The Hidden Cost of Volatility Drag

That said, let's dive into the world of MathFinance 345/Stat390.

The contract in question is a swap contract that involves exchanging one share of asset A for one share of asset B at time t = 1. The goal is to determine the fair market value q of the contract under two different approaches: an arbitrage argument and using the Fundamental Theorem.

Arbitrage Argument

To find the fair market value q, we need to compare the current prices of assets A and B in the market with their initial values. Let's assume that there is a riskless asset MoneyMarket with rate of return r.

The current price of asset A at time t = 1 is SA1, and the current price of asset B at time t = 1 is SB1. The fair market value q can be calculated as:

q = (SA1 - SB1) / (SB1 - SB0)

where SB0 is the initial value of asset B.

Using the Fundamental Theorem

Alternatively, we can use the Fundamental Theorem to find the fair market value q. This theorem states that under an arbitrage-free market with a riskless asset, the price of any security at time t = 1 can be expressed as:

SA1 / SB1 = (SB0 + v) / (SB0 - v)

where v is the volatility of asset B.

Portfolio Investment Implications

The fair market value q represents the expected payoff of the swap contract at expiration. However, it's essential to consider portfolio investment implications when evaluating this outcome. For instance, if we have a portfolio consisting of assets A and C with initial values S0A and S0C respectively, we can calculate the fair market value q in terms of these prices.

Risks and Opportunities

The fair market value q is sensitive to changes in volatility. In the context of asset B, higher volatility would increase the expected payoff of the swap contract, while lower volatility would decrease it.

Actionable Conclusion To maximize returns on our investment, we should aim to reduce volatility in our portfolio. One way to achieve this is by diversifying our holdings across different asset classes and sectors.

Put Options: A European Put with Strike K

That said, let's consider a European put option on an asset Stock with strike price K.

The payoff of the put option at time t = 1 can be expressed as:

P(t) = max(0, K - S(t))

where S(t) is the current value of the underlying asset at time t = 1.

To find the market price of a put with strike K in terms of S0, r, d1, and d2, we need to consider different scenarios for d1 < d2 < d3.

Replicating Portfolio

A replicating portfolio for the put option would consist of assets that perfectly replicate its payoff. In this case, the replicating portfolio would be a combination of stock A and MoneyMarket with initial values S0A and S0MM respectively.

The expected return on the replicating portfolio at time t = 1 can be calculated as:

E[P(t)] = (S0A - K) / (S0A + S0MM)

This result suggests that a put option with strike K can be replicated using a combination of stock A and MoneyMarket, where the initial value of the MoneyMarket is adjusted to match the expected payoff.

Incomplete Market

The market for options on asset Stock is incomplete because there are no liquid markets for these securities. This means that investors cannot easily buy or sell put options on this asset with certainty.

Example: No Replicating Portfolio

Let's consider a scenario where d1 = 0.2 and d3 = 0.5, and the initial value of Stock at time t = 0 is S0. The payoff of the put option would be:

P(t) = max(0, K - S(t)) = max(0, K - (S0 \ (1 - 0.2))) = max(0, K - 0.8K) = max(0, 0.2K)

To find the market price of this put option, we need to consider different scenarios for d2.

Example: No Unique Market Price

As mentioned earlier, the set of possible market prices of a put option is an interval of real numbers. This means that there are multiple equilibrium measures for the market that give different prices for the derivative security.

In conclusion, analyzing MathFinance 345/Stat390 requires careful consideration of various factors, including arbitrage arguments and portfolio investment implications. By understanding these concepts, we can better appreciate the complexities of financial markets and make more informed investment decisions.