Abstract: The goal of option pricing theory is to help the investors
to manage their money, enhance returns and control their financial
future by theoretically valuing their options. However, most of the
option pricing models have no analytical solution. Furthermore,
not all the numerical methods are efficient to solve these models
because they have nonsmoothing payoffs or discontinuous derivatives
at the exercise price. In this paper, we solve the American option
under jump diffusion models by using efficient time-dependent
numerical methods. several techniques are integrated to reduced
the overcome the computational complexity. Fast Fourier Transform
(FFT) algorithm is used as a matrix-vector multiplication solver,
which reduces the complexity from O(M2) into O(M logM).
Partial fraction decomposition technique is applied to rational
approximation schemes to overcome the complexity of inverting
polynomial of matrices. The proposed method is easy to implement
on serial or parallel versions. Numerical results are presented to prove
the accuracy and efficiency of the proposed method.
Abstract: The goal of option pricing theory is to help the
investors to manage their money, enhance returns and control their
financial future by theoretically valuing their options. Modeling
option pricing by Black-School models with jumps guarantees to
consider the market movement. However, only numerical methods
can solve this model. Furthermore, not all the numerical methods
are efficient to solve these models because they have nonsmoothing
payoffs or discontinuous derivatives at the exercise price. In this
paper, the exponential time differencing (ETD) method is applied
for solving partial integrodifferential equations arising in pricing
European options under Merton’s and Kou’s jump-diffusion models.
Fast Fourier Transform (FFT) algorithm is used as a matrix-vector
multiplication solver, which reduces the complexity from O(M2)
into O(M logM). A partial fraction form of Pad`e schemes is used
to overcome the complexity of inverting polynomial of matrices.
These two tools guarantee to get efficient and accurate numerical
solutions. We construct a parallel and easy to implement a version
of the numerical scheme. Numerical experiments are given to show
how fast and accurate is our scheme.
Abstract: Pricing financial contracts on several underlying assets
received more and more interest as a demand for complex derivatives.
The option pricing under asset price involving jump diffusion
processes leads to the partial integral differential equation (PIDEs),
which is an extension of the Black-Scholes PDE with a new integral
term. The aim of this paper is to show how basket option prices
in the jump diffusion models, mainly on the Merton model, can
be computed using RBF based approximation methods. For a test
problem, the RBF-PU method is applied for numerical solution
of partial integral differential equation arising from the two-asset
European vanilla put options. The numerical result shows the
accuracy and efficiency of the presented method.
Abstract: German electricity European options on futures using
Lévy processes for the underlying asset are examined. Implied
volatility evolution, under each of the considered models, is
discussed after calibrating for the Merton jump diffusion (MJD),
variance gamma (VG), normal inverse Gaussian (NIG), Carr, Geman,
Madan and Yor (CGMY) and the Black and Scholes (B&S) model.
Implied volatility is examined for the entire sample period, revealing
some curious features about market evolution, where data fitting
performances of the five models are compared. It is shown that
variance gamma processes provide relatively better results and that
implied volatility shows significant differences through time, having
increasingly evolved. Volatility changes for changed uncertainty, or
else, increasing futures prices and there is evidence for the need to
account for seasonality when modelling both electricity spot/futures
prices and volatility.
Abstract: This conference paper discusses a risk allocation problem for subprime investing banks involving investment in subprime structured mortgage products (SMPs) and Treasuries. In order to solve this problem, we develop a L'evy process-based model of jump diffusion-type for investment choice in subprime SMPs and Treasuries. This model incorporates subprime SMP losses for which credit default insurance in the form of credit default swaps (CDSs) can be purchased. In essence, we solve a mean swap-at-risk (SaR) optimization problem for investment which determines optimal allocation between SMPs and Treasuries subject to credit risk protection via CDSs. In this regard, SaR is indicative of how much protection investors must purchase from swap protection sellers in order to cover possible losses from SMP default. Here, SaR is defined in terms of value-at-risk (VaR). Finally, we provide an analysis of the aforementioned optimization problem and its connections with the subprime mortgage crisis (SMC).