4,506 research outputs found
A method for pricing American options using semi-infinite linear programming
We introduce a new approach for the numerical pricing of American options.
The main idea is to choose a finite number of suitable excessive functions
(randomly) and to find the smallest majorant of the gain function in the span
of these functions. The resulting problem is a linear semi-infinite programming
problem, that can be solved using standard algorithms. This leads to good upper
bounds for the original problem. For our algorithms no discretization of space
and time and no simulation is necessary. Furthermore it is applicable even for
high-dimensional problems. The algorithm provides an approximation of the value
not only for one starting point, but for the complete value function on the
continuation set, so that the optimal exercise region and e.g. the Greeks can
be calculated. We apply the algorithm to (one- and) multidimensional diffusions
and to L\'evy processes, and show it to be fast and accurate
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
Sequential Design for Optimal Stopping Problems
We propose a new approach to solve optimal stopping problems via simulation.
Working within the backward dynamic programming/Snell envelope framework, we
augment the methodology of Longstaff-Schwartz that focuses on approximating the
stopping strategy. Namely, we introduce adaptive generation of the stochastic
grids anchoring the simulated sample paths of the underlying state process.
This allows for active learning of the classifiers partitioning the state space
into the continuation and stopping regions. To this end, we examine sequential
design schemes that adaptively place new design points close to the stopping
boundaries. We then discuss dynamic regression algorithms that can implement
such recursive estimation and local refinement of the classifiers. The new
algorithm is illustrated with a variety of numerical experiments, showing that
an order of magnitude savings in terms of design size can be achieved. We also
compare with existing benchmarks in the context of pricing multi-dimensional
Bermudan options.Comment: 24 page
Some numerical methods for solving stochastic impulse control in natural gas storage facilities
The valuation of gas storage facilities is characterized as a stochastic impulse control problem with finite horizon resulting in Hamilton-Jacobi-Bellman (HJB) equations for the value function. In this context the two catagories of solving schemes for optimal switching are discussed in a stochastic control framework. We reviewed some numerical methods which include approaches related to partial differential equations (PDEs), Markov chain approximation, nonparametric regression, quantization method and some practitioners’ methods. This paper considers optimal switching problem arising in valuation of gas storage contracts for leasing the storage facilities, and investigates the recent developments as well as their advantages and disadvantages of each scheme based on dynamic programming principle (DPP
Pricing American Options by Exercise Rate Optimization
We present a novel method for the numerical pricing of American options based
on Monte Carlo simulation and the optimization of exercise strategies. Previous
solutions to this problem either explicitly or implicitly determine so-called
optimal exercise regions, which consist of points in time and space at which a
given option is exercised. In contrast, our method determines the exercise
rates of randomized exercise strategies. We show that the supremum of the
corresponding stochastic optimization problem provides the correct option
price. By integrating analytically over the random exercise decision, we obtain
an objective function that is differentiable with respect to perturbations of
the exercise rate even for finitely many sample paths. The global optimum of
this function can be approached gradually when starting from a constant
exercise rate.
Numerical experiments on vanilla put options in the multivariate
Black-Scholes model and a preliminary theoretical analysis underline the
efficiency of our method, both with respect to the number of
time-discretization steps and the required number of degrees of freedom in the
parametrization of the exercise rates. Finally, we demonstrate the flexibility
of our method through numerical experiments on max call options in the
classical Black-Scholes model, and vanilla put options in both the Heston model
and the non-Markovian rough Bergomi model
Executive stock option exercise with full and partial information on a drift change point
We analyse the optimal exercise of an executive stock option (ESO) written on
a stock whose drift parameter falls to a lower value at a change point, an
exponentially distributed random time independent of the Brownian motion
driving the stock. Two agents, who do not trade the stock, have differing
information on the change point, and seek to optimally exercise the option by
maximising its discounted payoff under the physical measure. The first agent
has full information, and observes the change point. The second agent has
partial information and filters the change point from price observations. This
scenario is designed to mimic the positions of two employees of varying
seniority, a fully informed executive and a partially informed less senior
employee, each of whom receives an ESO. The partial information scenario yields
a model under the observation filtration in which the
stock drift becomes a diffusion driven by the innovations process, an
-Brownian motion also driving the stock under
, and the partial information optimal stopping value
function has two spatial dimensions. We rigorously characterise the free
boundary PDEs for both agents, establish shape and regularity properties of the
associated optimal exercise boundaries, and prove the smooth pasting property
in both information scenarios, exploiting some stochastic flow ideas to do so
in the partial information case. We develop finite difference algorithms to
numerically solve both agents' exercise and valuation problems and illustrate
that the additional information of the fully informed agent can result in
exercise patterns which exploit the information on the change point, lending
credence to empirical studies which suggest that privileged information of bad
news is a factor leading to early exercise of ESOs prior to poor stock price
performance.Comment: 48 pages, final version, accepted for publication in SIAM Journal on
Financial Mathematic
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