Quote from yip1997:
I understand the dynamic replication, and so risk-neural valuation process, and it should use risk-free interest if deriving from this perspective.
However if you assume a pure brownian process, you should use the mean return if solved from this perspective. In reality, we use the standard deviation of stock return for sigma. Why don't we use the statistical mean of stock return for u.
Why can't we have free lunch statistically?
Yip, I've thought about this a bit more, and I'll try to explain better:
I've stolen this example from John Hull's textbook.
Assume the value of a stock is $20 and pays no dividend. At the end of three months, the stock price will be either $22 or $18. We need to value a European call option with a $21 strike and 3 months to expiry. This option can have only two possible values in three months: if the stock price is $22, the option is worth $1, if the stock price is $18, the option is worth nothing .
To price this call option, we can undertake static/dynamic replication whereby we set up a hypothetical trade consisting of the option and the stock, in a way such that there is no uncertainty about the value of the portfolio at the end of three months. Since the portfolio has no risk, the return earned by this portfolio must be the risk-free rate.
Imagine the trade initially is long X shares of stock, and short 1 call option. We can compute X so that the position is risk free at inception. If the stock goes up to $22 or down to $18, then the value of the portfolio is:
stock goes up = $22X − 1
stock goes down = $18X − 0
So, if we choose X = .25, then the value of the portfolio is
if stock goes up = $22X − 1 = $4.50
if stock goes down = $18X − 0 = $4.50
Whether the stock moves up or down, the value of the portfolio is $4.50. The correct value for X (eg.g the delta) is 0.25 to make this option a risk free equivalent to payouts from owning the stock
A risk-free portfolio must earn the risk free rate ("the law of one price" i.e 2 securities with the same pay-out must have the same price).
If the current risk-free rate is 12%, then the value of the position today must be the present value of $4.50 (the future value), or
4.50 Ã e^(−.12Ã.25) = 4.367
This, I hope, demonstrates why the risk free rate is used to calculate prices. The variance/std dev of stock movements is used when we aren't certain about the final payoff (e.g. when we aren't certain that at 3 months the stock price will be $18 or $21).