Risk-neutral Option Pricing


What is the rationale for why option pricing always uses risk-neutral probabilities/weights and the risk-free rate?

I understand that when a risk-free hedge is created, it should always yield a risk-free rate. The risk-free hedge is very clearly seen in the case of forwards, but I don’t see what the risk-free hedge would be in the case of options.

The text keeps drilling in the concept that option valuation should always be on a risk-neutral basis using the RFR, even in the structural model to debt valuation, but I just can’t see why this is the case.

I’m currently on the CDS reading, and they also mentioned in a footnote in this reading that risk-neutral and not true probabilities are used for valuation of the protection leg. I suppose this means that all derivatives are priced using risk-neutral probabilities and the risk-free rate, but I’m still not entirely clear why this is.

The short answer is that it’s done to prevent arbitrage.

It’s not really that different from, say, pricing a forward contract on a stock by increasing the spot price by the risk-free rate. There’s certainly no reason to believe that that’s what the stock’s price will be in the future, but if you priced it at any other value, there would be an arbitrage opportunity.

An interesting exercise would be to work out the transactions needed to capture the arbitrage profit if some other weights were used. If you can do that, you’ll see very quickly why those weights must be the ones chosen.