Upper Bound for European Call Option

[juhsu]

If X = strike price, the upper bound for a American put option is P <= X, which makes sense. For a European put option, you must add a time value component to the upper bound [p <= X*exp(-rt)] since you have to wait until the expiration date to receive proceeds from the sale of the underlying. This also makes sense.

The upper bound for both European and American calls is c, C <= Current Stock price. My question is why doesn't the upper bound for a European call option have a time value component like the European put if you have to wait until expiration to purchase the underlying?

 

[David Harper CFA FRM]

Hi @juhsu That's a good question, actually (I've had too much time to think about this sort of thing! ). It's because the strike price doesn't grow over time, but the minimum expectation of the stock price, S(0), is that it must at least grow by the risk-free rate to reach, at future time (t), S(0)*exp(rt). So, although the future upper bound on the put is (K), which discounts to K*exp(-rT), the future minimum value of the stock price is S(0)*exp(rt) which discounts at the risk-free rate to [S(0)*exp(rt)]*exp(-rT) = S(0). This is related to how I prefer to think about the minimum value of a European call option: if the stock must grow a the risk free rate, then the minimum future gain = S(0)*exp(rT) - K, which discounts to [S(0)*exp(rT) - K]*exp(-rT) = S(0)*exp(rT)*exp(-rT) - K*exp(-rT) = S(0) - K*exp(-rT). I hope that's helpful!

 

[brian.field]

I think that is a very interesting question and suggests that you are incredibly insightful @juhsu.

The upper bound for European calls is NOT the same as the upper bound for American calls. Indeed your intuition is precisely correct.

For American call options, the upper bound is the reference asset price, or the underlying's price.

For European calls, the upper bound is the prepaid forward price of the underlying asset.

So, the upper bound for European calls does contemplate or incorporate a risk-free discounting via the "prepaid" forward price.

 

[QuantMan2318]

Hi there @brian.field

I have always thought about these things from two angles. I think what David is trying to convey and what you are trying to say is basically approaching the
issue from opposite sides.

When we think of Call options in general, we can see that the value cannot exceed the value of the underlying Asset, therefore, in the case of American calls, they have to be the Spot price of the Asset at that point in time, which is the Expected Future Spot as David has pointed out. I think this value is taken as the value of the Underlying Asset in American Calls.

In the case of European Calls, as we have to exercise them only at maturity, the Expected Future Spot prices that we discount at the Rf rate becomes the Spot price that is used by Hull, Effectively, we can see that 'S' in European Calls is not the same as the 'S' used by Hull in American Calls.

I also think that the Prepaid Forward that you have elegantly pointed out is taken as equal to the current Spot price in the case of European Calls. Thus,
F*exp(-rf*t) = S
F = S*exp(rf*t), thus we are implicitly assuming that the Expected Future Spot prices are equal to the Forward prices.

This is just my intuition, what do you feel?

Thanks

 

[juhsu]

@David Harper CFA FRM Very helpful. Thanks!

@brian.field @QuantMan2318 Thanks for your responses. I think I understand what you guys are getting at.