Michael Schmutz

Multivariate Extension of Put-Call Symmetry

Multivariate analogues of the put-call symmetry can be expressed as certain symmetry properties of basket options and options on the maximum of several assets with respect to some (or all) permutations of the weights and the strike. The so-called self-dual distributions satisfying these symmetry conditions are completely characterized and their properties explored. It is also shown how to relate some multivariate asymmetric distributions to symmetric ones by a power transformation that is useful to adjust for carrying costs. Particular attention is devoted to the case of asset prices driven by Levy processes. Based on this, semistatic hedging techniques for multiasset barrier options are suggested.

Invariance properties of random vectors and stochastic processes based on the zonoid concept

Two integrable random vectors ξ and ξ* in IRd are said to be zonoid equivalent if, for each u∈IRd, the scalar products 〈ξ,u〉 and 〈ξ*,u〉 have the same first absolute moments. The paper analyses stochastic processes whose finite-dimensional distributions are zonoid equivalent with respect to time shift (zonoid stationarity) and permutation of time moments (swap-invariance). While the first concept is weaker than the stationarity, the second one is a weakening of the exchangeability property. It is shown that nonetheless the ergodic theorem holds for swap invariant sequences and the limits are characterized.

Semi-static hedging for certain Margrabe-type options with barriers

It turns out that in the bivariate Black-Scholes economy Margrabe type options exhibit symmetry properties leading to semi-static hedges of rather general barrier options. Some of the results are extended to variants obtained by means of Brownian subordination. In order to increase the liquidity of the hedging instruments for certain currency options, the duality principle can be applied to set up hedges in a foreign market by using only European vanilla options sometimes along with a risk-less bond. Since the semi-static hedges in the Black-Scholes economy are exact, closed form valuation formulas for certain barrier options can be easily derived.

Exchangeability type properties of asset prices

Let η = (η1,…,η n ) be a positive random vector. If its coordinates η i and η j are exchangeable, i.e. the distribution of η is invariant with respect to the swap π ij of its ith and jth coordinates, then Ef(η) = Ef(π ij η) for all integrable functions f. In this paper we study integrable random vectors that satisfy this identity for a particular family of functions f, namely those which can be written as the positive part of the scalar product 〈u, η〉 with varying weights u. In finance such functions represent payoffs from exchange options with η being the random part of price changes, while from the geometric point of view they determine the support function of the so-called zonoid of η. If the expected values of such payoffs are π ij -invariant, we say that η is ij-swap-invariant. A full characterisation of the swap-invariance property and its relationship to the symmetries of expected payoffs of basket options are obtained. The first of these results relies on a characterisation theorem for integrable positive random vectors with equal zonoids. Particular attention is devoted to the case of asset prices driven by Lévy processes. Based on this, concrete semi-static hedging techniques for multi-asset barrier options, such as weighted barrier swap options, weighted barrier quanto-swap options, or certain weighted barrier spread options, are suggested.

Intragroup transfers, intragroup diversification and their risk assessment

When assessing group solvency, an important question is to what extent intragroup transfers may be taken into account, as this determines to which extent diversification can be achieved. We suggest a framework to explicitly describe the families of admissible transfers that range from the free movement of capital to excluding any transactions. The constraints on admissible transactions are described as random closed sets. The paper focuses on the corresponding solvency tests that amount to the existence of acceptable selections of the random sets of admissible transactions.

Multiasset Derivatives and Joint Distributions of Asset Prices

Several of multiasset derivatives like basket options or options on the weighted maximum of assets exhibit the property that their prices determine uniquely the underlying asset distribution. Related to that the question how to retrieve this distributions from the corresponding derivatives quotes will be discussed. On the contrary, the prices of exchange options do not uniquely determine the underlying distributions of asset prices and the extent of this non-uniqueness can be characterised. The discussion is related to a geometric interpretation of multiasset derivatives as support functions of convex sets. Following this, various symmetry properties for basket, maximum and exchange options are discussed alongside with their geometric interpretations and some decomposition results for more general payoff functions.

A Stieltjes Approach to Static Hedges

Static hedging of complicated payoff structures by standard instruments becomes increasingly popular in finance. The classical approach is developed for quite regular functions, while for less regular cases, generalized functions and approximation arguments are used. In this note, we discuss the regularity conditions in the classical decomposition formula due to P. Carr and D. Madan (in Jarrow ed, Volatility, pp. 417–427, Risk Publ., London, 1998) if the integrals in this formula are interpreted as Lebesgue integrals with respect to the Lebesgue measure. Furthermore, we show that if we replace these integrals by Lebesgue–Stieltjes integrals, the family of representable functions can be extended considerably with a direct approach.