Quantitative finance uses data, mathematics and code to make a financial decision you can explain and check. The decision might be what price to quote for an option, how much risk two investments add to a portfolio, or whether a trading rule still earns anything after execution costs. The mathematics changes with the question.
Take an option that pays you for a stock finishing above 100 dollars. Imagine the stock costs 100 dollars today and can finish at either 120 or 90. The option pays 20 in the first state and zero in the second. How much would you pay for it today? Your first instinct might be to guess which outcome is more likely. In this small model, there is another way to start. Find stock and cash positions that produce the option's payment in both states.
If you hold two-thirds of a share, it will be worth 80 or 60 at maturity. Borrow an amount that will require repayment of 60 at maturity. After repayment the position is worth exactly 20 or zero. If cash grows by 2% over the period, borrowing that future 60 raises
60/1.02 today. Buying the shares costs
(2/3)100, so creating the
payoff costs about 7.84 dollars. The reason this price is useful is that you can trace it back to trades and see which assumptions it needs. If the model allowed more outcomes or trading were restricted, the same stock-and-cash construction might no longer replicate the payoff.
Other problems need a different kind of evidence. If you're estimating next month's portfolio risk, you need to decide how to represent joint moves and check your estimates on later data. If you're researching a trading idea, you need a baseline and a realistic account of what was known when each order would have been sent. The common thread is that you can point to the decision, the inputs and a check that might show your answer is wrong.