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Home/Financial Mathematics/Course 1

Course 1

Foundation

3-4 hours

Introduction to Financial Mathematics

Understanding uncertainty, capital markets, and the optimal allocation of resources across time

Learning Objectives

•Understand the fundamental problem of intertemporal allocation under uncertainty
•Master the concepts of states of the world, contingent claims, and complete markets
•Learn the structure and functions of the financial system
•Explore the historical development and key concepts of modern financial theory
•Understand the no-arbitrage principle and the law of one price

1. The Fundamental Problem in Financial Mathematics

Financial mathematics addresses a central economic problem: how should individuals and firms allocate resources across time when the future is uncertain? This involves making decisions today that affect consumption, investment, and risk exposure in future periods when outcomes are not known with certainty.

Definition 1.1: Intertemporal Allocation

Intertemporal allocation refers to the distribution of resources (consumption, savings, investment) across different time periods. The fundamental trade-off is between:

Present consumption: Immediate gratification and utility
Future consumption: Delayed gratification through savings and investment

In the absence of uncertainty, this is solved using the Fisher separation theorem. With uncertainty, we must also consider risk and the distribution of outcomes across possible future states.

Definition 1.2: States of the World

Let $S = \{s_1, s_2, \ldots, s_n\}$ represent the set of all possible states of the world at time $t=1$ , where each state $s_i$ represents a complete description of all relevant uncertainties resolved at that time.

Each state occurs with probability $\pi_i$ such that:

\sum_{i=1}^n \pi_i = 1, \quad \pi_i > 0 \text{ for all } i

This framework assumes a finite state space. In continuous-time models, we use probability measures on infinite state spaces.

Example 1.1: Simple Two-State Economy

Consider an economy with two periods (t=0 and t=1) and two possible states at t=1:

$s_1$ : Economic boom (probability $\pi_1 = 0.6$ )
$s_2$ : Economic recession (probability $\pi_2 = 0.4$ )

An investor must decide how to allocate wealth $W_0 = \$100,000$ at t=0 between consumption today and investments that pay off differently in the two states.

Suppose a risky asset pays:

X(s_1) = \$150, \quad X(s_2) = \$80

The expected payoff is:

\mathbb{E}[X] = 0.6 \times 150 + 0.4 \times 80 = 90 + 32 = \$122

The investor's optimal choice depends on their risk preferences (utility function), which we'll study in Course 3.

Remark 1.1: Completeness of State Description

In practice, the set of states $S$ must be sufficiently rich to capture all relevant sources of uncertainty affecting asset payoffs and investor welfare. This includes macroeconomic conditions, firm-specific outcomes, policy changes, technological innovations, etc.

The assumption of a finite state space is a simplification. Modern asset pricing often uses continuous-state models (e.g., stochastic differential equations) for greater realism.

2. Contingent Claims and Market Completeness

Definition 2.1: Contingent Claim

A contingent claim (or state-contingent security) is a financial contract that promises to pay a specified amount depending on which state of the world is realized. Formally, a contingent claim is a function:

X: S \to \mathbb{R}

where $X(s_i)$ represents the payoff if state $s_i$ occurs.

Definition 2.2: Arrow-Debreu Security

An Arrow-Debreu security (or elementary security) for state $s_i$ is a contingent claim that pays:

e_i(s_j) = \begin{cases} 1 & \text{if } j = i \\ 0 & \text{if } j \neq i \end{cases}

The price of the Arrow-Debreu security for state $s_i$ is denoted $q_i$ . These prices are called state prices.

Theorem 2.1: Spanning with Arrow-Debreu Securities

Any contingent claim $X$ can be replicated as a portfolio of Arrow-Debreu securities:

X = \sum_{i=1}^n X(s_i) \cdot e_i

Therefore, the no-arbitrage price of $X$ is:

P(X) = \sum_{i=1}^n q_i X(s_i)

Proof of Theorem 2.1:

Consider a portfolio holding $X(s_i)$ units of the Arrow-Debreu security $e_i$ for each $i = 1, \ldots, n$ .

If state $s_j$ occurs, the portfolio pays:

\sum_{i=1}^n X(s_i) \cdot e_i(s_j) = X(s_j) \cdot 1 + \sum_{i \neq j} X(s_i) \cdot 0 = X(s_j)

This holds for all $j = 1, \ldots, n$ , so the portfolio replicates $X$ exactly.

By the law of one price (no arbitrage), identical payoffs must have the same price:

P(X) = \sum_{i=1}^n X(s_i) \cdot q_i

∎

Definition 2.3: Complete Market

A market is complete if every contingent claim can be replicated (hedged) using available traded securities. Equivalently, a market with $n$ states is complete if there exist $n$ linearly independent traded assets.

In a complete market, any desired consumption plan across states can be achieved through appropriate portfolio choice.

Example 2.1: Complete Market with Two States

Consider two states ( $s_1$ , $s_2$ ) and two assets:

Risk-free bond: pays $1 in both states, price $B_0$
Risky stock: pays $S_1(s_1) = 110$ in state 1, $S_1(s_2) = 90$ in state 2, price $S_0 = 100$

The payoff matrix is:

\begin{pmatrix} 1 & 1 \\ 110 & 90 \end{pmatrix}

This matrix has rank 2 (linearly independent rows), so the market is complete.

Finding state prices: Let $q_1, q_2$ be the state prices. Then:

B_0 = q_1 + q_2

S_0 = 110q_1 + 90q_2

If $B_0 = 0.95$ (5% risk-free rate), solving gives:

q_1 = \frac{S_0 - 90B_0}{20} = \frac{100 - 85.5}{20} = 0.725

q_2 = B_0 - q_1 = 0.95 - 0.725 = 0.225

Remark 2.1: Market Incompleteness in Practice

Real financial markets are typically incomplete due to:

Infinitely many possible states (continuous distributions)
Limited number of traded securities
Uninsurable risks (labor income, health, natural disasters)
Transaction costs and trading constraints

Market incompleteness has important implications for risk sharing, asset pricing, and optimal portfolio choice.

3. Structure of the Financial System

The financial system consists of markets, institutions, and instruments that facilitate the flow of funds between savers (surplus units) and borrowers (deficit units), enable risk transfer, and support economic activity.

Definition 3.1: Financial Markets

Financial markets are organized venues or mechanisms where financial instruments are traded. They can be classified by:

Type of claim: Debt markets vs. Equity markets
Maturity: Money markets (short-term) vs. Capital markets (long-term)
Timing: Primary markets (new issues) vs. Secondary markets (existing securities)
Organization: Exchange-traded vs. Over-the-counter (OTC)

Definition 3.2: Financial Intermediaries

Financial intermediaries are institutions that channel funds from savers to borrowers, transforming financial claims in the process. Major types include:

Depository institutions: Commercial banks, savings banks, credit unions
Contractual savings institutions: Insurance companies, pension funds
Investment intermediaries: Mutual funds, hedge funds, private equity
Other intermediaries: Investment banks, finance companies, REITs

Proposition 3.1: Functions of Financial Intermediaries

Financial intermediaries perform several key economic functions:

Asset transformation: Transform assets with one set of characteristics (maturity, risk, liquidity) into assets with different characteristics
Information production: Screen and monitor borrowers, reducing information asymmetry
Risk transformation: Pool and diversify risks, provide risk-sharing opportunities
Transaction cost reduction: Achieve economies of scale in transaction processing
Liquidity provision: Offer more liquid claims to savers while holding less liquid assets

Example 3.1: Commercial Bank as Intermediary

A commercial bank demonstrates asset transformation:

Liabilities (sources of funds): Demand deposits (highly liquid, short maturity, low risk)
Assets (uses of funds): Commercial loans (illiquid, long maturity, higher risk)

The bank transforms short-term, liquid deposits into long-term, illiquid loans, earning a spread between loan interest rates and deposit rates.

Value added:

Savers get liquidity and safety
Borrowers get access to long-term funding
Bank earns profit from interest spread
Economy benefits from improved capital allocation

Definition 3.3: Financial Instruments

Financial instruments are contracts representing claims on future cash flows. Major categories:

Debt instruments: Bonds, loans, bills, notes (fixed claim)
Equity instruments: Common stock, preferred stock (residual claim)
Derivative instruments: Options, futures, swaps, forwards (claims contingent on underlying assets)
Hybrid instruments: Convertible bonds, callable bonds, etc.

4. Historical Development of Financial Theory

Modern financial mathematics emerged in the mid-20th century, building on centuries of economic thought and mathematical probability theory. Key milestones include:

1738: Daniel Bernoulli - Expected Utility Theory

Proposed that people maximize expected utility rather than expected value, introducing the concept of diminishing marginal utility and risk aversion. The famous St. Petersburg Paradox demonstrated the need for utility functions.

1900: Louis Bachelier - Random Walk Theory

In his PhD thesis "Théorie de la Spéculation," Bachelier modeled stock prices as Brownian motion, anticipating Einstein's work by five years. He derived an early version of the Black-Scholes formula for option pricing.

1952: Harry Markowitz - Portfolio Theory

Developed mean-variance portfolio optimization, showing how to construct efficient portfolios that maximize expected return for a given level of risk. Introduced the concept of diversification as a systematic risk management tool.

1958: Franco Modigliani & Merton Miller - Capital Structure Irrelevance

The M&M theorems showed that, under perfect markets, a firm's value is independent of its capital structure (debt-equity mix). This established the foundation for modern corporate finance theory.

1964: William Sharpe - Capital Asset Pricing Model (CAPM)

Derived a linear relationship between expected return and systematic risk (beta), providing a framework for asset pricing. The CAPM remains central to financial economics despite empirical challenges.

1973: Fischer Black, Myron Scholes, Robert Merton - Option Pricing

Derived a closed-form solution for European option prices using no-arbitrage arguments and dynamic hedging. The Black-Scholes-Merton model revolutionized derivatives markets and won the 1997 Nobel Prize.

1979: Daniel Kahneman & Amos Tversky - Prospect Theory

Challenged expected utility theory by documenting systematic deviations in human decision-making under risk. Prospect theory became the foundation of behavioral finance, earning Kahneman the 2002 Nobel Prize.

5. Fundamental Principles of Financial Mathematics

Theorem 5.1: No-Arbitrage Principle

There should exist no arbitrage opportunities in well-functioning markets. An arbitrage is a trading strategy that:

Requires no initial investment: $V_0 = 0$
Has no risk of loss: $V_1(s) \geq 0$ for all states $s$
Has positive probability of profit: $\mathbb{P}(V_1 > 0) > 0$

If arbitrage opportunities existed, rational investors would exploit them, causing prices to adjust until the opportunities disappeared.

Theorem 5.2: Law of One Price

Assets or portfolios with identical payoffs in all states must have the same price. Formally, if $X$ and $Y$ satisfy:

X(s) = Y(s) \text{ for all } s \in S

then necessarily:

P(X) = P(Y)

Proof of Theorem 5.2:

Suppose $P(X) > P(Y)$ despite $X(s) = Y(s)$ for all $s$ .

Arbitrage strategy:

Sell $X$ (receive $P(X)$ )
Buy $Y$ (pay $P(Y)$ )
Net cash flow at t=0: $P(X) - P(Y) > 0$

At t=1, for any state $s$ :

\text{Payoff} = Y(s) - X(s) = 0

This is an arbitrage: positive initial cash flow with zero risk. By the no-arbitrage principle, we must have $P(X) = P(Y)$ .

∎

Proposition 5.1: Risk-Neutral Valuation

Under no arbitrage, there exist risk-neutral probabilities $\{q_1, \ldots, q_n\}$ such that the price of any claim $X$ equals its discounted expected value:

P(X) = \frac{1}{1+r_f} \mathbb{E}^Q[X] = \frac{1}{1+r_f} \sum_{i=1}^n q_i X(s_i)

where $r_f$ is the risk-free rate and $\mathbb{E}^Q$ denotes expectation under the risk-neutral measure.

Remark 5.1: Physical vs. Risk-Neutral Probabilities

It's crucial to distinguish between:

Physical probabilities $\{\pi_1, \ldots, \pi_n\}$ : actual probabilities of states occurring
Risk-neutral probabilities $\{q_1, \ldots, q_n\}$ : artificial probabilities used for pricing

Risk-neutral probabilities adjust for risk preferences. They exist under no arbitrage but generally differ from physical probabilities unless investors are risk-neutral.

Example 5.1: Risk-Neutral Pricing

Consider the two-state example from earlier with $q_1 = 0.725$ , $q_2 = 0.225$ , and $r_f = 5.26\%$ .

Price a claim paying $X(s_1) = 50$ in state 1 and $X(s_2) = 30$ in state 2:

P(X) = q_1 X(s_1) + q_2 X(s_2) = 0.725 \times 50 + 0.225 \times 30

= 36.25 + 6.75 = 43

Alternatively, using the risk-free discount factor $B_0 = 0.95$ :

P(X) = \frac{\mathbb{E}^Q[X]}{1.0526} = \frac{0.725(50) + 0.225(30)}{1.0526} \approx 43

Note: Physical probabilities $\pi_1 = 0.6$ , $\pi_2 = 0.4$ would give expected value $42, but this is not the no-arbitrage price!

Key Takeaways

1.Financial mathematics addresses optimal allocation of resources across time and uncertain states
2.States of the world and contingent claims provide a formal framework for modeling uncertainty
3.Complete markets allow perfect risk sharing through Arrow-Debreu securities
4.Financial markets and intermediaries facilitate capital flow, risk transfer, and information aggregation
5.The no-arbitrage principle and law of one price are foundational to asset pricing theory
6.Risk-neutral valuation allows pricing contingent claims using adjusted probabilities

Frequently Asked Questions

What is the fundamental problem in financial mathematics?

Optimal allocation of resources across time and uncertain states. Individuals must decide how to distribute consumption and savings between present and future, facing uncertainty about future outcomes.

Why do we need probability theory in finance?

Financial decisions involve uncertainty about future events. Probability provides a mathematical framework to model and quantify this uncertainty, enabling rational decision-making under risk.

What is the difference between risk and uncertainty?

In Knight's distinction: risk refers to situations where probabilities are known (quantifiable), while uncertainty refers to situations where probabilities are unknown or unknowable.

What is arbitrage and why is it important?

Arbitrage is a risk-free profit opportunity from price discrepancies. The no-arbitrage principle is fundamental to asset pricing theory - it ensures market efficiency and provides pricing constraints.

What role do financial markets play in the economy?

Financial markets facilitate intertemporal resource allocation, enable risk sharing, provide price discovery, and improve information aggregation across economic agents.

Practice Quiz

Introduction to Financial Mathematics

Questions

Correct

Accuracy

In a two-period economy with states

s_1

and

s_2

, what does a state-contingent claim pay?

Medium

Not attempted

If an agent has utility function

u(c) = \ln(c)

, what type of preferences do they exhibit?

Medium

Not attempted

What is the no-arbitrage principle?

Easy

Not attempted

In a complete market with

n

states, how many linearly independent assets are needed?

Hard

Not attempted

What does it mean for a financial system to be 'informationally efficient'?

Medium

Not attempted

If state

s_1

has probability

\pi_1 = 0.6

and state

s_2

has probability

\pi_2 = 0.4

, what is the expected value of a claim paying

\

100

in

s_1

and

50

s_2

Easy

Not attempted

What is the primary function of financial intermediaries?

Easy

Not attempted

In the context of financial mathematics, what is a 'contingent claim'?

Medium

Not attempted

What does the 'law of one price' state?

Medium

Not attempted

Why is the separation of ownership and control important in modern finance?

Hard

Not attempted