While these maximum limits have to be explained separately for European and American options, we will first look into the upper and lower bounds of European call options.

Upper bounds of European call values:

Let us assume that the value of a call option is 65, and the underlying stock is trading at 60. In this scenario, one can write the call and sell the stock at 60, and make a profit of 5 per share. The call value on expiry cannot rise above the value of the underlying stock.

Now, assuming that the company announced a dividend of 5 per share. When this dividend is released, it decreases the value of shares that much. So, on the expiry date, it will have a value of 55 (60 – 5) in the spot market. Logically, the call value cannot go beyond 55 a share.

Going back to the first option value principle – the upper bound of a European call cannot go up above

the value of the underlying stock. If the dividend is declared, then the call option cannot go beyond the spot price of the stock minus the dividend amount.

Lower bounds of European call values:

Now let us see what the lowest value for a European call would be. It will be zero. It cannot fall any more than that agree? For a call value to be at zero, the value of the stock must also fall to zero.

Let’s assume that the value of the stock at Rs.102. A European one-year option call is available at a strike price of 108. If the risk-free rate is 8%, the present value of 108 after discounting 8% would be Rs.100.

In a worst-case scenario, the call cannot fall below Rs.2 (102 – 100). If it were to fall to say, Rs. 1, then one can buy the call option at a strike price of Rs.108 by paying Rs.1.By selling the stock at Rs.102, one can make a profit of Rs.101.

Out of Rs.101, if one were to invest 100 in risk-free bonds, he would get Rs.108 at the end of the year.

By using this Rs.108 to exercise the call option, he can get back his shares and make a risk-free profit of Rs.1

If all this was a bit confusing, you are to get a better understanding by going through the calculations.

Now let us take a look at the effect of dividend on options. Here is an example. One year call at a strike price of Rs.20 for a stock trading at Rs.50.

The dividend expected after a year is Rs.5 a share.

The value of the call option cannot go below the share value minus the expected dividend + the present value of strike price.

The present discounted value of Rs.5 at 8% interest would be Rs.4.63

The present discounted value of Rs.20 at 8% interest would be Rs. 18.52.

So, the lower bound value of a call cannot go below 50 minus (4.63 +18.52). Rs.26.85

The lower bound value of a European call option cannot go below the difference between the stock price and the present value of the option strike price.

If we know the dividend for sure, the value of the call cannot fall below the market value of the stock minus the present dividend and strike price values.