Parametric modeling of implied smile functions: a generalized SVI model

In this paper, we propose a parametric model of implied variance which is a natural generalization of the SVI model. The model improves the SVI by allowing more flexibly the negative curvature in the tails which is justified both theoretically and empirically. The fitting of the model, comparing with the other competing parametric models (SVI, SABR), to the implied volatility smile and the risk neutral density function is tested on SPX options.     

Tails,Fears, and Risk Premia

We show that the compensation for rare events accounts for a large fraction of the average equity and variance risk premia. Exploiting the special structure of the jump tails and the pricing thereof, we identify and estimate a new Investor Fears index. The index reveals large time‐varying compensation for fears of disasters. Our empirical investigations involve new extreme value theory approximations and high‐frequency intraday data for estimating the expected jump tails under the statistical probability measure, and short maturity out‐of‐the‐money options and new model‐free implied variation measures for estimating the corresponding risk‐neutral expectations.     

Option augmented density forecasts of market returns with monotone pricing kernel

Basic financial theory indicates that the ratio of the conditional density of the future value of a market index and the corresponding risk neutral density should be monotone, but a sizeable empirical literature finds otherwise. We therefore consider an option augmented density forecast of the market return obtained by transforming a baseline density forecast estimated from past excess returns so as to monotonize its ratio with a risk neutral density estimated from current option prices. To evaluate our procedure, we compare baseline and option augmented monthly density forecasts for the S&P 500 index over the period 1997–2013. We find that monotonizing the pricing kernel leads to a modest improvement in the calibration of density forecasts. Supplementary results supportive of this finding are given for market indices in France, Germany, Hong Kong, Japan and the UK.     

A semi-parametric approach to risk management

Abstract

The benchmark theory of mathematical finance is the Black–Scholes–Merton (BSM) theory, based on Brownian motion as the driving noise process for stock prices. Here the distributions of financial returns of the stocks in a portfolio are multivariate normal. Risk management based on BSM underestimates tails. Hence estimation of tail behaviour is often based on extreme value theory (EVT). Here we discuss a semi-parametric replacement for the multivariate normal involving normal variance–mean mixtures. This allows a more accurate modelling of tails, together with various degrees of tail dependence, while (unlike EVT) the whole return distribution can be modelled. We use a parametric component, incorporating the mean vector μ and covariance matrix Σ, and a non-parametric component, which we can think of as a density on [0,∞), modelling the shape (in particular the tail decay) of the distribution. We work mainly within the family of elliptically contoured distributions, focusing particularly on normal variance mixtures with self-decomposable mixing distributions. We discuss efficient methods to estimate the parametric and non-parametric components of our model and provide an algorithm for simulating from such a model. We fit our model to several financial data series. Finally, we calculate value at risk (VaR) quantities for several portfolios and compare these VaRs to those obtained from simple multivariate normal and parametric mixture models.     

A comparison of non-Gaussian VaR estimation and portfolio construction techniques

We propose a multivariate model of returns that accounts for four of the stylised facts of financial data: heavy tails, skew, volatility clustering, and asymmetric dependence with the aim of improving the accuracy of risk estimates and increasing out-of-sample utility of investors’ portfolios. We accommodate volatility clustering, the generalised Pareto distribution to capture heavy tails and skew, and the skewed-t copula to provide for asymmetric dependence. The proposed approach produces more accurate VaR estimates than seven competing approaches across eight data sets encompassing five asset classes. We show that this produces portfolios with higher utility, and lower downside risk than alternative approaches including mean–variance. We confirm that investors can substantially increase utility by accounting for departures from normality.     

The Variance Gamma Process and Option Pricing

A three parameter stochastic process, termed the variance gamma process, that generalizes Brownian motion is developed as a model for the dynamics of log stock prices. The process is obtained by evaluating Brownian motion with drift at a random time given by a gamma process. The two additional parameters are the drift of the Brownian motion and the volatility of the time change. These additional parameters provide control over the skewness and kurtosis of the return distribution. Closed forms are obtained for the return density and the prices of European options. The statistical and risk neutral densities are estimated for data on the S & P500 Index and the prices of options on this Index. It is observed that the statistical density is symmetric with some kurtosis, while the risk neutral density is negatively skewed with a larger kurtosis. The additional parameters also correct for pricing biases of the Black Scholes model that is a parametric special case of the option pricing model developed here.     

Retrieving risk neutral densities from European option prices based on the principle of maximum entropy

This paper suggests a new method of implementing the principle of maximum entropy to retrieve the risk neutral density of future stock, or any other asset, returns from European call and put prices. The method maximizes the entropy measure subject to risk neutral moment constraints in place of option prices used by previous studies. These moments can be retrieved from market option prices at each point of time, in a first step. Compared to other existing methods of retrieving the risk neutral density based on the principle of maximum entropy, the benefits of the method that the paper suggests is the use of all the available information provided by the market more efficiently. To evaluate the performance of the suggested method, the paper compares it to other risk neutral density estimation techniques by conducting a simulation study and carrying out some crucial empirical exercises.     

Multiple bids as a consequence of target management resistance: A count data approach

In this article, we focus on the question of target management resistance and the incidence of subsequent bids. A Poisson count data model is used where the dependent variable represents the number of bids (count) received and the independent variables comprise target management actions and firm specific characteristics. Of the target management actions considered, legal defense and the entry of a white knight are associated with additional bids. With respect to firm specific characteristics, we find that a high initial bid premium deters subsequent bids. Firm size is also significant and has an interesting relationship with the number of bids received. Larger target firms tend to receive more bids; however, the number of bids tails off for firms with assets exceedng $12 billion.     

Robust estimation of historical volatility and correlations in risk management

Financial time series have two features which, in many cases, prevent the use of conventional estimators of volatilities and correlations: leptokurtotic distributions and contamination of data with outliers. Other techniques are required to achieve stable and accurate results. In this paper, we review robust estimators for volatilities and correlations and identify those best suited for use in risk management. The selection criteria were that the estimator should be stable to both fractionally small departures for all data points (fat tails), and to fractionally large departures for a small number of data points (outliers). Since risk management typically deals with thousands of time series at once, another major requirement was the independence of the approach of any manual correction or data pre-processing. We recommend using volatility t-estimators, for which we derived the estimation error formula for the case when the exact shape of the data distribution is unknown. A convenient robust estimator for correlations is Kendall's tau, whose drawback is that it does not guarantee the positivity of the correlation matrix. We chose to use geometric optimization that overcomes this problem by finding the closest correlation matrix to a given matrix in terms of the Hadamard norm. We propose the weights for the norm and demonstrate the efficiency of the algorithm on large-scale problems.     

Pricing Reinsurance Contracts on FDIC Losses

This paper proposes a pricing model for the FDIC's reinsurance risk. We derive a closed‐form Weibull call option pricing model to price a call‐spread a reinsurer might sell to the FDIC. To obtain the risk‐neutral loss‐density necessary to price this call spread we risk‐neutralize a Weibull distributed FDIC annual losses by a tilting coefficient estimated from the traded call options on the BKX index. An application of the proposed approach yield reasonable reinsurance prices.     

Dark Markets,by Darrell Duffie

This paper develops a method for selecting and analysing stress scenarios for financial risk assessment, with particular emphasis on identifying sensible combinations of stresses to multiple factors. We focus primarily on reverse stress testing – finding the most likely scenarios leading to losses exceeding a given threshold. We approach this problem using a nonparametric empirical likelihood estimator of the conditional mean of the underlying market factors given large losses. We then scale confidence regions for the conditional mean by a coefficient that depends on the tails of the market factors to estimate the most likely loss scenarios. We provide rigorous justification for the confidence regions and the scaling procedure when the joint distribution of the market factors and portfolio loss is elliptically contoured. We explicitly characterize the impact of the heaviness of the tails of the distribution, contrasting a broad spectrum of cases including exponential tails and regularly varying tails. The key to this analysis lies in the asymptotics of the conditional variances and covariances in extremes. These results also lead to asymptotics for marginal expected shortfall and the corresponding variance, conditional on a market stress; we combine these results with empirical likelihood significance tests of systemic risk rankings based on marginal expected shortfall in stress scenarios.     

MIT Roundtable on Corporate Risk Management

Against the backdrop of financial crisis, a distinguished group of academics and practitioners discusses the contribution of financial management and innovation to corporate growth and value, along with the pitfalls and unintended consequences of such innovation. The main focus of most panelists is the importance of a capital structure and risk management approach that complement the strategy and operations of the business. Instructive examples are provided by Judy Lewent, former CFO and head of strategic planning at Merck, and Lakshmi Shyam‐Sunder, director of finance and risk management at the International Finance Corporation. But if these represent successful applications of finance theory, what about the large number of cases where the use of derivatives and other innovations has led to high leverage and apparent risk management failures? Part of the current trouble, as pointed out by Andrew Lo, can be attributed to the failure of risk managers and their models to account for highly improbable events—the so‐called fat tails of the distribution. But, as Robert Merton suggests in closing, there is a more comprehensive explanation for today's problems: the tendency of market participants to respond to potentially risk‐reducing financial innovation by increasing their risk‐taking in other areas. “What we have here,” says Merton,

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Estimating the basis risk of index-linked hedging strategies using multivariate extreme value theory

This paper studies the empirical quantification of basis risk in the context of index-linked hedging strategies. Basis risk refers to the risk of non-payment of the index-linked instrument, given that the hedger’s loss exceeds some critical level. The quantification of such risk measures from empirical data can be done in various ways and requires special consideration of the dependence structure between the index and the company’s losses as well as the estimation of the tails of a distribution. In this context, previous literature shows that extreme value theory can be superior to traditional methods with respect to estimating quantile risk measures such as the value at risk. Thus, the aim of this paper is to conduct an empirical analysis of basis risk using multivariate extreme value theory and extreme value copulas to estimate the underlying risk processes and their dependence structure in order to obtain a more adequate picture of basis risk associated with index-linked hedging strategies. Our results emphasize that the application of extreme value theory leads to better fits of the tails of the marginal distributions in the considered stock price sample and that traditional methods in regard to estimating marginal distributions tend to overestimate basis risk, while basis risk can in contrast be higher when taking into account extreme value copulas.     

VaR and ES for linear portfolios with mixture of generalized Laplace distributions risk factors

In this paper, we propose an explicit estimation of Value-at-Risk (VaR) and Expected Shortfall (ES) for linear portfolios when the risk factors change with a convex mixture of generalized Laplace distributions (M-GLD). We introduce the dynamics Delta-GLD-VaR, Delta-GLD-ES, Delta-MGLD-VaR and Delta-MGLD-ES, by using conditional correlation multivariate GARCH. The generalized Laplace distribution impose less restrictive assumptions during estimation that should improve the precision of the VaR and ES through the varying shape and fat tails of the risk factors in relation with the historical sample data. We also suggested some areas of application to measure price risk in agriculture, risk management and financial portfolio optimization.     

Extreme observations and risk assessment in the equity markets of MENA region: Tail measures and Value-at-Risk

The standard “delta-normal” Value-at-Risk methodology requires that the underlying returns generating distribution for the security in question is normally distributed, with moments which can be estimated using historical data and are time-invariant. However, the stylized fact that returns are fat-tailed is likely to lead to under-prediction of both the size of extreme market movements and the frequency with which they occur. In this paper, we use the extreme value theory to analyze four emerging markets belonging to the MENA region (Egypt, Jordan, Morocco, and Turkey). We focus on the tails of the unconditional distribution of returns in each market and provide estimates of their tail index behavior. In the process, we find that the returns have significantly fatter tails than the normal distribution and therefore introduce the extreme value theory. We then estimate the maximum daily loss by computing the Value-at-Risk (VaR) in each market. Consistent with the results from other developing countries [see Gencay, R. and Selcuk, F., (2004). Extreme value theory and Value-at-Risk: relative performance in emerging markets. International Journal of Forecasting, 20, 287–303; Mendes, B., (2000). Computing robust risk measures in emerging equity markets using extreme value theory. Emerging Markets Quarterly, 4, 25–41; Silva, A. and Mendes, B., (2003). Value-at-Risk and extreme returns in Asian stock markets. International Journal of Business, 8, 17–40], generally, we find that the VaR estimates based on the tail index are higher than those based on a normal distribution for all markets, and therefore a proper risk assessment should not neglect the tail behavior in these markets, since that may lead to an improper evaluation of market risk. Our results should be useful to investors, bankers, and fund managers, whose success depends on the ability to forecast stock price movements in these markets and therefore build their portfolios based on these forecasts.     

Managing extreme risks in tranquil and volatile markets using conditional extreme value theory

Financial risk management typically deals with low-probability events in the tails of asset price distributions. To capture the behavior of these tails, one should therefore rely on models that explicitly focus on the tails. Extreme value theory (EVT)-based models do exactly that, and in this paper, we apply both unconditional and conditional EVT models to the management of extreme market risks in stock markets. We find conditional EVT models to give particularly accurate Value-at-Risk (VaR) measures, and a comparison with traditional (Generalized ARCH (GARCH)) approaches to calculate VaR demonstrates EVT as being the superior approach both for standard and more extreme VaR quantiles.     

Developing a stress testing framework based on market risk models

The Basel 2 Accord requires regulatory capital to cover stress tests, yet no coherent and objective framework for stress testing portfolios exists. We propose a new methodology for stress testing in the context of market risk models that can incorporate both volatility clustering and heavy tails. Empirical results compare the performance of eight risk models with four possible conditional and unconditional return distributions over different rolling estimation periods. When applied to major currency pairs using daily data spanning more than 20 years we find that stress test results should have little impact on current levels of foreign exchange regulatory capital.     

The effects of board gender diversity on a firm's risk strategies

We study whether board gender diversity (BGD) affects corporate risk strategies. Specifically, we investigate the association between BGD and firms’ reputation risk and financial risk. Using S&P data from 1997 to 2013, we find that BGD is negatively associated with tax avoidance, suggesting firms with gender‐diverse boards are more cautious about potential reputation risks associated with aggressive tax strategies. However, we find that BGD is positively associated with firms’ financial risk. The combined findings illustrate that BGD aligns a firm's risk exposure closer to risk‐neutral shareholders’ preferences by reducing reputation risk exposure while enabling necessary financial risk exposure.     

Risk Management for Linear and Non-Linear Assets: A Bootstrap Method with Importance Resampling to Evaluate Value-at-Risk

Many empirical studies suggest that the distribution of risk factors has heavy tails. One always assumes that the underlying risk factors follow a multivariate normal distribution that is a assumption in conflict with empirical evidence. We consider a multivariate t distribution for capturing the heavy tails and a quadratic function of the changes is generally used in the risk factor for a non-linear asset. Although Monte Carlo analysis is by far the most powerful method to evaluate a portfolio Value-at-Risk (VaR), a major drawback of this method is that it is computationally demanding. In this paper, we first transform the assets into the risk on the returns by using a quadratic approximation for the portfolio. Second, we model the return’s risk factors by using a multivariate normal as well as a multivariate t distribution. Then we provide a bootstrap algorithm with importance resampling and develop the Laplace method to improve the efficiency of simulation, to estimate the portfolio loss probability and evaluate the portfolio VaR. It is a very powerful tool that propose importance sampling to reduce the number of random number generators in the bootstrap setting. In the simulation study and sensitivity analysis of the bootstrap method, we observe that the estimate for the quantile and tail probability with importance resampling is more efficient than the naive Monte Carlo method. We also note that the estimates of the quantile and the tail probability are not sensitive to the estimated parameters for the multivariate normal and the multivariate t distribution. The research of Shih-Kuei Lin was partially supported by the National Science Council under grants NSC 93-2146-H-259-023. The research of Cheng-Der Fuh was partially supported by the National Science Council under grants NSC 94-2118-M-001-028.     

The Variance Gamma Process and Option Pricing

A three parameter stochastic process, termed the variance gammaprocess, that generalizes Brownian motion is developed as amodel for the dynamics of log stock prices. Theprocess is obtainedby evaluating Brownian motion with drift at a random time givenby a gamma process. The two additional parameters are the driftof the Brownian motion and the volatility of the time change.These additional parameters provide control over the skewnessand kurtosis of the return distribution. Closed forms are obtainedfor the return density and the prices of European options.Thestatistical and risk neutral densities are estimated for dataon the S&P500 Index and the prices of options on this Index.It is observed that the statistical density is symmetric withsome kurtosis, while the risk neutral density is negativelyskewed with a larger kurtosis. The additional parameters alsocorrect for pricing biases of the Black Scholes model that isa parametric special case of the option pricing model developedhere.