Optimal static-dynamic hedges for exotic options under convex risk measures

We study the problem of optimally hedging exotic derivatives positions using a combination of dynamic trading strategies in underlying stocks and static positions in vanilla options when the performance is quantified by a convex risk measure. We establish conditions for the existence of an optimal static position for general convex risk measures, and then analyze in detail the case of shortfall risk with a power loss function. Here we find conditions for uniqueness of the static hedge. We illustrate the computational challenge of computing the market-adjusted risk measure in a simple diffusion model for an option on a non-traded asset.     

EXISTENCE OF STOCHASTIC EQUILIBRIUM WITH INCOMPLETE FINANCIAL MARKETS

This paper analyzes the aritrage-tree security markets and the general equilibrium ex-istence problem for a stochastic economy with incomplete financial markets. Information structure is given by an event tree. This paper restricts attention to puraly financial securities. It isassume that trading takes place in the sequence of spot markets and futures markets for securi-ties payable in units of account. Unlimited short-selling in securities is allowed. Financial markets may be incomplete, some consumption streams may be impossible to obtain by any tradingstrategy. Securities may be individually precluded from trade at arbitrary states and dates. Thesecurity price process is arbitrage-free the dividend process if and only if there exists a stochaticstate price (present value) process : the present value of the security prices at every vertex isthe present value of their dividend and capital values over the set of immediate successors ; thecurrent value of each security at every vertex is the present value of its future dividend streamover all succeeding vertices. The existence of such an equilibrium is proved under the followingcondition: continuous, weakly convex, strictly monotone and complete preferences, strictlypositive endowmenta and dividends processes.     

Liquidity Risk with Coherent Risk Measures

This paper concerns questions related to the regulation of liquidity risk, and proposes a definition of an acceptable portfolio. Because the concern is with risk management, the paper considers processes under the physical (rather than the martingale) measure. Basically, a portfolio is ‘acceptable’ provided there is a trading strategy (satisfying some limitations on market liquidity) which, at some fixed date in the future, produces a cash‐only position, (possibly) having positive future cash flows, which is required to satisfy a ‘convex risk measure constraint’.     

The challenge in managing new financial risks: adopting an heuristic or theoretical approach

The financial crisis began with the collapse of Lehman Brothers and the subprime asset backed securities debacle. Credit risk was turned into liquidity risk, resulting in a lack of confidence among financial institutions. In this article, we will propose a way to model liquidity risk and the credit risk in best practices. We will show that liquidity risk is a new type of risk and the current way to deal with it is based solely on observed variables without any theoretical link. We propose an heuristic approach to combine the numerous liquidity risk indicators with a logistic regression for the first time. In regards to credit risk, several articles prove that the best practice is to use an option model to appreciate this risk. We will present our methodology using stochastic diffusion for the interest rate because currently the yield curves aren’t liquid. This approach is more relevant because the basis model in prior publications has a constant interest rate or a forward rate. Both models allow a better understanding of liquidity and credit risks and the further development of research deals with the link between these two financial risks.     

On securitization,market completion and equilibrium risk transfer

We propose an equilibrium framework within which to price financial securities written on non-tradable underlyings such as temperature indices. We analyze a financial market with a finite set of agents whose preferences are described by a convex dynamic risk measure generated by the solution of a backward stochastic differential equation. The agents are exposed to financial and non-financial risk factors. They can hedge their financial risk in the stock market and trade a structured derivative whose payoff depends on both financial and external risk factors. We prove an existence and uniqueness of equilibrium result for derivative prices and characterize the equilibrium market price of risk in terms of a solution to a non-linear BSDE.     

Sensitivity measures for split-capital investment trusts

Statistical measures of risk based on historical data are usefultools in assessing risk for conventional investment-trust securities;but they are of limited use for securities of split-capitalinvestment trusts, and an alternative approach is proposed inthis paper. By differentiating formulae for the discounted cashflow, with respect to the underlying fundamental variables,‘sensitivity measures’ can be derived for most securitiesof split-capital investment trusts. These sensitivity measuresshow how the present value of expected future cash flows willvary as the real discount force changes, the real force of increasein the income (or capital value) of the underlying fund changes,and the estimated force of inflation changes.     

市场风险值VaR的算法与应用

进行金融风险管理时可以将风险划分为四类，即信用风险、经营风险、流动性风险和市场风险。其中市场风险是指金融市场价格（包括股票价格、利率、汇率和大宗可交易商品的价格）波动而引起的未来收益的不确定性。市场风险值VaR(Value at Risk）就是用来评价给定资产所面临的市场风险大小。本文介绍了VaR的定义、相关的计算方法和在证券投资决策中的应用实例。     

Credit spreads, endogenous bankruptcy and liquidity risk

In this paper, we consider a bond valuation model with both credit risk and liquidity risk to show that credit spreads are not negligible for short maturities. We adopt the structural approach to model credit risk, where the default triggering barrier is determined endogenously by maximizing equity value. As for liquidity risk, we assume that bondholders may encounter liquidity shocks during the lifetime of corporate bonds, and have to sell the bond immediately at the price, which is assumed to be a fraction of the price in a perfectly liquid market. Under this framework, we derive explicit expressions for corporate bond, firm value and bankruptcy trigger. Finally, numerical illustrations are presented.     

The diffusion of complex securities: The case of CAT bonds

Complex securities generally do not diffuse smoothly but by fits and starts in response to sudden shifts in demand, occurring as investors learn about the intrinsic value of the securities from their noisy performance. We use CAT bonds, a capital market-based alternative to CAT risk reinsurance, to illustrate the diffusion of a complex security that competes against a legacy financial product offered by financial intermediaries. We find that the diffusion of the security is highly path-dependent with the capricious ups and downs of its actual performance plus the competitive response of CAT reinsurers jointly determining its ultimate success or failure.     

Asset Pricing under Progressive Taxes and Existence of General Equilibrium

This paper shows that the existence of general equilibrium in a two-period economy with financial markets and progressive anonymous tax system is not at all problematic, provided securities are purely financial. We explore the concepts of weakly and strongly arbitrage-free security price for return and tax system, and prove arbitrage-free asset pricing theorems without short-sale restrictions. A general equilibrium is a set of current and future prices (contingent on uncertain events) and a set of individual plans such that all markets are cleared. The existence of such an equilibrium is proved under the following conditions: continuous, weakly convex, strictly monotone, complete preferences and strictly positive endowments.     

Convex risk measures for portfolio optimization and concepts of flexibility

Due to their axiomatic foundation and their favorable computational properties convex risk measures are becoming a powerful tool in financial risk management. In this paper we will review the fundamental structural concepts of convex risk measures within the framework of convex analysis. Then we will exploit it for deriving strong duality relations in a generic portfolio optimization context. In particular, the duality relationship can be used for designing new, efficient approximation algorithms based on Nesterov's smoothing techniques for non-smooth convex optimization. Furthermore, the presented concepts enable us to formalize the notion of flexibility as the (marginal) risk absorption capacity of a technology or (available) resources. This paper is dedicated to R.T. Rockafellar for his stimulating and impressive work in convex optimization for decades. We thank you for the insights and inspirations we gained from your fundamental research.     

Extending dynamic convex risk measures from discrete time to continuous time: A convergence approach

We present an approach for the transition from convex risk measures in a certain discrete time setting to their counterparts in continuous time. The aim of this paper is to show that a large class of convex risk measures in continuous time can be obtained as limits of discrete time-consistent convex risk measures. The discrete time risk measures are constructed from properly rescaled (‘tilted’) one-period convex risk measures, using a d-dimensional random walk converging to a Brownian motion. Under suitable conditions (covering many standard one-period risk measures) we obtain convergence of the discrete risk measures to the solution of a BSDE, defining a convex risk measure in continuous time, whose driver can then be viewed as the continuous time analogue of the discrete ‘driver’ characterizing the one-period risk. We derive the limiting drivers for the semi-deviation risk measure, Value at Risk, Average Value at Risk, and the Gini risk measure in closed form.     

On efficient portfolio selection using convex risk measures

In this article we systematically revisit the classic portfolio selection theory in both of its branches, the determination of the efficient financial positions among such a choice set and the selection of the financial position which maximizes some utility function whose functional form involves some ‘measure of risk’. We study these problems by considering certain classes of convex risk measures and we show that for these classes the solution of the utility maximization problems in reflexive spaces take the form of a zero-sum game between the investor and the market.     

Model spaces for risk measures

We show how risk measures originally defined in a model free framework in terms of acceptance sets and reference assets imply a meaningful underlying probability structure. Hereafter we construct a maximal domain of definition of the risk measure respecting the underlying ambiguity profile. We particularly emphasise liquidity effects and discuss the correspondence between properties of the risk measure and the structure of this domain as well as subdifferentiability properties.     

A characterization of vNM-stable sets for linear production games

We discuss linear production games or market games with a continuum of players which are represented as minima of finitely many nonatomic measures.?Within this context we consider vNM-Stable Sets according to von Neumann and Morgenstern. We classify or characterize all solutions of this type which are convex polyhedra, i.e., which are the convex hull of finitely many imputations. Specifically, in each convex polyhedral vNM-Stable Set (and not only in the symmetric ones), the different types of traders must organize themselves into cartels. The vNM-Stable Set is then the convex hull of the utility distributions of the cartels.?Using the results from the continuum, we obtain a similar characterization also for finite glove market games. Received December 1998/Revised version June 1999     

Convex risk measures on Orlicz spaces: inf-convolution and shortfall

We focus on, throughout this paper, convex risk measures defined on Orlicz spaces. In particular, we investigate basic properties of inf-convolutions defined between a convex risk measure and a convex set, and between two convex risk measures. Moreover, we study shortfall risk measures, which are convex risk measures induced by the shortfall risk. By using results on inf-convolutions, we obtain a robust representation result for shortfall risk measures defined on Orlicz spaces under the assumption that the set of hedging strategies has the sequential compactness in a weak sense. We discuss in addition a construction of an example having the sequential compactness.     

Dynamic conic hedging for competitiveness

The paper provides a new hedging methodology permitting systematic hedging choices with wide applications. Dynamic concave bid price, and convex ask price functionals from the recent literature are employed to construct new hedging strategies termed dynamic conic hedging. The primary focus of these strategies is to adopt positions maximizing a nonlinear conditional expectation expressed recursively as a concave current bid price for the one step ahead risk held or minimizing the convex current ask price for the risk promised. Risk management and hedging then have a new market value enhancing perspective different from the classical forms of risk mitigation, local variance minimization, or even expected utility maximization.     

To split or not to split: Capital allocation with convex risk measures

Convex risk measures were introduced by Deprez and Gerber [Deprez, O., Gerber, H.U., 1985. On convex principles of premium calculation. Insurance: Math. Econom. 4 (3), 179-189]. Here the problem of allocating risk capital to subportfolios is addressed, when convex risk measures are used. The Aumann-Shapley value is proposed as an appropriate allocation mechanism. Distortion-exponential measures are discussed extensively and explicit capital allocation formulas are obtained for the case that the risk measure belongs to this family. Finally the implications of capital allocation with a convex risk measure for the stability of portfolios are discussed. It is demonstrated that using a convex risk measure for capital allocation can produce an incentive for infinite fragmentation of portfolios.     

Measure-Valued Valuations and Mixed Curvature Measures of Convex Bodies

We prove a polynomial expansion for measure-valued functionals which are translation covariant on the set of convex bodies. The coefficients are measures on product spaces. We then apply this construction to the curvature measures of convex bodies and obtain mixed curvature measures for bodies in general relative position. These are used to generalize an integral geometric formula for nonintersecting convex bodies. Finally, we introduce support measures relative to a quite general structuring body B and describe connections between the different types of measures.     

Characterization of acceptance sets for co-monotone risk measures

We present a geometric characterization of acceptance sets for monotone, co-monotone and convex risk measures on finite state spaces. Geometrically, such acceptance sets can be represented by convex polygons with edges only on certain hyperplanes. We also provide some lower dimensional examples, and study acceptance sets for value at risk and expected shortfall.