能不能直接截图？

I first answer question2:

Yes, the correct expression should be: under Q measure, the discount process of stock price is a martingale.

next, for question 3:

In risk neutral probability, all the assets have an expected return or risk free rate (r), that's why this probability measure get this name. When pricing, you have to convert the probability space into the risk neutral measure. Why?

applying Ito formula to the option's price in the real world. You see that
dC(S,t)=Ctdt+CsdS+0.5*Css(dS)&^2 arrange it you may find that dC can be expressed as:

dC=()dt+()dW, the ()dt is call the drift term. You may find that under real measure, this term contains mu, the stock's expected return. Who knows the value of mu? nobody knows. This means that under real measure, we don't know the expected return of an asset clearly and it may vary from asset to asset. For stock is mu, for option is another.

If we have no idea of the asset's expected return, then how can we discount it? remember, if you write EP(max（St-k,0), you can not discount it with riskfree rate, because as we mentioned before, under P( real world) measure, the option don't grows at riskfree rate. So how do we solve this problem? risk neutral, we convert the measure into risk neural measure. This measure is really beautiful since under Q every asset grows at  risk free rate. So that‘s why we should calculate the expectation under Q measure.
As to the question 1:

This confuse many people when Black-scholes Option pricing formula first come into being. Because people find that the option price is not depend on the stock's expected return. I think there are many ways to explain this strange phenomenon. The more academic version is that through the dynamic hedging in a very very short period(see how BS construct the replication portfolio), the option's expected return is actually being"cancelled". A more understandable version is that, option( and also other derivatives) its prices depends on their underlying asset, say, stock's  price now(or some exotic ones also depends on the past). Actually option can be replicated with stock and bond. So that if you expected the stock price will rise in the future, this information is already contained in the stock price. The stock price will go up now. The same information will not be embodied in the option price again. This really makes sense, since the option can be replicated with stock, the price of the option is just the replication cost. The option's price is just depend on the stock price now. If every body think the stock will rise, this information will first embodied in the stock price and than affecting the option price. So I think this is quite reasonable and very straightforward.

There is a very similar interview question: There are two stock futures. One is a good stock the other is a junk stock which may go to bankrupt at any time. All the other conditions are the same, like the risk free rate, the future's maturity etc. The question is that, if the two stock's price now are the same, say 100, will there future price be the same?

The answer is yes! The reason is the same as just what I have mentioned. You may tell me that the junk companies price should worth a penny instead. However, as long as the price is 100, the option or the future's price is calculated on this price. If the company is expected to go bankrupt immediately, it is the stock price will go down making the option price goes down but the expectation will not be embodied in the option price just in the stock price.


Why the stock price now is at this level? Because it conbines many investors' expectation.

非常感谢！写的非常清楚！
有几个逻辑我还是不太理解，
1.q就是代表measure under real world，也就是说用真实的mu来discount，显然是不可能做到的。但是用mu来折现和对风险的偏好有什么关系呢？
2.因为用真实的mu不可能，所以就用r，用r就是Q－martingale，这个我现在可以理解了，可是为什么在风险中性的世界里就是risk-free rate呢？换句话说，向对于risk－neutral的概念是什么呢？为什么这个性质会决定用mu还是用r来折现呢？
3.第三个问题，第一种说法是用数学方法来解释，在推导的过程中mu消失了，是这样吧；第二种说stock的期望值高的话，本身就会反映在现在的股价里面，这个我不能完全接受：如果是这样的话，股价的变化应该是期望值的变化函数，才可以抵消期望值增大或减小对期权价格的影响，也就是说St随mu的变化可以量化的，可是这种期望值反应在股价上的效果完全是心理学上的东西，怎么可以用一个简单的函数表现出来呢？

I think the first two questions are the some thing.

The name "risk neutral" is very confusing. It has nothing to do with the investor's risk aversion. Only because we find that after change from P to Q， every asset grows in the r rate (see Girsanov Theorem). This is only a mathematical handling, you should not try to give it any real meaning. So we get measure Q, but how do we name it? Aha, yes, it has a good property that every asset has an expected return of r. When would this happen? only in the world of risk neutral. So let's call it risk neutral probability. That's it. Nothing complicated.

for question 3. If you are really interested in this topic. I would say, under Black-Scholes, the expected return has no impact on the option's price but for some other process like OU process which has some autocorrelation, it actually has some impact, but not directly by the drift but by the diffusion term.

The expectation we here is the probability term E() not people's expectation. We think people's expectation is affected by many factors and are random. Such psychology stuff is the so called Efficient market hypothesis.

If you can not accept this explanation, I will give you a more practical one. The option can be replicated using the information up to now. So that at maturity I can replicate the payoff. If I sell the option containing future information, lets say the option contains good information so it is more expensive. I can sell it to you and use the stock and bond to replicate it's payoff. Notice that when replicating the option, I do not care whether stock price will go up or go down. At maturity, in any situation, I can deliver you what ever you want. So if you pay me 100 for the option, I only use say 80 for replication, and at maturity, you and me are clear. I get 20 without any risk. This is the "future information premium" you pay me. However, you see that you do not get any benefit for pay 20.

For pricing issue, whatever drift you set for the stock price, the risk neutral measure will "kill" all of them. They will all be r under Q.
I think the replication explaination is more straight forword. I can use the information up to now to replicate any option. So if you pay me more, I will use part of the money to replicate it and deliver it to you at maturity and put the rest of the money in my pocket. That's it.

Good luck.

fantastic explanation! I also have a question regrading to this topic.
When we do stress testing or scenario analysis, we don't care about the exactly price that much. We are interested in the risk in different situation. Thus we use a lot of simulation.
Then when we do simulation, does dift term miu matters?
Do we estimate miu by historical data? or we use risk free rate?

Many Thanks