both VIX options and SPX options simultaneously. One drawback of this model is that calibration may be slow because no closed form solution for European options exists. In this paper, we apply modified versions of the second-order Monte Carlo scheme of Ninomiya and Victoir [Appl. Math. Finance, 2008, 15, 107–121], and compare these to the Euler–Maruyama scheme with full truncation of Lord et al. [Quant. Finance, 2010, 10(2), 177–194], demonstrating on the one hand that fast calibration of the DMR model is practical, and on the other that suitably modified Ninomiya–Victoir schemes are applicable to the simulation of much more complicated time-homogeneous models than may have been thought previously.     

A Non-Parametric Option Pricing Model: Theory and Empirical Evidence

In this paper, we propose an empirically-based, non-parametric option pricing model to evaluate S&P 500 index options. Given the fact that the model is derived under the real measure, an equilibrium asset pricing model, instead of no-arbitrage, must be assumed. Using the histogram of past S&P 500 index returns, we find that most of the volatility smile documented in the literature disappears.     

Financial markets in the laboratory: an experimental analysis of some stylized facts

This paper provides experimental evidence explaining a number of stylized facts associated with the behaviour of financial returns, in particular the fat tailed nature of their distribution and the persistence in their volatility. By means of a laboratory experiment, we investigate the effect of the quantity and quality of information present in a financial market upon its stylized facts, showing how both the quality and quantity of information might have an impact on volatility clustering and the emergence of fat tail returns.     

The relationship between implied and realized volatility: evidence from the Australian stock index option market

This paper examines the relationship between the volatility implied in option prices and the subsequently realized volatility by using the S&P/ASX 200 index options (XJO) traded on the Australian Stock Exchange (ASX) during a period of 5 years. Unlike stock index options such as the S&P 100 index options in the US market, the S&P/ASX 200 index options are traded infrequently and in low volumes, and have a long maturity cycle. Thus an errors-in-variables problem for measurement of implied volatility is more likely to exist. After accounting for this problem by instrumental variable method, it is found that both call and put implied volatilities are superior to historical volatility in forecasting future realized volatility. Moreover, implied call volatility is nearly an unbiased forecast of future volatility.

Mitigating Estimation Risk in Asset Allocation: Diagonal Models Versus 1/N Diversification

Recent literature suggests that optimal asset‐allocation models struggle to consistently outperform the 1/N naïve diversification strategy, which highlights estimation‐risk concerns. We propose a dichotomous classification of asset‐allocation models based on which elements of the inverse covariance matrix that a model uses: diagonal only versus full matrix. We argue that parsimonious diagonal‐only strategies, which use limited information such as volatility or idiosyncratic volatility, are likely to offer a good tradeoff between incorporating limited information while mitigating estimation risk. Evaluating five sets of portfolios over 1926–2012, we find that 1/N is generally not optimal when compared with these diagonal strategies.     

On optimal stratifications for multivariate distributions

Summary

In the present paper we study the problem of optimal stratifications for estimating the mean vector y of a given multivariate distribution F(x) with covariance matrix ζ both in cases of proportionate and of optimal (or generalized Neyman) allocations. It is noted that an “optimal stratification” is meant for one to make the covariance matrix of an unbiased estimator X for μ minimal, in the sense of semi-order defined below, in the symmetric matrix space. We show the existence of an optimal stratification and the necessary conditions for a stratification to be optimal. Besides we prove that an optimal stratification can be represented by a “hyperplane stratification” or a “quadratic hypersurface stratification” according to the proportionate or optimal (or generalized Neyman) allocation, and that the set of all optimal (or admissible) stratifications is a minimal complete class in the analogous sense of decision theory. Further we discuss the optimal stratification when a criterion based on a suitable real-valued function is adopted instead of the semi-order.     

A Stochastic Receding Horizon Control Approach to Constrained Index Tracking

This paper develops stochastic receding horizon control for a constrained index tracking problem. By modeling the asset dynamics in the problems as a linear system subject to state and control multiplicative noise, and approximating linear chance constraints with quadratic expectation constraints, we show that index tracking can be approached using stochastic receding horizon control. In particular, we use a closed loop version of stochastic receding horizon control where the on-line optimization is solved as a semi-definite program. Numerical examples demonstrate the computations involved in these problems and indicate that stochastic receding horizon control is a promising new approach to constrained index tracking. C. H. Sung completed this work while he was a graduate student in the Management Science and Engineering Department, Stanford University.     

Polynomial goal programming and the implicit higher moment preferences of US institutional investors in hedge funds

Polynomial goal programming (PGP) is a flexible method that allows investor preferences for different moments of the return distribution of financial assets to be included in the portfolio optimization. The method is intuitive and particularly suitable for incorporating investor preferences in higher moments of the return distribution. However, until now, PGP has not been able to meet its full potential because it requires quantification of “real” preference parameters towards those moments. To date, the chosen preference parameters have been selected somewhat “arbitrarily”. Our goal is to calculate implied sets of preference parameters using investors’ choices of and the importance they attribute to risk and performance measures. We use three groups of institutional investors—pension funds, insurance companies, and endowments—and derive implied sets of preference parameters in the context of a hedge fund portfolio optimization. To determine “real” preferences for the higher moments of the portfolio return distribution, we first fit implied preference parameters so that the PGP optimal portfolio is identical to the desired hedge fund portfolio. With the obtained economically justified sets of preference parameters, the well-established PGP framework can be employed more efficiently to derive allocations that satisfy institutional investor expectations for hedge fund investments. Furthermore, the implied preference parameters enable fund of hedge fund managers and other investment managers to derive optimal portfolio allocations based on specific investor expectations. Moreover, the importance of individual moments, as well as their marginal rates of substitution, can be assessed.     

Generic pricing of FX,inflation and stock options under stochastic interest rates and stochastic volatility

We consider the pricing of FX, inflation and stock options under stochastic interest rates and stochastic volatility, for which we use a generic multi-currency framework. We allow for a general correlation structure between the drivers of the volatility, the inflation index, the domestic (nominal) and the foreign (real) rates. Having the flexibility to correlate the underlying FX/inflation/stock index with both stochastic volatility and stochastic interest rates yields a realistic model that is of practical importance for the pricing and hedging of options with a long-term exposure. We derive explicit valuation formulas for various securities, such as vanilla call/put options, forward starting options, inflation-indexed swaps and inflation caps/floors. These vanilla derivatives can be valued in closed form under Schöbel and Zhu [Eur. Finance Rev., 1999, 4, 23–46] stochastic volatility, whereas we devise an (Monte Carlo) approximation in the form of a very effective control variate for the general Heston [Rev. Financial Stud., 1993, 6, 327–343] model. Finally, we investigate the quality of this approximation numerically and consider a calibration example to FX and inflation market data.     

Tracking bond indices in an integrated market and credit risk environment

Abstract

The management of credit risky assets requires simulation models that integrate the disparate sources of credit and market risk, and suitable optimization models for scenario analysis. In this paper we integrate Monte Carlo simulation models for credit risk with scenario optimization, and develop a methodology for tracking broadly defined corporate bond indices. Testing of the models shows that the integration of the multiple risk factors improves significantly the performance of tracking models. Good tracking performance can be achieved by optimizing strategic asset allocation among broad classes of corporate bonds. However, extra value is generated with a tactical model that optimizes bond picking decisions as well. It is also shown that adding small corporate bond holdings in portfolios that track government bond indices improves the risk/return characteristics of the portfolios. The empirical results to substantiate the findings of this study are obtained by backtesting the model over a recent 30 month period.     

A new closed-form solution as an extension of the Black–Scholes formula allowing smile curve plotting

In this paper, as a generalization of the Black–Scholes (BS) model, we elaborate a new closed-form solution for a uni-dimensional European option pricing model called the J-model. This closed-form solution is based on a new stochastic process, called the J-process, which is an extension of the Wiener process satisfying the martingale property. The J-process is based on a new statistical law called the J-law, which is an extension of the normal law. The J-law relies on four parameters in its general form. It has interesting asymmetry and tail properties, allowing it to fit the reality of financial markets with good accuracy, which is not the case for the normal law. Despite the use of one state variable, we find results similar to those of Heston dealing with the bi-dimensional stochastic volatility problem for pricing European calls. Inverting the BS formula, we plot the smile curve related to this closed-form solution. The J-model can also serve to determine the implied volatility by inverting the J-formula and can be used to price other kinds of options such as American options.