I'm sure this has been discussed before and so I apologise if anyone is getting bored of this question.i was asked to explain why you dont use the expected growth rate of the stock to price an option.I first stated that if you did you would make an arb profit from selling the call buying the put and buying a forward. He replied he wanted a concrete explanation rather than a proof by contradiction.I then replied that shorting a call and gonig long the replicating portfolio removes the risk and so is riskless hence should earn a risk free rate...Hence you can find the price of the option without using the expected growth rate of the stockHe wasnt satisfied...he said that tells him nothing..I then started digging myself into a grave....Anyone know what I should have said???

The expected growth rate is never known for sure. it could be different for you, for me and for Warren Buffet. If you are a speculator then you may have a view for the expected growth rate.Ergo your argument of:Self financing replicating portfolio + no-arbitrage => existence of an unique risk-neutral probability measure which "eliminates" the growth rate. The expected growth rate is the risk free rate under this risk-neutral p-measure.I dont know if he wanted an even more intuitive proof

QuoteAnyone know what I should have said???Tell him it can be seen in simple high school mathematics -- the binomialmodel with one time step. But, to understand it, you must *work through the arithmetic yourself*, not listen to a lecture about it.

Antoher way of proving the validity of the risk free rate over a growth rate is by the binominal option pricing model (portfolio replication to match the value of a call option). edit: Alan summed it nicely

player - i totally understand your frustration. an extremely intelligent person, who is not a quant, and who i don't want to embarass, told me that the standard risk-neutral forward is plain old wrong.he had examples of stocks with huge earnings potential which were undervalued, and if you calculate the 20 year forward it was ridiculous, etc, etc.i told him that i agreed that a 20 year option priced using the risk neutral forward was probably a huge buy. but it was not "wrong".we went back and forth and he got increasingly heated. i was polite and respectful but explained that the forward was not any kind of forecast but was a mathematical artifice used for arbitrage pricing. he started to get very annoyed and upset with me.finally i said "ok make me a market in the 20 year forward. if you are above or below the arbitrage price I can make riskless money off of you!" i then outlined the arbitrage strategy. he wasn't immediately convinced, but came around to seeing the sense in it, and he is now convinced of the sense of the risk-neutral forward.

I would say you can use the expected rate of return on the stock to price the option, but then you have to discount the expected future value at the expected rate of return on the option. This is self-evident and basic finance. However, since you don't know either rate, it's not very useful.If you accept that the option can be hedged risklessly, then it must have the same price for everyone, including a hypothetical risk neutral person who discounts everything at the risk-free rate. It's a neat trick that we can price it for him, and know that it must be the same price for us.