Once in a while, an issue of The Journal of Computational Finance is devoted to a workshop dedicated to an area of interest of the journal. The present issue is a example of this, as we report on the workshop "Models and Numerics in Financial Mathematics", which took place at the Lorentz Center in Leiden in the Netherlands on May 26-29, 2015. Click here to access the website for the workshop.
The workshop resulted in an intensive exchange of ideas and insights on various models and methods inspired by around twenty-five scientific presentations, as well as poster presentations by PhD students. Credit valuation adjustment, stochastic local volatility models, Hamilton-Jacobi-Bellman equations and backward stochastic differential equations (BSDEs), both for risk management and for financial derivatives pricing "in the wake of the financial crisis", were the main workshop themes. Through five high-quality scientific papers, some of the presented results are reported here.
The first research paper in this special issue is "The forward smile in local- stochastic volatility models" by Andrea Mazzon andAndrea Pascucci. Explicit expansion formulas are employed to approximate the values of forward start options in a multifactor local-stochastic volatility model. The expansion is based on polynomials and can be derived without numerical formulas. The quality of the approximation is then assessed by means of numerical comparisons.
In "From arbitrage to arbitrage-free implied volatilities" by Lech A. Grzelak and Cornelis W. Oosterlee, the issue's second paper, the stochastic collocation method is employed for defining an arbitrage-free density that is implied from the Hagan formula in the stochastic alpha beta rho (SABR) context. It is well-known that the density related to the Hagan formula for the implied volatilities is not always free of arbitrage. Based on a few collocation points on the implied survival distribution function and a projection of a Lagrange polynomial of an arbitrage-free variable, the resulting density is free of arbitrage. The proposed method is fast to implement, and the implied volatilities stay close to the ones obtained by the Hagan formula.
An arbitrage-free version of the SABR model is also the topic of our third paper: "Finite difference techniques for arbitrage-free SABR" by Fabien Le Floc'h and Gary Kennedy. This paper presents several second-order finite difference discretizations for the arbitrage-free SABR density problem to avoid undesirable oscillations in the numerical solution. The backward difference formula as well as the so-called Lawson-Swayne scheme are considered superior in terms of speed and stability.
The fourth paper in the issue, "Pricing swing options in electricity markets with two stochastic factors using a partial differential equation approach" by M. C. Calvo Garrido, M. Ehrhardt and C. Vázquez, also has accurate numerical discretizations as its main theme. Path-dependent swing options with multiple exercise rights lead to free boundary problems to be solved numerically.A finite element type discretization in the context of a Crank-Nicolson semi-Lagrange method is applied for this type of problem. Numerical experiments confirm the overall suitability of the proposed discretization.
Last but not least, the fifth paper in this special issue is "A mixed Monte Carlo and partial differential equation variance reduction method for foreign exchange options under the Heston-Cox-Ingersoll-Ross model" by Andrei Cozma and Christoph Reisinger. When currency is modeled by the Heston model, combined with the Cox- Ingersoll-Ross (CIR) dynamics for the domestic and foreign interest rate, numerical pricing can be done by means of Monte Carlo simulation. An inner Black-Scholes type expectation is, however, treated by means of a partial differential equation. This paper consists of theoretical and numerical contributions, in which variance reduction as well as insight into the method's complexity play prominent roles. Numerical results for a four-factor foreign exchange model show the efficiency of the numerical method developed.
We look back on a very successful and highly interesting Lorentz workshop, and we wish you very enjoyable reading of this issue of The Journal of Computational Finance.
Karel J. In 't Hout and Cornelis W. Oosterlee
The authors propose a method for determining an arbitrage-free density implied by the Hagan formula.
Pricing swing options in electricity markets with two stochastic factors using a partial differential equation approach
This paper considers the numerical valuation of swing options in electricity markets based on a two-factor model.
A mixed Monte Carlo and partial differential equation variance reduction method for foreign exchange options under the Heston–Cox–Ingersoll–Ross model
The paper concerns a hybrid pricing method build upon a combination of Monte Carlo and PDE approach for FX options under the four-factor Heston-CIR model.
This paper applies a variety of second-order finite difference schemes to the SABR arbitrage-free density problem and explores alternative formulations.
The Authors introduce a closed-form approximation for the forward implied volatilities.