# The Connections Between Physics and Finance

Douglas Scott

The connection between physics and finance goes back hundreds of years, and the names of the earliest physicists who studied finance may be surprising. In the 18^{th} century Bernoulli discovered Euler's constant e when investigating compound interest; more recently Jim Simons – of the Chern-Simons QFT fame – runs one of the most successful hedge funds of all time, and Nigel Goldenfield – who pioneered renormalization group theory in condensed matter – founded a financial-derivative software company.

Today many PhD graduates find a natural home in the financial industry. This talk will attempt to shed light as to why the fit is so natural, by exploring three facets of finance through the investment narrative: how does an investor choose their portfolio, how do they gauge the success of their investments, and finally how do they project and manage risk. I will spend the bulk of the talk on the pricing of derivative securities since this is my area of expertise, and the area that draws the most physicists.

First, I will discuss the types of problems that need to be solved, from choosing an investment, to forecasting financial risk, to hedging these risks by using derivative securities.

Second, the mathematical machinery required to solve these problems will be discussed, focusing on stochastic calculus, which has emerged as the lingua franca of financial security pricing. Stochastic calculus coupled with the concept of arbitrage freedom gives a very sound mathematical basis for the pricing of derivative securities.

Finally, I will discuss the models that are used to describe financial observables – such as prices, correlations, volatilities, and credit – and look at how the models have evolved as the important aspects of the data emerged.

When I first made the decision to move into finance, I was worried that the work would be a boring application of the mathematics I learned in my PhD. Happily, I was incorrect, as the field is deep with many interesting problems to study and novel mathematics to learn, and I hope to convey this in my talk.