Numeraire Invariance & Put-Call Parity: Unveiling Finance's Symmetry

Analysis: MathFinance 345/Stat390

Have you ever pondered what underlies the seemingly erratic dance of stock prices? It's not merely market sentiment or economic indicators; mathematics also plays a significant role. Today, we explore MathFinance 345/Stat390, delving into numeraire invariance, put-call parity, dividend-paying stocks, and more. So, let us grab our calculators and dive in!

Numeraire Invariance: The Rosetta Stone of Finance

At the core of financial mathematics lies the concept of numeraire invariance. Simply stated, it's the idea that the choice of numeraire doesn't affect final results in an arbitrage-free market.

Proving Proposition 1 involves demonstrating that the risk-neutral probability is indeed invariant under changes of numeraire. Here's a simplified step-by-step process:

1. Recall that the risk-neutral probability, Q, is defined as Q(A) = E^Q[IA] / E^Q[M], where IA is the indicator function for event A and M is the numeraire. 2. Change the numeraire to another asset, N. The new risk-neutral probability, QN, is then QN(A) = E^QN[IA] / E^QN[N]. 3. Using the change of numeraire formula, we have E^Q[N] = E^Q[N * M]. Substituting this into the equation for QN(A), we get QN(A) = E^Q[IA] / E^Q[M], which is exactly the same as Q(A).

Thus, the risk-neutral probability remains invariant under changes of numeraire. This might seem like a mere technicality, but it's crucial for deriving Black-Scholes pricing formulas and other fundamental results in financial mathematics.

Put-Call Parity: The Symmetry of Options

Now let's turn our attention to put-call parity, the beautiful symmetry that exists between options on the same underlying asset. Consider a single-period market with riskless bond BOND, risky stock STOCK, call option CALL with strike K, and put option PUT also with strike K.

Part (a): There exists a replicating portfolio in assets STOCK, CALL, and BOND for the asset PUT. Here's how you can construct it:

1. Buy one share of STOCK. 2. Short one CALL option. 3. Buy K shares of BOND.

The value of this portfolio at expiration is K if the stock price is less than or equal to K (i.e., the put option is in-the-money), and the stock price minus K otherwise, which matches the payoff of the PUT option.

Part (b): The t = 0 market price of PUT can be expressed as:

PUT = STOCK - CALL + Ke^-rT

where r is the riskless rate and T is the time to expiration. This formula follows directly from the replicating portfolio constructed in part (a).

Part (c): When r > 0, we must modify our answers slightly. The formula for PUT becomes:

PUT = STOCK - CALL + Ke^-rT

The main difference here is that now the riskless asset has a positive return, making the put option slightly more valuable due to the delayed payment of its strike price.

Stocks with Dividends: Capitalizing on Cash Flows

Dividends add another layer of complexity to stock pricing. When dealing with dividend-paying stocks, it's essential to account for these cash inflows when constructing replicating portfolios and deriving pricing formulas. This involves adjusting the stock price for the dividend payment at expiration:

Adjusted Stock Price = Stock Price - Dividend

Incorporating this adjustment into our previous formulas gives us:

Dividend-Adjusted PUT = Adjusted Stock Price - CALL + Ke^-rT