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Introduction

What is Mathematical Finance?

Finance concerns the allocation and pricing of assets—goods, stocks, etc.—and liabilities—debts, loans, bonds, etc.—in the presence of uncertainty and risk. This definition raises several fundamental questions:

  • What is trade, money, or pricing?
  • What are uncertainty and risk?
  • Which academic fields address these questions?
  • What role does mathematics—particularly stochastics—play in finance?

A Brief and Biased History of Trade, Money, and Finance

Trade of Goods and the Emergence of Money

  • Exchange of goods (around 150,000 BC):

    • Advantages: Better allocation of comparative advantages.
    • Question:
      • How is exchange value determined (bilateral agreement)?
      • Need for intermediaries.

  • Money as a medium of exchange (around 12,000 BC):

    • Advantages:
      • Solves the double coincidence problem, enabling efficient allocation.
      • Serves as a unit of account (numéraire).
      • Stores value over time.
    • Questions:
      • How is value established (multilateral agreements between numerous goods)? Law of demand and supply.
      • How does money preserve value over time?
      • Introduces the concept of the time value of money.

Assessing Uncertainty: The Rise of Financial Markets

  • Commodity markets (4,000 BC):
    Development of forward contracts to hedge against price fluctuations.

    • Question: How are prices determined?

  • Loans and banking systems (2,000 BC):
    Facilitated leveraged investments.

    • Question: How to price future payments?

  • Insurance (1400s):
    Emerged during overseas trading, with premiums exchanged for risk.

    • Sparked the beginnings of probability theory.

  • Stock markets (1600s):
    Enabled capital raising and introduced challenges such as dividend payments and stock price modeling (e.g., Bachelier model, Brownian motion).

  • Options (1600s, standardized in 1973):
    Put and call options provided bounded insurance against price movements.

  • Derivatives:
    Broader contracts written on assets, indices, interest rates, etc.

Pricing remains the central problem for all these financial instruments. While agreements between counterparties can set prices, mathematical models provide fair and robust valuations. A specialized subfield of mathematical finance also focuses on assessing financial risk.

Academic Fields Involved in Finance

  • Economics: Macroeconomics, microeconomics, decision theory.
  • Psychology: Behavioral finance.
  • Law.
  • Computer Science: Algorithmic trading, machine learning.
  • Mathematics:
    • Stochastics (modeling).
    • Statistics (calibration, machine learning...).
    • Optimization.
    • Functional analysis and partial differential equations (e.g., Black-Scholes model).