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

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 examine the pricing of aggregate volatility risk in the cross-section of stock returns. Consistent with theory, we find that stocks with high sensitivities to innovations in aggregate volatility have low average returns. In addition, we find that stocks with high idiosyncratic volatility relative to the Fama and French (1993) model have abysmally low average returns. This phenomenon cannot be explained by exposure to aggregate volatility risk. Size, book-to-market, momentum, and liquidity effects cannot account for either the low average returns earned by stocks with high exposure to systematic volatility risk or for the low average returns of stocks with high idiosyncratic volatility.

The Cross-Section of Volatility and Expected Returns

We examine the pricing of aggregate volatility risk in the cross-section of stock returns. Consistent with theory, we find that stocks with high sensitivities to innovations in aggregate volatility have low average returns. Stocks with high idiosyncratic volatility relative to the Fama and French (1993, Journal of Financial Economics 25, 2349) model have abysmally low average returns. This phenomenon cannot be explained by exposure to aggregate volatility risk. Size, book-to-market, momentum, and liquidity effects cannot account for either the low average returns earned by stocks with high exposure to systematic volatility risk or for the low average returns of stocks with high idiosyncratic volatility. IT IS WELL KNOWN THAT THE VOLATILITY OF STOCK RETURNS varies over time. While considerable research has examined the time-series relation between the volatility of the market and the expected return on the market (see, among others, Campbell and Hentschel (1992) and Glosten, Jagannathan, and Runkle (1993)), the question of how aggregate volatility affects the cross-section of expected stock returns has received less attention. Time-varying market volatility induces changes in the investment opportunity set by changing the expectation of future market returns, or by changing the risk-return trade-off. If the volatility of the market return is a systematic risk factor, the arbitrage pricing theory or a factor model predicts that aggregate volatility should also be priced in the cross-section of stocks. Hence, stocks with different sensitivities to innovations in aggregate volatility should have different expected returns. The first goal of this paper is to provide a systematic investigation of how the stochastic volatility of the market is priced in the cross-section of expected stock returns. We want to both determine whether the volatility of the market

https://www0.gsb.columbia.edu/faculty/rhodrick/crosssection.pdf

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

http://faculty.baruch.cuny.edu/lwu/890/pan2000.pdf

The jump-risk premia implicit in options: evidence from an integrated time-series study $

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.

An Overview of Value at Risk

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 hightlights 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.

/pdf/transform-analysis-and-asset-pricing-for-affine-jump-52dp8j3kcj.pdf

Transform Analysis and Asset Pricing for Affine Jump-Diffusions

This paper explores the nature of default arrival and recovery implicit in the term structures of sovereign CDS spreads. We argue that term structures of spreads reveal not only the arrival rates of credit events (λ Q ), but also the loss rates given credit events. Applying our framework to Mexico, Turkey, and Korea, we show that a single-factor model with λ Q following a lognormal process captures most of the variation in the term structures of spreads. The risk premiums associated with unpredictable variation in λ Q are found to be economically significant and co-vary importantly with several economic measures of global event risk, financial market volatility, and macroeconomic policy. THE BURGEONING MARKET FOR SOVEREIGN CREDIT DEFAULT SWAPS (CDS) contracts offers a nearly unique window for viewing investors’ risk-neutral probabilities of major credit events impinging on sovereign issuers, and their risk-neutral losses of principal in the event of a restructuring or repudiation of external debts. In contrast to many “emerging market” sovereign bonds, sovereign CDS contracts are designed without complex guarantees or embedded options. Trading activity in the CDS contracts of several sovereign issuers has developed to the point that they are more liquid than many of the underlying bonds. Moreover, in contrast to the corporate CDS market, where trading has been concentrated largely in the 5-year maturity contract, CDS contracts at several maturity points between 1 and 10 years have been actively traded for several years. As such, a full term structure of CDS spreads is available for inferring default and recovery information from market data. This paper explores in depth the time-series properties of the risk-neutral mean arrival rates of credit events (λ Q ) implicit in the term structure of sovereign CDS spreads. Applying our framework to Mexico, Turkey, and Korea, three countries with different geopolitical characteristics and credit ratings, we ∗ Pan is with the MIT Sloan School of Management and NBER. Singleton is with the Graduate

/pdf/default-and-recovery-implicit-in-the-term-structure-of-4f62yrulcm.pdf

Jun Pan

Papers

Transform analysis and asset pricing for affine jump-diffusions

The jump-risk premia implicit in options: evidence from an integrated time-series study $

An Overview of Value at Risk

Transform Analysis and Asset Pricing for Affine Jump-Diffusions

Default and Recovery Implicit in the Term Structure of Sovereign CDS Spreads