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Madan, and M. The fine structure of asset returns: An empirical investigation. Journal of Business, —, Chambers, C. Mallows, and B. A method for simulating stable random variables. Chen and H. Statistica Sinica, —, Cherubini, E. Luciano, and W. Copula Methods in Finance. Cheung and L. Dual multivariate auto-regressive modeling in state space for temporal signal separation.

Christoffersen and D. Backtesting value—at—risk: A duration—based approach. Journal of Financial Econometrics, — , A subordinated stochastic process model with finite variance for speculative prices. Econometrica, —, Signal Processing, —, Cont and P. Financial Modelling with Jump Processes. Corsi, S. Mittnik, C. Pigorsch, and U. The volatility of realized volatility. Working Paper, Center for Financial Studies, Tail behaviour of the euro.

Applied Economics, —, Dacorogna, U. Pictet, and C. Extremal forex returns in extremely large data sets. Extremes, —, An easily implemented generalised inverse Gaussian generator. Communications in Statistics: Simulations, —, Peng, and C. Using a bootstrap method to choose the sample fraction in tail index estimation. Tail index and quantile estimation with very high frequency data. Journal of Empirical Finance, —, Resnick, H. Extremal behaviour of solutions to a stochastic difference equation with applications to ARCH processes.

Stylized facts of nominal exchange rate returns. Ploeg, editor, The Handbook of International Macroeconomics, pages — Blackwell, Doganoglu and S. An approximation procedure for asymmetric stable Paretian densities. Computational Statistics, —, On the asymptotic normality of the maximum-likelihood es- timate when sampling from a stable distribution. Annals of Statistics, —, A note on kurtosis. Journal of the Royal Statistical Society, — , Eberlein and U. Hyperbolic distributions in finance. Bernoulli, — , Eberlein, U. Keller, and K. New insights into smile, mispricing, and value at risk: The hyperbolic model.

Eberlein and E. Generalized hyperbolic and inverse Gaus- sian distributions: Limiting cases and approximation of processes. Dalang, M. Dozzi, and F. Embrechts, C. Modelling Extremal Events for Insurance and Finance. Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation.

Mandelbrot and the stable Paretian hypothesis. The behavior of stock market prices. Foundations of Finance. Basic Books, Fama and R. Parameter estimates for symmetric stable distributions. Farmer and F.

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On the origin of power-law tails in price fluctuations. Quantitative Finance, 4:C7—C11, On Bayesian modeling of fat tails and skewness. Gabaix, P. Gopikrishnan, V. Plerou, and E. A unified econophysics explanation for the power-law exponents of stock market activity. Physica A, —88, Plerou, and H.

Selected Publications

A theory of power-law distributions in financial market fluctuations. Nature, —, Institutional investors and stock market volatility. Quarterly Journal of Economics, —, Galbraith and S. Circuit breakers and the tail index of equity returns. Gallant and G. Which moments to match?

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Gerber and E. Option pricing by Esscher transforms. Transac- tions of Society of Actuaries, —, Ghose and K. Implicit renewal theory and tails of solutions of random equations. Annals of Applied Probability, —, Gopikrishnan, M.

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Meyer, and H. Scaling of the distribution of fluctuations of financial market indices. Physical Review E, —, Gourieroux and J. Truncated maximum likelihood, goodness of fit tests and tail analysis. Gray and D. Empirical comparisons of distributional models for stock index returns. Journal of Business Finance and Accounting, —, Probability: A Graduate Course. Haas, S. Mittnik, and M. Mixed normal conditional heteroskedas- ticity. More evidence on the distribution of security returns. Journal of Finance, —, Hall, W. Brorsen, and S. The distribution of futures prices: A test of the stable Paretian and mixture of normals hypotheses.

Journal of Financial and Quantitative Analysis, —, On some simple estimates of an exponent of regular variation. Series B, —42, A new approach to the economic analysis of nonstationary time series and the business cycle. Autoregressive conditional density estimation. International Eco- nomic Review, —, Similarities in the distribution of stock market price changes between the eighteenth and twentieth centuries. Journal of Business, —79, Hazan, Z. Landsman, and U. Robustness via a mixture of exponen- tial power distributions.

Computational Statistics and Data Analysis, —, He and T. A simple general approach to inference about the tail of a distribution. Hols and C. The limiting distribution of extremal exchange rate returns. Modeling heteroskedasticity in daily foreign-exchange rates. Hsu, R. Miller, and D. On the stable Paretian behavior of stock-market prices. Huisman, K. Koedijk, C. Kool, and F. Tail-index estimates in small samples. The tail-fatness of FX returns reconsidered.

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Jones and M. A skew extension of the t-distribution, with ap- plications. McGraw-Hill, Kaizoji and M. Empirical laws of a stock price index and a stochastic model. Advances in Complex Systems, —, Stable densities under change of scale and total variation inequalities.

A common error concerning kurtosis. Journal of the American Statistical Association, , Journal of Econometrics, —76, Kearns and A. Estimating the density tail index for financial time series. Random difference equations and renewal theory for products of random matrices.

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Acta Mathematica, —, Koedijk, M. Schafgans, and C. The tail index of exchange rate returns. Koedijk, P. Stork, and C. Differences between foreign exchange rate regimes: The view from the tails. Journal of International Money and Finance, —, Kogon and D. Characteristic function based estimation of stable distribution parameters. Adler, R.

Feldman, and M. Asymmetric power distribution: Theory and applications to risk management. Models of stock returns: A comparison. Regression-type estimation of the parameters of stable laws. Neumann, M. Stock returns and hyper- bolic distributions. Mathematical and Computer Modelling, —15, Techniques for verifying the accuracy of risk management models. Journal of Derivatives, —84, Lau, H. Lau, and J. The distribution of stock returns: New evidence against the stable model. Leadbetter, G. Lindgren, and H. Stochastic volatility as a simple generator of apparent financial power laws and long memory.

Quantitative Finance, —, Lee and B. Econometric Theory, —52, Ling and M. Statistics and Probability Letters, —, The asymptotic distribution of extreme stock market returns. Jour- nal of Business, —, The choice of the distribution of asset returns: How extreme value theory can help?

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  6. Annals of Economics and Finance, 13, , We analyze return predictability for the Chinese stock market, including the aggregate market portfolio and the components of the aggregate market, such as portfolios sorted on industry, size, book-to-market and ownership concentration. Considering a variety of economic variables as predictors, both in-sample and out-of-sample tests highlight significant predictability in the aggregate market portfolio of the Chinese stock market and substantial differences in return predictability across components.

    Among industry portfolios, Finance and insurance, Real estate, and Service exhibit the most predictability, while portfolios of small-cap, low book-to-market ratio and low ownership concentration firms also display considerable predictability. Two key findings provide economic explanations for component predictability: i based on a novel out-of-sample decomposition, time-varying systematic risk premiums captured by the conditional CAPM model largely account for component predictability; ii industry concentration significantly explain differences in return predictability across industries, consistent with the information-flow frictions emphasized by Hong, Torous, and Valkanov The modern portfolio theory pioneered by Markowitz is widely used in practice and extensively taught to MBAs.

    Since the combinations are theory-based, our study may be interpreted as reaffirming the usefulness of the Markowitz theory in practice. Journal of Financial Economics, 99, , Employing a forecast combination approach based on a variety of economic variables and lagged component returns as predictors, we find significant evidence of out-of-sample return predictability for nearly all component portfolios. The pattern of component return predictability is enhanced during business-cycle recessions, linking component return predictability to the real economy.

    Considering various component-rotation investment strategies, we show that out-of-sample component return predictability can be exploited to substantially improve portfolio performance. Journal of Portfolio Management, 37, , , Cross Sectional Asset Pricing Tests. A major problem in finance is to understand why different financial assets earn vastly different returns on average. In this paper, we survey various econometric approaches that have been developed to empirically examine various asset pricing models used to explain the difference in cross section of security returns.

    The approaches range from regressions to the generalized method of moments, and the associated asset pricing models are both conditional and unconditional. In addition, we review some of the major empirical studies. Annual Review of Financial Economics, 2, , Bayesian Portfolio Analysis. This paper reviews the literature on Bayesian portfolio analysis. Information about events, macro conditions, asset pricing theories, and security-driving forces can serve as useful priors in selecting optimal portfolios.

    Moreover, parameter uncertainty and model uncertainty are practical problems encountered by all investors. The Bayesian framework neatly accounts for these uncertainties, whereas standard statistical models often ignore them. We review Bayesian portfolio studies when asset returns are assumed both independently and identically distributed as well as predictable through time. We cover a range of applications, from investing in single assets and equity portfolios to mutual and hedge funds. We also outline existing challenges for future work.

    Economic objectives are often ignored when estimating parameters, though the loss of doing so can be substantial. This paper proposes a way to allow Bayesian priors to reflect the objectives. In terms of out-of-sample performance, the Bayesian rules under the objective-based priors can outperform substantially some of the best rules developed in the classical framework.

    Journal of Financial and Quantitative Analysis, 45, , First draft, September, Stock market predictability is of considerable interest in both academic research and investment practice. In this paper, we tighten this bound by a squared factor of the correlation between the default pricing kernel and the state variables of the economy.

    Since the correlation can be substantially smaller than one, our bound can be much tighter than Ross's. An empirical application illustrates that while Ross's bound is not binding, our bound does. Economics Letters, , , Fabozzi and Dashan Huang. In this paper we provide a survey of recent contributions to robust portfolio strategies from operations research and finance to the theory of portfolio selection.

    Our survey covers results derived not only in terms of the standard mean-variance objective, but also in terms of two of the most popular risk measures, mean-VaR and mean-CVaR developed recently. In addition, we review optimal estimation methods and Bayesian robust approaches. Annals of Operations Research, , , In this paper, we use a simple model to illustrate that the existence of a large, negative wealth shock and insufficient insurance against such a shock can potentially explain both the limited stock market participation puzzle and the low-consumption-high-savings puzzle that are widely documented in the literature.

    We then conduct an extensive empirical analysis on the relation between household portfolio choices and access to private insurance and various types of government safety nets, including social security and unemployment insurance. The empirical results demonstrate that a lack of insurance against large, negative wealth shocks is strongly correlated with lower participation rates and higher saving rates.

    Overall, the evidence suggests an important role of insurance in household investment and savings decisions. Journal of Financial Economics, 96, , While a host of economic variables have been identified in the literature with the apparent in-sample ability to predict the equity premium, Welch and Goyal find that these variables fail to deliver consistent out-of-sample forecasting gains relative to the historical average.

    Arguing that substantial model uncertainty and instability seriously impair the forecasting ability of individual predictive regression models, we recommend combining individual model forecasts to improve out-of-sample equity premium prediction. Combining delivers statistically and economically significant out-of-sample gains relative to the historical average on a consistent basis over time.

    We provide two empirical explanations for the benefits of the forecast combination approach: i combining forecasts incorporates information from numerous economic variables while substantially reducing forecast volatility; ii combination forecasts of the equity premium are linked to the real economy.

    Top on the Most Read List as of June Review of Financial Studies, 23, , The Longer working paper version. Financial Analysts Journal, 66 1 , , The Black-Litterman model is a popular approach for asset allocation by blending an investor's proprietary views with the views of the market. However, their model ignores the data-generating process whose dynamics can have significant impact on future portfolio returns.

    This paper extends, in two ways, the Black-Litterman model to allow Bayesian learning to exploit all available information-- the market views, the investor's proprietary views as well as the data. Our framework allows practitioners to combine insights from the Black-Litterman model with the data to generate potentially more reliable trading strategies and more robust portfolios.

    Further, we show that many Bayesian learning tools can now be readily applied to practical portfolio selections in conjunction with the Black-Litterman model. Journal of Portfolio Management, 36 1 , , The median is a better measure than the mean in evaluating the long-term value of a portfolio. However, the standard plug-in estimate of the median is too optimistic. It has a substantial upward bias that can easily exceed a factor of two. In this paper, we provide an unbiased forecast of the median of the long-term value of a portfolio. In addition, we also provide an unbiased forecast of an arbitrary percentile of the long-term portfolio value distribution.

    This allows us to construct the likely range of the long-term portfolio value for any given confidence level. Finally, we provide an unbiased forecast of the probability for the long-term portfolio value falling into a given interval. Our unbiased estimators provide a more accurate assessment of the long-term value of a portfolio than the traditional estimators, and are useful for long-term planning and investment.

    Financial Analysts Journal, 65 4 , , In this paper, we analyze the usefulness of technical analysis, specifically the widely used moving average trading rule from an asset allocation perspective. We show that when stock returns are predictable, technical analysis adds value to commonly used allocation rules that invest fixed proportions of wealth in stocks. When there is uncertainty about predictability which is likely in practice, the fixed allocation rules combined with technical analysis can outperform the prior-dependent optimal learning rule when the prior is not too informative.

    Moreover, the technical trading rules are robust to model specification, and they tend to substantially outperform the model-based optimal trading strategies when there is uncertainty about the model governing the stock price. Journal of Financial Economics, 92, , The fundamental law of active portfolio management tells an active manager how to transform his alpha forecasts into the valued-added of his active portfolio by using a linear strategy with active positions proportional to the forecasts.

    This linear strategy is conditionally optimal because it is optimal each period conditional on the forecasts at that time. However, the unconditional value-added the valued-added over the long haul or over multiple periods is what usually the manager strives earnestly for. Under this unconditional objective, the linear strategy can approach zero value-added if the forecasts or signals have a high kurtosis. To overcome this problem, we provide an investment strategy that maximizes the unconditional value-added with the optimal use of conditional information.

    Our strategy is nonlinear in the forecasts, but has a simple economic interpretation. When the alpha forecasts are high, we invest less aggressively than the linear strategy, and when the forecasts are low, we invest more aggressively. In this way, we tend to smooth our value-added over time, and hence, on a risk-adjusted basis, our long-term unconditional value-added will in most cases be substantially higher than that based on the linear strategy, particularly when the alpha forecasts experience high kurtosis.

    Journal of Portfolio Management, 35 1 , , The fundamental law of active portfolio management pioneered by Grinold provides profound insights on the value creation process of managed funds. However, a key weakness of the law and its various extensions is that they ignore the estimation risk associated with the parameter inputs of the law.

    We show that the estimation errors have a substantial impact on the value-added of an actively managed portfolio, and they can easily destroy all the value promised by the law if they are not dealt with carefully. For bettering the chance of active managers to beat benchmark indices, we propose two methods, scaling and diversification, that can be used effectively to minimize the impact of the estimation errors significantly.

    Journal of Portfolio Management, 34 4 , , In this paper, we provide a model-free test for asymmetric correlations in which stocks move more often with the market when the market goes down than when it goes up. We also provide such tests for asymmetric betas and covariances. In addition, we evaluate the economic significance of incorporating asymmetries into investment decisions. When stocks are sorted by size, book-to-market and momentum, we find strong evidence of asymmetry for both the size and momentum portfolios, but no evidence for the book-to-market portfolios.

    Moreover, the asymmetries can be of substantial economic importance for an investor with a disappointment aversion preference of Ang, Bekaert and Liu If the investors's felicity function is of the power utility form and if his coefficient of disappointment aversion is between 0. Review of Financial Studies, 20, , Optimal Portfolio Choice with Parameter Uncertainty. In this paper, we analytically derive the expected loss function associated with using sample means and covariance matrix of returns to estimate the optimal portfolio. Our analytical results show that the standard plug-in approach that replaces the population parameters by their sample estimates can lead to very poor out-of-sample performance.

    We further show that with parameter uncertainty, holding the sample tangency portfolio and the riskless asset is never optimal. An investor can benefit by holding some other risky portfolios that help reduce the estimation risk. In particular, we show that a portfolio that optimally combines the riskless asset, the sample tangency portfolio, and the sample global minimum-variance portfolio dominates a portfolio with just the riskless asset and the sample tangency portfolio, suggesting that the presence of estimation risk completely alters the theoretical recommendation of a two-fund portfolio.

    Journal of Financial and Quantitative Analysis, 42, , While existing studies use almost exclusively this procedure, we show that alternative two-pass methods can have better small sample performance. In addition, we provide tractable GMM approaches that accommodate conditional heteroscedasticity of the data. Moreover, the risk premium estimates and t-ratios of the Fama and MacBeth procedure provide no information on whether the model is misspecified or not, and they can be misleadingly interpreted if the model is indeed misspecified.

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    We not only provide formal model misppecification tests, but also how that various estimation methods are useful in detecting model misppecification. Journal of Financial Economics, 84, , Using Bootstrap to Test Portfolio Efficiency. To facilitate wide use of the bootstrap method in finance, this paper shows by intuitive arguments and by simulations how it can improve upon existing tests to allow less restrictive distributional assumptions on the data and to yield more reliable higher-order accurate asymptotic inference.

    In particular, we apply the method to examine the efficiency of CRSP value-weighted stock index, and to test the well-known Fama and French three-factor model. We find that existing tests tend to over-reject. Annals of Economics and Finance, 7, , Portfolio Optimization under Asset Pricing Anomalies. Fama and French find that the SMB and the HML factors explain much of the cross-section stock returns that are unexplained by the CAPM, whereas Daniel and Titman show that it is the characteristics of the stocks that are responsible rather than the factors.

    But both arguments are largely based only on expected return comparisons, and little is known about how important each of the two explanations matters to an investor's investment decisions in general and portfolio optimization in particular. In this paper, we show that a mean-variance maximizing investor who exploits the asset pricing anomaly of the CAPM can achieve substantial economic gains than simply holding the market index.

    Indeed, using Japanese data over the period , we find that the optimized portfolio constructed from characteristics-based model and based on the first largest stocks is the best performing one and has monthly returns more than 0. In many applications, the correlation is small, and hence our bound can be substantially tighter than Hansen-Jagannathan's. Another example is applying the new bound, with the growth rate of consumption as a state variable, to the 25 size and book-to-market sorted portfolios used by Fama and French , then it is more than times greater than the Hansen-Jagannathan bound.

    Journal of Business, 79, , As the usual normality assumption is firmly rejected by the data, investors encounter a data-generating process DGP uncertainty in making investment decisions. In this paper, we propose a novel way to incorporate uncertainty about the DGP into portfolio analysis. We find that accounting for fat tails leads to nontrivial changes in both parameter estimates and optimal portfolio weights, but the certainty—equivalent losses associated with ignoring fat tails are small.

    This suggests that the normality assumption works well in evaluating portfolio performance for a mean-variance investor. Journal of Financial Economics, 72, , This paper characterizes the forces that determine time-variation in expected international asset returns. We offer a number of innovations. By using the latent factor technique, we do not have to prespecify the sources of risk. We solve for the latent premiums and characterize their time-variation.

    We find evidence that the first factor premium resembles the expected return on the world market portfolio. However, the inclusion of this premium alone is not sufficient to explain the conditional variation in the returns. We find evidence of a second factor premium which is related to foreign exchange risk. Our sample includes new data on both international industry portfolios and international fixed income portfolios.

    We find that the two latent factor model performs better in explaining the conditional variation in asset returns than a prespecified two factor model.

    Power Laws in Economics and Finance

    Finally, we show that differences in the risk loadings are important in accounting for the cross-sectional variation in the international returns. Annals of Economics and Finance, 3, , This paper characterizes the rate of convergence of discrete-time multinomial option prices. We show that it depends on the smoothness of option payoff function, and is much lower than commonly believed because the payoff functions are often all-or-nothing type and not continuously differentiable.

    We propose two methods, one of which is to smooth the payoff function, that help to yield the same rate of convergence as smooth payoff functions. Mathematical Finance, 10, , We find that beta, adjusted for infrequent trading or not, fails to explain the cross-section of monthly expected returns, but does a much better job for horizons over half- and one-year. Annals of Economics and Finance, 1, , A new framework is proposed to find the best linear combinations of economic variables that optimally forecast security factors. In particular, we obtain such combinations from Chen et al.

    Journal of Business 59, , five economic variables, and obtain a new GMM test for the APT which is more robust than existing tests. In addition, by using Fama and French's five factors, we test whether fewer factors are sufficient to explain the average returns on 25 stock portfolios formed on size and book-to-market.

    While inconclusive in-sample, a three-factor model appears to perform better out-of-sample than both four- and five-factor models. Journal of Financial Markets, 2, , Testing Multi-beta Pricing Models. This paper presents a complete solution to the estimation and testing of multi-beta models by providing a small sample likelihood ratio test when the usual normality assumption is imposed and an almost analytical GMM test when the normality assumption is relaxed.

    Using 10 size portfolios from January to December , we reject the joint efficiency of the CRSP value-weighted and equal-weighted indices. We also apply the tests to analyze a new version of Fama and French [Fama, E. Common risk factors in the returns on stocks and bonds. Journal of Financial Economics 33, 3—] three-factor model in addition to two standard ones, and find that the new version performs the best. Journal of Empirical Finance, 6, , In this paper, we point out that the widely used stochastic discount factor SDF methodology ignores a fully specified model for asset returns.

    As a result, it suffers from two potential problems when asset returns follow a linear factor model. The first problem is that the risk premium estimate from the SDF methodology is unreliable. The second problem is that the specification test under the SDF methodology has very low power in detecting misspecified models.

    Traditional methodologies typically incorporate a fully specified model for asset returns, and they can perform substantially better than the SDF methodology. Journal of Finance, 54, , Monte Carlo simulation is widely used in practice to value exotic options for which analytical formulas are not available. When valuing those options that depend on extreme values of the underlying asset, convergence of the standard simulation is slow as the time grid is refined, and even a daily simulation interval produces unacceptable errors. This article suggests approximating the extreme value on a subinterval by a random draw from the known theoretical distribution for an extreme of a Brownian bridge on the same interval.

    This approach provides reliable option values and retains the flexibility of simulations, in that it allows great freedom in choosing a price process for the underlying asset or a joint process for the asset price, its volatility, and other asset prices.

    Financial Analysts Journal, 53, , Within the past few years several articles have suggested that returns on large equity portfolios may contain a significant predictable component at horizons 3 to 6 years. Impact Factor 1. Issue Please enter a valid issue for volume. About this Journal.

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