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on Risk Management |
| By: | Jean-Philippe Bouchaud (Science & Finance, Capital Fund Management; CEA Saclay;); Didier Sornette (UCLA; Science & Finance, Capital Fund Management); Giulia Iori |
| Abstract: | We present a theory of option pricing and hedging, designed to address non-perfect arbitrage, market friction and the presence of `fat' tails. An implied volatility `smile' is predicted. We give precise estimates of the residual risk associated with optimal (but imperfect) hedging. |
| JEL: | G10 |
| URL: | http://d.repec.org/n?u=RePEc:sfi:sfiwpa:500039&r=rmg |
| By: | Marc Potters (Science & Finance, Capital Fund Management); Jean-Philippe Bouchaud (Science & Finance, Capital Fund Management; CEA Saclay;) |
| Abstract: | We reconsider the problem of option pricing using historical probability distributions. We first discuss how the risk-minimisation scheme proposed recently is an adequate starting point under the realistic assumption that price increments are uncorrelated (but not necessarily independent) and of arbitrary probability density. We discuss in particular how, in the Gaussian limit, the Black-Scholes results are recovered, including the fact that the average return of the underlying stock disappears from the price (and the hedging strategy). We compare this theory to real option prices and find these reflect in a surprisingly accurate way the subtle statistical features of the underlying asset fluctuations. |
| JEL: | G10 |
| URL: | http://d.repec.org/n?u=RePEc:sfi:sfiwpa:500036&r=rmg |
| By: | Jean-Philippe Bouchaud (Science & Finance, Capital Fund Management; CEA Saclay;) |
| Abstract: | Estimating and controlling large risks has become one of the main concern of financial institutions. This requires the development of adequate statistical models and theoretical tools (which go beyond the traditionnal theories based on Gaussian statistics), and their practical implementation. Here we describe three interrelated aspects of this program: we first give a brief survey of the peculiar statistical properties of the empirical price fluctuations. We then review how an option pricing theory consistent with these statistical features can be constructed, and compared with real market prices for options. We finally argue that a true `microscopic' theory of price fluctuations (rather than a statistical model) would be most valuable for risk assessment. A simple Langevin-like equation is proposed, as a possible step in this direction. |
| JEL: | G10 |
| URL: | http://d.repec.org/n?u=RePEc:sfi:sfiwpa:500042&r=rmg |
| By: | Ronald J. Balvers (Division of Economics and Finance, West Virginia University); Yangru Wu (Department of Finance and Economics, Rutgers University) |
| Abstract: | A number of studies have separately identified mean reversion and momentum, but this paper considers these effects jointly: Potential for mean reversion and momentum is combined optimally into one indicator, interpretable as a risk-adjusted expected return. Combination momentum-contrarian strategies, used to select from among 18 developed equity markets at a monthly frequency, outperform both pure momentum and pure contrarian strategies. A key assumption is that, among developed markets, only global equity price index shocks can have permanent components, as would be reasonable in a production-based asset-pricing context, given that production levels converge across developed countries. The results hold with basic risk corrections and continue to hold after transactions costs are included. They reveal that it is important to control for mean reversion in exploiting momentum and vice versa. |
| Keywords: | Mean Reversion, Momentum, International Asset Pricing, Investment Strategies |
| JEL: | G10 G15 |
| URL: | http://d.repec.org/n?u=RePEc:wvu:wpaper:04-11&r=rmg |
| By: | Jean-Philippe Bouchaud (Science & Finance, Capital Fund Management; CEA Saclay;); Jean-Pierre Aguilar (Science & Finance, Capital Fund Management); Didier Sornette (UCLA; Science & Finance, Capital Fund Management); Christian Walter |
| Abstract: | We propose a method of optimization of asset allocation in the case where the stock price variations are supposed to have "fat" tails represented by power laws. Generalizing over previous works using stable Lévy distributions, we distinguish three distinct components of risk described by three different parts of the distributions of price variations: unexpected gains (to be kept), harmless noise inherent to financial activity, and unpleasant losses, which is the only component one would like to minimize. The independent treatment of the tails of distributions for positive and negative variations and the generalization to large events of the notion of covariance of two random variables provide explicit formulae for the optimal portfolio. The use of the probability of loss (or equivalently the Value-at-Risk), as the key quantity to study and minimize, provides a simple solution to the problem of optimization of asset allocations in the general case where the characteristic exponents are different for each asset. |
| JEL: | G10 |
| URL: | http://d.repec.org/n?u=RePEc:sfi:sfiwpa:500044&r=rmg |
| By: | Marc Potters (Science & Finance, Capital Fund Management); Jean-Philippe Bouchaud (Science & Finance, Capital Fund Management; CEA Saclay;); Jean-Pierre Aguilar (Science & Finance, Capital Fund Management) |
| Abstract: | When the available statistical information is imperfect, it is dangerous to follow standard optimisation procedures to construct an optimal portfolio, which usually leads to a strong concentration of the weights on very few assets. We propose a new way, based on generalised entropies, to ensure a minimal degree of diversification. |
| JEL: | G10 |
| URL: | http://d.repec.org/n?u=RePEc:sfi:sfiwpa:500045&r=rmg |
| By: | Ronald J. Balvers (Division of Economics and Finance, West Virginia University); Yangru Wu (Department of Finance and Economics, Rutgers University) |
| Abstract: | If transitory profitable trading opportunities exist, filter rules are used to mitigate transaction costs. We use a dynamic programming framework to design an optimal filter which maximizes after-cost expected returns. The filter size depends crucially on the degree of persistence of trading opportunities, transaction cost, and standard deviation of shocks. Applying our theory to daily dollar-yen exchange trading, we find that the optimal filter can be economically significantly different from a naïve filter equal to the transaction cost. The candidate trading strategies generate positive returns that disappear after accounting for transaction costs. However, when the optimal filter is used, returns after costs remain positive and are higher than for naïve filters. |
| Keywords: | Transaction Costs, Filter Rules, Trading Strategies, Foreign Exchange |
| JEL: | G10 G15 G11 |
| URL: | http://d.repec.org/n?u=RePEc:wvu:wpaper:04-12&r=rmg |
| By: | Guidolin, Massimo; Timmermann, Allan G |
| Abstract: | This Paper characterizes the term structure of risk measures such as Value at Risk (VaR) and expected shortfall under different econometric approaches including multivariate regime switching, GARCH-in-mean models with student-t errors, two-component GARCH models and a non-parametric bootstrap. We show how to derive the risk measures for each of these models and document large variations in term structures across econometric specifications. An out-of-sample forecasting experiment applied to stock, bond and cash portfolios suggests that the best model is asset- and horizon specific but that the bootstrap and regime switching model are best overall for VaR levels of 5% and 1%, respectively. |
| Keywords: | nonlinear econometric models; simulation models; term structure of risk |
| JEL: | G12 |
| Date: | 2004–09 |
| URL: | http://d.repec.org/n?u=RePEc:cpr:ceprdp:4645&r=rmg |
| By: | Norden, Lars; Weber, Martin |
| Abstract: | This Paper analyses the empirical relationship between credit default swap, bond and stock markets during the period 2000-02. Focusing on the intertemporal comovement, we examine weekly and daily lead-lag relationships in a vector autoregressive model and the adjustment between markets caused by cointegration. First, we find that stock returns lead CDS and bond spread changes. Second, CDS spread changes Granger cause bond spread changes for a higher number of firms than vice versa. Third, the CDS market is significantly more sensitive to the stock market than the bond market and the magnitude of this sensitivity increases when credit quality becomes worse. Finally, the CDS market plays a more important role for price discovery than the corporate bond market. |
| Keywords: | credit derivatives; credit risk; credit spreads; lead-lag relationship |
| JEL: | C32 G10 G14 |
| Date: | 2004–10 |
| URL: | http://d.repec.org/n?u=RePEc:cpr:ceprdp:4674&r=rmg |
| By: | Acharya, Viral V; Pedersen, Lasse Heje |
| Abstract: | This Paper solves explicitly a simple equilibrium asset pricing model with liquidity risk – the risk arising from unpredictable changes in liquidity over time. In our liquidity-adjusted capital asset pricing model, a security’s required return depends on its expected liquidity as well as on the covariances of its own return and liquidity with market return and market liquidity. In addition, the model shows how a negative shock to a security’s liquidity, if it is persistent, results in low contemporaneous returns and high predicted future returns. The model provides a simple, unified framework for understanding the various channels through which liquidity risk may affect asset prices. Our empirical results shed light on the total and relative economic significance of these channels. |
| Keywords: | asset pricing; frictions; liquidity; liquidity risk; transaction costs |
| JEL: | G00 G10 G12 |
| Date: | 2004–10 |
| URL: | http://d.repec.org/n?u=RePEc:cpr:ceprdp:4718&r=rmg |
| By: | Patrick de Fontnouvelle; John Jordan; Eric Rosengren |
| Abstract: | Quantification of operational risk has received increased attention with the inclusion of an explicit capital charge for operational risk under the new Basle proposal. The proposal provides significant flexibility for banks to use internal models to estimate their operational risk, and the associated capital needed for unexpected losses. Most banks have used variants of value at risk models that estimate frequency, severity, and loss distributions. This paper examines the empirical regularities in operational loss data. Using loss data from six large internationally active banking institutions, we find that loss data by event types are quite similar across institutions. Furthermore, our results are consistent with economic capital numbers disclosed by some large banks, and also with the results of studies modeling losses using publicly available "external" loss data. |
| JEL: | G2 |
| Date: | 2005–02 |
| URL: | http://d.repec.org/n?u=RePEc:nbr:nberwo:11103&r=rmg |
| By: | Jean-Philippe Bouchaud (Science & Finance, Capital Fund Management; CEA Saclay;); Benoit Pochard (Centre de mathematiques appliquees, Ecole Polytechnique, Palaiseau, FRANCE) |
| Abstract: | We propose a versatile Monte-Carlo method for pricing and hedging options when markets are inco;plete, for an arbitrary risk criterion (chosen here to be the expected shortfall), for a large class of stochastic processes, and in the presence of transaction costs. We illustrate the method on plain vanilla options when the price returns follow a Student-t distribution. We show that in the presence of fat tails, our strategy allows to significantly reduce extreme risks, and generically leads to low Gamma hedging. Similarly, the inclusion of transaction costs reduces the Gamma of the optimal strategy. |
| JEL: | G10 |
| URL: | http://d.repec.org/n?u=RePEc:sfi:sfiwpa:500029&r=rmg |
| By: | Marc Potters (Science & Finance, Capital Fund Management); Jean-Philippe Bouchaud (Science & Finance, Capital Fund Management; CEA Saclay;); Lorenzo Cornalba |
| Abstract: | We consider the problem of option pricing and hedging when stock returns are correlated in time. Within a quadratic-risk minimisation scheme, we obtain a general formula, valid for weakly correlated non-Gaussian processes. We show that for Gaussian price increments, the correlations are irrelevant, and the Black-Scholes formula holds with the volatility of the price increments on the scale of the re-hedging. For non-Gaussian processes, further non trivial corrections to the `smile' are brought about by the correlations, even when the hedge is the Black-Scholes Delta-hedge. We introduce a compact notation which eases the computations and could be of use to deal with more complicated models. |
| JEL: | G10 |
| URL: | http://d.repec.org/n?u=RePEc:sfi:sfiwpa:500030&r=rmg |
| By: | Marc Potters (Science & Finance, Capital Fund Management); Jean-Philippe Bouchaud (Science & Finance, Capital Fund Management; CEA Saclay;); Dragan Sestovic |
| Abstract: | We propose a new `hedged' Monte-Carlo (HMC) method to price financial derivatives, which allows to determine simultaneously the optimal hedge. The inclusion of the optimal hedging strategy allows one to reduce the financial risk associated with option trading, and for the very same reason reduces considerably the variance of our HMC scheme as compared to previous methods. The explicit accounting of the hedging cost naturally converts the objective probability into the `risk-neutral' one. This allows a consistent use of purely historical time series to price derivatives and obtain their residual risk. The method can be used to price a large class of exotic options, including those with path dependent and early exercise features. |
| JEL: | G10 |
| URL: | http://d.repec.org/n?u=RePEc:sfi:sfiwpa:500031&r=rmg |
| By: | Marc Potters (Science & Finance, Capital Fund Management); Jean-Philippe Bouchaud (Science & Finance, Capital Fund Management; CEA Saclay;); Laurent Laloux (Science & Finance, Capital Fund Management); Pierre Cizeau (Science & Finance, Capital Fund Management) |
| Abstract: | We show that results from the theory of random matrices are potentially of great interest to understand the statistical structure of the empirical correlation matrices appearing in the study of price fluctuations. The central result of the present study is the remarkable agreement between the theoretical prediction (based on the assumption that the correlation matrix is random) and empirical data concerning the density of eigenvalues associated to the time series of the different stocks of the S&P500 (or other major markets). In particular the present study raises serious doubts on the blind use of empirical correlation matrices for risk management. |
| JEL: | G10 |
| URL: | http://d.repec.org/n?u=RePEc:sfi:sfiwpa:500051&r=rmg |
| By: | Marc Potters (Science & Finance, Capital Fund Management); Jean-Philippe Bouchaud (Science & Finance, Capital Fund Management; CEA Saclay;); Laurent Laloux (Science & Finance, Capital Fund Management); Pierre Cizeau (Science & Finance, Capital Fund Management) |
| JEL: | G10 |
| URL: | http://d.repec.org/n?u=RePEc:sfi:sfiwpa:500053&r=rmg |