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Financial Mathematics — Definition, Formula & Examples

Financial mathematics is the branch of applied math that puts a fair price on money, loans, investments, and risk. It starts with interest and the time value of money and extends to random models of stock prices.

Financial mathematics, also called mathematical finance, is the field that models financial markets using probability theory, stochastic calculus, and optimization. Its central principle is no-arbitrage: you cannot make a riskless profit from nothing. Under suitable market assumptions, an asset's price equals the expected value of its discounted future payoffs under a risk-neutral probability measure.

Key Formula

P=∑k=1nCk(1+r)tkP = \sum_{k=1}^{n} \frac{C_k}{(1+r)^{t_k}}
Where:
  • PP = price today (present value) of the asset
  • CkC_k = cash flow received at the k-th payment date
  • tkt_k = time of the k-th payment, in years
  • rr = annual discount (interest) rate, as a decimal
  • nn = number of cash flows

Worked Example

Problem: A bond pays $100 at the end of year 1 and $1,100 at the end of year 2. The annual interest rate is 10%, compounded once per year. What is the fair price of the bond today?
Step 1: Discount the year-1 payment back one year.
1001.1≈90.91\frac{100}{1.1} \approx 90.91
Step 2: Discount the year-2 payment back two years.
1100(1.1)2=11001.21≈909.09\frac{1100}{(1.1)^2} = \frac{1100}{1.21} \approx 909.09
Step 3: Add the present values. The exact values sum to a whole number.
P=1001.1+11001.21=1000P = \frac{100}{1.1} + \frac{1100}{1.21} = 1000
Answer: The fair price is $1,000. This makes sense: the bond's 10% coupon rate equals the market rate, so it sells at its face value.

Why It Matters

Financial mathematics is the core of actuarial exams such as the SOA's Exam FM and of careers in quantitative finance and banking. The same discounting idea, combined with random models like geometric Brownian motion, leads to the Black–Scholes formula for pricing stock options.

Common Mistakes

Mistake: Adding cash flows from different dates directly, for example saying the bond above is worth $100 + $1,100 = $1,200.
Correction: A dollar later is worth less than a dollar now. Discount each cash flow to the same date before you add the values.

Related Terms