Applying lie theory to value credit derivatives with time dependent parameters

Abstract

Over the past few decades, the valuation of credit derivatives suffers from the main drawback that their models do not have closed-form solutions and require extensive use of numerical techniques to solve time-dependent partial differential equations (PDEs). Using the Lie group theory from abstract algebra we significantly simplify the valuation of the credit derivatives with time-dependent parameters. Unlike simulation or numerical based methods, our method obtains option values relatively quickly and more efficient. Exploiting the well-defined algebraic structure of the pricing partial differential equations of the default-risky bonds, the new method enables us to derive analytical closed-form pricing formulae very straightforwardly. Even though It has been pointed out that such a pricing problem is rather formidable and defies the conventional approach for the single-asset Black-Scholes model with time-dependent parameters (Bos and Ware [3]), within the framework of the lie algebraic approach, the generalization of other options is very simple and straightforward.

Keywords:

• Lie theory
• Credit derivative
• Abstract algebra
• Time-dependent PDE
• Financial engineering