I have a quick question regarding risk-neutrality. In the Black-Scholes framework, the expected return on the stock is irrelevant to the price of the option. I'm having trouble wrapping my head around the following scenario: imagine call options expiring in several days on two identical stocks, except one has a expected growth rate of 50% while the other has an expected growth rate of 1%. The call options should have the exact same price. This seems bizarre, since the former option has a much better chance of expiring in the money. If someone could clear this up for me I'd greatly appreciate it.P.S. If this is true, is speculation with options entirely pointless? That is, does it make any sense to buy options as a means to bet on a stock going up or down?

its all about the cost of replication - if the cost is the same then the price is the same. if stock A has an expected rate of return of 50% over a few days then it is unlikely that its volatility would be the same as stock B with a 1% expected rate of return over a few days.

It just seems bizarre that one would pay the same amount for the two call options. I know this situation is highly unlikely, but am simply thinking about it as a theoretical exercise. As an investor, I would think the option on stock A is much more valuable.

as i said it is unlikely that you will pay the same.. if it is known with certainty that stock A is going to go up 50% then it will instantenously go up 50% - the fact that it hasnt means that there is also a probability that it will go down such that the expected value of the stock is its current price. the market will not price the two options equally.

Oh ok. Gotcha. Thanks for clearing this up for me.

You can consider a call on 1 stock of A and q stocks of B such that their values at ) are the same. The only assumption is equal volatilities. On the other hand BS is a theoretical approach that teaches us what is the call option price and so it is legitimate to state such a problem to check how sensible the BS price interpretation in theory. The other question is how to apply it? But it is also would be use your theoretical interpretation.

not to belabour the point but lets consider the case of a stock in a binomial world where its currently trading at 100. if there is a 60% chance that it will go to 150 in a few days (50% return) then with a 40% probability it could go to 2.5 (-97.5% return). the 100 strike call must be worth 33.05085 to be consistent with this. so you sell the option for 33.05085 buy .338983 of the stock as hedge and no matter which of the two states is achieved then you are hedged.

And use the same construction and assume the same states but one time 90% and 10% and other 10% and 90% for the same states. In both cases you are hedged with the same premium. But why you need to be hedged when you almost sure lost. In continuous setting B&S did not make an assumption that two stocks do not have the same initial price with equal volatility. That is they teach all of us to pay the same premium for real return within -infinity to +infinity. In other words future cash flows with the same volatility and expected return within -infinity to +infinity have the same spot price. Then what is their undestanding of the price regardless of derivatives?

QuoteOriginally posted by: listAnd use the same construction and assume the same states but one time 90% and 10% and other 10% and 90% for the same states. In both cases you are hedged with the same premium. But why you need to be hedged when you almost sure lost. In continuous setting B&S did not make an assumption that two stocks do not have the same initial price with equal volatility. That is they teach all of us to pay the same premium for real return within -infinity to +infinity. In other words future cash flows with the same volatility and expected return within -infinity to +infinity have the same spot price. Then what is their undestanding of the price regardless of derivatives?i dont think the premium can be the same and this is why. in a 90/10 world the two state stock prices cannot be 150/2.5 as that is not consistent with a todays price of 100. ditto in a 10/90 world. the terminal stock prices cannot be independently selected. if you fix the up probability and the up stock price then the down stock price must change to be consistent with today's price.

Here is another way of looking at things.Forget options for the moment. Consider just spot and future. By the arbitrage theorem they are related by the risk-free rate. Just like with BS the expected growth rate makes no difference.Madness, eh? No. If there is a valid expected growth rate, the market may know about it. They will know that spot or future (or both) is under-priced and adjust both accordingly. Even if the market doesn't know that growth rate, it will act on its best guess. In other words, the market's expected future growth rate will be already priced in and therefore, relative to current spot, be neutralised (i.e. reduced to the risk-free rate).Clearly then there is no information to be had from spot and future about the expected growth rate. But if you think you have a better estimate than the market as a whole, you can go long or short as appropriate with either spot or future. But the market won't tell you if you are right or wrong until after the event.Can options help? If people use options for leverage on expected return, then you'd think there might be some extra information in options market data. But not with BS. You need a different model. Can a better model give you this information? Perhaps. But not with constant volatility. Any information that is there will be hidden in the volatility smile/skew.

I think what was really confusing me was that I misinterpreted the fact that the prices must be the same that the expected return on the options must also be the same. Now I realize that this certainly need not be the case. I revisited Black and Scholes original paper and their CAPM-based derivation helped clear things up.

That's right. In fact, there is a very nice relation between the expected return on the stock and the expected return on the call under GBM: see this thread

there are 2 different things model and real world market. In BS model we do not have market. We have just one stock price replaced by a SDE. Do we have definition of the option price? rather not than yes. They use some general ideas that change in value of the portfolio with option and risk free bond shold be equal. That might make sense? But it leads to the conclusionthat different future cash flows have equal spot price and that is substential contradiction. Where does mistake come from? It was not highlighted the assumption that the option pricing define price as expected value and therefore option price is nonrandom. That in turn has lead to use BS price as fair, perfect and i.e.On the other hand there is no evidence to exagerate expected reduction. Another option pricing interpretation stems from the fact that considering stock ( actually the solution of SDE) for particular scenario omega we are in deterministic setting. Therefore if at maturity S(T) > K for the chosen omega then option price should provide the same return as the stock. If S(T) < K the option price is 0. This leads to the arbitrge free definition of the option price as a random function depending on initial spot data ( t, x ). When market establish price of the option as $c at t that means that market participants weight the risk to loose $c if S(T)<K and to gain profit when S(T)>K and establish the price$c. The distribution of S(T) is relevant. In theory it could be log-normal or other. Besides the methematical derivation of the BS equation is incorrect itself and in Hull's book in particular.