Probability Distribution Of Returns Research Articles

Probability distribution of returns in the Heston model with stochastic volatility*

We study the Heston model, where the stock price dynamics is governed by a geometrical (multiplicative) Brownian motion with stochastic variance. We solve the corresponding Fokker-Planck equation exactly and, after integrating out the variance, find an analytic formula for the time-dependent probability distribution of stock price changes (returns). The formula is in excellent agreement with the Dow-Jones index for the time lags from 1 to 250 trading days. For large returns, the distribution is exponential in log-returns with a time-dependent exponent, whereas for small returns it is Gaussian. For time lags longer than the relaxation time of variance, the probability distribution can be expressed in a scaling form using a Bessel function. The Dow-Jones data for 1982-2001 follow the scaling function for seven orders of magnitude.

Volatility clustering and scaling for financial time series due to attractor bubbling.

A microscopic model of financial markets is considered, consisting of many interacting agents (spins) with global coupling and discrete-time heat bath dynamics, similar to random Ising systems. The interactions between agents change randomly in time. In the thermodynamic limit, the obtained time series of price returns show chaotic bursts resulting from the emergence of attractor bubbling or on-off intermittency, resembling the empirical financial time series with volatility clustering. For a proper choice of the model parameters, the probability distributions of returns exhibit power-law tails with scaling exponents close to the empirical ones.

Business cycle asymmetries in stock returns: Evidence from higher order moments and conditional densities

Markov switching models with time-varying means, variances and mixing weights are applied to characterize business cycle variation in the probability distribution and higher order moments of stock returns. This allows us to provide a comprehensive characterization of risk that goes well beyond the mean and variance of returns. Several mixture models with different specifications of the state transition are compared and we propose a new mixture of Gaussian and student- t distributions that captures outliers in returns. The models produce very similar expected returns and volatilities but imply very different time series for conditional skewness, kurtosis and predictive density. Consistent with economic theory, the gains in predictive accuracy from considering two-state mixture models rather than a single-state specification are higher for small firms than for large firms.

A linked-modeling framework to estimate maize production risk associated with ENSO-related climate variability in Argentina

A risk assessment framework is presented to characterize the vulnerability of agricultural production systems to El Niño-southern oscillation (ENSO)-related climate variability. The framework was applied to current maize production systems in two locations (Pergamino and Pilar) in the Pampas of central-eastern Argentina. Climatic, agronomic, and economic models were linked to produce probability distributions of farm-level yields and net returns by ENSO phase. Generally, an enhanced chance of higher (lower) simulated maize yields existed during warm (cold) ENSO events. However, regional differences existed: the effect of warm events on yields was more marked in Pilar, but Pergamino showed a proportionally stronger response to cold events. The modeling framework allowed the exploration of outcomes of high and low scenarios of soil water availability at planting time and ENSO phase. High initial soil water availability in Pilar offset increased yield risks from dry conditions associated with cold ENSO events. Fluctuations of output prices were shown to have considerable influence on the risk associated with ENSO-related climate variability. Despite these general results, there was considerable overlap in yields and net returns for the various ENSO phases. This overlap has significant implications for the adoption of ENSO forecasts in agriculture. The risk assessment framework developed here is a necessary precursor to risk management studies that prescribe or describe possible responses to expected climate scenarios.

Business Cycle Asymmetries in Stock Returns: Evidence from Higher Order Moments and Conditional Densities

Markow switching models with time-varying means, variances and mixing weights are applied to charakterize business cycle variation in the probability distribution and higher order moments of stock returns. This allows us to provide a comprehensive characterization of risk that goes well beyond the mean and variance of returns. Several mixture models with different specifications of the state transition are compared and we propose a new mixture of Gaussian and student-t distributions that captures outliers in returns. The models produce very similar expected returns and volatilities but imply very different time series for conditional skewness, kurtosis and predictive density. Consistent with economic theory, the gains in predictive accuracy from considering two-state mixture models rather than a single-state specification are higher for small firms than for large firms.

Multifractal returns and hierarchical portfolio theory

We extend and test empirically the multifractal model of asset returns based on a multiplicative cascade of volatilities from large to small time scales. Inspired by an analogy between price dynamics and hydrodynamic turbulence, it models the time scale dependence of the probability distribution of returns in terms of a superposition of Gaussian laws, with a log-normal distribution of the Gaussian variances. This multifractal description of asset fluctuations is generalized into a multivariate framework to account simultaneously for correlations across time scales and between a basket of assets. The reported empirical results show that this extension is pertinent for financial modelling. Two sources of departure from normality are discussed: at large time scales, the distinction between discretely and continuously discounted returns leads to the usual log-normal deviation from normality; at small time scales, the multiplicative cascade process leads to multifractality and strong deviations from normality. By perturbation expansions of the cumulants of the distribution of returns, we are able to quantify precisely the interplay and crossover between these two mechanisms. The second part of the paper applies this theory to portfolio optimization. Our multiscale description allows us to characterize the portfolio return distribution at all time scales simultaneously. The portfolio composition is predicted to change with the investment time horizon (i.e. the time scale) in a way that can be fully determined once an adequate measure of risk is chosen. We discuss the use of the fourth-order cumulant and of utility functions. While the portfolio volatility can be optimized in some cases for all time horizons, the kurtosis and higher normalized cumulants cannot be simultaneously optimized. For a fixed investment horizon, we study in detail the influence of the number of rebalancing of the portfolio. For the large risks quantified by the cumulants of order larger than two, the number of periods has a non-trivial influence, in contrast with Tobin's result valid in the mean-variance framework. This theory provides a fundamental framework for the conflicting optimization involved in the different time horizons and quantifies systematically the trade-offs for an optimal inter-temporal portfolio optimization.

Home Bias in International Stock Return Expectations

Despite the advantages of international portfolio diversification, actual equity portfolio holdings reveal a strong bias towards domestic stocks. One hypothesis is that this bias can be explained by stock return expectations expressed in probability judgments. To test that hypothesis and to analyze the underlying effects that might cause distortions in investors' expectations, we conducted a cross country study in Germany and the U.S. comparing participants' judgments about an identical set of German and U.S. stocks. Results show that both test groups feel more competent about domestic stocks. The asymmetric perception of competence is connected with an asymmetric assessment of probabilities. For both test groups subjective probability distributions of stock returns are significantly less dispersed and more optimistic for stocks associated with high competence levels than for stocks associated with low competence levels.

THE DISTRIBUTION OF RETURNS OF STOCK PRICES

We perform a phenomenological study of stock price fluctuations of individual companies. We systematically analyze two different databases covering securities from the three major US stock markets. We consider (i) the trades and quotes (TAQ) database, for which we analyze 40 million records for 1000 US companies for the 2-year period 1994–95, and (ii) the Center for Research and Security Prices (CRSP) database, for which we analyze 35 million daily records for approximately 16,000 companies in the 35-year period 1962–96. We study the probability distribution of returns over varying time scales — from 5 min up to 4 years. For time scales from 5 min up to approximately 16 days, we find that the tails of the distributions can be well described by a power-law decay, characterized by an exponent α ≈ 3 — well outside the stable Lévy regime 0 < α < 2. For time scales greater than 16 days, we observe results consistent with a slow convergence to Gaussian behavior.

Assessing the Impact of Price and Yield Risk in the Southern U.S. Catfish Industry

The study characterizes catfish price and yield risk by empirically determining the probability distribution of stochastic price and yield variables. The impact of the stochastic variables is then evaluated in the context of a catfish enterprise budget. Probability distributions of net returns are evaluated using stochastic dominance criteria. Results indicated that the distributions of yields and catfish prices appear to determine the shape of the distribution of residual net returns, and that the beta distribution most appropriately described the expected net returns for all combinations of production systems and farm sizes. If a beta distribution describes reality, but the risk averse producer assumes his expected returns are normally distributed, then he or she will underestimate the probability of obtaining low net returns. Thus, under an assumption of normality, producers may make decisions that place their operations in greater financial peril than they may otherwise would if they understood the true distribution of their expected net returns. Results also showed that the single-batch production system for small size farms was the most inefficient technology/size combination.

Estimation Risk and Portfolio Selection in the Lower Partial Moment

Portfolio selection models generally assume that the investor knows the parameters of the probability distribution of security returns. In practise the investor must, however, employ estimates of the necessary parameters. In this paper we investigate the effect of estimation risk on the efficient frontier in the lower partial moment framework. The results of the average difference between the actual and estimated portfolios show that the estimated portfolios are biased predictors of the actual portfolios. However, the estimates of the optimal portfolios can be improved. If our concern is the uncertainty in the optimal portfolio weights, then a bootstrap approach should be used to improve the optimizations. On the other hand, if our concern is related to the risk and portfolio mean returns of the optimized portfolios, then a James-Stein approach should be used . ∗I thank seminar participants at Lund University, at the Nordic Econometric meeting , Uppsala, 1999, and at the 28th Annual Conference of Economists, Melbourne, 1999, for valuable comments.

A quantile regression neural network approach to estimating the conditional density of multiperiod returns

This paper presents a new approach to estimating the conditional probability distribution of multiperiod financial returns. Estimation of the tails of the distribution is particularly important for risk management tools, such as Value-at-Risk models. Using daily exchange rates, a new approach is compared to GARCH-based quantile estimates. The results suggest that the new method offers a useful alternative for estimating the conditional density.

Scaling of the distribution of price fluctuations of individual companies.

We present a phenomenological study of stock price fluctuations of individual companies. We systematically analyze two different databases covering securities from the three major U.S. stock markets: (a) the New York Stock Exchange, (b) the American Stock Exchange, and (c) the National Association of Securities Dealers Automated Quotation stock market. Specifically, we consider (i) the trades and quotes database, for which we analyze 40 million records for 1000 U.S. companies for the 2-yr period 1994-95; and (ii) the Center for Research and Security Prices database, for which we analyze 35 million daily records for approximately 16,000 companies in the 35-yr period 1962-96. We study the probability distribution of returns over varying time scales Delta t, where Delta t varies by a factor of approximately 10(5), from 5 min up to approximately 4 yr. For time scales from 5 min up to approximately 16 days, we find that the tails of the distributions can be well described by a power-law decay, characterized by an exponent 2.5 < proportional to < 4, well outside the stable Lévy regime 0 < alpha < 2. For time scales Delta t >> (Delta t)(x) approximately equal to 16 days, we observe results consistent with a slow convergence to Gaussian behavior. We also analyze the role of cross correlations between the returns of different companies and relate these correlations to the distribution of returns for market indices.

Risk Aversion and the Bid-Ask Spread

This paper studies the properties of bid and ask prices posted by a monopolistic market maker, without parametric assumptions about the utility function of the market maker or about the probability distribution of the return of the risky asset. We first prove that the two prices can be higher or lower than the expected value of the asset, and that the spread is decreasing in the initial wealth when the market maker exhibits decreasing absolute risk aversion. We conclude by some examples illustrating the fact that almost all shapes can be obtained for the bid–ask spread (as a function of the inventory) depending on the probability distribution of the payment of the risky asset.

Scaling of the distribution of fluctuations of financial market indices.

We study the distribution of fluctuations of the S&P 500 index over a time scale deltat by analyzing three distinct databases. Database (i) contains approximately 1 200 000 records, sampled at 1-min intervals, for the 13-year period 1984-1996, database (ii) contains 8686 daily records for the 35-year period 1962-1996, and database (iii) contains 852 monthly records for the 71-year period 1926-1996. We compute the probability distributions of returns over a time scale deltat, where deltat varies approximately over a factor of 10(4)-from 1 min up to more than one month. We find that the distributions for deltat<or= 4 d (1560 min) are consistent with a power-law asymptotic behavior, characterized by an exponent alpha approximately 3, well outside the stable Lévy regime 0<alpha<2. To test the robustness of the S&P result, we perform a parallel analysis on two other financial market indices. Database (iv) contains 3560 daily records of the NIKKEI index for the 14-year period 1984-1997, and database (v) contains 4649 daily records of the Hang-Seng index for the 18-year period 1980-1997. We find estimates of alpha consistent with those describing the distribution of S&P 500 daily returns. One possible reason for the scaling of these distributions is the long persistence of the autocorrelation function of the volatility. For time scales longer than (deltat)x approximately 4 d, our results are consistent with a slow convergence to Gaussian behavior.

Risk Analysis of Planting Date, Seeding Rate, and Grain Sorghum Hybrid Maturity in Kansas Using Simulated Yields

Many variables influence yields of grain sorghum [Sorghum bicolor (L.) Moench] and make it difficult to provide appropriate recommendations to managers concerning planting dates, seeding rates, and hybrid maturity classes. General recommendations in extension publications provide a wide range of possible planting dates and seeding rates and usually suggest planting the latest maturing hybrid within the limitations of projected moisture availability and average length of growing season. Most research related to these issues focuses on just one variable, while holding others constant. This study used simulated grain sorghum yields to provide an economic risk analysis of planting date, seeding rate, and hybrid maturity for grain sorghum production in Kansas. The results showed that optimal planting dates, seeding rates, and maturity classes vary by location and risk preference. In general, the results indicate that extremely risk-averse managers would choose somewhat later planting dates, earlier maturing hybrids, and lower seeding rates than less risk-averse or risk-preferring managers.

Predicting the Present Value Distribution of a Forest Plantation Investment

Abstract Forestry investment decisions may be based on the probability distribution of financial return in addition to a point estimate of mean return. This study describes an approach to predicting the present value distribution of a plantation investment using actual data on timber price and yield. Changes in stumpage price are modeled with a lognormal diffusion process called geometric Brownian motion (GBM). Timber yield is modeled with a variant of GBM that includes an age-dependent growth component. Model parameters are estimated with time-series observations of loblolly pine (Pinus taeda L.) price and yield in the southeastern United States. Because GBM models have lognormally distributed errors, present value distributions are skewed with extremely long right-hand tails. The median and quartiles of the distribution provide a better measure of central tendency and spread than do the mean and standard deviation. A median-maximizing feedback cutting rule does not perform any better than a median-maximizing fixed rotation age suggesting that no economic gain can be obtained by monitoring timber price and yield under the assumptions of our models. The forecast error, measured by the distance between quartiles, is about twice the size of the median present value. System error is the primary cause, and error in the price process contributes more to the variability in present value than does error in the yield process. Parameter uncertainty increases forecast error 15 to 35%. The large forecast error raises the question of whether better predictive models can be built or whether the present value of a plantation investment is inherently uncertain. For. Sci. 42(3):378-388.

Long-Term Contracts, Short-Term Investment and Monitoring

term contracts, asymmetric information between investors and firms can make it impossible to implement profitable long-term projects. The paper characterizes the structure of optimal, renegotiation-proof contracts for unmonitored and monitored finance. Monitoring by investors, although itself subject to distorting incentive constraints, is shown to be able to overcome the short-term bias of investment and thus to lengthen the firms' planning horizon. I. INTRODUCTION An important question in the literature on corporate finance in the recent years has been whether the dependence on outside finance can force firms to undertake inefficient amounts of investment. Little work has been done, however, on the impact of outside finance on the quality of investment. The present paper studies this issue in the context of a question which has attracted considerable attention in the financial press and in the economic policy debate: can the dependence on outside finance lead a firm to undertake inefficient myopic investments?' To do this, the paper studies a dynamic model of financial contracting that allows one to characterize the choice of a firm's investment horizon. It points to information asymmetries as responsible for investment myopia, discusses the costs and benefits of monitoring as a reaction to it, and characterizes the optimal dynamic debt structure for unmonitored, as well as for monitored finance. According to standard financial theory, the market sees through the corporate veil and encourages the choice of efficient projects. In response to this, an important recent literature has developed more explicit theories of financial contracts and their role for the behaviour of imperfectly informed and strategically acting economic agents.2 This paper builds on this literature and analyses the problem of optimal financial contracting for a firm that wants to raise capital for a risky long-term investment project. The framework used to study the issue is a simple two-period contracting model, complicated by the interplay of hidden-characteristics and hidden-action problems on the side of the firm. More specifically, the probability distribution of project returns is assumed to depend on the intrinsic quality of the project, as well as on the investment horizon and effort chosen

Whole farm economic evaluation of East Coast fever immunization strategies on farms in the Uasin Gishu District of Kenya

Abstract East Coast feve (ECF) is a major constraint limiting livestock production in Eastern, Central, and Southern Africa. The infection and treatment method (ITM) has proven effective in research station and field trials. The work described in this paper was to quantify the economic payoffs to case study farms that adopt alternative ITM strategies and current ECF control methods. A whole farm simulation model was used to estimate the financial and economic payoffs to ITM and to estimate the probability distribution of returns for alternative ECF control strategies. Stochastic dominance was used to rank the alternative ECF controls for risk averse decision makers. The whole farm simulation model used for the analysis has the advantage of incorporating risk aspects involved in farm production, and ranking alternative strategies, as well as estimating the farmer's premium for risk preference associated with alternative ECF control strategies.

Invisible Parameters in Option Prices

ABSTRACTThis 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.

Field measurements of the roughness of fault surfaces

We recorded the height of a granitic fault surface as a function of position along one‐dimensional profiles. We show that the profiles exhibit an “anisotropic” scaling invariance: self‐affinity. The difference between the maximum and the minimum height, and the standard deviation of the height, over a length L are proportional to Lζ, where ζ ≈ 0.84. Other properties such as the Return Probability distribution or the Power Spectrum of the profile comfort this result. This self‐affine property is in good agreement with recent works on artificial fractured surfaces. Previous studies at field scale are consistent with this concept.