Why Option Pricing Uses Risk-Neutral Rather Than Historical Returns

# How Are Option Model Assumptions Justified In Practice


# How Are Option Model Assumptions Justified In Practice












I am reading this article, and I am wondering how comments like

> there may be a 50/50 chance that the underlying asset price can increase or decrease by 30 percent in one period.

are reconciled with obviously non-zero and potentially asymmetric typical historical return distributions in the underlying?

## Answer by demully (score 0)

https://quant.stackexchange.com/a/57391

Apologies for the delay in response.

Yes, the option will indeed be priced as if there was an equal chance of an x% rise or fall in the underlying.

Strictly speaking, this is an equal chance of an outcome x% higher or lower than the underlying's forward price (not its spot price). However, the difference is often marginal in many cases, and usually a small fraction of the price risk posed by the underlying's volatility.

This is called the "risk-neutral measure", which is (the default) used to price financial derivatives. As such, this is entirely distinct from the observed historical distribution of the underlying's returns; as well as from the expectation-based "risk-premium measures" that are the defaults used to value financial assets, in the valuation chapters of standard textbooks.

So we could very well have a situation where historical returns were 4% on average; where you think the current price represents a 3% future return while I think it will be 5%; you think the market should return 4% and I think it should be 3%; and we trade an option that assumes it's zero! To be clear, there is no contradiction in any of this. There are just different multiple different measures of "return" out there (and I've omitted some above ;-)

The reason that derivatives assume zero is quite simple. If it priced for anything else, then investors could "cheat" and arbitrage their brokers/counterparties. This does mean that an option priced for equal upside/downside might have an expected positive/negative bias. But this is simply a function of the expected bias in the underlying itself. But if you give me an option that is differently biased to the underlying, then I can get a free lunch.

Let's take a simplified example. Let's imagine the bond market was a quarterly coin flip, with a 75% probability of a +1% payoff and a 25% chance of a -1% payoff. That's a 2% expected annual return with (approx.) 4% volatility.

Now instead of buying my bonds costing 100 apiece, I could hold cash and buy bond futures. If interest rates were at say 1%, then the future will cost me 0.99. My 99.01 today (ie 100 cash less future) * 1.01 (interest) = 100 (cost of the bond), while the future will deliver the same returns as if I actually bought the bonds.

Note that the future does not make any assumptions about the fairness of the underlying bond market. It is as biased as the underlying, to make the outcome for the buyers of the bonds the same as those holding cash and buying the futures.

Suppose you believe that the future should be worth more because the bond market pays off more. So you'd be a buyer up to say, a future price of 2. Fine. I'll borrow 100, buy the bonds, and sell the futures for 2. In a year's time, I'll have 100 of bonds and 2.02 cash in bank (whatever the bond market actually did will be net nil for me, shorting the futures); repay my loan at 101, leaving me 1.02 in profit with no risk taken.

Now exactly the same argument can be made for options on the underlying as for the futures above. Because call-put parity requires long call plus short put to be equal to the forward. Price either to reflect the actual risks of the underlying, and you'll simply create the potential for a free lunch.

So the argument might go that a quarterly call delivered a payoff 1 with 75% probability, so it should cost 0.75. Meanwhile, the same put delivers a payoff of 1 with 25% probability, so it should cost 0.25. So then I would simply borrow to buy the bonds, and sell a call to buy a put. I'm borrowing 100 - 0.75 + 0.25 = 99.5. In three months time, I will have exactly 100 whatever happens in the market. Less 99.5 principal and 0.25 quarterly interest equals 0.25 riskless 3m profit, or ~1 annualised arbitrage for nil risk.

Thus financial derivatives must assume that the underlying is a "fair bet", even if it might be evidently "unfair" in reality, and historically. Thus the derivatives will end up carrying the same bias as the underlying, which is fair between the two. If the derivatives themselves represented a fair bet, then investors can cheat, by gaming the different risk profiles between the underlying and its derivatives.

This is why there is a "risk-neutral measure" that is different from the historical or economic/valuations measure of asset risk.