Search

In this paper we give easy-to-implement closed-form expressions for European and Asian Greeks for general L2-payoff functions and underlying assets in an exponential Lévy process model with nonvanishing Brownian motion part. The results are based on Hilbert space valued Malliavin Calculus and extend previous results from the literature. Numerical experiments suggest, that in the case of a continuous payoff function, a combination of Malliavin Monte Carlo Greeks and the finite difference method has a better convergence behavior, whereas in the case of discontinuous payoff functions, the Malliavin Monte Carlo method clearly is the superior method compared to the finite difference approach, for first- and second order Greeks. Reduction arguments from the literature based on measure change imply that the expressions for the Greeks in this paper also hold true for generalized Asian options in particular for fixed and floating strike Asian options.

We study the impact of dependence uncertainty on the expectation of the product of $d$ random variables, $\mathbb{E}(X_1X_2\cdots X_d)$ when $X_i \sim F_i$ for all~$i$. Under some conditions on the $F_i$, explicit sharp bounds are obtained and a numerical method is provided to approximate them for arbitrary choices of the $F_i$. The results are applied to assess the impact of dependence uncertainty on coskewness. In this regard, we introduce a novel notion of "standardized rank coskewness," which is invariant under strictly increasing transformations and takes values in $[-1,\ 1]$.

We consider a completely specified factor model for a risk vector X = (X 1, . . ., Xd ), where the joint distributions of the components of X with a risk factor Z and the conditional distributions of X given Z are specified. We extend the notion of *-product of d-copulas as introduced for d = 2 and continuous factor distribution in Darsow et al. [6] and Durante et al. [8] to the multivariate and discontinuous case. We give a Sklar-type representation theorem for factor models showing that these *-products determine the copula of a completely specified factor model. We investigate in detail approximation, transformation, and ordering properties of *-products and, based on them, derive general orthant ordering results for completely specified factor models in dependence on their specifications. The paper generalizes previously known ordering results for the worst case partially specified risk factor models to some general classes of positive or negative dependent risk factor models. In particular, it develops some tools to derive sharp worst case dependence bounds in subclasses of completely specified factor models.

For the class of (partially specified) internal risk factor models we establish strongly simplified supermodular ordering results in comparison to the case of general risk factor models. This allows us to derive meaningful and improved risk bounds for the joint portfolio in risk factor models with dependence information given by constrained specification sets for the copulas of the risk components and the systemic risk factor. The proof of our main comparison result is not standard. It is based on grid copula approximation of upper products of copulas and on the theory of mass transfers. An application to real market data shows considerable improvement over the standard method.

In the framework of continuous-time market models with specified pricing density, optimal payoffs under increasing law-invariant preferences are known to be anti-monotonic with the pricing density. Consequently, optimal portfolio selection problems can be reformulated as optimization problems on real functions under monotonicity conditions. We solve two basic types of these optimization problems, which makes it possible to obtain in a fairly unified way the optimal payoff for several portfolio selection problems of interest. In particular, we completely solve the optimal portfolio selection problem for an investor with preferences as in cumulative prospect theory or as in Yaari’s dual theory. Extending previous work, we also characterize optimal payoffs when the payoff is required to have a fixed copula with some benchmark (state-dependent constraint). Specifically, we show that if one can determine the optimal payoff under a concave law-invariant objective, then one can also determine the optimal payoff when adding the state-dependent constraint. In the final part of the paper, we consider an extension to (incomplete) market models in which the pricing density is not completely specified. When a sufficient number of payoffs have a known market price, we show that optimal payoffs are anti-monotonic to some pricing density that we explicitly derive from these market prices. As examples, we deal with some exponential Levy market models and some market models involving Ito processes.

The study of worst case scenarios for risk measures (e.g. the Value at Risk) when the underlying risk vector (or portfolio of risks) is not completely specified is a central topic in the literature on robust risk measurement. In this paper we discuss partially specified factor models as introduced in Bernard et al. (2017) in more detail for the class of additive factor models which admit more explicit results. These results allow to describe in more detail the reduction of risk bounds obtainable by this method in dependence on the degree of positive resp. negative dependence induced by the systematic risk factors. The insight may help in applications of this reduction method to get a better qualitative impression on the range of influence of the partially specified factor structure.

Conditionally comonotonic risk vectors have been proved in [4] to yield worst case dependence structures maximizing the risk of the portfolio sum in partially specified risk factor models. In this paper we investigate the question how risk bounds depend on the specification of the pairwise copulas of the risk components Xiwith the systemic risk factor. As basic toolwe introduce a new ordering based on sign changes of the derivatives of copulas. This together with discretization by n-grids and the theory of supermodular transfers allows us to derive concrete ordering criteria for the maximal risks.

The study of worst-case scenarios for risk measures (e.g., Value-at-Risk) when the underlying risk (or portfolio of risks) is not completely specified is a central topic in the literature on robust risk measurement. In this paper, we tackle the open problem of deriving upper bounds for strictly concave distortion risk measures on moment spaces. Building on early results of Rustagi (1957, 1976), we show that in general this problem can be reduced to a parametric optimization problem. We completely specify the sharp upper bound (and corresponding maximizing distribution function) when the first moment and any other higher moment are fixed. Specifically, in the case of a fixed mean and variance, we generalize the Cantelli bound for (Tail) Value-at-Risk in that we express the sharp upper bound for a strictly concave distorted expectation as a weighted sum of the mean and standard deviation.

We consider the problem of determining risk bounds for the Value at Risk for risk vectors X where besides the marginal distributions also information on the distribution or on the expectation of some functionals Tj(X), 1 ≤ j ≤ m, is available. In particular this formulation includes the case where information on subgroup sums or maxima or on the correlations or covariances is available. Based on the method of dual bounds we obtain improved risk bounds compared to the marginal case. In general the explicit calculation of the dual bounds poses a challenge. We discuss various forms of relaxation of these bounds which are accessible and in some cases even lead to sharp bounds.

Dieses Kapitel gibt eine Einfuhrung zur Bestimmung (bzw. Auswahl) von Optionspreisen uber Minimumdistanz-Martingale sowie zum Bepreisen und Hedgen uber Nutzenfunktionen. Daruberhinaus wird das Problem der Portfoliooptimierung behandelt. Anhand von exponentiellen Levy-Modellen werden fur eine Reihe von Standard-Nutzenfunktionen diese Verfahren im Detail charakterisiert.

Dieses Kapitel gibt eine Einfuhrung in Grundbegriffe und Grundideen der Optionspreisbestimmung im Rahmen von Modellen in diskreter Zeit. So lasst sich z. B. auf einfache Weise das grundlegende No-Arbitrage-Prinzip und die sich durch ein geeignetes Hedging-Argument daraus ergebende Festlegung des fairen (arbitragefreien) Preises erlautern. Dieser Zugang erlaubt es mit geringem technischen Aufwand wesentliche Begriffe und Methoden einzufuhren und z. B. mittels Approximation die Black-Scholes-Formel herzuleiten.

Dieses Kapitel ist der Bestimmung von optimalen Hedging-Strategien durch das Kriterium der varianz-minimalen Strategie gewidmet. In unvollstandigen Marktmodellen ist nicht jeder Claim H hedgebar. Eine naturliche Frage ist daher: Wie gut ist H hedgebar?