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- Monte Carlo simulation (2)
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- Libor market model (1)
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Project
- E5 (22)
Application Area
- E (22)
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.
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 derive a link between the short rate and a new index constructed in a multiasset
economy. This uses two structural assumptions: The volatility structure
of the assets is rigidly spherical , and the short rate function is homogeneous of
degree 0. We give clear motivations for the assumptions, and our main result is
economically intuitive and testable from observed data. A preliminary empirical
study illustrates how one can test such results.
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.
In a rather general setting of Itô-Lévy processes we study a class of transforms (Fourier for example) of the state variable of a process which are holomorphic in some disc around time zero in the complex plane. We show that such transforms are related to a system of analytic vectors for the generator of the process, and we state conditions which allow for holomorphic extension of these transforms into a strip which contains the positive real axis. Based on these extensions we develop a functional series expansion of these transforms in terms of the constituents of the generator. As application, we show that for multidimensional affine Itô-Lévy processes with state dependent jump part the Fourier transform is holomorphic in a time strip under some stationarity conditions, and give log-affine series representations for the transform.
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 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.