An Elementary Introduction to Mathematical Finance

Mathematical finance uses mathematical tools to model and analyze financial markets. At its core lies the concept of present value: A dollar today is worth more than a dollar tomorrow, primarily due to the potential for investment and the presence of risk. Discounting future cash flows using an appropriate interest rate is fundamental to pricing assets.

Interest rates are the prices of borrowing money. Simple interest is calculated only on the principal amount, while compound interest is calculated on the principal and accumulated interest, leading to exponential growth. This distinction is crucial when comparing different investment options.

Probability theory plays a vital role. Expected value, calculated by multiplying each possible outcome by its probability and summing the results, provides a measure of the average outcome. Variance and standard deviation quantify the dispersion of possible outcomes, representing risk. Higher volatility, indicated by a higher standard deviation, generally implies higher risk.

One of the cornerstones is the time value of money. Annuities are a series of equal payments made at regular intervals. Perpetuities are annuities that continue indefinitely. Formulas exist to calculate the present value and future value of both annuities and perpetuities, allowing for comparisons of different payment streams.

Derivatives, such as options and futures, derive their value from an underlying asset. Options give the holder the *right*, but not the obligation, to buy (call option) or sell (put option) the underlying asset at a specified price (strike price) on or before a specified date (expiration date). Futures contracts obligate the buyer to purchase and the seller to sell the underlying asset at a predetermined price on a future date. These instruments allow for hedging risk (reducing exposure to price fluctuations) and speculation (profiting from anticipated price movements).

The Black-Scholes model is a widely used formula for pricing European options (options that can only be exercised at expiration). It uses the current stock price, strike price, time to expiration, risk-free interest rate, and volatility of the underlying asset to calculate the theoretical price of the option. While simplified and based on assumptions that may not always hold in reality, it provides a benchmark for option pricing.

Efficient Market Hypothesis (EMH) posits that asset prices fully reflect all available information. In its strongest form, EMH suggests that it's impossible to consistently outperform the market because prices instantly adjust to new information. Weaker forms allow for the possibility of some market inefficiencies, providing opportunities for informed investors. The validity and extent of EMH remain debated among academics and practitioners.

Mathematical finance provides a framework for understanding and navigating the complexities of financial markets. While these are basic concepts, they form the foundation for more advanced topics like portfolio optimization, risk management, and quantitative trading.