This year’s December issue of The Journal of Computational Finance consists of four quite different papers. They discuss option pricing models as well as portfolio optimization, and all are based on different numerical approaches. Two of our papers focus on the partial differential equation (PDE) formulation of the option pricing problem. Another looks at its probabilistic formulation and the accompanying stochastic differential equations (SDEs). The remaining paper works with a utility function and mean–variance portfolio optimization. All of the papers contain very interesting findings.
The first paper in this issue is “A hybrid tree/finite-difference approach for Heston–Hull–White-type models” by Maya Briani, Lucia Caramellino and Antonino Zanette. Here, a hybrid PDE solution method is proposed for a hybrid SDE system for equity with correlated stochastic equity volatility as well as a stochastic interest rate model. The hybrid solution method enables us to obtain efficient European and American option prices. As a by-product, the authors introduce a new simulation scheme that can be adopted for Monte Carlo option valuation. This paper is based on a detailed examination of the numerical scheme by means of various numerical experiments.
Thorsten Hens and János Mayer from the University of Zurich provide our second paper: “Cumulative prospect theory and mean–variance analysis: a rigorous comparison”. In this work, a numerical optimization approach is presented to solve optimal portfolio selection problems involving objective functions from cumulative prospect theory. Two different asset allocations are compared: one in which the mean–variance efficient frontier is computed and subsequently maximized based on prospect theory, and another in which direct maximization of the cumulative prospect theory utility has taken place. The differences between the two allocations are found to be significant for asset allocation data from pension funds. The observed differences are explained on the basis of the prospect theory preference for positive skewness.
Pricing multidimensional financial derivatives with stochastic volatilities using the dimensional-adaptive combination technique”, this issue’s third paper, also features a PDE option valuation formulation, where the curse of dimensionality (in the case of high-dimensional option pricing problems) is handled by means of the sparse grid technique. In this paper, Janos Benk and Dirk Pflüger advocate a dimensional-adaptive sparse grid. This dimensional adaptivity enables the authors to spend a higher grid resolution only on those problem dimensions that are most relevant. Convergence is shown for option pricing problems in up to six dimensions.
This issue’s fourth and final paper is “Volatility risk structure for options depending on extrema” by Tomonori Nakatsu. Here, a decomposition formula is presented to calculate the option Vega for various exotic lookback and barrier options. The asset dynamics in this paper is the constant elasticity of variance process, which may be seen as a perturbed diffusion process. It is the Lamperti transformation that enables the author to calculate directional derivatives with respect to the diffusion coefficient. These form the basis for the decomposition of the option Vega into an extrema sensitivity, a terminal sensitivity and a drift sensitivity.
I wish you very enjoyable reading of this issue of The Journal of Computational Finance.
Cornelis W. Oosterlee
CWI – Dutch Center for Mathematics and Computer Science, Amsterdam
In this paper, the authors study a hybrid tree/finite-difference method, which allows us to obtain efficient and accurate European and American option prices in the Heston–Hull– White and Heston–Hull–White2d models.
This paper proposes a numerical optimization approach that can be used to solve portfolio selection problems including several assets and involving objective functions from cumulative prospect theory (CPT).
Pricing multidimensional financial derivatives with stochastic volatilities using the dimensional-adaptive combination technique
In this paper, the authors present a new and general approach to price derivatives based on the Black–Scholes partial differential equation (BS-PDE) in a multidimensional setting.
In this paper, the authors give a decomposition formula to calculate the vega index (sensitivity with respect to changes in volatility) for options with prices that depend on the extrema (maximum or minimum) and terminal value of the underlying stock…