3,373 research outputs found
Efficient and accurate log-L\'evy approximations to L\'evy driven LIBOR models
The LIBOR market model is very popular for pricing interest 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
L\'evy-driven LIBOR model and aim at developing accurate and efficient
log-L\'evy 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, swaptions and sticky ratchet
caps show that the approximations perform very well. In addition, we also
consider the log-L\'evy approximation of annuities, which offers good
approximations for high volatility regimes.Comment: 32 pages, 21 figures. Added an example of a path-dependent option
(sticky ratchet caplet). Forthcoming in the Journal of Computational Financ
Markov Functional Market Model nd Standard Market Model
The introduction of so called Market Models (BGM) in 1990s has developed
the world of interest rate modelling into a fresh period. The obvious
advantages of the market model have generated a vast amount of research
on the market model and recently a new model, called Markov functional
market model, has been developed and is becoming increasingly popular.
To be clearer between them, the former is called standard market model
in this paper.
Both standard market models and Markov functional market models are
practically popular and the aim here is to explain theoretically how each
of them works in practice. Particularly, implementation of the standard
market model has to rely on advanced numerical techniques since Monte
Carlo simulation does not work well on path-dependent derivatives. This
is where the strength of the Longstaff-Schwartz algorithm comes in. The
successful application of the Longstaff-Schwartz algorithm with the standard
market model, more or less, adds another weight to the fact that the
Longstaff-Schwartz algorithm is extensively applied in practice
The least squares method for option pricing revisited
It is shown that the the popular least squares method of option pricing
converges even under very general assumptions. This substantially increases the
freedom of creating different implementations of the method, with varying
levels of computational complexity and flexible approach to regression. It is
also argued that in many practical applications even modest non-linear
extensions of standard regression may produce satisfactory results. This claim
is illustrated with examples
Risk Managing Bermudan Swaptions in the Libor BGM Model
This article presents a novel approach for calculating swap vega per bucket in the Libor BGM model. We show that for some forms of the volatility an approach based on re-calibration may lead to a large uncertainty in estimated swap vega, as the instantaneous volatility structure may be distorted by re-calibration. This does not happen in the case of constant swap rate volatility. We then derive an alternative approach, not based on re-calibration, by comparison with the swap market model. The strength of the method is that it accurately estimates vegas for any volatility function and at a low number of simulation paths. The key to the method is that the perturbation in the Libor volatility is distributed in a clear, stable and well understood fashion, whereas in the re-calibration method the change in volatility is hidden and potentially unstable.central interest rate model, Libor BGM model, swaption vega, risk management, swap market model, Bermudan swaption
Smoking Adjoints: fast evaluation of Greeks in Monte Carlo calculations
This paper presents an adjoint method to accelerate the calculation of Greeks by Monte Carlo simulation. The method calculates price sensitivities along each path; but in contrast to a forward pathwise calculation, it works backward recursively using adjoint variables. Along each path, the forward and adjoint implementations produce the same values, but the adjoint method rearranges the calculations to generate potential computational savings. The adjoint method outperforms a forward implementation in calculating the sensitivities of a small number of outputs to a large number of inputs. This applies, for example, in estimating the sensitivities of an interest rate derivatives book to multiple points along an initial forward curve or the sensitivities of an equity derivatives book to multiple points on a volatility surface. We illustrate the application of the method in the setting of the LIBOR market model. Numerical results confirm that the computational advantage of the adjoint method grows in proportion to the number of initial forward rates
Risk managing bermudan swaptions in the libor BGM model
This article presents a novel approach for calculating swap vegaper bucket in the Libor BGM model. We show that for some forms of thevolatility an approach based on re-calibration may lead to a large uncertaintyin estimated swap vega, as the instantaneous volatility structure maybe distorted by re-calibration. This does not happen in the case of constantswap rate volatility. We then derive an alternative approach, not based onre-calibration, by comparison with the swap market model. The strength ofthe method is that it accurately estimates vegas for any volatility functionand at a low number of simulation paths. The key to the method is thatthe perturbation in the Libor volatility is distributed in a clear, stable andwell understood fashion, whereas in the re-calibration method the change involatility is hidden and potentially unstable.risk management;libor BGM model;central interest rate model;bermudan swaptions;swap market model
Skewed Libor Market Model and Gaussian HJM explicit approaches to rolled deposit options
A simple exotic option (floor on rolled deposit) is studied in the shifted log-normal Libor Market (LMM) and Gaussian HJM models. The shifted log-normal LMM exhibits a controllable volatility skew. An explicit approach is used for both models. Using approximations the price in the LMM is obtained without Monte Carlo simulation. The more precise approximation uses a twisted version of the perdictor-corrector adapted to explicit solutions. The results of the approximation are surprisingly good.Libor Market Model; Heath-Jarrow-Morton; skew; smile; explicit solution; approximation; Bond Market Model; option on composition; existence results
Monte Carlo Greeks for financial products via approximative transition densities
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.Comment: 24 page
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