Question

Please prove American inequality boundary in the options market.

Answer 1

Boundary conditions

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

For all options contracts, the minimum boundary value is always zero, since options cannot be priced at negative money. Meanwhile, maximum boundary values will differ depending on the whether the option is a call or put, and if it is an American or European style option.

Understanding Boundary Conditions

Before the introduction of binomial tree pricing models and the Black-Scholes model, investors and traders relied heavily on boundary conditions to set the minimum and maximum possible values for the call and put options that they were pricing. These boundary conditions change according to whether the option is American or European, since American options can be exercised early. This ability to exercise at any point prior to the expiration date affects the way the price is calculated, and American options will trade at a premium relative to equivalent European options by virtue of this feature.

Minimum and Maximum Boundary Conditions

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. If the price of the underlying asset is greater than the price indicated in the call option then the investor would not exercise the option, since exercising the option would result in the investor paying more than the market price. This is the case for both a European call and an American call.

The maximum value of a put option is reached when the underlying asset has no worth, such as in the case of a company's bankruptcy when the underlying security is a stock. For a European put option, the maximum value computed as is the present value of the exercise price. This is because European options cannot be exercised at any point, and instead can only be exercised at expiration at a specified price. The value of an American option must be at least as great as a European option

While technically the maximum value of an asset could be set at infinity – an asset could increase in value with no ceiling – this is considered impractical. The value of the underlying asset is likely to fall within a reasonable boundary that can be modeled with standard deviations or other stochastic methods.