Why Option Prices Use Risk-Neutral Rather Than Real-World Probabilities

# Should the price of a vanilla option be the Weighted average probabilistic payoff?


# Should the price of a vanilla option be the Weighted average probabilistic payoff?












I was reading Ricardo Rebonato's "Volatility and Correlation: The Perfect Hedger and the Fox".

In chapter two he is discussing pricing a contingent claim via replication and then also by what he calls "Naive expectation", this is where I had a question. I had always thought that if you were given the true probability distribution of the underlying, the price of an vanilla option would be the integral from the strike to either infinity or zero (depending on call or put) discounted by the risk free rate. Essentially, the price of an option = the probabilistic weighted average payoff at expiration (ignore early exercise for the moment). Is this not true?

My reading of Rebonato's book seems to imply that it's not. He says this in chapter 2:

"One could be tempted to speculate, on the basis of the extra piece of information we have now that the 'fair price' of the contingent claim today should be equal to the weighted average of the two possible outcomes C1 and C2, appropriately discounted by the rate implied by the riskless bond...A moment's reflection, however, shows that the expectation calculated on the basis of the probabilities we know to apply in the real world to events w1 and w2 must in general produce a value different from the fair price obtained using the replication strategy."

He further goes on to say that since the probabilities in no way depend on the replication strategy, it would be an enormous coincidence if the prices actually lined up like this.

I was hoping someone could clarify this a bit for me. Is my understanding of probabilities and options pricing correct and I am just misinterpreting what Rebonato is saying here?

## Answer by MrLCh (score 1, accepted)

https://quant.stackexchange.com/a/80297

Maybe take a step back from options and look at futures. Say you know that an asset (currently worth $100\\\$$ returns (in expectation) $6\%$ over the next year (or you know the real world probability distribution) and the risk free rate (which you can borrow and lend at) is $4\%$.

What would the price of the future on that asset be?

By using the expectation of the real world distribution you would get $106 \\\$$. This is what the asset is worth in expectation. But this would open up arbitrage:

If you value the future at $106\\\$$, everybody is happy to sell it to you. The seller would then borrow $100\\\$$ and buy the asset. Then sell the asset to you for $106\\\$$ in one year. Using the proceeds he would proceed to pay off the debt (and interested) ($-104\\\$$) and nets a risk free profit of $2$.

In this case you would have to use the risk neutral probability distribution (with mean equal to the risk free rate), despite your knowledge that this is not the "real world" mean.

The case for options is not that straightforward. For example Black-Scholes needs some more restrictions on the underlying distribution to allow dynamic hedging, but the underlying idea stays the same. There are other scenarios, where you can show that you need to use a risk neutral distribution and not the real world/physical one.

## Answer by dm63 (score 2)

https://quant.stackexchange.com/a/80289

You have the correct interpretation. The naive expectation does not produce the correct price. An enormous amount of literature in this subject is available on this site and elsewhere. Look for ‘risk-neutral pricing’.