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Mathematical Finance's Blog

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GRAPHICAL MODELS FOR CORRELATED DEFAULTS

February 15, 2012 Comments (0)

A simple graphical model for correlated defaults is proposed, with explicit formulas for the loss distribution. Algebraic geometry techniques are employed to show that this model is well posed for default dependence: it represents any given marginal distribution for single firms and pairwise correlation matrix. These techniques also provide a calibration algorithm based on maximum likelihood estimation. Finally, the model is compared with standard normal copula model in terms of tails of the...

MODIFIED LELAND’S STRATEGY FOR A CONSTANT TRANSACTION COSTS RATE

February 15, 2012 Comments (0)

In 1985 Leland suggested an approach to price contingent claims under proportional transaction costs. Its main idea is to use the classical Black–Scholes formula with a suitably adjusted volatility for a periodical revision of the portfolio whose terminal value approximates the pay‐off. Unfortunately, if the transaction costs rate does not depend on the number of revisions, the approximation error does not converge to zero as the frequency of revisions tends to infinity. In the present paper,...

THE MULTIVARIATE supOU STOCHASTIC VOLATILITY MODEL

February 15, 2012 Comments (0)

Using positive semidefinite supOU (superposition of Ornstein–Uhlenbeck type) processes to describe the volatility, we introduce a multivariate stochastic volatility model for financial data which is capable of modeling long range dependence effects. The finiteness of moments and the second‐order structure of the volatility, the log‐ returns, as well as their “squares” are discussed in detail. Moreover, we give several examples in which long memory effects occur and study how the model as well...

NO MARGINAL ARBITRAGE OF THE SECOND KIND FOR HIGH PRODUCTION REGIMES IN DISCRETE TIME PRODUCTION–INVESTMENT MODELS WITH PROPORTIONAL TRANSACTION COSTS

February 15, 2012 Comments (0)

We consider a class of production–investment models in discrete time with proportional transaction costs. For linear production functions, we study a natural extension of the no‐arbitrage of the second kind condition introduced by Rásonyi. We show that this condition implies the closedness of the set of attainable claims and is equivalent to the existence of a strictly consistent price system under which the evaluation of future production profits is strictly negative. This allows us to discuss...

A CONSISTENT PRICING MODEL FOR INDEX OPTIONS AND VOLATILITY DERIVATIVES

February 15, 2012 Comments (0)

We propose a flexible framework for modeling the joint dynamics of an index and a set of forward variance swap rates written on this index. Our model reproduces various empirically observed properties of variance swap dynamics and enables volatility derivatives and options on the underlying index to be priced consistently, while allowing for jumps in volatility and returns. An affine specification using Lévy processes as building blocks leads to analytically tractable pricing formulas for...

COOPERATIVE GAMES WITH GENERAL DEVIATION MEASURES

February 15, 2012 Comments (0)

Cooperative games with players using different law‐invariant deviation measures as numerical representations for their attitudes towards risk in investing to a financial market are formulated and studied. As a central result, it is shown that players (investors) form a coalition (cooperative portfolio) that behaves similar to a single player (investor) with a certain deviation measure. An explicit formula for that deviation measure is obtained. An approach to optimal risk sharing among...

RECOVERING PORTFOLIO DEFAULT INTENSITIES IMPLIED BY CDO QUOTES

February 15, 2012 Comments (0)

We propose a stable nonparametric algorithm for the calibration of “top‐down” pricing models for portfolio credit derivatives: given a set of observations of market spreads for collateralized debt obligation (CDO) tranches, we construct a risk‐neutral default intensity process for the portfolio underlying the CDO which matches these observations, by looking for the risk‐neutral loss process “closest” to a prior loss process, verifying the calibration constraints. We formalize the problem in...

A NONZERO‐SUM GAME APPROACH TO CONVERTIBLE BONDS: TAX BENEFIT, BANKRUPTCY COST, AND EARLY/LATE CALLS

February 15, 2012 Comments (0)

Convertible bonds are hybrid securities that embody the characteristics of both straight bonds and equities. The conflicts of interest between bondholders and shareholders affect the security prices significantly. In this paper, we investigate how to use a nonzero‐sum game framework to model the interaction between bondholders and shareholders and to evaluate the bond accordingly. Mathematically, this problem can be reduced to a system of variational inequalities and we explicitly derive the...

GREED, LEVERAGE, AND POTENTIAL LOSSES: A PROSPECT THEORY PERSPECTIVE

February 15, 2012 Comments (0)

This paper quantifies the notion of greed, and explores its connection with leverage and potential losses, in the context of a continuous‐time behavioral portfolio choice model under (cumulative) prospect theory. We argue that the reference point can serve as the critical parameter in defining greed. An asymptotic analysis on optimal trading behaviors when the pricing kernel is lognormal and the S‐shaped utility function is a two‐piece CRRA shows that both the level of leverage and the...

POSITIVE ALPHAS, ABNORMAL PERFORMANCE, AND ILLUSORY ARBITRAGE

February 15, 2012 Comments (0)

Jensen’s alpha is well known to be a measure of abnormal performance in the evaluation of securities and portfolios where abnormal performance is defined to be an expected return that exceeds the equilibrium risk adjusted rate. It is also well known that in estimating Jensen’s alpha, a nonzero value can be obtained by using incorrect factors or not employing time varying betas. This paper makes two additional contributions to the performance evaluation literature. First, we show that a stronger...