The upper bound price of a European put cannot be more than Rs.833.50, this being the present value of the strike price. If the cost of the put were to be above Rs.833.50, let us say Rs.860, then :–

one can immediately sell a put option to get Rs.860 and invest Rs. 833.50 at 8% to receive Rs.900 after the end of one year. The Rs 26.50 difference is the spot profit. (860-833.50).

It does not make any difference, even if the dividend on the stock is known. In the case of upper bound European Put option, the only rule to remember is that it cannot exceed the current value of strike price.

One must note that, even in a worst-case scenario, the maximum loss that a put writer is likely to incur is the strike price. This loss is reduced by investing the strike value in risk-free investments at 8% returns.

Another principle is that a European put option cannot have a higher value or equal value to the present strike price. Here the dividend factor is not relevant.

Lower bound of European puts.

A put option cannot have its price lower than the difference between the present value of the strike price and the price of the stock.

To give an example, let us assume that the stock price is Rs 70, and the strike price is Rs 75. Let us also take the present value of the strike price as Rs.73. Now, the cost of a put option cannot be lower than Rs 3. It is the difference between the present strike price value and the stock price.

If it were less than Rs 3, the investor could buy the put. He can borrow to the extent of the present value of strike price and use the money to buy the stock at the current market price making a profit in the process.

In brief, the principles for upper and lower bounds of options are:

• The upper bound value of a call cannot go above the value of the underlying stock. When the quantum of dividend is known, the price of the call option cannot increase more than the spot price of the stock less the present dividend value.

• The lower bound value of a call can never go below the difference in stock price and the present value of the strike price. When the quantum of dividend is known, the call price cannot fall below the spot price of the stock less the present dividend value, less the current value of the strike value.

• A put option cannot have an upper bound value that is greater than or equal to the present strike price value. Here the dividend factor is not relevant.

• A put option cannot have a price that is lower than the difference between the current value of the strike price and the stock price.

• A put option cannot have a price that is lower than the difference between the present value of the strike price and the stock price.