Sample-path solutions for simulation optimization problems and stochastic variational inequalities

inequality;simulation;optimization

Indifference Pricing and Hedging in a Multiple-Priors Model with Trading Constraints

This paper considers utility indifference valuation of derivatives under model uncertainty and trading constraints, where the utility is formulated as an additive stochastic differential utility of both intertemporal consumption and terminal wealth, and the uncertain prospects are ranked according to a multiple-priors model of Chen and Epstein (2002). The price is determined by two optimal stochastic control problems (mixed with optimal stopping time in the case of American option) of forward-backward stochastic differential equations. By means of backward stochastic differential equation and partial differential equation methods, we show that both bid and ask prices are closely related to the Black-Scholes risk-neutral price with modified dividend rates. The two prices will actually coincide with each other if there is no trading constraint or the model uncertainty disappears. Finally, two applications to European option and American option are discussed.Comment: 28 pages in Science China Mathematics, 201

Variational inequalities in Hilbert spaces with measures and optimal stopping problems

We study the existence theory for parabolic variational inequalities in weighted $L^2$ spaces with respect to excessive measures associated with a transition semigroup. We characterize the value function of optimal stopping problems for finite and infinite dimensional diffusions as a generalized solution of such a variational inequality. The weighted $L^2$ setting allows us to cover some singular cases, such as optimal stopping for stochastic equations with degenerate diffusion coefficient. As an application of the theory, we consider the pricing of American-style contingent claims. Among others, we treat the cases of assets with stochastic volatility and with path-dependent payoffs.Comment: To appear in Applied Mathematics and Optimizatio

Optimal Investment with Stopping in Finite Horizon

In this paper, we investigate dynamic optimization problems featuring both stochastic control and optimal stopping in a finite time horizon. The paper aims to develop new methodologies, which are significantly different from those of mixed dynamic optimal control and stopping problems in the existing literature, to study a manager's decision. We formulate our model to a free boundary problem of a fully nonlinear equation. Furthermore, by means of a dual transformation for the above problem, we convert the above problem to a new free boundary problem of a linear equation. Finally, we apply the theoretical results to challenging, yet practically relevant and important, risk-sensitive problems in wealth management to obtain the properties of the optimal strategy and the right time to achieve a certain level over a finite time investment horizon

Reduced basis methods for pricing options with the Black-Scholes and Heston model

In this paper, we present a reduced basis method for pricing European and American options based on the Black-Scholes and Heston model. To tackle each model numerically, we formulate the problem in terms of a time dependent variational equality or inequality. We apply a suitable reduced basis approach for both types of options. The characteristic ingredients used in the method are a combined POD-Greedy and Angle-Greedy procedure for the construction of the primal and dual reduced spaces. Analytically, we prove the reproduction property of the reduced scheme and derive a posteriori error estimators. Numerical examples are provided, illustrating the approximation quality and convergence of our approach for the different option pricing models. Also, we investigate the reliability and effectivity of the error estimators.Comment: 25 pages, 27 figure

The Effect of Non-Smooth Payoffs on the Penalty Approximation of American Options

This article combines various methods of analysis to draw a comprehensive picture of penalty approximations to the value, hedge ratio, and optimal exercise strategy of American options. While convergence of the penalised solution for sufficiently smooth obstacles is well established in the literature, sharp rates of convergence and particularly the effect of gradient discontinuities (i.e., the omni-present `kinks' in option payoffs) on this rate have not been fully analysed so far. This effect becomes important not least when using penalisation as a numerical technique. We use matched asymptotic expansions to characterise the boundary layers between exercise and hold regions, and to compute first order corrections for representative payoffs on a single asset following a diffusion or jump-diffusion model. Furthermore, we demonstrate how the viscosity theory framework in [Jakobsen, 2006] can be applied to this setting to derive upper and lower bounds on the value. In a small extension to [Bensoussan & Lions, 1982], we derive weak convergence rates also for option sensitivities for convex payoffs under jump-diffusion models. Finally, we outline applications of the results, including accuracy improvements by extrapolation.Comment: 34 Pages, 10 Figure