This course describes and examines financial derivatives such as forwards, futures and options. Drawing on real world financial markets experience and applications, and from classical texts and publications of impact and these innovative in the field. We review the original motivations for the creation, use of such financial instruments, & discuss the various instruments and strategies in real markets. We then present the financial mathematics of the evolution of such financial derivatives. In detail, we present the derivation of mathematical formula that describes generally derivatives & specifically address issues inherent to European style options, floating strike options, and early exercise uncertainty in American style options. From a wealth portfolio level of description to the trajectory of a random increment & the statistics of the underlying asset the derivative is written on. We present in detail the traditional and modern sophisticated derivations, techniques and computing methods utilized to mathematically describe & quantify, and which are furthermore used to successfully apply trading of these financial instruments. The course material is intended to be supplemented by published materials and with freeware applications written in say Spreadsheets, Matlab, or the .nb Mathematica 'notebook' languages etc. these readily available, and where interest regarding a particular presented topic may inspire further inquiry by the inquisitive.
What am I going to get from this course?
- Introduction to the world of financial mathematics as typified by financial derivatives
- Hedging strategies & various derivatives uses. Introduction In Detail.
- A brief review of financial markets & uncertainty.
- Statistics & statistical distributions & their properties.
- The rxpected value of an observable <x(t)>, as the statistically weighted sum or integral.
- Deriving statistics from information theoretic arguments. The observable as information.
- Deriving the equations of temporal evolution of the probability density functions PDFs of statistics. How to obtain the PDEs partial differential equations of evolution of the statistical distributions a.k.a. the PDFs probability density functions.
- Deriving the stochastic differential equations SDEs of the PDEs partial differential equations. The meaning of 'equivalent descriptions at the micro & macro evolution levels.
- The portfolio of assets & derivatives & the maximization of its efficiency. The Black-Scholes equation, a backwards Fokker-Planck type PDE.
- The European Style Black-Scholes equation.
- The European Style options Valuated by Alternative methods.
- The American Style option.
- Solving the American Style option early exercise problem discrete & continuous models & computation