August 31st, 2011, 11:13 am
QuoteOriginally posted by: listbut what is the price ? is it something between 0 and 50 ?-------------------------------------------------------------------------------example 1if the real probability of 50 = 1 and prob of 200 = 0. Then with prob 1 call option equal to 0example 2if the real probability of 50 = 0 and prob of 200 = 1. Then with prob 1 call option equal to 50example 3if the real probability of 50 is close to 1 say 0.999 and prob of 200 = 0.001 . Then with prob 0. 999 call option will be equal to 0. This 3 example shows that to say something about the price we need to know the stock distribution. If you do not know it we could not say anything about call option price on theoretical level.Examples 1 & 2 violate the no arbitrage assumption since the stock is deterministic. In 1 I short the stock and go long the bond to get free money and reverse this in example 2, therefore you cannot use the BS framework as the price is not uniquely determined. There are an infinite number of combinations that will give the option payoff. e.g. in example 2 I can buy $5 worth of stock and $90 of bond and have a payoff of $100, this implied a price of $95 or buy $100 dollars of stock and sell $100 of the bond and get a payoff of $100, this implies a price of $0 (the arbitrage situation), you can't say the call has a price of $50 since this is not the only portfolio giving the desired payoff.In 3 this is not true anymore, there is now a risk (albeit small) arising from the stock, I can now give you a unique price that if you think it should be more I can sell you the option at that price and hedge buy buying the stock and get a guaranteed profit, if you think it's lower I can buy the option from you and hedge the option by selling the stock to get a guaranteed profit. Even if the small probability event occurs I do not lose money.The problem seems to be that you think the random scenario price should converge to the deterministic scenario one (which actually doesn't exist) as the probability of an event approaches 1, you have not justification to think that since that situation cause a violation of the BS assumptions.
Last edited by
ACD on August 30th, 2011, 10:00 pm, edited 1 time in total.