Title: Navigating Derivatives in an Arbitrage-Free World

Unraveling the Swap Contract Puzzle

Are you ready to dive into the intricacies of financial derivatives? Let's start with a Swap Contract that promises an exchange of assets at a future date. But how do we determine its fair market value?

Arbitrage Argument: The Key to Fair Value

By employing an arbitrage argument, we can find the fair market value of the contract, q. If there's no way to make a riskless profit without investing any initial capital, then the market is in equilibrium and our Swap Contract is priced correctly.

Fundamental Theorem: A Deeper Look

Using the Fundamental Theorem, we can provide a more sophisticated approach to pricing this contract. This method helps us understand how the uncertainty surrounding the share prices of assets A and B at time t=1 affects the fair market value, q.

European Put Options: Hedging Strategies Explained

European put options offer investors the right to sell an asset at a predetermined price on or before a specific expiration date. Let's explore how to price these contracts and find replicating portfolios.

Pricing Formula Revealed

In this two-scenario market, we derive a formula for the market price of a put with strike K in terms of S0, r, d1, d2—the share prices of Stock at t = 0 and t = 1 in the two scenarios.

Replicating Portfolio Construction

A replicating portfolio for the put consists of Bond and Stock. By carefully selecting the appropriate combination, we can mimic the payoff structure of the put option.

Incomplete Markets: When Derivatives Go Awry

An incomplete market is one where there's no way to replicate every derivative security using just the available assets. Let's examine such a scenario and its implications for investors.

An Incomplete Market Unveiled

In this example, we explore an incomplete market with two assets: Bond and Stock, and three scenarios. The market is incomplete because it's impossible to find a replicating portfolio for certain derivative securities.

Derivative Securities With No Replicating Portfolio

There are indeed derivatives for which there exists no replicating portfolio in the assets Bond and Stock. These securities can cause headaches for investors, as their prices may not be uniquely determined in an equilibrium.

Markets with Infinite Scenarios: The Limits of Arbitrage Pricing

When dealing with markets containing infinite scenarios, the Fundamental Theorem of Arbitrage Pricing might not always hold true. Let's delve into an example that illustrates this limitation.

A Counterexample to the Rule

In a market with three assets (A1, A2, and B) and infinte scenarios, we demonstrate a situation where there are no arbitrages and no equilibrium measure.

Conclusion: Embrace Derivatives With Caution

As we've seen, derivatives such as Swap Contracts and European put options can provide valuable hedging strategies for investors. However, they also come with their own set of complexities. Understanding the intricacies of these financial instruments is crucial to making informed investment decisions.