2,730 research outputs found
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 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 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
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