Computational Methods in Financial Mathematics Course Project

This course project is made up of two parts. Part one is an investigation and implementation of pricing of financial derivatives using numerical methods for the solution of partial differential equations. Part two is an introduction of Monte Carlo methods in financial engineering. The name of course is MA573:Computational Methods in Financial Mathematics, spring 2009, given by Professor Marcel Blais

Course Summary of Computational Methods of Financial Mathematics

Most realistic financial derivatives models are too complex to allow explicit analytic solutions. The computational techniques used to implement those models fall into two broad categories: finite difference methods for the solution of partial differential equations (PDEs) and Monte Carlo simulation. Accordingly, the course consists of two sections. The first half of the course focuses on finite difference methods. The following topics are discussed; Parabolic PDEs, Black-Scholes PDE for European and American options; binomial and trinomial trees; explicit, implicit and Crank- Nicholson finite difference methods; far boundary conditions, convergence, stability, variance bias; early exercise and free boundary conditions; parabolic PDEs arising from fixed income derivatives; implied trees for exotic derivatives, adapted trees for interest rate derivatives. The second half of the course focuses on Monte Carlo. The following topics are discussed; Random number generation and testing; evaluation of expected payoff by Monte Carlo simulation; variance reduction techniques�antithetic variables, importance sampling, martingale control variables; stratification, low-discrepancy sequences and quasi-Monte Carlo methods; efficient evaluation of sensitivity measures; methods suitable for multifactor and term-structure dependent models. Computational Methods of Financial Mathematics is taught by Marcel Blais, a professor at Worcester Polytechnic Institute

Application of Lie symmetries to Solving Partial Differential Equations associated with the Mathematics of Finance

In financial markets one is sometimes confronted with a complicated system of partial differential equations arising from some physical important problem, and the discovery of the explicit solution of the problem can result with very useful information. That is, the explicit solutions of the financial market models can be used as benchmarks for testing numerical methods of physical experiments. This fact is evidenced by the work of economists Black and Scholes, the Black-Scholes model, whereby they deduced the financial models from solving a linear parabolic partial diiierential equation that were then used in the iinance literature as the main vehicle for pricing contingent claims such as call and put options, together with all other financial derivatives. Due to their work a rich arsenal of methods of theory of partial differential equations were suddenly available for mathematicians working in the area of mathematical finance. Adopting their approach of deducing prices of contingent claim via solving the associated PDE models, we apply the algorithmic quantitative theory of Lie, the Lie symmetry analysis, to derive and solve the models associated with interest rate derivatives whose price dynamics comprise of partial differential equations in their set up. The interest rate derivative model that we consider is of great importance because it deviates from the usual models that are depended on the usual Vasicek model which has a disadvantage of producing negative interest rates. Our interest rate derivative PDE model is depended on the functional interest rate model that satisfies all properties of an interest rate model and produces positive interest rates upon certain restriction put on the co-domain. We obtain their Lie point symmetries and transformations that we then use to deduce their exact group-invariant solutions. In particular, we analyse a zero-coupon bond pricing PDE model and obtain its various reductions that we then use to solve and produce the pricing models for the aforementioned contingent claim. A systematic reductions on optimal Lie algebra is further performed to obtain optimal invariant solutions of the model as well. The resulting analytical expressions in both cases can then be used to add to the minute number of pricing models for the interest rate derivatives instruments in the literature; also play a vital role as benchmarks to verify real world data that is analysed numerically by numerical methods in financial markets

``String'' formulation of the Dynamics of the Forward Interest Rate Curve

We propose a formulation of the term structure of interest rates in which the forward curve is seen as the deformation of a string. We derive the general condition that the partial differential equations governing the motion of such string must obey in order to account for the condition of absence of arbitrage opportunities. This condition takes a form similar to a fluctuation-dissipation theorem, albeit on the same quantity (the forward rate), linking the bias to the covariance of variation fluctuations. We provide the general structure of the models that obey this constraint in the framework of stochastic partial (possibly non-linear) differential equations. We derive the general solution for the pricing and hedging of interest rate derivatives within this framework, albeit for the linear case (we also provide in the appendix a simple and intuitive derivation of the standard European option problem). We also show how the ``string'' formulation simplifies into a standard N-factor model under a Galerkin approximation.Comment: 24 pages, European Physical Journal B (in press