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# An Application of Data Science and Mathematics in Finance

An Application of Data Science and Mathematics in Finance

Imagine that one day, you see people are queuing up in front of Bank A; so you ask the staff at the counter, you are told that they are offering anyone (regardless of their credit history) a loan of \$100,000 at a fixed annual rate at 2%. You then look around, the Bank B next door offers 1-year term deposit with a fixed annual rate at 3% for the same amount (\$100,000). After 5 minutes’ waiting, you sign for the loan from Bank A and immediately, you deposit the borrowed \$100,000 in Bank B, expecting to earn 1,000 dollar based on (3%-2%) x 100,000 one year from now. Your lucky day!

In Finance, without investments, an opportunity that allows one to gain profits from simultaneously buying and selling something is called Arbitrage Opportunity. The above opportunity, which involves borrowing through a loan from Bank A and lending through a deposit at Bank B, represents an arbitrage opportunity in Financial Economists’ parlance. However, one can easily see this type of opportunities won’t last long. In Financial markets, such arbitrage opportunity should not exist and could never last because of the force of the so-called Invisible Hands (Adam Smith) and so-called Efficient Market Hypothesis (Eugene Fama).

In this study, we applied mathematics (a bit of calculus and probability theory) and economic principles to derive so-called No-arbitrage conditions, which basically sets out the basic relationship between CDS contracts of different maturities should follow. Violating the conditions we derived, nonsensical probability (negative or greater-than-1 probability) and arbitrage opportunities will emerge. Based on the mathematical conditions derived in our research paper, after examining 24 million credit default swap (CDS) contracts, we identified 2,416 pairs of CDS contracts as arbitrage opportunities, which are not supposed to exist according to many literature, including those from Nobel Laureate Eugene Fama.

Everyday, hundreds of thousands of investors watch the market closely, why wasn’t there a queue of investors (like the one above in front of the bank) to take advantage of such apparent opportunities reported in our research? In this paper, we present how to identify such arbitrage opportunities, empirical evidence and explanations regarding why there wasn’t a queue.

Here is an example from two CDS contracts selling CDS protection on Microsoft (Please read the paper). The green line indicates the value of the trades. On 03/12/2008, the green line starts at zero, indicating that no investment is needed; also, throughout the time, it is above zero, which means positive profits persist for the trades, thus, an arbitrage opportunity. Further details are in the paper: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3282399