This proposed research requests funding for the development of high performance computational tools for the fast and accurate calibration of pricing models for financial and energy option contracts using available market data. The results of this research will be used to determine the parameters of various option pricing models that relax the restrictive assumptions of the widely used Black-Scholes-Merton model. For options of the European type, we investigate algorithms for solving optimization problem constrained by discretized partial integro-differential equations. For options of the American type with early exercise features, we develop efficient solution techniques for mathematical programs with complementarity constraints. We also investigate various methods for solving linear complementarity systems arising from the semi-discretization of parabolic variational inequalities for the valuation of American type options.
If successful, the results of the proposed research will lead to the accurate calibration of financial and energy option pricing models, a practical problem that is of fundamental importance to many industrial constituents, including financial firms, pension and mutual fund managers, market participants, energy generators, and commodity producers, who use option contracts as hedging tools to safeguard against large price fluctuations in interest rates, currency exchange rates, equity prices, energy and commodity prices. Improved pricing models in which risks associated with the underlying financial variables are modeled in more appropriate ways will yield better investment decisions and will help reduce arbitrage opportunities and stabilize the financial markets. The results of our research will also benefit the mathematical programming field and other service industries where similar model calibration problems are important.
Mathematical models are used routinely in the finance and energy industry for the pricing of derivatives securities -- most prominently, option contracts. All these models contain certain critical parameters as inputs that are intrinsic to the underlying assets. The fast and accurate determination of these parameters is key to the valuation of derivative contracts in the models. The main outcome of this project is the development of an optimization approach to fit these models, taking advantage of the special problem structures. The algorithm can be used to derive prices that respect and closely trace the observed market prices so as to increase the generalization power of the theoretical models for the purpose of obtaining more market-consistent prices. Our methodology has been tested on several pricing models of financial derivatives of the American type. The objective functions of the optimization problems are discrepancy measures between the theoretical model prices and the observed market prices along with a regularization term employed to enhance the robustness of the calibration process. Constraints on the model parameters are easily included in the calibration to account for their physical characteristics and numerical ranges. For options of the American type, the calibration problem is formulated as a mathematical program with complementarity constraints. These constraints are introduced to model the early exercise feature of the American options. New methods for solving these problems have been developed. They are two-phase iterations that consist of an active set prediction phase and a subspace phase. The algorithms enjoy favorable convergence properties under weaker assumptions than those assumed for competing methods. The active set prediction phase employs matrix splitting iterations that are tailored to the structure of the (nonconvex) bound-constrained problems and the (asymmetric) linear complementarity problems studied in this project. The pactical impact of this research can be seen in the improved pricing of financial and energy contracts, a problem that is of fundamental importance to many industries, including energy generators, financial firms, pension funds, market participants, and commodity producers, who use these contracts as hedging tools to safeguard against large price fluctuations of the underlying assets. Our improved pricing models allow better investment decisions and help to reduce arbitrage opportunities. The results of our research also impact the mathematical programming field, as they show how to incorporate second order information to accelerate convergence of the optimization algorithms.
