"Double or Nothing: Martingales' Unexpected Twist"

Double or Nothing? The Unexpected Twist in Martingales

Ever played a game where you double your bet each time you lose, hoping to win it all back? It's an exciting strategy, but as we'll see with the Double or Nothing martingale, it might not pay off as expected.

In MathFinance 345/Stat390, we're exploring this intriguing paradox. The sequence (Zn)n≥0 is defined such that Zn = 2ξnZn−1 for all n ≥1, where ξi are independent Bernoulli random variables with success parameter 1/2. This sequence forms a martingale relative to the usual filtration.

The Optional Stopping Formula: Not Always True

The Optional Stopping Theorem suggests that EZ0 = EZτ when τ is a stopping time and Zn is a martingale. However, our Double or Nothing martingale challenges this assumption. In fact, we can prove that EZ0 ≠ EZτ.

Here's why: The sequence Zn tends towards infinity as n increases. Therefore, the probability of reaching zero (and thus stopping) decreases over time. Consequently, EZτ becomes exceedingly small, unlike EZ0 which equals 1 due to the initial condition Z0 = 1.

Arbitrage Opportunities in Infinite Period Markets

Now let's explore the economic implications with an infinite period market. Consider a homogeneous binary market with a risky asset (Stock) and a riskless asset (Bond). The share price St evolves as in a T-period homogeneous binary market, with constants d < 1 < u such that u −1 + d = 1.

In Problem 2, we're tasked with showing this market permits arbitrage. By constructing a self-financing portfolio θ whose value Vθt evolves like the Double or Nothing martingale Zn, we can create an arbitrage opportunity. This challenges the economic assumption that efficient markets shouldn't allow arbitrages in infinite periods.

Early Exercise: The American Put Option Strategy

Next up, let's discuss early exercise with American put options. In certain markets and circumstances, exercising a put option early might yield a higher expected payoff. For instance, in a homogeneous 2-period binary market, you could get a better return by exercising the put at t = 1 when the partial scenario is −.

This strategy hinges on the riskless asset's rate of return r being greater than zero. If r = 0, there's no advantage to early exercise. So, keep an eye on interest rates and market conditions for potential arbitrage opportunities.

First-Passage Time Distribution: Navigating Barrier Options

Lastly, we'll delve into first-passage time distribution with Bernoulli random variables ξi and Sn = Pn i=1 ξi. The purpose is to find the distribution of τ, the first time that the "random walk" reaches level 1.

By showing that Yn = zSn/ϕ(z)n is a martingale relative to the natural filtration, we can calculate E(1/ϕ(z)τ) = 1/z when z ≥1. For any 0 < ζ < 1, Eζτ = (you figure out what goes here). Using calculus and the fact that Eζτ = P∞ n=1 ζnP{τ = n}, we can find a formula for P{τ = n}.