It is well known that solutions to many contemporary financial problems can be obtained only by numerical methods. As the underlying financial models and instruments have become more complicated compared to their predecessors a few decades ago, advanced computational methods are needed nowadays to solve the associated problems. This special issue collects a variety of papers that elaborate on or make use of such advanced numerical methods, dedicated to valuing derivatives.
Concerning the underlying asset price models, various papers consider models with jumps, starting with the classical Merton model and going further on to the multi-asset Lévy models with fully correlated jump processes. Under these models, financial option values are acquired by either solving a partial (integro-)differential equation, that is P(I)DE, or by integrating the probability density function for the asset price employing integral transforms (Fourier) and tree methods, or using Monte Carlo. All these main approaches can be found in the present special issue.
The series of papers begins with a very useful and readable introduction into multilevel Monte Carlo methods for option valuation, which outlines the basic ideas and theoretical results available in the literature. The second paper introduces a test set of option valuation problems proposed to serve as a benchmark for modern and future numerical methods, and also provides a first comparison using this benchmark.
The topics considered in the 12 subsequent papers include, but are not limited to:
Discretizations of P(I)DEs with finite difference and finite volume methods, as well as meshless methods
Options with various types of payoffs (European, American, Bermudan, Asian, barrier, etc.)
Equity, credit, and commodity derivatives
Models with stochastic volatility, jumps, regime switching, or nonlinearity
Optimal decision-making for option valuation
GPU computing in finance
We thank all the authors for their contributions to this special issue and the referees for their diligent and timely reviews. We are also grateful to Professor A.Q.M. Khaliq, Editor-in-Chief of IJCM, for his deep attention, close involvement, and encouragement. Finally, we wish everyone a fruitful and enjoyable reading!