The June 2017 Journal of Computational Finance is, again, an interesting issue in terms of the numerical techniques its authors employ. However, before we launch into an overview of the featured papers, I would like to welcome Professor Andrea Pascucci, from the University of Bologna, Italy, to our editorial board. Professor Pascucci is an expert on, among other topics, the derivation of asymptotic formulas and corresponding efficient valuation techniques when dealing with involved processes in computational finance. It is great to have someone with his level of expertise on board.
The current issue is, as usual, diverse in the numerical techniques it presents, and modern in terms of the financial applications it considers. We encounter a Monte Carlo type particle method in the context of a local correlation model; error analysis for Fourier type option pricing methods; a finite difference partial differential equation (PDE) method in the context of credit valuation adjustments (CVAs); and a comparison of different numerical methods in the context of a Gaussian term structure model.
We begin with “Calibration of local correlation models to basket smiles”. Here, Julien Guyon from Bloomberg L.P. introduces a new family of local correlation models in order to accurately calibrate to the implied volatility smile of basket options, which may arise in a variety of asset classes. State-dependent correlations allow for better hedging. By incorporating this model into the framework of the Monte Carlo particle method, calibrated local correlation models can be built. The technique is generalized to models based on stochastic interest rates, stochastic dividend yields,
local stochastic volatility and local correlation.
The second contribution to this issue is “Error analysis in Fourier methods for option pricing” by Fabián Crocce, Juho Häppölä, Jonas Kiessling and Raúl Tempone. In this paper the errors made when using a Fourier method to price European options under exponential Lévy processes are analyzed and bounded. The bound is analyzed in detail, and it is also used to determine different method parameters for the numerical technique to converge in a robust and efficient way.
Our third paper is “Efficient estimation of sensitivities for counterparty credit risk with the finite difference Monte Carlo method” by Cornelis S. L. de Graaf, Drona Kandhai and Peter M. A. Sloot. In this paper Monte Carlo forward asset path generation is combined with the numerical solution of a PDE to compute the exposure and potential future exposure related to the option contract by an iteration backward in time. The asset model considered is based on stochastic volatility and a stochastic interest rate, which results in a three-dimensional PDE (plus time as the fourth problem
dimension). The proposed finite difference Monte Carlo method is particularly useful for the efficient computation of CVA sensitivities.
In “Smile with the Gaussian term structure model”, Abdelkoddousse Ahdida, Aurélien Alfonsi and Ernesto Palidda introduce an affine extension of the linear Gaussian term structure model that gives rise to an implied volatility smile. A Wishartdriven model for interest rates is proposed, in which the parameters and state variables may be clearly interpreted with regard to the corresponding yield curve dynamics. Different numerical models have been compared with this model; these range from Fourier and Laplace transform based methods via perturbation theory based efficient
pricing methods to a second-order stochastic differential equation discretization scheme, which forms the basis of a Monte Carlo method.
I wish you very enjoyable reading of this issue of The Journal of Computational Finance.
CWI – Dutch Center for Mathematics and Computer Science, Amsterdam
The authors build a whole family of local correlation models by combining the particle method with a new, simple idea.
The authors provide a bound for the error committed when using a Fourier method to price European options, when the underlying follows an exponential Lévy dynamic.
Efficient estimation of sensitivities for counterparty credit risk with the finite difference Monte Carlo method
This paper presents a natural extension of the LGM that keeps the affine structure and generates an implied volatility smile.