| Description |
In this paper I will explain the mathematical theory behind financial derivative pricing. I will begin with the assumption that our financial market resides in a discretetime setting. Although the ultimate goal of the paper is to construct a pricing strategy in the continuous world, it is important to build an intuitive sense of what the actual mechanics are of the pricing strategy, before some of the more complex mathematics is introduced. I will demonstrate in the discrete world that because of arbitrage, we can price contingent claims by the use of a replicating portfolio strategy. Key probabilistic concepts such as change of measure, martingales, and conditional expectation will be introduced here and examples of their financial relevance will be expanded upon in depth. In the next half of the paper the transition to the continuous world will be made and with it Brownian motion will be introduced. I will show that a more than adequate model for a stock process can be found in geometric Brownian motion. The same pricing strategy that was used for the discrete world is again used here in the continuous world except the mathematics will be a little more complex. Ito's Calculus and Several theorems such as the C-M-0 theorem and the martingale representation theorem will be introduced here. Finally it will be shown, through the use of these tools, that the value of any contingent claim can be found by taking its expected value under the measure that makes the discounted stock price a martingale. This is the primary result of the paper. To end, the famous Black-Scholes equation for pricing a European call option will be derived by the use of the major results of the paper. |
