Analysis: MathFinance 345/Stat390 Homework 1

In this analysis, we will explore two approaches to evaluating MathFinance 345/Stat390 Homework 1: an arbitrage argument and the Fundamental Theorem. We will also examine put options on stock and their replicating portfolios in different scenarios.

Arbitrage Argument

Imagine a portfolio consisting of a basket of stocks with known prices at time t = 0, denoted as SA t and SB t , respectively. Let's say there is an upfront payment q to enter into a swap contract between these assets. At time 1, you exchange one share of asset A for one share of asset B under a put option with strike K.

To find the fair market value q of this contract, we can use two methods: arbitrage argument and Fundamental Theorem. First, using the arbitrage argument, we assume that there is no risk-free interest rate r , implying that the expected return on assets A and B is equal to their respective prices at time 0.

Next, applying the Fundamental Theorem of Arithmetic Mean - Constant Proportion (AMCP) for Put-Call Parity, we derive an equation relating q , SA t , SB t , K , and r . This equation represents the expected value of q in terms of the given variables.

Example: Calculating the Arbitrage Value

Suppose SA 0 = 100 and SB 0 = 90, with a riskless rate of return r = 5%. Using the arbitrage argument, we can calculate the expected value of q as follows:

q = (SB 1 - SA 1)K / (SA 1 - SB 1)

Substituting the given values, we get:

q ≈ (90 - 100)K / (-10) q ≈ 9K

However, applying AMCP for Put-Call Parity yields a different result. Using the equation:

q = K(SA 0 + SB 0) / (SA 1 - SB 1)

we find that q ≈ K(190) / 20 q ≈ 95K

Replicating Portfolios

Now, let's consider put options on stock with different strike prices K . We'll examine the replicating portfolio for each option in scenarios ω1 and ω2 , where SA t = d1 < d2 and SB t = di .

For example, in scenario ω1 (d1 < d2), we can use a portfolio consisting of:

20 shares of Stock 80 shares of MoneyMarket

The expected value of the put option is given by:

E[put] = K(d1 - SA) / SB + q(0)

where q is the upfront payment to enter into the contract. To find the optimal weights, we use the mean-variance optimization method.

Example: Replicating Portfolio for Put Option in Scenario ω2

In scenario ω2 (d2 < di ), a replicating portfolio consists of:

80 shares of Stock 20 shares of MoneyMarket

Using the same expected value equation as before, we can calculate the optimal weights for each asset.

Incomplete Market

The MathFinance 345/Stat390 Homework 1 market is incomplete when there are three scenarios ω1 , ω2 , and ω3 with different share prices.