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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.
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.
In this paper we develop several regression algorithms for solving general stochastic optimal control problems via Monte Carlo. This type of algorithms is particulary useful for problems with a high-dimensional state space and complex dependence structure of the underlying Markov process with respect to some control. The main idea behind the algorithms is to simulate a set of trajectories under some reference measure and to use the Bellman principle combined with fast methods for approximating conditional expectations and functional optimization. Theoretical properties of the presented algorithms are investigated and the convergence to the optimal solution is proved under mild assumptions. Finally, we present numerical results for the problem of pricing a high-dimensional Bermudan basket option under transaction costs in a financial market with a large investor.
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 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.
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.
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).