The Hidden Cost of Volatility Drag

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

Numeraire Invariance: A Replicating Portfolio in Arbitrage-Free Markets

In a single-period market with a riskless asset BOND, we can show that there exists a replicating portfolio in STOCK, CALL, and BOND for the asset PUT. This is because the expected value of PUT is given by the discounted sum of its payoffs over all possible scenarios, which is equivalent to the sum of the discounted expected values of STOCK and CALL.

To illustrate this, consider a single-period market with the following payoff structure:

Stock: S(t=1) = ST + δST Call option: C(0,T;K,S(t=1)) = 0 if K > S(t=1), otherwise C(0,T;K,S(t=1)) Put option: P(0,T;K,S(t=1)) = 0 if K < S(t=1), otherwise P(0,T;K,S(t=1)) = -K

Assuming the rate of return on BOND is r = 0, we can compute the expected value of PUT as follows:

E[PUT] = ∑[P(t) \ P(t)] = (1 + δ)(ST + δST) / ST - K(1 + δ)T

This shows that a replicating portfolio in STOCK and CALL can be used to replicate the payoff of PUT.

Put-Call Parity: A Formula for the Market Price of PUT

As we have shown, there exists a replicating portfolio in STOCK, CALL, and BOND for the asset PUT. To deduce a formula for the market price of PUT in terms of the market prices of STOCK and CALL, let's consider the following:

Let F0 be the forward price of STOCK at time T.

Then, we can write:

F0 = S(0) + ∫[0,T] δS(t)f(t)dt - K

where f(t) is a non-negative function representing the dividend paid by STOCK at each time t. Rearranging this equation, we get:

F0 = (1 + δ)ST / ω

This shows that the forward price F0 of asset STOCK can be expressed in terms of its initial value S0 and market prices of CALL.

Portfolio Involvement: Putting it into Practice

To apply our analysis to practical portfolio management, consider a T-period market with BOND whose rate of return is r = 0. Let STOCK be a risky asset that pays dividends at each time t = 1, 2, ..., T, where δSt shares of BOND are received.

We can construct an equilibrium distribution π for the market as follows:

π(ω) = (1 + δT)X / (∑[ω \ ST]) - ω

where X is a vector representing all possible market scenarios.

The forward price F0 of asset STOCK can then be computed as:

F0 = ∑[ω \ S(0)] / (∑[ω \ X])

This shows that the forward price F0 of asset STOCK depends on its initial value S0 and market prices of CALL.

Stocks with Dividends: A 10-Year Backtest

Consider a T-period market with BOND whose rate of return is r = 0. Let STOCK be a risky asset that pays dividends at each time t = 1, 2, ..., T, where δSt shares of BOND are received.

We can construct an equilibrium distribution π for the market as follows:

π(ω) = (1 + δT)X / (∑[ω \ ST]) - ω

where X is a vector representing all possible market scenarios.

To perform a 10-year backtest, we can use the following formula to compute the forward price F0 of asset STOCK:

F0 = ∑[ω \ S(0)] / (∑[ω \ X])

This shows that the forward price F0 of asset STOCK depends on its initial value S0 and market prices of CALL.

The Arbitrage Price of a Floor Contract

Consider a homogeneous, T-period binary market with BOND whose rate of return is r = 0. Let STOCK be a risky asset that follows equations (26)-(27) from the notes. Assume that there exists a floor contract FLOOR that pays one share of BOND at every time t = 1, 2, ..., T when the share price of STOCK is below its initial value S0.

The arbitrage price of one FLOOR can be computed as follows:

Arbitrage Price = ∑[t \ (S(0) - ω)] / (∑[ω \ X])

where ω represents all possible market scenarios, and X is a vector representing the dividend payments from STOCK over time.

This shows that the arbitrage price of one FLOOR depends on its initial value S0 and market prices of CALL.

Actionable Conclusion

In conclusion, our analysis of MathFinance 345/Stat390 has shown that there are several key concepts to consider when dealing with arbitrage-free markets. Specifically:

Numeraire invariance is a fundamental property of financial markets. Put-call parity provides a formula for the market price of PUT. Portfolio involvement requires careful consideration of dividend payments from risky assets. Stocks with dividends can be used to construct equilibrium distributions and compute forward prices.

By applying these concepts, investors can better understand the risks and opportunities in arbitrage-free markets.