Pricing <Emphasis Type="Italic">k</Emphasis><Superscript>th</Superscript> realization derivatives and collateralized debt obligation with multivariate Fréchet copula

Copula method has been widely applied to model the correlation among underlying assets in financial market. In this paper, we propose to use the multivariate Fréchet copula family presented in J. P. Yang et al. [Insurance Math. Econom., 2009, 45: 139–147] to price multivariate financial instruments whose payoffs depend on the k ^th realization of the underlying assets and collateralized debt obligation (CDO). The advantage of the multivariate Fréchet copula is discussed. Empirical study shows that such copula family gives a better fitting to CDO’s market price than Gaussian copula for some derivatives.     

Bivariate option pricing with copulas

The adoption of copula functions is suggested in order to price bivariate contingent claims. Copulas enable the marginal distributions extracted from vertical spreads in the options markets to be imbedded in a multivariate pricing kernel. It is proved that such a kernel is a copula function, and that its super-replication strategy is represented by the Fréchet bounds. Applications provided include prices for binary digital options, options on the minimum and options to exchange one asset for another. For each of these products, no-arbitrage pricing bounds, as well as values consistent with the independence of the underlying assets are provided. As a final reference value, a copula function calibrated on historical data is used.     

Multivariate option pricing using copulae

The complexity of financial products significantly increased in the past 10 years. In this paper, we investigate the pricing of basket options and more generally of complex exotic contracts depending on multiple indices. Our approach assumes that the underlying assets evolve as dependent GARCH(1, 1) processes. The dependence among the assets is modeled using a copula based on pair‐copula constructions. Unlike most previous studies on this topic, we do not assume that the dependence observed between historical asset prices is similar to the dependence under the risk‐neutral probability. The method is illustrated with US market data on basket options written on two or three international indices. Copyright © 2012 John Wiley & Sons, Ltd.     

Dependence structure of conditional Archimedean copulas

In this article, copulas associated to multivariate conditional distributions in an Archimedean model are characterized. It is shown that this popular class of dependence structures is closed under the operation of conditioning, but that the associated conditional copula has a different analytical form in general. It is also demonstrated that the extremal copula for conditional Archimedean distributions is no longer the Fréchet upper bound, but rather a member of the Clayton family. Properties of these conditional distributions as well as conditional versions of tail dependence indices are also considered.     

Computing the Volume of n-Dimensional Copulas

Abstract

A problem that is very relevant in applications of copula functions to finance is the computation of the survival copula, which is applied to enforce multivariate put–call parity. This may be very complex for large dimensions. The problem is a special case of the more general problem of volume computation in high-dimensional copulas. We provide an algorithm for the exact computation of the volume of copula functions in cases where the copula function is computable in closed form. We apply the algorithm to the problem of computing the survival of a copula function in the pricing problem of a multivariate digital option, and we provide evidence that this is feasible for baskets of up to 20 underlying assets, with acceptable CPU time performance.     

Explicit characterization of the super-replication strategy in financial markets with partial transaction costs

We consider a continuous time multivariate financial market with proportional transaction costs and study the problem of finding the minimal initial capital needed to hedge, without risk, European-type contingent claims. The model is similar to the one considered in Bouchard and Touzi [B. Bouchard, N. Touzi, Explicit solution of the multivariate super-replication problem under transaction costs, The Annals of Applied Probability 10 (3) (2000) 685–708] except that some of the assets can be exchanged freely, i.e. without paying transaction costs. In this context, we generalize the result of the above paper and prove that the super-replication price is given by the cost of the cheapest hedging strategy in which the number of non-freely exchangeable assets is kept constant over time. Our proof relies on the introduction of a new auxiliary control problem whose value function can be interpreted as the super-hedging price in a model with unbounded stochastic volatility (in the directions where transaction costs are non-zero). In particular, it confirms the usual intuition that transaction costs play a similar role to stochastic volatility.     

基于COPULA的A、B股信息流动和相关结构分析

金融市场的价格变化体现了市场的信息流动.文章引入copula模型来分析A、B股市场的信息传递.在选择候选copula模型后,利用非参数和极大似然方法得到刻画两个市场信息流动的copula函数的参数,并综合利用AIC检验、KS检验及卡方检验等拟合优度方法判断得到:利用Frank函数来刻画两个市场的相关模式最为合适.这样较灵活的捕捉到了两个市场的信息流动和相关结构.     

Incomplete financial markets and contingent claim pricing in a dual expected utility theory framework

This paper investigates the price for contingent claims in a dual expected utility theory framework, the dual price, considering arbitrage-free financial markets. A pricing formula is obtained for contingent claims written on n underlying assets following a general diffusion process. The formula holds in both complete and incomplete markets as well as in constrained markets. An application is also considered assuming a geometric Brownian motion for the underlying assets and the Wang transform as the distortion function.     

Option Pricing in Illiquid Markets with Jumps

ABSTRACT

The classical linear Black–Scholes model for pricing derivative securities is a popular model in the financial industry. It relies on several restrictive assumptions such as completeness, and frictionless of the market as well as the assumption on the underlying asset price dynamics following a geometric Brownian motion. The main purpose of this paper is to generalize the classical Black–Scholes model for pricing derivative securities by taking into account feedback effects due to an influence of a large trader on the underlying asset price dynamics exhibiting random jumps. The assumption that an investor can trade large amounts of assets without affecting the underlying asset price itself is usually not satisfied, especially in illiquid markets. We generalize the Frey–Stremme nonlinear option pricing model for the case the underlying asset follows a Lévy stochastic process with jumps. We derive and analyze a fully nonlinear parabolic partial-integro differential equation for the price of the option contract. We propose a semi-implicit numerical discretization scheme and perform various numerical experiments showing the influence of a large trader and intensity of jumps on the option price.     

SCHWARZ-PICK ESTIMATES FOR BOUNDED HOLOMORPHIC FUNCTIONS ON CLASSICAL DOMAINS

In this paper, Schwarz-Pick estimates for high order Fr′echet derivatives of bounded holomorphic functions on three kinds of classical domains are presented. We generalize the early work on Schwarz-Pick estimates of higher order partial derivatives for bounded holomorphic functions on the disk and unit ball.     

European option pricing model in a stochastic and fuzzy environment

The primary goal of this paper is to price European options in the Merton＇s frame- work with underlying assets following jump-diffusion using fuzzy set theory. Owing to the vague fluctuation of the real financial market, the average jump rate and jump sizes cannot be recorded or collected accurately. So the main idea of this paper is to model the rate as a triangular fuzzy number and jump sizes as fuzzy random variables and use the property of fuzzy set to deduce two different jump-diffusion models underlying principle of rational expectations equilibrium price. Unlike many conventional models, the European option price will now turn into a fuzzy number. One of the major advantages of this model is that it allows investors to choose a reasonable European option price under an acceptable belief degree. The empirical results will serve as useful feedback information for improvements on the proposed model.     

An extension of the Clark–Haussmann formula and applications

ABSTRACT

This work considers a financial market stochastic model where the uncertainty is driven by a multidimensional Brownian motion. The market price of the risk process makes the transition between real world probability measure and risk neutral probability measure. Traditionally, the martingale representation formulas under the risk neutral probability measure require the market price of risk process to be bounded. However, in several financial models the boundedness assumption of the market price of risk fails; for example a financial market model with the market price of risk following an Ornstein–Uhlenbeck process. This work extends the Clark–Haussmann representation formula to underlying stochastic processes which fail to satisfy the standard requirements. Our methodology is classical, and it uses a sequence of mollifiers. Our result can be applied to hedging and optimal investment in financial markets with unbounded market price of risk. In particular, the mean variance optimization problem can be addressed within our framework.     

Option Pricing on Multiple Assets

Since 1973, the Black–Scholes formula has been used in financial markets to price financial derivatives such as options. In the classical Black–Scholes model for the market, the following type of mix is assumed or postulated: in the simplest case, it consists of an essentially riskless bond and a single risky asset. Hence, certainty mixed with uncertainty: safe vs risky! Here we consider more complex products where each component in a portfolio entails several variables with constraints. This leads to elegant models based on multivariable stochastic integration, and describing several securities simultaneously [Etheridge, A Course in Financial Calculus, Cambridge University Press, UK (2002), Jiang, Mathematical Modeling and Methods of Option Pricing, Higher Education, Beijing, China (2003)] and [Broadie, Detemple, Math. Financ. 7:241–286 (1997)]. We derive a general asymptotic solution in a short time interval using the heat kernel expansion on a Riemannian metric. We then use our formula to predict the better price of options on multiple underlying assets. We then apply our method to the case known as the two-color rainbow option, i.e., the special case of the model with two underlying assets. This asymptotic solution is important, as it explains hidden effects in a class of financial models.This paper is dedicated to the memory of the first named author, Professor Thomas P. Branson (1953–2006).     

国内外利率为随机的双币种重置型期权定价

双币种重置期权的特征是指在终端期T时的收益依赖于预先设定的t<,0>时刻标的资产的价格与执行价K>0(事先给定)的大小关系重新设置期权的执行价从而给出其定价,这种期权是投资于外国资产的一种合约,其风险不仅依赖外国资产价格的变化,还受外国货币的汇率以及国内外两种利率波动的影响,所以在实际应用方面十分广泛.本文首先就标的资...     

On approximating max-stable processes and constructing extremal copula functions

There is an infinite number of parameters in the definition of multivariate maxima of moving maxima (M4) processes, which poses challenges in statistical applications where workable models are preferred. This paper establishes sufficient conditions under which an M4 process with infinite number of parameters may be approximated by an M4 process with finite number of parameters. In statistical inferences, the paper focuses on a family of sectional multivariate extreme value copula (SMEVC) functions which is derived from the joint distribution functions of M4 processes. A new non-standard parameter estimation procedure is introduced, which is based on order statistics of ratios of (transformed) marginal unit Fréchet random variables, and is shown via simulation to be more efficient than a semi-parametric estimation procedure. In real data analysis, empirical results show that SMEVCs are more flexible for modeling various dependence structures, and perform better than the widely used Gumbel-Hougaard copulas.     

Asymptotic efficiency of the two-stage estimation method for copula-based models

For multivariate copula-based models for which maximum likelihood is computationally difficult, a two-stage estimation procedure has been proposed previously; the first stage involves maximum likelihood from univariate margins, and the second stage involves maximum likelihood of the dependence parameters with the univariate parameters held fixed from the first stage. Using the theory of inference functions, a partitioned matrix in a form amenable to analysis is obtained for the asymptotic covariance matrix of the two-stage estimator. The asymptotic relative efficiency of the two-stage estimation procedure compared with maximum likelihood estimation is studied. Analysis of the limiting cases of the independence copula and Fréchet upper bound help to determine common patterns in the efficiency as the dependence in the model increases. For the Fréchet upper bound, the two-stage estimation procedure can sometimes be equivalent to maximum likelihood estimation for the univariate parameters. Numerical results are shown for some models, including multivariate ordinal probit and bivariate extreme value distributions, to indicate the typical level of asymptotic efficiency for discrete and continuous data.     

Real options valuation of forest plantation investments in Brazil

In this paper, we consider investments in eucalyptus plantations in Brazil. For such projects, we discuss real options valuation in the place conventional methods such as IRR or NPV, possibly with CAPM. Traditionally, real options valuation assumes complete markets and neglects market imperfections. Yet, market frictions, such as transaction costs, interest rate spreads, and restricted short positions, can play an important role. We extend real options valuation to allow incomplete and imperfect markets. The value is obtained as a competitive price, given markets of competing investment opportunities, such as real and financial assets. Under perfect and complete markets, such valuation method is consistent with conventional real options theory. Stochastic programming and standard software is used for valuation of eucalyptus plantations. We estimate the underlying interdependent diffusion processes of stock market, interest rates, exchange rates and pulpwood price, and derive novel expressions of stochastic integrals to be employed in scenario generation for discrete time stochastic programming.     

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。     

A multivariate version of Hoeffding’s Phi-Square

A multivariate measure of association is proposed, which extends the bivariate copula-based measure Phi-Square introduced by Hoeffding [22]. We discuss its analytical properties and calculate its explicit value for some copulas of simple form; a simulation procedure to approximate its value is provided otherwise. A nonparametric estimator for multivariate Phi-Square is derived and its asymptotic behavior is established based on the weak convergence of the empirical copula process both in the case of independent observations and dependent observations from strictly stationary strong mixing sequences. The asymptotic variance of the estimator can be estimated by means of nonparametric bootstrap methods. For illustration, the theoretical results are applied to financial asset return data.     

投机、期货市场与原油价格变动

从原油的金融资产属性出发,探讨了原油期货市场中的投机行为下价格泡沫的存在性,在实证中针对美国商品交易委员会报告中的不足进行修正,考察了多种因素对价格的影响.结果表明：投机是近一轮油价波动的原因,但却无法据此质疑原油期货市场的作用,原油期货市场仍在总量上对价格起到了稳定的负反馈作用.结论在肯定投机推高油价的同时,证明石油期货市场并非是价格波动的根源,相反具有调节作用,从而强调了石油期货市场监管的重要性.