This course could be more aptly described as "Introduction to Stochastic Calculus With Applications in Quantitative Finance". We will spend a significant amount of time on the basics of continuous time stochastic processes which can be represented as the sum of an ordinary Riemann integral and a "stochastic integral" with respect to a Wiener process.

To do this in a 100% exact fashion, with rigorous proofs, requires as prerequisite a graduate level course in a measure theoretically founded probability theory. But this is an undergraduate level class, and we will thus have to take a different approach.

• Develop the basics of abstract integration and probability with focus on conditional expectations and continuous time stochastic processes. Most proofs will only be outlined or even entirely omitted, but many precise definitions and formulations of important theorems and propositions will be given and continually referenced later on.
• Study a discrete time version (the binomial) model of the pricing of an option, in a market which rules out arbitrage, to gain some intuition concerning the continuous time approach.
• Become proficient at working with the Itô formula. Many financial math problems can be solved with help of that formula.
• A major example of the above is the pricing of financial derivatives such as a European call option.
• No knowledge of ordinary and partial differential equations is assumed, and yet we will learn how to associate with certain PDEs (partial differential equations) so called diffusion processes. Those stochastic processes lead to stochastic representations of the solutions of the underlying PDE. We will study in detail how this works for the pricing of a European call, and thus become acquanted with the famous Black Scholes model.
• A possible misconception: I have heard from David Biddle, who is involved in undergraduate advising, that many of you believe this course is specifically meant for students who want an actuarial degree. This is definitely not the case: