23,240 research outputs found
Sequential Monte Carlo pricing of American-style options under stochastic volatility models
We introduce a new method to price American-style options on underlying
investments governed by stochastic volatility (SV) models. The method does not
require the volatility process to be observed. Instead, it exploits the fact
that the optimal decision functions in the corresponding dynamic programming
problem can be expressed as functions of conditional distributions of
volatility, given observed data. By constructing statistics summarizing
information about these conditional distributions, one can obtain high quality
approximate solutions. Although the required conditional distributions are in
general intractable, they can be arbitrarily precisely approximated using
sequential Monte Carlo schemes. The drawback, as with many Monte Carlo schemes,
is potentially heavy computational demand. We present two variants of the
algorithm, one closely related to the well-known least-squares Monte Carlo
algorithm of Longstaff and Schwartz [The Review of Financial Studies 14 (2001)
113-147], and the other solving the same problem using a "brute force" gridding
approach. We estimate an illustrative SV model using Markov chain Monte Carlo
(MCMC) methods for three equities. We also demonstrate the use of our algorithm
by estimating the posterior distribution of the market price of volatility risk
for each of the three equities.Comment: Published in at http://dx.doi.org/10.1214/09-AOAS286 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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
Sequential Monte Carlo Methods for Option Pricing
In the following paper we provide a review and development of sequential
Monte Carlo (SMC) methods for option pricing. SMC are a class of Monte
Carlo-based algorithms, that are designed to approximate expectations w.r.t a
sequence of related probability measures. These approaches have been used,
successfully, for a wide class of applications in engineering, statistics,
physics and operations research. SMC methods are highly suited to many option
pricing problems and sensitivity/Greek calculations due to the nature of the
sequential simulation. However, it is seldom the case that such ideas are
explicitly used in the option pricing literature. This article provides an
up-to date review of SMC methods, which are appropriate for option pricing. In
addition, it is illustrated how a number of existing approaches for option
pricing can be enhanced via SMC. Specifically, when pricing the arithmetic
Asian option w.r.t a complex stochastic volatility model, it is shown that SMC
methods provide additional strategies to improve estimation.Comment: 37 Pages, 2 Figure
Multi-keyword multi-click advertisement option contracts for sponsored search
In sponsored search, advertisement (abbreviated ad) slots are usually sold by
a search engine to an advertiser through an auction mechanism in which
advertisers bid on keywords. In theory, auction mechanisms have many desirable
economic properties. However, keyword auctions have a number of limitations
including: the uncertainty in payment prices for advertisers; the volatility in
the search engine's revenue; and the weak loyalty between advertiser and search
engine. In this paper we propose a special ad option that alleviates these
problems. In our proposal, an advertiser can purchase an option from a search
engine in advance by paying an upfront fee, known as the option price. He then
has the right, but no obligation, to purchase among the pre-specified set of
keywords at the fixed cost-per-clicks (CPCs) for a specified number of clicks
in a specified period of time. The proposed option is closely related to a
special exotic option in finance that contains multiple underlying assets
(multi-keyword) and is also multi-exercisable (multi-click). This novel
structure has many benefits: advertisers can have reduced uncertainty in
advertising; the search engine can improve the advertisers' loyalty as well as
obtain a stable and increased expected revenue over time. Since the proposed ad
option can be implemented in conjunction with the existing keyword auctions,
the option price and corresponding fixed CPCs must be set such that there is no
arbitrage between the two markets. Option pricing methods are discussed and our
experimental results validate the development. Compared to keyword auctions, a
search engine can have an increased expected revenue by selling an ad option.Comment: Chen, Bowei and Wang, Jun and Cox, Ingemar J. and Kankanhalli, Mohan
S. (2015) Multi-keyword multi-click advertisement option contracts for
sponsored search. ACM Transactions on Intelligent Systems and Technology, 7
(1). pp. 1-29. ISSN: 2157-690
An Analysis of the Heston Stochastic Volatility Model: Implementation and Calibration using Matlab
This paper analyses the implementation and calibration of the Heston
Stochastic Volatility Model. We first explain how characteristic functions can
be used to estimate option prices. Then we consider the implementation of the
Heston model, showing that relatively simple solutions can lead to fast and
accurate vanilla option prices. We also perform several calibration tests,
using both local and global optimization. Our analyses show that
straightforward setups deliver good calibration results. All calculations are
carried out in Matlab and numerical examples are included in the paper to
facilitate the understanding of mathematical concepts.Comment: 34 page
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
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