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In this article we propose a novel approach to reduce the computational complexity
of the dual method for pricing American options. We consider a sequence of
martingales that converges to a given target martingale and decompose the original
dual representation into a sum of representations that correspond to dierent levels
of approximation to the target martingale. By next replacing in each representation
true conditional expectations with their Monte Carlo estimates, we arrive at what
one may call a multilevel dual Monte Carlo algorithm. The analysis of this algorithm
reveals that the computational complexity of getting the corresponding target upper
bound, due to the target martingale, can be signicantly reduced. In particular, it
turns out that using our new approach, we may construct a multilevel version of the
well-known nested Monte Carlo algorithm of Andersen and Broadie (2004) that is,
regarding complexity, virtually equivalent to a non-nested algorithm. The performance
of this multilevel algorithm is illustrated by a numerical example.
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.
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).
Primal-dual linear Monte Carlo algorithm for multiple stopping - An application to flexible caps
(2012)
In this paper we consider the valuation of Bermudan callable derivatives with
multiple exercise rights. We present in this context a new primal-dual linear
Monte Carlo algorithm that allows for ecient simulation of lower and upper price
bounds without using nested simulations (hence the terminology). The algorithm
is essentially an extension of a primal{dual Monte Carlo algorithm for standard
Bermudan options proposed in Schoenmakers et al. (2011), to the case of multiple
exercise rights. In particular, the algorithm constructs upwardly a system of dual
martingales to be plugged into the dual representation of Schoenmakers (2010).
At each level the respective martingale is constructed via a backward regression
procedure starting at the last exercise date. The thus constructed martingales are
nally used to compute an upper price bound. At the same time, the algorithm
also provides approximate continuation functions which may be used to construct
a price lower bound. The algorithm is applied to the pricing of
exible caps
in a Hull and White (1990) model setup. The simple model choice allows for
comparison of the computed price bounds with the exact price which is obtained
by means of a trinomial tree implementation. As a result, we obtain tight price
bounds for the considered application. Moreover, the algorithm is generically
designed for multi-dimensional problems and is tractable to implement.
We study minimal supersolutions of backward stochastic differential equations. We show the existence and uniqueness of the minimal supersolution, if the generator is jointly lower semicontinuous, bounded from below by an affine function of the control variable, and satisfies a specific normalization property. Semimartingale convergence is used to establish the main result.
We study a nonlinear operator defined via minimal supersolutions of backward stochastic differential equations with generators that are monotone in y, convex in z, jointly lower semicontinuous, and bounded below by an affine function of the control variable. We show existence, uniqueness, monotone convergence, Fatou’s Lemma and lower semicontinuity of this functional. We provide a comparison principle for the underlying minimal supersolutions of BSDEs, which we illustrate by maximizing expected exponential utility.
We provide results on the existence and uniqueness of equilibrium in dynamically incomplete financial markets in discrete time. Our framework allows for heterogeneous agents, unspanned random endowments and convex trading constraints. In the special case where all agents have preferences of the same type and all random endowments are replicable by trading in the financial market we show that a one-fund theorem holds and give an explicit expression for the equilibrium pricing kernel. If the underlying noise is generated by finitely many Bernoulli random walks, the equilibrium dynamics can be described by a system of coupled backward stochastic difference equations, which in the continuous-time limit becomes a multi-dimensional backward stochastic differential equation. If the market is complete in equilibrium, the system of equations decouples, but if not, one needs to keep track of the prices and continuation values of all agents to solve it. As an example we simulate option prices in the presence of stochastic volatility, demand pressure and short-selling constraints.
We consider a class of generalized capital asset pricing models in continuous time with a finite number of agents and tradable securities. The securities may not be sufficient to span all sources of uncertainty. If the agents have exponential utility functions and the individual endowments are spanned by the securities, an equilibrium exists and the agents’ optimal trading strategies are constant. Affine processes, and the theory of information-based asset pricing are used to model the endogenous asset price dynamics and the terminal payoff. The derived semi-explicit pricing formulae are applied to numerically analyze the impact of the agents’ risk aversion on the implied volatility of simultaneously-traded European-style options.
The LIBOR market model is very popular for pricing inter-
est rate derivatives, but is known to have several pitfalls. In addition, if
the model is driven by a jump process, then the complexity of the drift
term is growing exponentially fast (as a function of the tenor length). In
this work, we consider a Levy-driven LIBOR model and aim at developing accurate and efficient log-Levy approximations for the dynamics of
the rates. The approximations are based on truncation of the drift term
and Picard approximation of suitable processes. Numerical experiments
for FRAs, caps and swaptions show that the approximations perform
very well. In addition, we also consider the log-Levy approximation of
annuities, which offers good approximations for high volatility regimes.
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 with-
out 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).
Recently, there is a growing trend to offer guarantee products where the investor is allowed to shift her account/investment value between multiple funds. The switching right is granted a finite number per year, i.e. it is American style with multiple exercise possibilities. In consequence, the pricing and the risk management is based on the switching strategy which maximizes the value of the guarantee put option. We analyze the optimal stopping problem in the case of one switching right within different model classes and compare the exact price with the lower price bound implied by the optimal deterministic switching time. We show that, within the class of log-price processes with independent increments, the stopping problem is solved by a deterministic stopping time if (and only if) the price process is in addition continuous. Thus, in a sense, the Black & Scholes model is the only (meaningful) pricing model where the lower price bound gives the exact price. It turns out that even moderate deviations from the Black & Scholes model assumptions give a lower price bound which is really below the exact price. This is illustrated by means of a stylized stochastic volatility model setup.
In this paper we consider the optimal stopping problem for general dynamic monetary utility functionals. Sufficient conditions for the Bellman principle and the existence of optimal stopping times are provided. Particular attention is payed to representations which allow for a numerical treatment in real situations. To this aim, generalizations of standard evaluation methods like policy iteration, dual and consumption based approaches are developed in the context of general dynamic monetary utility functionals. As a result, it turns out that the possibility of a particular generalization depends on specific properties of the utility functional under consideration.
We compute the length of geodesics on a Riemannian manifold by regular polynomial interpolation of the global solution of the eikonal equation related to the line element $ds^2=g_ijdx^idx^j$ of the manifold. Our algorithm approximates the length functional in arbitrarily strong Sobolev norms. Error estimates are obtained where the geometric information is used. It is pointed out how the algorithm can be used to get accurate approximations of solutions of linear parabolic partial differential equations leading to obvious applications in finance, physics and other sciences.
We derive global analytic representations of fundamental solutions for a class of linear parabolic systems with full coupling of first order derivative terms where coefficients may depend on space and time. Pointwise convergence of the global analytic expansion is proved. This leads to analytic representations of solutions of initial-boundary problems of first and second type in terms of convolution integrals or convolution integrals and linear integral equations. The results have both analytical and numerical impact. Analytically, our representations of fundamental solutions of coupled parabolic systems may be used to define generalized stochastic processes. Moreover, some classical analytical results based on a priori estimates of elliptic equations are a simple corollary of our main result. Numerically, accurate, stable and efficient schemes for computation and error estimates in strong norms can be obtained for a considerable class of Cauchy- and initial-boundary problems of parabolic type. Furthermore, there are obvious and less obvious applications to finance and physics.
We present two approximation methods for pricing of CMS spread options in Libor market models. Both approaches are based on approximating the underlying swap rates with lognormal processes under suitable measures. The first method is derived straightforwardly from the Libor market model. The second one uses a convexity adjustment technique under a linear swap model assumption. A numerical study demonstrates that both methods provide satisfactory approximations of spread option prices and can be used for calibration of a Libor market model to the CMS spread option market.
The Real Multiple Dual
(2009)
In this paper we present a dual representation for the multiple stopping
problem, hence multiple exercise options. As such it is a natural generalization of the
method in Rogers (2002) and Haugh and Kogan (2004) for the standard stopping
problem for American options. We consider this representation as the real dual as it is
solely expressed in terms of an infimum over martingales rather than an infimum over
martingales and stopping times as in Meinshausen and Hambly (2004). For the multiple
dual representation we present three Monte Carlo simulation algorithms which require
only one degree of nesting.
In a rather general setting of multivariate stochastic volatility market models we derive global iterative probabilistic schemes for computing the free boundary and its Greeks for a generic class of American derivative models using front-fixing methods. Establishment of convergence is closely linked to a proof of global regularity of the free boundary surface.
In this paper we carry over the concept of reverse probabilistic representa-
tions developed in Milstein, Schoenmakers, Spokoiny (2004) for diffusion pro-
cesses, to discrete time Markov chains. We outline the construction of reverse
chains in several situations and apply this to processes which are connected
with jump-diffusion models and finite state Markov chains. By combining
forward and reverse representations we then construct transition density esti-
mators for chains which have root-N accuracy in any dimension and consider
some applications.
We present a generic non-nested Monte Carlo procedure for computing true upper bounds for Bermudan products, given an approximation of the Snell envelope. The pleonastic ``true'' stresses that, by construction, the estimator is biased above the Snell envelope. The key idea is a regression estimator for the Doob martingale part of the approximative Snell envelope, which preserves the martingale property. The so constructed martingale may be employed for computing dual upper bounds without nested simulation. In general, this martingale can also be used as a control variate for simulation of conditional expectations. In this context, we develop a variance reduced version of the nested primal-dual estimator (Anderson & Broadie (2004)) and nested consumption based (Belomestny & Milstein (2006)) methods . Numerical experiments indicate the efficiency of the non-nested Monte Carlo algorithm and the variance reduced nested one.
In this paper we propose a Libor model with a high-dimensional specially structured system of
driving CIR volatility processes. A stable calibration procedure which takes into account
a given local correlation structure is presented. The calibration algorithm is FFT based, so fast and easy
to implement.
We consider regular polynomial interpolation algorithms on recursively defined sets of interpolation points which approximate global solutions of arbitrary well-posed systems of linear partial differential equations. Convergence of the "limit" of the recursively constructed family of polynomials to the solution and error estimates are obtained from a priori estimates for some standard classes of linear partial differential equations, i.e. elliptic and hyperbolic equations. Another variation of the algorithm allows to construct polynomial interpolations which preserve systems of linear partial differential equations at the interpolation points. We show how this can be applied in order to compute higher order terms of WKB-approximations of fundamental solutions of a large class of linear parabolic equations. The error estimates are sensitive to the regularity of the solution. Our method is compatible with recent developments for solution of higher dimensional partial differential equations, i.e. (adaptive) sparse grids, and weighted Monte-Carlo, and has obvious applications to mathematical finance and physics.
In this paper we introduce efficient Monte Carlo estimators for the valuation
of high-dimensional derivatives and their sensitivities (”Greeks”).
These estimators are based on an analytical, usually approximative representation
of the underlying density. We study approximative densities
obtained by the WKB method. The results are applied in the context of
a Libor market model.
In this paper we lay the foundation for a numerical algorithm to
simulate high-dimensional coupled FBSDEs under weak coupling or
monotonicity conditions. In particular we prove convergence of a
time discretization and a Markovian iteration. The iteration
differs from standard Picard iterations for FBSDEs in that the
dimension of the underlying Markovian process does not increase
with the number of iterations. This feature seems to be
indispensable for an efficient iterative scheme from a numerical
point of view. We finally suggest a fully explicit numerical
algorithm and present some numerical examples with up to
10-dimensional state space.
We show that pricing a big class of relevant options by hedging
and no-arbitrage can be extended beyond semimartingale models. To
this end we construct a subclass of self-financing portfolios that
contains hedges for these options, but does not contain arbitrage
opportunities, even if the stock price process is a
non-semimartingale of some special type.
Moreover, we show that the option prices depend
essentially only on a path property of the stock price process,
viz. on the quadratic variation. As a consequence, we can
incorporate many stylized facts to a pricing model without
changing the option prices.
We introduce a forward scheme to simulate backward SDEs. Compared
to existing schemes, we avoid high order nestings of conditional
expectations backwards in time. In this way the error, when
approximating the conditional expectation, in dependence of the
time partition is significantly reduced. Besides this generic
result, we present an implementable algorithm and provide an error
analysis for it. Finally, we demonstrate the strength of the new
algorithm by solving some financial problems numerically.
In Kolodko & Schoenmakers (2004) and Bender & Schoenmakers (2004) a policy iteration was introduced which allows to achieve tight lower approximations of the price for early exercise options via a nested Monte-Carlo simulation in a Markovian setting. In this paper we enhance the algorithm by a scenario selection method. It is demonstrated by numerical examples that the scenario selection can significantly reduce the number of actually performed inner simulations, and thus can heavily speed up the method (up to factor 10 in some examples). Moreover, it is shown that the modified algorithm retains the desirable properties of the original one such as the monotone improvement property, termination after a finite number of iteration steps, and numerical stability.
We propose a valuation method for callable structures in a multi-factor Libor model which are path-dependent in the sense that, after calling, one receives a sequence of cash-flows in the future, instead of a well specified cash-flow at the calling date. The method is based on a Monte Carlo procedure for standard Bermudans recently developed in Kolodko & Schoenmakers (2004), and is applied to the cancelable snowball interest rate swap. The proposed procedure is quite generic, straightforward to implement, and can be easily adapted to other related path-dependent products.
We present a new iterative procedure for solving the multiple stopping
problem in discrete time and discuss the stability of the algorithm.
The algorithm produces monotonically increasing approximations of the
Snell envelope, which coincide with the Snell envelope after finitely many
steps. Contrary to backward dynamic programming, the algorithm allows
to calculate approximative solutions with only a few nestings of conditional
expectations and is, therefore, tailor-made for a plain Monte-Carlo
implementation.
In this project we propose the use of some widespread prediction techniques in the last few years for modeling derivatives. In order to do that, we have reviewed the state-of-the-art of the prediction models dealing with stochastic processes. In the oil futures sector, Schwartz suggested a model in which the oil futures price was split in two factors: the long-term equilibrium price and the short-term variations. As a result, we propose a Hull-White discrete-time two-factor interest rate model, whose factors are the short and the long term.
We apply theoretical results of S. Peng on supersolutions for BS-DEs
to the problem of finding optimal superhedging strategies in a
Black-Scholes market under constraints. Constraints may be imposed
simultaneously on wealth process and portfolio. They may be nonconvex,
time-dependent, and random. Constraints on the portfolio may
e.g. be formulated in terms of the amount of money invested, the portfolio
proportion, or the number of shares held.
The CreditRisk model launched by CSFB in 1997 is widely used by practitioners in the banking sector as a simple means for the quantification of credit
risk, primarily of the loan book. We present an alternative numerical recursion scheme for CreditRisk, equivalent to an algorithm recently proposed by
Giese, based on well-known expansions of the logarithm and the exponential
of a power series. We show that it is advantageous to the Panjer recursion
advocated in the original CreditRisk
document, in that it is numerically stable. The crucial stability arguments are explained in detail. Furthermore, the
computational complexity of the resulting algorithm is stated.
We introduce a new Monte Carlo method for constructing the exercise
boundary of an American option in a generalized Black-Scholes framework.
Based on a known exercise boundary, it is shown how to price and hedge the
American option by Monte Carlo simulation of suitable probabilistic represen-
tations in connection with the respective parabolic boundary value problem.
The methods presented are supported by numerical simulation experiments.