Option valuation: Upper and lower bounds Part III
by J Victor on September 9th, 2012We have already discussed the upper limits and lower limits of European options in our earlier posts. Now, here we discuss the remaining part – upper and lower limits of American options.
Upper bounds of American calls:
We know that an American call option can be exercised at any time during the contract period. The principle of upper bounds of american calls are the same as we saw in upper bounds of European call values. The difference in excersisability will not make any impact on the upper bound values .So, the upper bound value of an american 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.It’s straight forward and needs no further explanation.
Lower bounds of American calls:
What would be the lowest value for an american call? An american call without known dividends cannot have a value that’s lower than the difference between stock price and the excercise price. Since the calls are executable at any point of time, the present value of excercise price need not be calculated for caculating the lower bound.So, if the underlying stock price is trading at 100 and if the srike price of it’s american call is at 98 , the call value cannot be less than Rs 2. ie, stock pricestrike price.
Suppose now, that the call price is less than Rs 2 , say Re.1. A trader can immediately buy a call for Rs 1 and sell the stock for Rs 100 and since it is an american call, excercise the call immediately ( that is, buy the stock at excercise oprice of Rs 98 ) and make a profit of Rs 1 per share.
Connecting the points discuused above,it’s logical that
 If the stock price is ’0′ , then the value of american call must be ’0′.
 The lower bound of an american call is the difference between stock price and the excercise price. The minimum value would be ’0′.
 An american call cannot be less than an european call. American call = European call for a non dividend paying stock.
 The early exercise decisions of american calls are based the principle of time value of money.
Upper bounds of American puts:
The american put cannot have a value that’s greater than the strike price.So, if the underlying stock is at Rs 70 with strike price at Rs 75, the value of the put cannot be greater than 75. That’s simple to understand. The existence or non existane of dividends in the underlying stock desn’t make any difference in this case. The upper bound will always be the strike price in the case of american puts.
Lower bounds of American puts:
The lower bound of an american put would be the difference between strike price and the stock price. It cannot be less than that. For example, if the underlying stock is trading at Rs 70 and the strike price id at Rs 75, the value of put canot be less than Rs 5. If the dividend is known with certainity , the value of put cannot be lower than the strike price – ( stock price + the present value of dividend), provided the stock is trading cumdividend.
Connecting the points discussed above, it’s clear that 
 If the stock price is ’0′, then the value of american put must be it’s excercise price. It cannot go higher than that.
 The lower bound of an american put is the difference between the exercise price and the stock price at start. The minimum value would be ’0′.
 The maximum value of man american put is it’s strike price.
 The value of american put would be equal to the corresponding european put.
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