Spot and derivative pricing in the EEX power market

Using spot and futures price data from the German EEX Power market, we test the adequacy of various one-factor and two-factor models for electricity spot prices. The models are compared along two different dimensions: (1) We assess their ability to explain the major data characteristics and (2) the forecasting accuracy for expected future spot prices is analyzed. We find that the regime-switching models clearly outperform its competitors in almost all respects. The best results are obtained using a two-regime model with a Gaussian distribution in the spike regime. Furthermore, for short and medium-term periods our results underpin the frequently stated hypothesis that electricity futures quotes are consistently greater than the expected future spot, a situation which is denoted as contango.     

A Simple Option-Pricing Formula

A simple option-pricing formula based on the Weibull distribution is introduced. The simplicity of the algebraic form and ease of implementation are comparable to those of Black-Scholes. Application to S&P 500 options shows that the pricing biases present in the Black-Scholes model are eliminated. Prices produced by the presented model generally lie within or close to the bid-ask spread. For long-term options (over one year), the Weibull formula exhibits significantly higher precision than the Black-Scholes formula does. While a rigorous comparison of all available models is necessary, the simplicity and precision of the proposed model are its main advantages over the existing models.     

Empirical Bayes credibility

Abstract

A credibility estimator is Bayes in the restricted class of linear estimators and may be viewed as a linear approximation to the (unrestricted) Bayes estimator. When the structural parameters occurring in a credibility formula are replaced by consistent estimators based on data from a collective of similar risks,we obtain an empirical credibility estimator, which is a credibility counterpart of empirical Bayes estimators. Empirical credibility estimators are proposed under various model assumptions, and sufficient conditions for asymptotic optimality are established.     

Inference in models with adaptive learning

Identification of structural parameters in models with adaptive learning can be weak, causing standard inference procedures to become unreliable. Learning also induces persistent dynamics, and this makes the distribution of estimators and test statistics non-standard. Valid inference can be conducted using the Anderson-Rubin statistic with appropriate choice of instruments. Application of this method to a typical new Keynesian sticky-price model with perpetual learning demonstrates its usefulness in practice.     

Novel advancements in the Markov chain stock model: analysis and inference

In this paper we propose further advancements in the Markov chain stock model. First, we provide a formula for the second order moment of the fundamental price process with transversality conditions that avoid the presence of speculative bubbles. Second, we assume that the process of the dividend growth is governed by a finite state discrete time Markov chain and, under this hypothesis, we are able to compute the moments of the price process. We impose assumptions on the dividend growth process that guarantee finiteness of price and risk and the fulfilment of the transversality conditions. Subsequently, we develop non parametric statistical techniques for the inferential analysis of the model. We propose estimators of price, risk and forecasted prices and for each estimator we demonstrate that they are strongly consistent and that properly centralized and normalized they converge in distribution to normal random variables, then we give also the interval estimators. An application that demonstrate the practical implementation of methods and results to real dividend data concludes the paper.     

Cramér–Von Mises distance estimation for some positive infinitely divisible parametric families with actuarial applications

Cramér–Von Mises (CVM) inference techniques are developed for some positive flexible infinitely divisible parametric families generalizing the compound Poisson family. These larger families appear to be useful for parametric inference for positive data. The methods are based on inverting the characteristic functions. They are numerically implementable whenever the characteristic function has a closed form. In general, likelihood methods based on density functions are more difficult to implement. CVM methods also lead to model testing, with test statistics asymptotically following a chi-square distribution. The methods are for continuous models, but they can also handle models with a discontinuity point at the origin such as the case of compound Poisson models. Simulation studies seem to suggest that CVM estimators are more efficient than moment estimators for the common range of the compound Poisson gamma family. Actuarial applications include estimation of the stop loss premium, and estimation of the present value of cash flows when interest rates are assumed to be driven by a corresponding Lévy process.     

Security bid/ask dynamics with discreteness and clustering: Simple strategies for modeling and estimation

This paper proposes a dynamic model of bid and ask quotes that incorporates a stochastic cost of market-making, discreteness (restriction of quotes to a fixed grid) and clustering (the tendency of quotes to lie on `natural' multiples of the tick size). The Gibbs sampler provides a convenient vehicle for estimation. The model is estimated for daily and intradaily US Dollar/Deutschemark Reuters quotes.     

Co-movements of index options and futures quotes

We report evidence that the co-movements of index options and index futures quotes differ sharply from perfect correlation in periods with option trades. In half-hour intervals with (without) option trades 25% (12%) of call option quote changes have either the opposite sign or are larger in magnitude than the corresponding index futures quote changes. We calibrate a stochastic volatility model that allows for trade and no-trade periods using real data and simulate the joint co-movements of index quotes and option quotes in this model. We show that for trade intervals the observed co-movements differ from the benchmark case established by our simulations approximately three times too often. We provide empirical evidence that market microstructure effects – specifically, stale quotes and aggressive quotes – explain the majority of the deviations from the benchmark. Our findings are relevant for techniques that use estimates of local co-movements as inputs to price or hedge options.     

A long-memory integer-valued time series model,INARFIMA, for financial application

A model to account for the long-memory property in a count data framework is proposed and applied to high-frequency stock transactions data. By combining features of the INARMA and ARFIMA models, an Integer-valued Auto Regressive Fractionally Integrated Moving Average (INARFIMA) model is proposed. The unconditional and conditional first- and second-order moments are given. The CLS, FGLS and GMM estimators are discussed. In its empirical application to two stock series for AstraZeneca and Ericsson B, we find that both series have a fractional integration property.     

Trades and Quotes: A Bivariate Point Process

This article formulates a bivariate point process to jointlyanalyze trade and quote arrivals. In microstructure models,trades may reveal private information that is then incorporatedinto new price quotes. This article examines the speed of thisinformation flow and the circumstances that govern it. A jointlikelihood function for trade and quote arrivals is specifiedin a way that recognizes that an intervening trade sometimescensors the time between a trade and the subsequent quote. Modelsof trades and quotes are estimated for eight stocks using Tradeand Quote database (TAQ) data. The essential finding for thearrival of price quotes is that information flow variables,such as high trade arrival rates, large volume per trade, andwide bid–ask spreads, all predict more rapid price revisions.This means prices respond more quickly to trades when informationis flowing so that the price impacts of trades and ultimatelythe volatility of prices are high in such circumstances.     

A hyperbolic diffusion model for stock prices

In the present paper we consider a model for stock prices which is a generalization of the model behind the Black–Scholes formula for pricing European call options. We model the log-price as a deterministic linear trend plus a diffusion process with drift zero and with a diffusion coefficient (volatility) which depends in a particular way on the instantaneous stock price. It is shown that the model possesses a number of properties encountered in empirical studies of stock prices. In particular the distribution of the adjusted log-price is hyperbolic rather than normal. The model is rather successfully fitted to two different stock price data sets. Finally, the question of option pricing based on our model is discussed and comparison to the Black–Scholes formula is made. The paper also introduces a simple general way of constructing a zero-drift diffusion with a given marginal distribution, by which other models that are potentially useful in mathematical finance can be developed.     

THE DEALERS' PRICE/SIZE QUOTE AND MARKET LIQUIDITY

We model trading in a competitive securities market where informed traders and liquidity traders transact with dealers. The dealers' entire published quote is modeled: bid-ask prices and the number of shares the dealer is willing to buy/sell at these prices (i.e., size quotes). We argue that size quotes are a more informative indicator of market liquidity than the bid-ask spread's adverse-selection component. Moreover, the size quotes reveal several market characteristics that cannot be inferred from the bid-ask spread's adverse-selection component alone. The model generates a number of empirically testable predictions that clarify certain key elements of market liquidity.     

Inferring Public and Private Information from Trades and Quotes

We propose a new model that uses nonsynchronous, ultra‐high frequency data to analyze the sequential impact of trades and quotes on the price process. Private information is related to the impact of trades and public information to the impact of quotes. The model is extended to include various other factors that affect public and private information. For 20 active Nasdaq stocks, private information causes, on average, 9.43% of daily stock price movements. Additionally, quotes are more informative when (1) many dealers set the best price and (2) traditional market makers rather than Electronic Communication Networks set the best price.     

Generalized Pareto processes and fund liquidity risk

Motivated by the modelling of liquidity risk in fund management in a dynamic setting, we propose and investigate a class of time series models with generalized Pareto marginals: the autoregressive generalized Pareto process (ARGP), a modified ARGP and a thresholded ARGP. These models are able to capture key data features apparent in fund liquidity data and reflect the underlying phenomena via easily interpreted, low-dimensional model parameters. We establish stationarity and ergodicity, provide a link to the class of shot-noise processes, and determine the associated interarrival distributions for exceedances. Moreover, we provide estimators for all relevant model parameters and establish consistency and asymptotic normality for all estimators (except the threshold parameter, which is to be estimated in advance). Finally, we illustrate our approach using real-world fund redemption data, and we discuss the goodness-of-fit of the estimated models.     

FURTHER EXAMINATION OF PRICE DISCOVERY ON THE NYSE AND REGIONAL EXCHANGES

I re‐examine price discovery on the New York Stock Exchange (NYSE) and regional exchanges. I employ three common‐trend cointegration models to analyze the equilibrium dynamics between the NYSE and regional exchanges for the thirty Dow stocks. The overall results show that whether the regional exchanges free‐ride on the NYSE in obtaining equilibrium prices depends on whether trade prices or quotes are examined. The regional exchanges play a significant (though less important) role in the price‐discovery process for trade prices. However, the contributions of regional exchanges in price discovery of quotes are negligible. I explain the inconsistency between the results using quotes and those using trades. I also highlight the problems of using either quotes or trades in examining this free‐riding hypothesis and suggest future research on the different informativeness of trades on the NYSE and regional exchanges. JEL classification: G20, C32.     

Skewness and Stock Option Prices

Abstract

In the classical Black-Scholes model, the logarithm of the stock price has a normal distribution, which excludes skewness. In this paper we consider models that allow for skewness. We propose an option-pricing formula that contains a linear adjustment to the Black-Scholes formula. This approximation is derived in the shifted Poisson model, which is a complete market model in which the exact option price has some undesirable features. The same formula is obtained in some incomplete market models in which it is assumed that the price of an option is defined by the Esscher method. For a European call option, the adjustment for skewness can be positive or negative, depending on the strike price.     

The geometric chain-ladder

Abstract

The log normal reserving model is considered. The contribution of the paper is to derive explicit expressions for the maximum likelihood estimators. These are expressed in terms of development factors which are geometric averages. The distribution of the estimators is derived. It is shown that the analysis is invariant to traditional measures for exposure.     

Invisible Parameters in Option Prices

This paper characterizes contingent claim formulas that are independent of parameters governing the probability distribution of asset returns. While these parameters may affect stock, bond, and option values, they are “invisible” because they do not appear in the option formulas. For example, the Black-Scholes ( 1973 ) formula is independent of the mean of the stock return. This paper presents a new formula based on the log-negative-binomial distribution. In analogy with Cox, Ross, and Rubinstein's ( 1979 ) log-binomial formula, the log-negative-binomial option price does not depend on the jump probability. This paper also presents a new formula based on the log-gamma distribution. In this formula, the option price does not depend on the scale of the stock return, but does depend on the mean of the stock return. This paper extends the log-gamma formula to continuous time by defining a gamma process. The gamma process is a jump process with independent increments that generalizes the Wiener process. Unlike the Poisson process, the gamma process can instantaneously jump to a continuum of values. Hence, it is fundamentally “unhedgeable.” If the gamma process jumps upward, then stock returns are positively skewed, and if the gamma process jumps downward, then stock returns are negatively skewed. The gamma process has one more parameter than a Wiener process; this parameter controls the jump intensity and skewness of the process. The skewness of the log-gamma process generates strike biases in options. In contrast to the results of diffusion models, these biases increase for short maturity options. Thus, the log-gamma model produces a parsimonious option-pricing formula that is consistent with empirical biases in the Black-Scholes formula.     

Options markets, self-fulfilling prophecies, and implied volatilities

This paper answers the following often asked question in option pricing theory: if the underlying asset's price does not satisfy a lognormal distribution, can market prices satisfy the Black-Scholes formula just because market participants believe it should? In complete markets, if the underlying asset's objective distribution is not lognormal, then the answer is no. But, in an incomplete market, if the underlying asset's objective distribution is not lognormal and all traders believe it is, then the answer is yes! The Black-Scholes formula can be a self-fulfilling prophecy. The proof of this second assertion consists of generating an economy where self-confirming beliefs sustain the Black-Scholes formula as an equilibrium. An asymmetric information model is provided, where the underlying asset's price has stochastic volatility and drift. This model is distinct from the existing pricing models in the literature, and it provides new empirical implications concerning Black-Scholes implied volatilities and the bid/ask spread. Similar to stochastic volatility models, this model is consistent with the implied volatility “smile” pattern in strike prices. In addition, it is consistent with implied volatilities being biased predictors of future volatilities.     

Modeling daily price limits

This paper characterizes the behavior of observed asset prices under price limits and proposes the use of two-limit truncated and Tobit regression models to analyze regression models whose dependent variable is subject to price limits. Through a proper arrangement of the sample, these two models, the estimation of which is easy to implement, are applied only to subsets of the sample under study, rather than the full sample. Using the estimation of simple linear regression model as an example, several Monte Carlo experiments are conducted to compare the performance of the maximum likelihood estimators (MLEs) based on these two models and a generalized method of moments (GMM) estimator developed by K. C. John Wei and R. Chiang. The results show that under different price limits and various distributional assumptions for the error terms, the MLEs based on the two-limit Tobit and truncated regression models and the GMM estimator perform reasonably well, while the naive OLS estimator is downward biased. Overall, the MLE based on the two-limit Tobit model outperforms the other estimators.