Option valuation: Upper bounds and lower bounds Part II
by J Victor on August 19th, 2012Having learned the upper and lower limits of European calls, next we’ll look at the upper and lower bounds of European put options.
Upper bound of European puts.
Let’s take an example – the stock of HFDC is trading at 800 right now. 1year put options on this stock are available at a strike price of 900. If we calculate the present value of 900 at 8% risk free interest rate, we’ll get 833.50 as the answer.
Logically, the upper bound price of a European put cannot exceed that 833.50 which is the present value of the strike. If price of the put is above 833.50, say 860, then –
You can immediately sell a put and get 860 and Invest 833.50 at 8% to get back 900 at the end of one year. The difference of Rs 26.50 is your profit ion the spot. (860833.50).
Now, if the dividend on stock is known, it doesn’t make any difference. The only rule to be remembered in case of upper bound European out prices is that it cannot exceed the present value of the strike price.
Not that, in the worst case the maximum loss that a put writer will suffer is the strike price. This loss is mitigated by investing the present value of strike at 8% risk free investments.
So that brings us to the third principle – An European put cannot have a greater value than or equal to the present value of the strike price. The dividend factor is irrelevant here.
Lower bound of European puts.
A European put cannot have a price that’s lower than the difference between the present value of the strike price and the stock price.
For example assume that the stock price is Rs 60 and the strike price is Rs 65. Let’s also assume that the present value of strike price is 63. In this case, the value of a European put cannot be lower than Rs 3. That is, the difference between the present value of the strike price and the stock price.
If it is less than Rs 3, an investor can buy the put , borrow the present value of strike price and use it to buy the stock at current market price and profit from the deal.
Now, we summarize the basic principles for upper and lower bounds of European options:
 The upper bound value of an European call can never rise beyond the value of the underlying stock. When the dividend is known with certainty, the call values cannot rise beyond the spot value of the stock less present value of the dividend.
 The lower bound value of an European call can never fall below the difference between stock value and the present value of strike price. When the dividend is known with certainty, the call values cannot fall below the spot value of the stock minus present value of the dividend minus present value of the strike value.
 An European put cannot have an upper bound value greater than or equal to the present value of the strike price. The dividend factor is irrelevant here.
 A European put cannot have a price that’s lower than the difference between the present value of the strike price and the stock price.
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