For better or for worse, Mathematics has always been a tool of modern Economics and Finance. It is not surprising that an important fraction of those who pursue an academic career in Economics come from highly mathematical disciplines. A few days ago I stumbled with a paper that describes arbitrage opportunities as gauge fields. For some of you the analogy may seem senseless, but for me it is beautiful. In this post I develop a small example of how physics ideas can be extended to Finance.
Before any Finance, let’s talk about fields (not the one with grass in the classic Windows wallpaper). A vector field consists in assigning a vector to every point in a space. Gravity can be seen as a field. Wherever you are, you always have a large arrow pulling you downwards to the centre of the earth. The length of this arrow (the magnitude) decreases as you move away at a ratio proportional to the inverse of the distance squared.
In a more modern framework, gravity can be seen as a “curvature” in space-time. This curvature induces a potential (the ability to move) which increases as objects approach to massive bodies.
At this point you might be thinking, Where is the Economics behind all this Physics? The idea of curvature or force fields has a beautiful and simple correspondence in Finance.
Money across time and currencies
We all know 100 USD are not equivalent to 100 EUR, and 100 USD today are not equivalent to 100 USD in a year from now. Exchange rates allow us to convert between currencies, and discount factors (interest rates) between time. Consider an economy in which time is continuous, and there exists a measure of currencies of size . We can describe money (in some currency) at different points in time as all points in a ray starting at the origin . Monetary values in this economy are points inside a disc of radius equal to the horizon time. The inclination of each ray corresponds to the currency analysed.
If exchange rates and interest rates (in each currency) are chosen appropriately, every point in the disk is equivalent to each other. In economic terms it means that there are no arbitrage opportunities. As an example, suppose the interest rate in USD is 1% and the interest rate in EUR is 2%. Let’s also suppose that the exchange rate at time is 1.1 EUR/USD and at is 1 EUR/USD.
You can obtain an arbitrage profit as follows:
Borrow USD at a 1% interest rate at . Exchange those USD into euros at a rate of 1.1 EUR/USD, to get 1.1 EUR. Invest those euros at a 2% interest rate which will give you EUR in the future. Now is time to pay your credit, exchange your euros into dollars to obtain USD. The amount you owe to the bank is
Your profit will be:
If there are no arbitrage opportunities, every point in the disk is equivalent to each other. In mathematical words any point in the disk is invariant to rotations and translations, in physics words every point is gauge invariant.
Arbitrage can be seen as a vector field or curvature in which we assign to every point a vector. Each point has a vector that points to another point in the disc. This pair of points (different interest rates and exchange rates) will give a potential arbitrage opportunity. Arbitrage can be seen as a curvature in the 2D disk. The degree of curvature measures the strength of the arbitrage opportunity and therefore the size of the potential investor’s profit. As time goes by, investors’ pressure will correct any price anomalies.
In conclusion, this is an interesting and peculiar way to visualise arbitrage opportunities. Not only we can visualize the direction of the potential arbitrage opportunity, but also predict the speed at which investors exploit this potential. If the curvature is small, the incentives to profit from arbitrage opportunities decrease due to transaction costs. If the curvature is small, profits will compensate transaction costs and more investors are willing to profit from these opportunities.
I believe in the cooperation among disciplines. If a tool works, why don’t borrow it?