In this paper we study BSDEs arising from a special class of backward stochastic partial differential equations (BSPDEs) that is intimately related to utility maximization problems with respect to arbitrary utility functions. After providing existence and uniqueness we discuss the numerical realizability. Then we study utility maximization problems on incomplete financial markets whose dynamics are governed by continuous semimartingales. Adapting standard methods that solve the utility maximization problem using BSDEs, we give solutions for the portfolio optimization problem which involve the delivery of a liability at maturity. We illustrate our study by numerical simulations for selected examples. As a byproduct we prove existence of a solution to a very particular quadratic growth BSDE with unbounded terminal condition. This complements results on this topic obtained in [6,7,8].
We extend some recent works by Delong and Imkeller concerning Backward
stochastic differential equations with time delayed generators (delay
BSDE). We provide sharper a priori estimates and show that the solution
of a delay BSDE is in $L^p$. We introduce decoupled systems of SDE and
delay BSDE (which we term delay FBSDE) and give sufficient conditions
for the variational differentiability of their solutions. We connect
these derivatives to the Malliavin derivatives of such delay FBSDE via
the usual representation formulas which in turn give access to several
path regularity results. In particular we prove an extension of the
$L2$-path regularity result for delay FBSDE.
With an emphasis on generators with quadratic growth in the control variable we consider
measure solutions of BSDE, a solution concept corresponding to the notion of risk neutral
measure in mathematical finance. In terms of measure solutions, solving a BSDE reduces
to martingale representation with respect to an underlying filtration. Measure solutions
related to measures equivalent to the historical one provide classical solutions. We derive
the existence of measure solutions in scenarios in which the generating functions are just
continuous, of at most linear growth in the control variable (corresponding to generators of
at most quadratic growth in the usual sense), and with a random bound in the time parameter
whose stochastic integral is a BMO martingale. Our main tools include a stability property
of sequences of measure solutions, for which a limiting solution is obtained by means of the
weak convergence of measures.
Optimal dual martingales, their analysis and application to new algorithms for Bermudan products
(2012)
In this paper we introduce and study the concept of optimal and surely
optimal dual martingales in the context of dual valuation of Bermudan
options, and outline the development of new algorithms in this context.
We provide a characterization theorem, a theorem which gives conditions
for a martingale to be surely optimal, and a stability theorem concerning martingales which are near to be surely optimal in a sense. Guided
by these results we develop a framework of backward algorithms for constructing such a martingale. In turn this martingale may then be utilized
for computing an upper bound of the Bermudan product. The methodology is pure dual in the sense that it doesn’t require certain (input)
approximations to the Snell envelope.
In an Ito-Levy environment we outline a particular regression based
backward algorithm which allows for computing dual upper bounds without nested Monte Carlo simulation. Moreover, as a by-product this algorithm also provides approximations to the continuation values of the
product, which in turn determine a stopping policy. Hence, we may obtain lower bounds at the same time.
In a first numerical study we demonstrate a backward dual regression algorithm in a Wiener environment that is easy to implement and
is regarding accuracy comparable with the method of Belomestny et. al.
(2009).
In this paper, we study the dual representation for generalized multiple stopping problems,
hence the pricing problem of general multiple exercise options. We derive a dual representation which allows for cashflows which are subject to volume constraints modeled by
integer valued adapted processes and refraction periods modeled by stopping times. As
such, this extends the works by Schoenmakers (2010), Bender (2011a), Bender (2011b),
Aleksandrov and Hambly (2010), and Meinshausen and Hambly (2004) on multiple exercise
options, which either take into consideration a refraction period or volume constraints, but
not both simultaneously. We also allow more flexible cashflow structures than the additive
structure in the above references. For example some exponential utility problems are covered
by our setting. We supplement the theoretical results with an explicit Monte Carlo algorithm
for constructing confidence intervals for the price of multiple exercise options and exemplify
it by a numerical study on the pricing of a swing option in an electricity market.
We consider the problem of numerical approximation for forward-backward stochastic
differential equations with drivers of quadratic growth (qgFBSDE). To illustrate the significance
of qgFBSDE, we discuss a problem of cross hedging of an insurance related financial
derivative using correlated assets. For the convergence of numerical approximation schemes for
such systems of stochastic equations, path regularity of the solution processes is instrumental.
We present a method based on the truncation of the driver, and explicitly exhibit error estimates
as functions of the truncation height. We discuss a reduction method to FBSDE with globally
Lipschitz continuous drivers, by using the Cole-Hopf exponential transformation. We finally
illustrate our numerical approximation methods by giving simulations for prices and optimal
hedges of simple insurance derivatives.