Binomial Option Valuation Model

Binomial option valuation models. Yikes! The name sounds scary and complicated. But in reality, it's not as bad as you might think. Envision a newly planted tree, just breaking out of the ground. It's a fresh start, and there are endless opportunities. Do you think that tree has a predetermined path?

That's it!

Actually, no.

A tree doesn't have an endless path once it breaks the soils. It's impacted by numerous variables. And only after you're standing in front of a fully grown tree can you easily see the numerous paths taken to get to one single point in the ground. Essentially, you can trace the tip of the tallest branch all the way back to the ground. Similarly, the binomial-option valuation model comes in to analyze all different paths a security can take.

The binomial-option valuation model is a valuable tool for pricing a security with a large range of possibilities. The core of this approach should be very familiar. It centers on the law of one price and arbitrage. Recall that two securities with the same future cash flows should have the same current price, regardless of what happens. And just like forward and swap contracts, option contracts are valued from an arbitrageur's perspective. Keep in mind the two key rules: don't use your own money, and don't take any price risk.

In valuing a call or put option contract of any sort, the contract expiration gives you the starting point for calculating the value because a future price is specified at the initiation of the contract. Why's that?

Actually, that's not it. While cash flows will be zero, the price specified at initiation could differ from the price at expiration.

That's not it. The intrinsic value could be positive, negative, or zero. It just depends on the prices.

That's right! At the end of the contract, the exercise price will help determine the intrinsic value, so it's a given amount, and that helps price the options by working back from expiration to initiation. The exercise values for a call _c__{_T_} or put _p__{_t_} are $$\displaystyle c_T = MAX(0,S_T - X)$$ $$\displaystyle p_T = MAX(0,X - S_T)$$. _X_ is the strike price, and _S__{_T_} is the underlying price at expiration. If during the contract period an option price deviates from these formulas, then an arbitrage opportunity exists. But at expiration, there's no uncertainty: the option can be valued. And that value is made up of a price component and a time component, with the time value always being nonnegative.

To sum it up: [[summary]]

No

Yes

The cash flows will be zero

The intrinsic value will be zero

The exercise price will help determine the intrinsic value

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