# Why the expected return rate of a stock has nothing to do with its option price?

OK, I admit that this is a frequently asked question. But I couldn't find a satisfying answer after I read the explanations of books, went through the derivations of B-S formula, and searched answers online. My question is that, I can understand the derivation of the B-S formula, but what is the intuition that the expected return rate of a stock has nothing to do with its option price?

Suppose I have two stocks A and B, the price is the same today, both worth 20 dollars. Stock A has a expected return of 0.5 dollars/week, a volatility of 50%; stock B has a expected return of 10 dollars/week, a volatility of 1%. For call options with strike price 40 dollars expiring in 1 month(4 weeks), how can the option price for stock A is greater than that for B, since stock A is expected to worth only 22 dollars while B is worth 60 dollars? Any intuitive explanations?
I can also make examples where stock A has vanishingly expected return and B has infinitely large expected return.

Because you can hedge. Once you have delta hedged, the pay-off is symmetric about up and down moves so drift doesn't matter.

Also the delta-hedged call and the delta hedged put have to have the same value since they have the same pay-off. (Put-call parity) Yet any argument that the call should be worth more because of drift says that the put should be worth less. Since they are worth the same, the drift must not matter.

The whole thing takes a lot of getting used to. A large part of my motivation for writing my first book (concepts and practice of mathematical finance) was to get my head around this issue and then to explain how I did.

• Hi @Mark, nice to see your post here! I read some of your articles before. Regarding the first point you made, yes, because of delta hedging, either drift enters the option price or not, both cases are all OK. For the 2nd point, since the put-call parity, $C-P=F$, where $F=S(t)-e^{-r(T-t)}K$, if $S(t)=\alpha dt + \sigma dt$, and the options are priced as $dC^\prime = dC + 1/2\alpha dt$, $dP^\prime = dP - 1/2\alpha dt$, the put-call parity still holds. So in this case, it's OK for the drift term to matter. – Allanqunzi Apr 28 '15 at 7:05
• but $C=F+P$ so the drift has to affect both $C$ and $P$ in the same direction – Mark Joshi Apr 28 '15 at 9:00
• But why not the drift affects $C$ in the same direction as that for the sum of $F+P$, at least this is what $C=F+P$ suggests. – Allanqunzi Apr 28 '15 at 16:03
• but $C$ is in the money when $P$ is out and vice versa so an argument that $C$ is worth more because of drift makes $P$ worth both less and more. – Mark Joshi Apr 28 '15 at 21:51

I think to gain intution you have to understand that the same agents that value the stocks will value the options. And agents compensate for volatility by demanding higher expected returns. Therefore you should ask: Why are stocks priced as they are in the first place?

In your example, the stock with higher volatility has much lower expected return. This can only happen systematically if agents are 'risk loving' or temporarily if expected returns or volatility change suddenly (e.g. due to significant news).

In the first case the higher option price is intuitive because for 'risk lovers' volatility is a good thing. In the second case the imbalance will be restored by spot price moves (e.g. the second spot will drop until its expected return offers sufficient compensation for the risk). Then balance in option markets will also be restored by different spots in the pricing formula.

• I think your answer is based more on philosophical grounds than empirical observations. I currently do not have the time to check what you have said but I cannot be convinced that stocks with high volatility will tend to have high returns. Do you have some evidence towards your claim? – baibo Dec 29 '19 at 13:22

Yes and No.

In the absence of arbitragers, the price of the option will be different for each speculator based on their drift expectations (and each speculator has a risk in his position and will limit his ability to trade large sizes to avoid bankruptcy) and the option price will converge to priced off a supply-and-demand driven drift expectation.

However, if there are arbitragers who can delta-hedge an option to create a risk-free position, then if the drift of the combined delta-hedged option position is different from risk-free rate, the arbitrage can become infinitely rich in no time. In real world, we don't see arbitrage becoming infinitely rich, and the only way to avoid such a situation (at least in academic settings) that an arbitrager can make money is to assume that the option is priced using a drift of risk-free rate.

In short, the options are priced so that arbitragers don't make money. It doesn't mean that the actual drift expectations are the risk-free rate.

Subtle difference.

In practice, however, volatility is not constant. It boils down to choosing between two parameters - the volatility and the drift rate. Practitioners, prefer to assume that the drift rate is static and create a volatility surface (volatility varies with moneyness and maturity), but in an alternative world, one can assume that the volatility is constant and there is a risk-free rate surface (i.e. the risk-free rate to borrow for hedging an option varies with the moneyness and maturity).

Practically, it is very difficult to get a measurement of a stock's true drift while there are very well-documented processes to estimate volatility. It is therefore very convenient mathematically to select the risk neutral pricing measure that eliminates idiosyncratic drift.

At its heart, Black Scholes constructs a dynamic, replicating portfolio for an option on a stock. Consider an option with strike K = 0. How would you replicate that call? Just buy the stock! Drift in no way factored into your replication strategy.

By constructing a replicating portfolio out of the security which has drift, we are implicitly taking that drift into account in order to properly replicate the next future instantaneous states.

"Why the expected return rate of a stock has nothing to do with its option price?"

It has everything to do with the option price! The option price is a function of the stock price. If the expected rate of return on the stock price declines, the stock price will decline as will the option price.

"Suppose I have two stocks A and B, the price is the same today, both worth 20 dollars. Stock A has a expected return of 0.5 dollars/week, a volatility of 50%; stock B has a expected return of 10 dollars/week, a volatility of 1%."

Why on earth would these two stocks have the same price? Stock A should clearly be worth far less!

• Not sure why I'm down voted...OP and most of the responses (excepting Kiwiakos excellent response) are missing the point. – user9403 Apr 30 '15 at 0:26
• This is the most intuitive explanation +1 – Jay Na Feb 18 '17 at 17:22
• I upvoted your answer. Intuitive +1 – James LT Dec 1 '17 at 21:28

It is an artifact of sorts. You can explain it via hedging, but it comes down to the fact that the stock has some and some volatility $\sigma$, and 'equilibrium' drift $\mu$, and when you go to price the option (using B-S)the drift cancels out.

In the end, it supposes that stocks are fairly priced in a competitive market and the $\sigma$ justifies the $\mu$, so in the end it all works out. Think of it this way: there is (1) some stock pricing model relating $\sigma$ to $\mu$, and (2) some option model relating the same two variables. So, it is sort of like eliminating one variable in simultaneous equations.

Two things:
No. 1: Who tells you that A and B have those expected returns? If everybody in the market agreed the prices would be different because the view on expected returns and risk are included in the prices of the stocks and those prices are included in the option pricing formula.

No. 2: A complicating factor is the non-linearity of options so to give you an intuition let's keep things simple and just look at a linear derivative because the big idea is the same:

You want to price a derivative on stock A. The product just pays the current price of A in \$.

Now, how would you price it? Would you think about expected returns or your risk preferences? No, you won't, you would just take the current price of A and perhaps add some spread. Therefore the expected returns and risk preferences did not matter (=risk neutrality) because this product is derived (= derivative) from an underlying product (=underlying).

This is because all of the different perceived expected returns and risk preferences of the market participants are already included in the price of the underlying and the derivative can be hedged with the underlying continuously (at least this is what is often taken for granted). As soon as the price of derivative diverges from the original price a shrewd trader would just buy/sell the underlying and sell/buy the derivative to pocket a risk free profit - and the price will soon come back again... The same of course also goes for a derivative on stock B.

So, you see, the basic concept of risk neutrality is quite natural and easy to grasp. Of course, the devil is in the details... but that is another story.

See also my answer to a similar question here: Why Drifts are not in the Black Scholes Formula

This is only true under very narrow assumptions such as a log-normal likelihood function. Your intuition is correct. Consider the simple case of a very long-term short put being written on a firm paying liquidating dividends. Now, most people would immediately point out that this violates the assumptions of Black-Scholes, but that is sort of my point. Black-Scholes is, at best, a very fragile model. Empirically it does not work. Not only has there never been a successful validation study, the end of the original article discloses that it failed empirical testing.

Furthermore, there are greater and more material problems with Black-Scholes than the one you mention. It is a Frequentist model, which means it cannot be a coherent model under the statistical definition of coherence and the Bayesian model does not resemble it, implying any solution would be an inadmissible solution as well. There is also a non-existence proof on the parameter estimators. The model is derived under the assumption that the parameters are known. However, it is known by theorem that this class of problems has no Frequentist or Likelihoodist solution if the parameters are not known, hence all of the anomalies in the literature. It isn't that it is a bad or good model, it is a model that cannot be measured. Even if it is true, the proof is vacuous. A model that cannot be measured isn't a model.

Stay away from black-Scholes. It assumes that returns to the underlying asset are normally distributed. This is objectively false. Even when you're using it to price a large index (where in theory the central limit theorem should ensure normally distributed returns), it breaks down because the central limit theorem requires each asset in the index to have a defined distribution (even if it is arbitrary and you don't know what it is) with measurable first and second moments

How do you measure the moments? By taking a sample of an asset's past returns? Firms are always changing, so a sample may be biased if it's too big, and inaccurate if it's too small. Do firms even have moments? Even if a firm's returns have an underlying distribution (and considering that hedge funds have turned supercomputers on the S&P to try to machine-learn a distribution and have found nothing, it's quite possible that they don't--or that their distribution is constantly changing too quickly to make a good estimate), there's no reason to believe that their distribution has defined moments (For example, the Cauchy distribution has no moments. Any estimate of a Cauchy distribution's expected value will be wildly inaccurate). That's not even touching on the fact that the CLT requires the use of independent assets--and estimating and compensating for correlations between assets is an entirely different beast with it's own (significant) set of flaws. Since I don't feel like writing a whole manifesto, I'll let you figure them out yourself.

When you can't take accurate moments of an asset, the black-scholes breaks down. Black-Scholes won the nobel prize because if it's usefulness in understanding the intuition behind the options market (it is a great way to think about it in an economics class). But you should never, ever, ever use it (or any mathematics, honestly) to make a real decision. As Warren Buffet said, he loves quantitative trading because it gives him idiots to trade against.