A maximum likelihood approach to volatility estimation for a Brownian motion using high,low and close price data

Abstract

Volatility plays an important role in derivatives pricing, asset allocation, and risk management, to name but a few areas. It is therefore crucial to make the utmost use of the scant information typically available in short time windows when estimating the volatility. We propose a volatility estimator using the high and the low information in addition to the close price, all of which are typically available to investors. The proposed estimator is based on a maximum likelihood approach. We present explicit formulae for the likelihood of the drift and volatility parameters when the underlying asset is assumed to follow a Brownian motion with constant drift and volatility. Our approach is to then maximize this likelihood to obtain the estimator of the volatility. While we present the method in the context of a Brownian motion, the general methodology is applicable whenever one can obtain the likelihood of the volatility parameter given the high, low and close information. We present simulations which indicate that our estimator achieves consistently better performance than existing estimators (that use the same information and assumptions) for simulated data. In addition, our simulations using real price data demonstrate that our method produces more stable estimates. We also consider the effects of quantized prices and discretized time.     

Design principles for long‐lived Internet agents

Prior research on intelligent Internet agents has failed to address the needs of long‐lived data‐collecting agents, focusing instead on short‐lived transaction agents. Transaction agents typically run for a few seconds and retrieve information for a single transaction. With the staggering growth of electronic commerce, researchers and practitioners will want to design long‐lived data‐collecting agents that intelligently search for, retrieve, interpret, categorize, and store vast amounts of related information each time that they run. Such agents can run over the course of days rather than seconds and can be used by practitioners for decision support applications or by researchers as part of an empirical research methodology. This paper proposes a framework for agent sophistication, and emphasizes a number of design concepts for long‐lived Internet agents, including intelligence, validation, concurrency, recovery, monitoring, and interactivity. These concepts are used in the development of an illustrative tool called Electronic Data Retrieval LexicaL Agent (eDRILL), an object‐oriented data‐collecting agent. eDRILL is designed using the Unified Modeling Language (UML) and is written in Java. It gathers research data from an online auction. © 2000 John Wiley & Sons, Ltd.     

Project selection and execution in teams

We use a mechanism‐design approach to study a team whose members select a joint project and exert individual efforts to execute it. Members have private information about the qualities of alternative projects. Information sharing is obstructed by a trade‐off between adaptation and motivation. We determine the conditions under which first‐best project and effort choices are implementable and show that these conditions can become relaxed as the team grows in size. We also characterize the second‐best mechanism and find that it may include a “motivational bias,” that is, a bias in favor of the team's initially preferred project, and higher‐than‐optimal effort by uninformed team members.     

Defined Benefit or Defined Contribution? A Study of Pension Choices

We solve an empirically parameterized life‐cycle model of consumption and pension choices to show how expected earnings growth and risk affect the benefits of final‐salary defined benefit (DB) pension plans, relative to pension plans that are defined contribution (DC) in nature. We use micro data on the pension choices of individuals to provide evidence consistent with the model predictions: (1) individuals who expect a higher growth rate of earnings are more likely to choose DB final‐salary schemes, and (2) individuals who face a higher variance of persistent income shocks are less likely to choose DB final‐salary schemes. We control for cohort and age fixed effects in the empirical analysis.     

Firm reputation and horizontal integration

We study effects of horizontal integration on firm reputation in an environment where customers observe only imperfect signals about firms' effort/quality choices. Horizontal integration leads to a larger market base for the merged firm, and thus helps reputation building with more effective punishments and better monitoring by eliminating idiosyncratic shocks of individual markets. But it allows the merged firm to deviate only in a subset of markets, which hinders reputation building by making it more difficult for consumers to monitor its quality. We show that these effects give rise to a reputation‐based theory of the optimal firm size and derive its comparative statics.     

Robust estimation of historical volatility and correlations in risk management

Financial time series have two features which, in many cases, prevent the use of conventional estimators of volatilities and correlations: leptokurtotic distributions and contamination of data with outliers. Other techniques are required to achieve stable and accurate results. In this paper, we review robust estimators for volatilities and correlations and identify those best suited for use in risk management. The selection criteria were that the estimator should be stable to both fractionally small departures for all data points (fat tails), and to fractionally large departures for a small number of data points (outliers). Since risk management typically deals with thousands of time series at once, another major requirement was the independence of the approach of any manual correction or data pre-processing. We recommend using volatility t-estimators, for which we derived the estimation error formula for the case when the exact shape of the data distribution is unknown. A convenient robust estimator for correlations is Kendall's tau, whose drawback is that it does not guarantee the positivity of the correlation matrix. We chose to use geometric optimization that overcomes this problem by finding the closest correlation matrix to a given matrix in terms of the Hadamard norm. We propose the weights for the norm and demonstrate the efficiency of the algorithm on large-scale problems.     

Why are first‐year accounting studies inclusive?

This study is motivated by the increasing diversity among first‐year accounting students and the increasing number of first‐year accounting students whose majors are not in accounting related areas in UK universities. The main contribution of this study is that it uses student data over four consecutive academic years from a first‐year accounting course at a UK university to provide empirical evidence in support of the theoretical framework proposed by Rankin et al. (2003). Our results show the effects of metacognitive knowledge and content knowledge on academic performance as well as highlighting the inclusiveness of the first‐year accounting course. For instance, regardless of the choices of secondary subjects, students who have good prior academic achievement are the best performers on the first‐year accounting course. The influence of content knowledge on academic performance is strongly felt when the assessments of the course changed from a 100 per cent final exam to a combination of mid‐term coursework and a final exam. The results suggest that well‐designed mid‐term coursework is academically beneficial to accounting students, especially non‐native English‐speaking students.     

Parallel imports and price controls

Price controls create opportunities for international arbitrage. Many have argued that such arbitrage, if tolerated, will undermine intellectual property rights and dull the incentives for investment in research‐intensive industries such as pharmaceuticals. We challenge this orthodox view and show, to the contrary, that the pace of innovation often is faster in a world with international exhaustion of intellectual property rights than in one with national exhaustion. The key to our conclusion is to recognize that governments will make different choices of price controls when parallel imports are allowed by their trade partners than they will when they are not.     

Estimating a continuous hedonic‐choice model with an application to demand for soft drinks

Using micro‐level scanner data, I study empirically the consumer demand for soft drinks, which is characterized by multiple‐product, multiple‐unit purchasing behavior. I develop a continuous hedonic‐choice model to investigate how consumers choose the best basket of products to satisfy various needs. My model's embedded‐characteristics approach both helps to reduce the dimensionality problem in model estimation and generates flexible substitution patterns. Hence, the model is useful in application to data with many product choices that are correlated with each other at the individual level. The estimation results show that interesting substitutability and even a form of complementarity exist among soft drinks.     

Participant‐focused analysis: explanatory power of the classic psychometric paradigm in risk perception

Typical psychometric paradigm factors appear to have greater explanatory power for individual participants than previously envisaged. It is possible to acquire interpretable information about single participants using two factors (catastrophic potential and social and personal exposure) from aggregated participant‐focused data. Our results suggest that the classical psychometric model originated by Fischhoff and Slovic in the early 1980s to explain differences among hazards may also be capable of accounting for differences among participants. While socio‐demographic conditions on their own do not have substantial explanatory power, they are statistically significant and appear to dictate the position of participants within the factor space obtained using a participant‐focused analysis. One of the principal criticisms of the psychometric paradigm has been its lack of interpretability when using disaggregated data, but incorporating socio‐demographic variables overcomes this limitation.     

The SIML Estimation of Integrated Covariance and Hedging Coefficient Under Round-off Errors,Micro-market Price Adjustments and Random Sampling

For estimating the integrated volatility and covariance by using high frequency data, Kunitomo and Sato (Math Comput Simul 81:1272–1289, 2011; N Am J Econ Finance 26:289–309, 2013) have proposed the separating information maximum likelihood (SIML) method when there are micro-market noises. The SIML estimator has reasonable finite sample properties and asymptotic properties when the sample size is large when the hidden efficient price process follows a Brownian semi-martingale. We shall show that the SIML estimation is useful for estimating the integrated covariance and hedging coefficient when we have round-off errors, micro-market price adjustments and noises, and when the high-frequency data are randomly sampled. The SIML estimation is consistent, asymptotically normal in the stable convergence sense under a set of reasonable assumptions and it has reasonable finite sample properties with these effects.     

The rules of standard‐setting organizations: an empirical analysis

This article empirically explores standard‐setting organizations' policy choices. Consistent with our earlier theoretical work, we find (i) a negative relationship between the extent to which an SSO is oriented to technology sponsors and the concession level required of sponsors and (ii) a positive correlation between the sponsor friendliness of the selected SSO and the quality of the standard. We also develop and test two extensions of the earlier model: the presence of provisions mandating royalty‐free licensing is negatively associated with disclosure requirements, and the relationship between concessions and user friendliness is weaker when there is only a limited number of SSOs.     

Non-parametric estimation of historical volatility

Evolving volatility is a dominant feature observed in most financial time series and a key parameter used in option pricing and many other financial risk analyses. A number of methods for non-parametric scale estimation are reviewed and assessed with regard to the stylized features of financial time series. A new non-parametric procedure for estimating historical volatility is proposed based on local maximum likelihood estimation for the t-distribution. The performance of this procedure is assessed using simulated and real price data and is found to be the best among estimators we consider. We propose that it replaces the moving variance historical volatility estimator.     

The Cyclical Reaction of Fiscal Policies in the Euro Area: The Role of Modelling Choices and Data Vintages*

We survey the recent empirical literature concerning the cyclical response of fiscal policies in the euro area, finding large differences in results. We show that these differences are heavily influenced by the choices made in modelling fiscal behaviour. We make a case for the use, in assessing policies and in the EMU context, of the standard modelling choice where the discretionary reaction of fiscal policy is directly estimated. Models where the overall reaction to the cycle – which includes the effects of both discretionary actions and automatic stabilisers – is directly estimated tend to suggest either strong pro‐cyclical or strong counter‐cyclical discretionary reactions, depending on how this component is identified. Within the standard model and for the years 1994 to 2008, we show that results are significantly affected by the data vintage (ex post or real‐time). With ex post data, we find an unambiguous indication of a‐cyclicality. Using real‐time data, we find that the output gap matters. However, depending on whether we assess policy intentions or actual policies, euro‐area governments' behaviour radically changes. A plausible interpretation is that in the implementation phase, governments loosen their fiscal stance, giving in to political pressures that are proportional to the scale of the economic difficulties in bad times and the size of the ‘growth dividend’ in good times.     

Robust Measures of Earnings Surprises

Event studies of market efficiency measure earnings surprises using the consensus error (CE), given as actual earnings minus the average professional forecast. If a subset of forecasts can be biased, the ideal but difficult to estimate parameter‐dependent alternative to CE is a nonlinear filter of individual errors that adjusts for bias. We show that CE is a poor parameter‐free approximation of this ideal measure. The fraction of misses on the same side (FOM), which discards the magnitude of misses, offers a far better approximation. FOM performs particularly well against CE in predicting the returns of U.S. stocks, where bias is potentially large.     

Menu simplification for portfolio selection under short-sales constraints

We introduce a risk-reduction-based procedure to identify a subset of funds with a resulting opportunity set that is at least as good as the original menu when short-sales are imposed. Relying on Wald tests for mean-variance spanning, we show that the better results for the subset can be explained by a higher concentration of covariance entries between its assets, ultimately leading to smaller Frobenius norms of the associated matrices. With data on US-defined contribution plans, where participants have limited financial literacy, tend to be overwhelmed and prefer to make decisions among fewer choices, we obtain a 75% average reduction.     

Resource Allocation Auctions within Firms

There is growing interest in the use of markets within firms. Proponents have noted that markets are a simple and efficient mechanism for allocating resources in economies in which information is dispersed. In contrast to the use of markets in the broader economy, the efficiency of an internal market is determined in large part by the endogenous contractual incentives provided to the participating, privately informed agents. In this paper, we study the optimal design of managerial incentives when resources are allocated by an internal auction market, as well as the efficiency of the resulting resource allocations. We show that the internal auction market can achieve first‐best resource allocations and decisions, but only at an excessive cost in compensation payments. We then identify conditions under which the internal auction market and associated optimal incentive contracts achieve the benchmark second‐best outcome as determined using a direct revelation mechanism. The advantage of the auction is that it is easier to implement than the direct revelation mechanism. When the internal auction mechanism is unable to achieve second‐best, we characterize the factors that determine the magnitude of the shortfall. Overall, our results speak to the robust performance of relatively simple market mechanisms and associated incentive systems in resolving resource allocation problems within firms.     

Go for broke or play it safe? Dynamic competition with choice of variance

We consider a differential game in which the joint choices of the two players influence the variance, but not the mean, of the one‐dimensional state variable. We show that a pure strategy perfect equilibrium in stationary Markov strategies (ME) exists and has the property that patient players choose to play it safe when sufficiently ahead and to take risks when sufficiently behind. We also provide a simple condition that implies both players choose risky strategies when neither one is too far ahead, a situation that ensures a dominant player emerges “quickly.”     

Asymmetries in Price‐Setting Behavior: New Microeconometric Evidence from Switzerland

In this paper, we follow the recent empirical literature that has specified reduced‐form models for price setting that are closely tied to (S, s) ‐pricing rules. Our contribution to the literature is twofold. First, we propose an estimator that relaxes distributional assumptions on the unobserved heterogeneity. Second, we use the estimator to examine the prevalence of positive price changes in a low‐inflation environment. Our model estimates suggest that, if inflation falls from 0.9% to zero, the share of positive price changes in all price changes falls from 63.6% to 56.2%.     

The effects of risk‐taking tendency on risk choice and pre‐ and post‐decisional information selection

Ideally, people seek and select information about unfamiliar risks with which they are confronted, before they make a risk choice. This study investigated what happens when people do not have this opportunity. The main question was how risk‐taking tendency influences intuitive risk decisions and how this impacts subsequent information search and subsequent choices. In the present study, participants had to make a choice about an unfamiliar risk, either before or after they had had the opportunity to search for (risk‐promoting or risk‐averse) information. In the condition where they could only seek for information after they had made a choice, they had to reconsider their first choice and make a second risk choice. We expected that (1) risk‐taking tendency would impact people's risk choices, but only in the situation where they have little information. On the basis of cognitive dissonance theory, it was furthermore predicted that (2) risk‐taking tendency and (3) initial risk choice would affect risk information selection. Furthermore, we predicted that (4) the first risk choice and (5) the risk information selected would influence the subsequent risk choice. The results suggest that if people make a first, intuitive decision about an unknown risk, risk‐taking tendency has an effect on the choice, but that this does not happen when people can first select information. Risk‐taking tendency did not influence information selection, but initial choice did (although in another way than we expected). Furthermore, both the first risk choice and the risk information selected affected subsequent risk choices. These findings suggest that people often make initial intuitive decisions that are influenced by personality characteristics, and that are subsequently difficult to change.