Title: Exploring the Intricacies of Multiperiod Models and Trees: Understanding the Principle of Numeraire Invariance

Deciphering the Challenges in Derivative Securities Pricing

In the financial realm, the constraints of one-period models become evident when dealing with the pricing and hedging of derivative securities. Real-world trading occurs in continuous time, necessitating frequent portfolio adjustments for optimal hedging of numerous derivatives. Nevertheless, discrete multiperiod models consisting of a large number of very short periods can offer useful numerical approximations (Source Material: Lecture 2).

The Principle of Numeraire Invariance: A Streamlining Technique

The Fundamental Theorem, established in Lecture 1, presumes the existence of a riskless asset—an asset whose share price remains constant under all market scenarios at maturity (t = 1). However, this assumption can be relaxed to require only an asset with a strictly positive share price under all scenarios. This weaker premise is a result of the principle of numeraire invariance, a potent tool for simplifying arbitrage pricing problems (Source Material: Lecture 2).

In essence, the principle of numeraire invariance asserts that the choice of "currency" used to measure prices does not affect the outcome. Share prices can be expressed using any freely traded asset whose share price remains strictly positive under all market scenarios. This permits us to switch between different numéraires without altering the arbitrage-free property of a market (Source Material: Lecture 2).

A Mathematical View on Numeraire Invariance

In a one-period market with finitely many market scenarios, the principle of numeraire invariance can be expressed as follows. Given a set of possible market scenarios Ω and a set of freely traded assets A1, A2, ..., AK, we consider two markets, M1 and M2. Both markets share the same set of scenarios ω and freely traded assets, but in market M1, asset Aj's share price is Sj 0 at t = 0, while in scenario ω, it is Sj 1(ω) at t = 1. In market M2, the corresponding prices are tilde Sj 0 and tilde Sj 1(ω), where (1) tilde Sj 0 = Sj 0/S1 0 and tilde Sj 1(ω) = Sj 1(ω)/S1 1(ω). Asset A1 is riskless in market M2, given that its share price remains positive under every market scenario (Source Material: Lecture 2).

Proposition 1 posits that if asset A1's share price in market M1 remains strictly positive under all scenarios, then there is no arbitrage in market M1 if and only if there is no arbitrage in market M2. Consequently, if market M1 is free of arbitrage, then market M2 has an equilibrium distribution tilde π, since it features a riskless asset (Source Material: Lecture 2).

Multiperiod Models: Managing Complexity

Considering multiperson markets with finitely many traded assets A1, A2, ..., AK, we can utilize the principle of numeraire invariance to simplify pricing formulas and arguments (Source Material: Lecture 2). This results in more concise expressions and reasoning (Source Material: Lecture 2).