Large deviations of realized volatility

This paper studies large and moderate deviation properties of a realized volatility statistic of high frequency financial data. We establish a large deviation principle for the realized volatility when the number of high frequency observations in a fixed time interval increases to infinity. Our large deviation result can be used to evaluate tail probabilities of the realized volatility. We also derive a moderate deviation rate function for a standardized realized volatility statistic. The moderate deviation result is useful for assessing the validity of normal approximations based on the central limit theorem. In particular, it clarifies that there exists a trade-off between the accuracy of the normal approximations and the path regularity of an underlying volatility process. Our large and moderate deviation results complement the existing asymptotic theory on high frequency data. In addition, the paper contributes to the literature of large deviation theory in that the theory is extended to a high frequency data environment.     

Continuous time trading of a small investor in a limit order market

We provide a mathematical framework to model continuous time trading of a small investor in limit order markets. We show how elementary strategies can be extended in a suitable way to general continuous time strategies containing orders with infinitely many different limit prices. The general limit buy order strategies are predictable processes with values in the set of nonincreasing demand functions. It turns out that our strategy set of limit and market orders is closed, but limit orders can turn into market orders when passing to the limit, and any element can be approximated by a sequence of elementary strategies.     

Bayesian modeling of financial returns: A relationship between volatility and trading volume

The modified mixture model with Markov switching volatility specification is introduced to analyze the relationship between stock return volatility and trading volume. We propose to construct an algorithm based on Markov chain Monte Carlo simulation methods to estimate all the parameters in the model using a Bayesian approach. The series of returns and trading volume of the British Petroleum stock will be analyzed. Copyright © 2009 John Wiley & Sons, Ltd.     

Sample path Large Deviations and optimal importance sampling for stochastic volatility models

Sample path Large Deviation Principles (LDP) of the Freidlin–Wentzell type are derived for a class of diffusions, which govern the price dynamics in common stochastic volatility models from Mathematical Finance. LDP are obtained by relaxing the non-degeneracy requirement on the diffusion matrix in the standard theory of Freidlin and Wentzell. As an application, a sample path LDP is proved for the price process in the Heston stochastic volatility model.     

Estimation of integrated volatility in stochastic volatility models

In the framework of stochastic volatility models we examine estimators for the integrated volatility based on the pth power variation (i.e. the sum of pth absolute powers of the log‐returns). We derive consistency and distributional results for the estimators given high‐frequency data, especially taking into account what kind of process we may add to our model without affecting the estimate of the integrated volatility. This may on the one hand be interpreted as a possible flexibility in modelling, for example adding jumps or even leaving the framework of semimartingales by adding a fractional Brownian motion, or on the other hand as robustness against model misspecification. We will discuss possible choices of p under different model assumptions and irregularly spaced data. Copyright © 2005 John Wiley & Sons, Ltd.     

Neuro-dynamic trading methods

Investment strategies are usually based on forecasting models, and these are optimized with respect to past predictive performance. However, the main goal of most investors is the optimization of a risk-adjusted performance measure, such as the well-known Sharpe index. This issue has been approached by a few different studies within the area of Neurocomputing. The present paper briefly describes and empirically compares some of the models and methods proposed in those studies. Such adaptive methods can be computationally demanding, and convergence to high-quality solutions can be difficult to achieve, yet they can be very useful in automated trading systems, namely for portfolio management. In particular, the Q-learning algorithm, when combined with neural networks for value function approximation, seems to be a reasonably competitive approach, although not overall superior to alternative ones.     

Option hedging by an influential informed investor

This work extends the study of hedging problems in markets with asymmetrical information: an agent is supposed to possess an additional information on market prices, unknown to the common investor. The financial hedging problem for the influential and informed trader is modeled by a forward–backward stochastic differential equation, to be solved under an initial enlargement of the Brownian filtration. An existence and uniqueness theorem is proved under standard assumptions. The financial interpretation is derived, in terms of investment strategy for the informed and influential agent, as well as the conclusions concerning the general influenced market, in terms of completeness of the market. An example of such influenced and informed model is provided. Copyright © 2011 John Wiley & Sons, Ltd.     

Heterogeneous trading agents

In this article, we present a multiagent system (MAS) simulation of a financial market and investigate the requirements to obtain realistic data. The model consists of autonomous, interactive agents that buy stock on a financial market. Transaction decisions are based on a number of individual and collective elements, the former being risk aversion and a set of decision rules reflecting their anticipation of the future evolution of prices and dividends and the latter the information arriving on the market influencing the decision making process of each trader. We specifically look at this process and the following observations hold: The market behavior is determined by the information arriving at the market and agent heterogeneity is required in order to obtain the right statistical properties of the price and return time series. The observed results are not sensitive to changes in the parameter values. © 2003 Wiley Periodicals, Inc.     

A note on constant proportion trading strategies

We consider constant proportion (CP) trading strategies when there are multiple underlying securities and use a recently derived expression for the terminal wealth of a CP strategy to address two issues. First, we characterize the performance of a CP strategy relative to the performance of the corresponding buy-and-hold strategy. We then explain the performance of leveraged ETFs which have been criticized for not performing as expected, particularly during the financial crisis of 2008.     

A trading mechanism contingent on several indices

We introduce a trading mechanism where the execution of an order on a security can be made contingent on the relation between the clearing price of the security and the clearing price of one or several indices. A mechanism similar to ours, but limited to only one index, was implemented on the Tel Aviv Stock Exchange. We argue that it is in some cases crucial to make the execution of an order contingent on several indices. Our mechanism consists of a particular implementation of a double-sided multi-unit combinatorial auction with substitutes (or DMCS auction), which we introduced in an earlier article.     

Fuzzy portfolio model with different investor risk attitudes

We propose a fuzzy portfolio model designed for efficient portfolio selection with respect to uncertain or vague returns. Although many researchers have studied the fuzzy portfolio model, no researcher has yet attempted a behavioral analysis of the investor in the fuzzy portfolio model. To address this problem, we examined investor risk attitudes—risk-averse, risk-neutral, or risk-seeking behaviors—to discover an efficient method for fuzzy portfolio selection. In this study, we relied on the advantages of possibilistic mean–standard deviation models that we believed would fit the risk attitudes of investors. Thus, we developed a fuzzy portfolio model that focuses on different investor risk attitudes so that fuzzy portfolio selection for investors who possess different risk attitudes can be achieved more easily. Finally, we presented a numerical example of a portfolio selection problem to illustrate ways to address problems presented by a variety of investor risk attitudes.     

Derivatives trading for insurers

We investigate optimal strategies for a constant absolute risk aversion (CARA) insurer to manage its business risk through not only equity investment and proportional reinsurance but also trading derivatives of the equity. We obtain the optimal strategies in closed-form and quantify the value of derivatives trading by means of certainty-equivalence. Some numerical examples and sensitivity analysis are presented to illustrate our theoretical results. Our numerical results show that, unlike standard CRRA investors, the gain from trading derivatives to a CARA insurer is small and the insurer needs to expose itself to a relatively large position to fully enjoy the gain.     

The emissions trading paradox

This article considers the price history of CO₂ allowances in the EU Emission Trading Scheme. Since European Emissions Trading started in 2005, the prices of allowances have varied between less than one and thirty Euro per ton of CO₂. This previously unpredicted volatility and, more notably, a significant price crash in May 2005 led to the hypothesis that electricity producers might use their market power to influence the prices of allowances. Besides market power, the combination of information asymmetry and price interdependencies (between prices of primary goods – especially electricity – and allowances) plays an important role in explaining the emissions trading paradox. The model presented will show that banking can lead to such a price crash if market participators act rationally. Furthermore, in such a scenario banking can be profitable for sellers at the cost of buyers.     

Some results on quadratic hedging with insider trading

We consider the hedging problem in an arbitrage-free incomplete financial market, where there are two kinds of investors with different levels of information about the future price evolution, described by two filtrations F and G=F∨σ(G) where G is a given r.v. representing the additional information. We focus on two types of quadratic approaches to hedge a given square-integrable contingent claim: local risk minimization (LRM) and mean-variance hedging (MVH). By using initial enlargement of filtrations techniques, we solve the hedging problem for both investors and compare their optimal strategies under both approaches.

In particular, for LRM, we show that for a large class of additional non trivial r.v.s G both investors will pursue the same locally risk minimizing portfolio strategy and the cost process of the ordinary agent is just the projection on F of that of the insider. For the MVH approach, we study also some general stochastic volatility model, including Hull and White, Heston and Stein and Stein models. In this more specific setting and for r.v.s G which are measurable with respect to the filtration generated by the volatility process, we obtain an expression for the insider optimal strategy in terms of the ordinary agent optimal strategy plus a process admitting a simple feedback-type representation.     

Periodic autoregressive stochastic volatility

This paper proposes a stochastic volatility model (PAR-SV) in which the log-volatility follows a first-order periodic autoregression. This model aims at representing time series with volatility displaying a stochastic periodic dynamic structure, and may then be seen as an alternative to the familiar periodic GARCH process. The probabilistic structure of the proposed PAR-SV model such as periodic stationarity and autocovariance structure are first studied. Then, parameter estimation is examined through the quasi-maximum likelihood (QML) method where the likelihood is evaluated using the prediction error decomposition approach and Kalman filtering. In addition, a Bayesian MCMC method is also considered, where the posteriors are given from conjugate priors using the Gibbs sampler in which the augmented volatilities are sampled from the Griddy Gibbs technique in a single-move way. As a-by-product, period selection for the PAR-SV is carried out using the (conditional) deviance information criterion (DIC). A simulation study is undertaken to assess the performances of the QML and Bayesian Griddy Gibbs estimates in finite samples while applications of Bayesian PAR-SV modeling to daily, quarterly and monthly S&P 500 returns are considered.     

Estimation of the instantaneous volatility

This paper is concerned with the estimation of the volatility process in a stochastic volatility model of the following form: dX _t =  a _t dt + σ _t dW _t , where X denotes the log-price and σ is a càdlàg semi-martingale. In the spirit of a series of recent works on the estimation of the cumulated volatility, we here focus on the instantaneous volatility for which we study estimators built as finite differences of the power variations of the log-price. We provide central limit theorems with an optimal rate depending on the local behavior of σ. In particular, these theorems yield some confidence intervals for σ _t .     

The pricing of derivatives on assets with quadratic volatility

The basic model of financial economics is the Samuelson model of geometric Brownian motion because of the celebrated Black-Scholes formula for pricing the call option. The asset's volatility is a linear function of the asset value and the model guarantees positive asset prices. In this paper, it is shown that the pricing partial differential equation can be solved for level-dependent volatility which is a quadratic polynomial. If zero is attainable, both absorption and negative asset values are possible. Explicit formulae are derived for the call option: a generalization of the Black-Scholes formula for an asset whose volatiliy is affine, the formula for the Bachelier model with constant volatility, and new formulae in the case of quadratic volatility. The implied Black-Scholes volatilities of the Bachelier and the affine model are frowns, the quadratic specifications imply smiles.