What are Boundary Conditions?
Let me explain boundary conditions to you directly: they are the maximum and minimum values that show where the price of an option has to fall. You use them to estimate what an option might cost, but remember, the real price could be higher or lower than these boundaries.
For every options contract, the minimum boundary is always zero because you can't price an option at a negative value. The maximum boundaries vary based on whether it's a call or put option, and if it's American or European.
Key Takeaways
- Boundary conditions were used to set minimum and maximum possible values for call and put options prior to the introduction of binomial tree and Black-Scholes pricing models.
- Boundary conditions change according to whether the option is American or European, since American options can be exercised prior to expiry.
- The absolute minimum value for an option is zero, since an option cannot be sold for a negative amount of money.
- The maximum value in a boundary condition is set to the current value of the underlying asset.
Understanding Boundary Conditions
Before binomial tree pricing models and the Black-Scholes model came along, investors and traders like you relied on boundary conditions to establish the minimum and maximum possible values for call and put options they were pricing. These conditions adjust depending on if the option is American or European, because American options allow early exercise.
This early exercise feature impacts how the price is calculated, and that's why American options trade at a premium compared to similar European options.
Minimum and Maximum Boundary Conditions
The absolute minimum for any option is zero—you can't sell it for negative money. The maximum in these conditions is the current value of the underlying asset. If the underlying asset's price is higher than the call option's strike, you wouldn't exercise it, as it'd mean paying more than market price. This holds for both European and American calls.
For put options, the maximum value hits when the underlying asset is worthless, like in a company bankruptcy if it's a stock. For a European put, it's the present value of the exercise price, since you can only exercise at expiration. An American option's value is at least as much as a European one's.
While an asset's value could theoretically go to infinity with no upper limit, that's not practical. In reality, the underlying asset's value stays within reasonable boundaries that you can model using standard deviations or other stochastic methods.
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