Option valuation: Upper bounds and lower bounds – Part 1
by J Victor on August 12th, 2012What are upper and lower bounds of options?
One important principle while valuing options is that at any time, the value of a call or a put cannot exceed certain limits – on the higher side as well as on the lower side. In option lingo, the maximum limit up to which an option value can go on the higher side is commonly referred to as ‘upper bounds of an option’ and the maximum limit below which an option value cannot fall is called the ‘lower bounds of an option’.
These maximum limits will have to be discussed for European and American options separately. We first take up the upper and lower bounds of European calls.
Upper bounds of European call values:
Let’s assume that the call value of an option is 55 and the underlying stock is trading at 50 in the spot market. In such a scenario, anybody can write the call and sell the stock on spot, and take home the difference of 5 per share. Hence, it’s clear that the call value at expiry cannot rise beyond the value of the underlying stock.
Now, let’s further assume that the company has announced a dividend of 5 per share. Dividend, when paid, decreases the value of shares to that extent. Hence on expiry, the stock will be valued at 45 (50 – 5) in the spot market and logically, the call value cannot exceed 45 per share.
This brings us to the first principle in option value – 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.
Lower bounds of European call values:
What would be the lowest value for an European call? It should be zero. It cannot fall below that, Right? For the call value to be at zero, the stock value should also fall to zero. If the stock value is above zero (say 2) the minimum value of call should be the present value of Rs 2 (strike price).
Let’s try an example – assume that the stock value is at 102. One year European option call at strike price 108 is available. If the risk free rate is considered to be 8%, the present value of 108 discounted at 8% would be 100. In this case, the value of the call cannot fall below 2 (102 – 100) If it falls to (say, Re 1) then –
You can buy the call at strike 108 – you pay Re 1
You sell the stock at 102 – You get 102. Your net gain = 101
From that 101, you invest 100 in risk free bonds and get 108 at the year end.
Use that 108 to exercise the call and get back your shares.
Get a profit of Rs 1, risk free – immediately.
May be that was slightly confusing. Go through the calculation one more time and you’ll get it.
As a next step, here also, we need to discus the effect of dividends. Let’s take another example where the stock value is at 50 and one year calls at strike price 20 are available. The dividend to be received a year later is estimated at Rs 5 per share. In this case, the value of the call cannot fall below the share value Less (present value of dividend expected + present value of strike value)
The present value of 5 discounted at 8% would be = 4.63
The present value of 20 discounted at 8% would be =18.52
Hence, the lower bound value of a call cannot fall below 50(4.63 +18.52) = 26.85
This brings us to the second principle in option value – 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.
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