In this paper we study both market risks and nonmarket risks, without complete markets assumption, and discuss methods of measurement of these risks. We present and justify a set of four desirable properties for measures of risk, and call the measures satisfying these properties “coherent.” We examine the measures of risk provided and the related actions required by SPAN, by the SEC=NASD rules, and by quantile-based methods. We demonstrate the universality of scenario-based methods for providing coherent measures. We offer suggestions concerning the SEC method. We also suggest a method to repair the failure of subadditivity of quantile-based methods.

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Coherent Measures of Risk

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Convex Analysisの二,三の進展について

A new approach to optimizing or hedging a portfolio of nancial instruments to reduce risk is presented and tested on applications. It focuses on minimizing Conditional Value-at-Risk (CVaR) rather than minimizing Value-at-Risk (VaR), but portfolios with low CVaR necessarily have low VaR as well. CVaR, also called Mean Excess Loss, Mean Shortfall, or Tail VaR, is anyway considered to be a more consistent measure of risk than VaR. Central to the new approach is a technique for portfolio optimization which calculates VaR and optimizes CVaR simultaneously. This technique is suitable for use by investment companies, brokerage rms, mutual funds, and any business that evaluates risks. It can be combined with analytical or scenario-based methods to optimize portfolios with large numbers of instruments, in which case the calculations often come down to linear programming or nonsmooth programming. The methodology can be applied also to the optimization of percentiles in contexts outside of nance.

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Optimization of conditional value-at-risk

Scoring rules assess the quality of probabilistic forecasts, by assigning a numerical score based on the predictive distribution and on the event or value that materializes. A scoring rule is proper if the forecaster maximizes the expected score for an observation drawn from the distributionF if he or she issues the probabilistic forecast F, rather than G ≠ F. It is strictly proper if the maximum is unique. In prediction problems, proper scoring rules encourage the forecaster to make careful assessments and to be honest. In estimation problems, strictly proper scoring rules provide attractive loss and utility functions that can be tailored to the problem at hand. This article reviews and develops the theory of proper scoring rules on general probability spaces, and proposes and discusses examples thereof. Proper scoring rules derive from convex functions and relate to information measures, entropy functions, and Bregman divergences. In the case of categorical variables, we prove a rigorous version of the ...

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Strictly Proper Scoring Rules, Prediction, and Estimation

We present a consumptionâ€ based model that explains a wide variety of dynamic asset pricing phenomena, including the procyclical variation of stock prices, the longâ€ horizon predictability of excess stock returns, and the countercyclical variation of stock market volatility. The model captures much of the history of stock prices from consumption data. It explains the shortâ€ and longâ€ run equity premium puzzles despite a low and constant riskâ€ free rate. The results are essentially the same whether we model stocks as a claim to the consumption stream or as a claim to volatile dividends poorly corelated with consumption. The model is driven by an independently and identically distributed consumption growth process and adds a slow â€ moving external habit to the standard power utility function. These features generate slow countercyclical variation in risk premia. The model posits a fundamentally novel description of risk premia. Investors fear stocks primarily because they do poorly in recessions unrelated to the risks of longâ€ run average consumption growth.

By Force of Habit: A Consumption-Based Explanation of Aggregate Stock Market Behavior

"Dynamic Asset Pricing Theory" is a textbook for doctoral students and researchers on the theory of asset pricing and portfolio selection in multiperiod settings under uncertainty. The asset pricing results are based on the three increasingly restrictive assumptions: absence of arbitrage, single-agent optimaltiy, and equilibrium. These results are unified with two key concepts, state prices and martingales. Technicalities are given relatively little emphasis so as to draw connections between these concepts and to make plain the similarities between discrete and continuous-time models. For simplicity, all continuous-time models are based on Brownian motion. Applications include term structure models, derivative valuation and hedging methods, and dynamic programming algorithms for portfolio choice and optimal exercise of American options. Numerical methods covered include Monte Carlo simulation and finite-difference solvers for partial differential equations. Each chapter provides extensive problem exercises and notes to the literature. This second edition is substantially longer, while still retaining the consciseness for which the first edition was praised. All chapters from the first edition have been revised. Two new chapters have been added on term structure modeling and on derivative securities. References have been updated throughout. With this new edition, "Dynamic Asset Pricing Theory" remains the definitive textbook in the field.

https://epge.fgv.br/we/MD/Financas/2012?action=AttachFile&do=get&target=DuffieChap1.pdf

Dynamic Asset Pricing Theory

This article presents convenient reduced-form models of the valuation of contingent claims subject to default risk, focusing on applications to the term structure of interest rates for corporate or sovereign bonds. Examples include the valuation of a credit-spread option. This article presents a new approach to modeling term structures of bonds and other contingent claims that are subject to default risk. As in previous “reduced-form” models, we treat default as an unpredictable event governed by a hazard-rate process. 1 Our approach is distinguished by the parameterization of losses at default in terms of the fractional reduction in market value that occurs at default. Specifically, we fix some contingent claim that, in the event of no default, pays X at time T . We take as given an arbitrage-free setting in which all securities are priced in terms of some short-rate process r and equivalent martingale measure Q [see Harrison and Kreps (1979) and Harrison and Pliska (1981)]. Under this “risk-neutral” probability measure, we letht denote the hazard rate for default at time t and let Lt denote the expected fractional loss in market value if default were to occur at time t , conditional

Modeling Term Structures of Defaultable Bonds

This paper presents a consistent and arbitrage-free multifactor model of the term structure of interest rates in which yields at selected fixed maturities follow a parametric muitivariate Markov diffusion process with “stochastic volatility.” the yield of any zero-coupon bond is taken to be a maturity-dependent affine combination of the selected “basis” set of yields. We provide necessary and sufficient conditions on the stochastic model for this affine representation. We include numerical techniques for solving the model, as well as numerical techniques for calculating the prices of term-structure derivative prices. the case of jump diffusions is also considered.

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A yield-factor model of interest rates

In the setting of ‘‘affine’’ jump-diffusion state processes, this paper provides an analytical treatment of a class of transforms, including various Laplace and Fourier transforms as special cases, that allow an analytical treatment of a range of valuation and econometric problems. Example applications include fixed-income pricing models, with a role for intensity-based models of default, as well as a wide range of option-pricing applications. An illustrative example examines the implications of stochastic volatility and jumps for option valuation. This example highlights the impact on option ‘smirks’ of the joint distribution of jumps in volatility and jumps in the underlying asset price, through both jump amplitude as well as jump timing.

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Transform analysis and asset pricing for affine jump-diffusions

This article gives a broad and accessible overview o f models o f value at risk (WR), a popular measure o f the market risk of afinancialfirm’s “book,” the list ofpositions in various instruments that expose the firm to financial risk. Roughly speaking, the value at risk o f a portjolio is the loss in market value over a given time period, such as one day or two weeks, that is exceeded with a small probability, such as 1%. We focus narrowly on the market risk associated with changes in the prices or rates of underlying traded instruments. Traditionally, this would include such aspects o f credit risk as the risk o f changes in the spreads ofpublicly traded corporate and sovereign bonds. In order to maintain a narrow focus, however, I/aR does not traditionally include, and we do not review here, the risk ofdefdult on long-term derivative contracts. We begin with models o f the distribution ofunderlying market returns and ofvolatility, emphasizing the roles o f price jumps and o f stochastic volatility in determining the ‘ffatness” o f the tails o f the distributions o f returns in various markets. We then turn to methods for approximating the value at risk o f derivatives, using numerical approximations based on delta andgamma. Estimation o f the value at risk o f a portjolio ofpositions is then discussed, beginning with the choice of which risk factors to include. A n extensive numerical example illustrates the accuracy of various alternative methods, allowing for correlated jumps in the underlying market prices. Finally, we review “scenario analysis,” which treats the potential losses associated with particular scenarios, such as a given parallel shiji o f a yield curve or a given change in volatility.

Darrell Duffie

Papers

Dynamic Asset Pricing Theory

Modeling Term Structures of Defaultable Bonds

A yield-factor model of interest rates

Transform analysis and asset pricing for affine jump-diffusions

An Overview of Value at Risk