Upper and Lower Bounds of Options

We will derive upper and lower bounds of call and put options for both European and American options. These bounds depend on only one assumption that risk-free rate is above zero, (r ≥ 0) . These bounds are not affected by other factors like specific stock price, dividends expected, strike price, time to maturity or volatility of the stock. If the price of these options is below the lower bound or above the upper bound, then there exist an arbitrage opportunities and it would be possible to make riskless profit on the market.

Both American and European call option gives a right to the option holder to purchase one share stock for a certain price. The option itself cannot be worth more than the stock.

In case this condition does not hold, we can make profit by selling an option and buying a stock.

Both American and European put option gives the holder the right to sell one share of stock for the strike price. The put option cannot be worth more than the strike price.

However, since European option can be exercised at maturity only, the European put option cannot be worth more than the present value of the strike price. In this case, the actual price of the European put option is even lower than of American option and equals to

In case an European put option is worth more than the present value of the strike price, an arbitrageur can get riskless profit by simply selling the option for this price and investing proceeds at the risk-free interest rate.

Firstly, all options never have a negative value.

c ≥ 0 ,   C ≥ 0 ,   p ≥ 0 ,   P ≥ 0

American call and put options are valuable as least as corresponding European options.

The American option option is worth at least its exercise price now. However, by exercising at maturity, as with European call options, we get even more in present terms. Thus, the lower bound of American and European call options is equal to the difference between current stock price and the present value of the strike price. For both call types of options the lower bound is

If the price of the call option is below this price, we can construct a portfolio and gain an arbitrage profit without any risk. This example has been considered previously.

However, this lower bound does not apply for the European call option if there is expected dividend payment during the life of an option. The dividend payment of the stock decrease the value of the European call option. It also pushes down the potential value of the American call option too. However, in case of American call option, the dividend payment does not make an impact on the value directly, since an option can be exercised prior the dividend payment date.

The lower bound for European put options equals to

In case this condition is violated, there exist an arbitrage opportunity reviewed previously. The value of the put option is equal to at least its exercise price now. In case of American put option, the lower bound is higher. Since an American put option can be exercised now and should be worth at least as the difference between current stock and strike prices. Hence,

There exist also special cases for American options. Specifically, American options with more time to maturity are at least as valuable as the ones with shorter period. Please note that we consider only options with the same starting date (t = 0) . Hence, if (T > t) . then

This assumption does not hold for European options. Firstly, if we consider two identical European options with the same starting and maturity date, it is obvious – both options can be exercised at maturity only and their value should be identical. At the same time, if we consider two European options with the same starting date, the one with shorter period might be worth even more because of number of factors: expected dividend payments, the price of the stock at specific date, risk-free interest rate, stock volatility changes and so on.