"Arbitrage in MathFinance: Swap & Options Analysis"

Arbitrage Opportunities Explored: MathFinance 345/Stat390 Analysis

The efficiency of financial markets in the absence of arbitrage opportunities is a fascinating topic. Today, we examine this concept through the lens of recent assignments from MathFinance 345/Stat390. Grab your coffee and let's dive in!

Swap Contract: Determining Fair Market Value

We begin with a swap contract between buyer X and seller Y, involving an upfront payment `q` from X to Y followed by a share exchange at time 1. Our task is to find the fair market value `q` using arbitrage arguments and the Fundamental Theorem.

Arbitrage Argument: If `q` is too high, one could buy shares of asset A today and wait until tomorrow for profit. Conversely, if `q` is too low, selling shares of asset B tomorrow would yield a riskless gain. Thus, the fair value `q` should eliminate arbitrage opportunities.

Fundamental Theorem: Alternatively, we can use the Fundamental Theorem to find `q`. The expected value of the contract at time 0, discounted at rate `r`, equals the present value of expected payoffs. This gives us an equation to solve for `q`.

Put Options: Deriving Market Price and Replicating Portfolio

Next, we consider European put options on an asset Stock with two possible scenarios at time 1 (`ω1` and `ω2`), where share prices are `d1 < d2`. Our goal is to derive a formula for the market price of a put with strike `K`.

Using a riskless hedge argument, the market price of the put is `(S0 - K)e^(-r) + E[e^(−r)d1|ω=ω2] − d2e(−r)`, where `E[e^(−r)d1|ω=ω2]` represents the expected payoff in scenario `ω2`.

Moreover, this put can be replicated using a portfolio consisting of `Δ` shares of Stock and `-Δ*K/(B(0))` bonds, with `Δ` determined such that the portfolio value matches the put's market price.

Incomplete Markets: Derivative Security without Replicating Portfolio

We then examine an incomplete market scenario with two freely traded assets (Bond and Stock) and three scenarios (`ω1`, `ω2`, `ω3`). Here, we find a derivative security for which no replicating portfolio can be formed.

To illustrate this, consider a derivative security paying 1 unit of asset A if scenario `ωi` occurs. Since market prices of assets A and B do not uniquely determine an equilibrium measure, this derivative security has no arbitrage price. Thus, no replicating portfolio exists for it.

Infinite Scenarios: When the Fundamental Theorem Fails

Lastly, we challenge the Fundamental Theorem in a market containing infinitely many scenarios. It turns out that the theorem doesn't always hold in such markets.

Consider a market with three traded assets (A1, A2, B), where asset B is riskless and share prices are given by `S1_t` and `S2_t`. With an infinite set of scenarios `Ω`, we can prove that the Fundamental Theorem fails if certain conditions are not met.