Available at: https://digitalcommons.calpoly.edu/theses/2014

#### Date of Award

6-2019

#### Degree Name

MS in Mathematics

#### Department

Mathematics

#### Advisor

Dana Paquin

#### Abstract

In this thesis, we will develop the fundamental properties of financial mathematics, with a focus on establishing meaningful connections between martingale theory, stochastic calculus, and measure-theoretic probability. We first consider a simple binomial model in discrete time, and assume the impossibility of earning a riskless profit, known as arbitrage. Under this no-arbitrage assumption alone, we stumble upon a strange new probability measure *Q*, according to which every risky asset is expected to grow as though it were a bond. As it turns out, this measure *Q* also gives the arbitrage-free pricing formula for every asset on our market. In considering a slightly more complicated model over a finite probability space, we see that *Q* once again makes its appearance. Finally, in the context of continuous time, we build a framework of stochastic calculus to model the trajectories of asset prices on a finite time interval. Under the absence of arbitrage once more, we see that *Q* makes its return as a Radon-Nikodym derivative of our initial probability measure. Finally, we use the properties of *Q* and a stochastic differential equation that models the dynamics of the assets of our market, known as the Ito formula, in order to derive the classic Black-Scholes Equation.

#### Award received:

Outstanding Thesis Award - Department of Mathematics

#### Included in

Finance Commons, Other Applied Mathematics Commons, Other Mathematics Commons, Partial Differential Equations Commons, Probability Commons