Slide Deck 2 (No pauses)

ACTSC 446 – Part II Discrete-time Models

• Instructor: Zachary Van Oosten

• Term: Winter 2025

Course Objectives

• Aim of the Course: Price derivatives, which requires a model of stock prices.

• Key Model: The simplest model is the binomial model.

Course Outline

1. The One-Period Binomial Model

2. The Multi-Period Binomial Model

3. Option Pricing in the Binomial Model

4. Dividends

5. Exotic Options

6. General Discrete Market Models

One-Period Binomial Model

Model Description

• Contains two key components:

  1. Something to represent randomness.

  2. Something to represent the time-value of money.

• Time Frame: Only one period (two dates: t=0 and t=1).

• Stock Prices:

  • Price today: S0 (observable).

  • Price at end of period: S1 (random).

Price Movement

• At time t=1, stock price can:

  1. Increase to S1 = u S0 with probability p.

  2. Decrease to S1 = d S0 with probability 1-p.

• Constraints:

  • Must satisfy: 0 < d < u.

Risk-Free Asset

• Risk-free asset (like a bond) has:

  • Price today: B0 = 1 (also given).

  • Price at t=1: B1 = B0 (1 + r), with risk-free rate r > 0.

Portfolios & Arbitrage

• Portfolio Vector: θ = (x, y), where:

  • x = number of bonds held.

  • y = units of stock held (can be fractional or negative).

Value Process:

• At t=0: Vθ 0 = x + y S0 (constant)

• At t=1: Vθ 1 = x(1 + r) + y S1 (random).

Arbitrage Opportunities

• Arbitrage opportunity exists if:

  • Vθ 0 ≤ 0

  • Vθ 1 ≥ 0 with probability P (Vθ 1 > 0) > 0.

• In a well-functioning market, arbitrage opportunities are assumed to be absent.

Completeness Condition

• The model is arbitrage-free if and only if:

  • d < 1 + r < u.

Risk-Neutral Probabilities

• Define qu and qd = 1 - qu such that 1 + r = qu u + qd d.

• Under risk-neutral measure:

  • S0 = (1 / (1 + r)) * EQ[S1].

Multi-Period Binomial Model

Model Description

• Time Frame: Extends to T periods

• At time t+1: St+1 = St Zt+1, with Zt+1 following the same distribution as Z.

Trading Strategy

• Defined as a stochastic process of positions in assets over time.

• Self-Financing Portfolio: Indicates that portfolio value at time t maintains the same conditions as prior time.

Portfolios & Arbitrage

• Multi-Period Arbitrage: A trading strategy θ is arbitrage if:

  • S0 · θ0 ≤ 0

  • ST · θT > 0 (where ST is the price vector at T).

General Discrete Market Models

Framework Overview

• Market model definition: Expands the binomial to include multiple risky assets and states of the world.

• Use of state-price vectors to determine pricing in the context of replicated securities.

Fundamental Theorem of Asset Pricing

• Arbitrage-free conditions aligned with state-price vectors:

  1. Existence of state-price vector (Ψ) implies no arbitrage.

  2. Risk-Neutral Probability: A modified probability measure ensuring expected discounted prices align with security prices.

Option Pricing in the Binomial Model

European Options Pricing

• Risk-Neutral Price determined by discounted expected payoff under risk-neutral measure.

  • Two key results:

    • Recursive valuation: ΠX(t) = e^{-r(T-t)} EQ[ΠX(T)|St]

    • Pricing formula: ΠX(0) = e^{-rT} EQ[X].

Exotic Options

• Characteristics: More complex payoff structures compared to standard options.

• Includes varieties like Asian, Lookback, and Barrier options, each with unique pricing methodologies.

Dividends and Their Implications

Dividends Handling

• Options are not entitled to dividends but must factor in how dividends affect stock pricing.

• Types of Dividends: Continuous and discrete, with distinct effects on pricing models.

Practical Implications

• Build models that incorporate dividend expectations to inform option pricing strategies.

Conclusion

• The course covers critical concepts in discrete-time models emphasizing option pricing, market completeness, and the implications of dividends, aiming to equip students with a thorough understanding of how to approach pricing derivatives in mathematical finance.