arXiv subjects

Martin Schweizer

Publications and source records attributed to Martin Schweizer.

7 recordsLinked to original sources

arXiv2026

Cash-invariant hull representation of divergence preferences

Uniformly weighted divergence preferences (UWDP) introduced in Maccheroni et al. (2006) are an important class of risk-averse preferences that contain as a special case the monotone mean--variance utility. UWDP are characterised by the lowest expected value of an act in $L^\infty$ under an adversarially chosen probability measure combined with the divergence of this measure. Our main result provides an alternative, computationally friendlier formula, which establishes in full generality that UWDP are the translation-invariant hull of state-independent expected utility over $L^0$. Some consequences of the new representation are studied.

q-fin.MF↗

arXiv2025

Dynamically optimal portfolios for monotone mean--variance preferences

Monotone mean-variance (MMV) utility is the minimal modification of the classical Markowitz utility that respects rational ordering of investment opportunities. This paper provides, for the first time, a complete characterization of optimal dynamic portfolio choice for the MMV utility in asset price models with independent returns. The task is performed under minimal assumptions, weaker than the existence of an equivalent martingale measure and with no restrictions on the moments of asset returns. We interpret the maximal MMV utility in terms of the monotone Sharpe ratio (MSR) and show that the global squared MSR arises as the nominal yield from continuously compounding at the rate equal to the maximal local squared MSR. The paper gives simple necessary and sufficient conditions for mean-variance (MV) efficient portfolios to be MMV efficient. Several illustrative examples contrasting the MV and MMV criteria are provided.

q-fin.PM↗

arXiv2024

New Stochastic Fubini Theorems

The classic stochastic Fubini theorem says that if one stochastically integrates with respect to a semimartingale $S$ an $\eta(dz)$-mixture of $z$-parametrized integrands $\psi^z$, the result is just the $\eta(dz)$-mixture of the individual $z$-parametrized stochastic integrals $\int\psi^z{d}S.$ But if one wants to use such a result for the study of Volterra semimartingales of the form $ X_t =\int_0^t \Psi_{t,s}dS_s, t \geq0,$ the classic assumption that one has a fixed measure $\eta$ is too restrictive; the mixture over the integrands needs to be taken instead with respect to a stochastic kernel on the parameter space. To handle that situation and prove a corresponding new stochastic Fubini theorem, we introduce a new notion of measure-valued stochastic integration with respect to a general multidimensional semimartingale. As an application, we show how this allows to handle a class of quite general stochastic Volterra semimartingales.

math.PR↗

arXiv2012

Mean-variance hedging via stochastic control and BSDEs for general semimartingales

We solve the problem of mean-variance hedging for general semimartingale models via stochastic control methods. After proving that the value process of the associated stochastic control problem has a quadratic structure, we characterize its three coefficient processes as solutions of semimartingale backward stochastic differential equations and show how they can be used to describe the optimal trading strategy for each conditional mean-variance hedging problem. For comparison with the existing literature, we provide alternative equivalent versions of the BSDEs and present a number of simple examples.

math.PR↗

arXiv2012

Cone-Constrained Continuous-Time Markowitz Problems

The Markowitz problem consists of finding in a financial market a self-financing trading strategy whose final wealth has maximal mean and minimal variance. We study this in continuous time in a general semimartingale model and under cone constraints: Trading strategies must take values in a (possibly random and time-dependent) closed cone. We first prove existence of a solution for convex constraints by showing that the space of constrained terminal gains, which is a space of stochastic integrals, is closed in L^2. Then we use stochastic control methods to describe the local structure of the optimal strategy, as follows. The value process of a naturally associated constrained linear-quadratic optimal control problem is decomposed into a sum with two opportunity processes L^{\pm} appearing as coefficients. The martingale optimality principle translates into a drift condition for the semimartingale characteristics of L^{\pm} or equivalently into a coupled system of backward stochastic differential equations for L^{\pm}. We show how this can be used to both characterise and construct optimal strategies. Our results explain and generalise all the results available in the literature so far. Moreover, we even obtain new sharp results in the unconstrained case.

q-fin.PM↗

arXiv2005

Dynamic exponential utility indifference valuation

We study the dynamics of the exponential utility indifference value process C(B;α) for a contingent claim B in a semimartingale model with a general continuous filtration. We prove that C(B;α) is (the first component of) the unique solution of a backward stochastic differential equation with a quadratic generator and obtain BMO estimates for the components of this solution. This allows us to prove several new results about C_t(B;α). We obtain continuity in B and local Lipschitz-continuity in the risk aversion α, uniformly in t, and we extend earlier results on the asymptotic behavior as α\searrow0 or α\nearrow\infty to our general setting. Moreover, we also prove convergence of the corresponding hedging strategies.

math.PR↗

arXiv2005

Number of paths versus number of basis functions in American option pricing

An American option grants the holder the right to select the time at which to exercise the option, so pricing an American option entails solving an optimal stopping problem. Difficulties in applying standard numerical methods to complex pricing problems have motivated the development of techniques that combine Monte Carlo simulation with dynamic programming. One class of methods approximates the option value at each time using a linear combination of basis functions, and combines Monte Carlo with backward induction to estimate optimal coefficients in each approximation. We analyze the convergence of such a method as both the number of basis functions and the number of simulated paths increase. We get explicit results when the basis functions are polynomials and the underlying process is either Brownian motion or geometric Brownian motion. We show that the number of paths required for worst-case convergence grows exponentially in the degree of the approximating polynomials in the case of Brownian motion and faster in the case of geometric Brownian motion.

math.PR↗