Option Basics -- X

Upper, lower bounds

Anup Menon

AMONG the important properties of options prices are that at any particular point in time, the value of a call option or a put option cannot exceed a particular price.

If options prices fluctuate above the upper bound or below the lower bound then investors have an opportunity to market profits through arbitrage. The concept is better understood with a example.

First let us consider the upper bound. The upper bound is the maximum possible price for a call option or a put option at any point in time. Call options and put options have different upper bounds. In the case of a call option, in any circumstances, an options cannot be worth more than the stock price. Therefore, the value of a call option should be lower or equal to the stock price. If there is a violation of this rule, then arbitrageurs will enter and make a riskless profit by buying the stock and selling the call option.

In the case of a put option, the upper bound is the strike price at which the contract has been entered. In any circumstances, the value of a put will be lower than or at most equal to the strike price of the option. If this condition is violated then an investor can make use of the arbitrage opportunity by writing the option and investing the proceeds at the risk-free rate of interest.

As is the case with upper bounds, call and put options have different lower bounds. The lower bound for a call option is the difference between the stock price and the discounted value of the strike price at the risk-free rate of interest. If the call price violates this rule, investors can arbitrage by buying the call option and selling the stock short. The lower bound for the put is the reverse of that of the call. It would be the difference between the discounted value of the strike price at the risk free rate against the stock price. If this rule is violated the investors can profit by borrowing money and buying a put and the stock. This would lead to a positive cash flow on maturity.