# Boundary conditions

## What are boundary conditions?

Boundary conditions are the maximum and minimum values ​​that indicate where the option price must lie. Boundary conditions are used to estimate the pricing of options, but the actual price of an option may be higher or lower than the price set as the boundary condition.

For all options contracts, the minimum boundary value is always zero because options cannot be negatively priced. At the same time, the maximum boundary value will vary depending on whether the option is call or bear, and whether it is an American or European-style option.

### key takeaways

• Before the introduction of binary trees and the Black-Scholes pricing model, boundary conditions were used to set minimum and maximum possible values ​​for calls and puts.
• The boundary conditions vary depending on whether the option is American or European, as American options can be exercised before expiration.
• The absolute minimum for an option is zero because options cannot be sold with a negative number.
• The maximum value in the boundary conditions is set to the current value of the underlying asset.

## Understanding Boundary Conditions

Before the introduction of the binary tree pricing model and the Black-Scholes model, investors and traders relied heavily on boundary conditions to set the minimum and maximum possible values ​​for the calls and puts they were pricing. These boundary conditions vary depending on whether the option is American or European, because American options can be exercised early.

This ability to exercise at any time before expiration affects how prices are calculated, and with this feature, American-style options trade at a premium relative to equivalent European options.

## Min and Max Boundary Conditions

The absolute minimum for an option is zero because options cannot be sold with a negative number. The maximum value in the boundary conditions is set to the current value of the underlying asset. If the price of the underlying asset is higher than the price indicated in the call option, the investor will not exercise the option because exercising the option would result in the investor paying more than the market price. This is the case for both European calls and US calls.

The maximum value of a put option is reached when the underlying asset has no value, such as in the case of a company bankruptcy where the underlying security is a stock. For European put options, the calculated maximum value is the present value of the strike price. This is because European-style options cannot be exercised at any time, but can only be exercised at the specified price upon expiration. American-style options must be at least as valuable as European-style options.

Although technically the asset’s maximum value can be set to infinityAssets can increase in value without a capThis is considered impractical. The value of the underlying asset is likely to fall within reasonable bounds that can be modeled with standard deviation or other stochastic methods.

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