We discuss a nonlinear multigrid method for a linear complementarity problem. The convergence is improved by a recombination of iterants. The problem under consideration deals with option pricing from mathematical finance. Linear complementarity problems arise from so-called American-style options. A 2D convectiondiffusion type operator is discretized with the help of second order upwind discre...

In this paper, the American exchange option (AEO) valuation problem is modelled as a free boundary problem. The critical stock price for an AEO is satisfied an integral equation implicitly. When the remaining time is large enough, an asymptotic formula is provided for pricing an AEO. The numerical results reveal that our asymptotic pricing formula is robust and accurate for the long-term AEO. K...

A general framework is developed to treat optimal control problems for a generalized BlackScholes model, which is used for option pricing. The volatility function is retrieved from a set of market observations. The optimal volatility function is found by minimizing the cost functional measuring the discrepancy between the model solution (pricing) and the observed market price, via the unconstra...

This paper deals with an option pricing model which can be thought of as a hybrid stochastic and local volatility model. This model is built on the local volatility term of the well-known constant elasticity of variance (CEV) model multiplied by a stochastic volatility term driven by a fast mean-reverting Ornstein-Uhlenbeck process. An asymptotic formula for European option price is derived to ...

The goal of this paper is to show that the jump-diffusion models are an essential and easy-to-learn tool for option pricing and risk management, and that they provide an adequate description of stock price fluctuations and market risks. We try to give an overview of the field without focusing on technical details. After introducing several widely used jump-diffusion models, we discuss Fourier t...

In this paper it is shown how symmetry methods can be used to find exact solutions for European option pricing under a time-dependent 3/2-stochastic volatility model View the MathML source. This model with A(t) constant has been proven by many authors to outperform the Heston model in its ability to capture the behaviour of volatility and fit option prices. Further, singular perturbation techni...

Market prices of corporate bond spreads and of credit default swap (CDS) rates do not match each other. In this paper, we argue that the liquidity premium, the cheapest-to-deliver (CTD) option and actual market segmentation explain the pricing differences. Using the European transaction data from Reuters and Bloomberg, we estimate the liquidity premium that is timevarying and firm-specific. We ...

The price of an option can under some assumptions be determined by the solution of the Black–Scholes partial differential equation. Often options are issued on more than one asset. In this case it turns out that the option price is governed by the multi-dimensional version of the Black–Scholes equation. Options issued on a large number of underlying assets, such as index options, are of particu...

We study pricing under the local volatility. Our research is mainly intended for pedagogical purposes. In the first part of our work we study the local volatility modeling. We derive the local volatility formula in terms of the European call prices and in terms of the market implied volatilities. We propose and calibrate to the DAX option data a functional form for the implied volatility which ...