That said, let's dive into the world of infinite period markets and explore why most investors miss a crucial pattern.

Why Most Investors Miss This Pattern

In finite period markets, it's easy to see why investors overlook an important principle: the risk premium. When trading in finite periods, there's less time to absorb market volatility, making it harder for investors to estimate true risk premia. However, in infinite period markets, this assumption breaks down.

The Risk Premium in Infinite Period Markets

Consider a homogeneous binary market with a risky asset Stock and a riskless asset Bond. In finite period markets, the share price St of Stock evolves according to a T-period homogeneous binomial model, while St+1(ω1ω2 ... ωt+) = uSt(ω1ω2 ... ωt). This means that the market's expected return is still discounted by r/2.

The Risk Premium in Infinite Period Markets

In contrast, an infinite period binary market allows for a self-f financing portfolio θ. This means that investors can create a perpetual bond with yield r > 0 and value Vθt evolving as the double-or-nothing martingale Zn. By definition, an arbitrage is a portfolio whose value does not change over time.

The Arbitrage Opportunity

That said, what happens if we construct a self-f financing portfolio θ that permits an arbitrage? We can show that in certain markets and circumstances, it may be better to exercise the put option early rather than waiting until t = 2. This is particularly true when u < d, with u −1 u −d = 1 −d.

The American Put Option

In a homogeneous 2-period binary market, we can demonstrate that for some values of u, d, r, and K, exercising the put option at t = 1 gives a higher expected payoff than waiting until t = 2. This is because early exercise allows investors to take advantage of the time value of money, which is more pronounced in infinite period markets.

First-Passage Time Distribution

To further illustrate this point, let's consider an infinite sequence of independent Bernoulli- random variables ξ1, ξ2, . . . and define τ = min{n : Sn = 1} to be the first time that the "random walk" reaches the level 1. The distribution of τ can be found using a martingale approach.

First-Passage Time Distribution

That said, what's interesting is that if z > 0, the sequence Yn = zSn/ϕ(z)n is a martingale relative to the natural filtration. Moreover, we have E(1/ϕ(z)τ) = 1/z, which suggests that early exercise in an infinite period market can be advantageous.

Conclusion

In conclusion, investing in infinite period markets requires a different mindset than finite period markets. By understanding the risk premium and arbitrage opportunities in these markets, investors can make more informed decisions about when to buy or sell. Remember, the midterm exam will be held on October 24 during the regular class period in Kent 107.