The problem itself doesn't mention BS model.

It just asks you to use risk-neutral valuation to price this derivative.

Remember,risk-neutral valuation is a very important tool. The BS model can be solved by risk-neutral valuation.

faint to death...

Black-Scholes "IS" risk neutral pricing. That's why they were awarded Nobel prize, not for option pricing, but for risk neutral pricing.

Or do you have your way to solve it using risk-averse or risk-seeking valuation? You'll be nominated.

[此贴子已经被作者于2007-9-7 1:50:26编辑过]

Did I say BS model can be solved using risk-averse or risk-seeking valuation?

The BS model's contribution is not for option pricing, but for its partial-derivative equation. The pricing formulation generated by risk-neutral valuation satisfies the BS equation.

[此贴子已经被作者于2007-9-7 5:03:28编辑过]

No, you just said "risk-neutral valuation is a very important tool. The BS model can be solved by risk-neutral valuation".

This sounds like "The BS model can also be solved by some other tools, although maybe less important than risk-neutral valuation".

So I wonder if you can solve it by risk-averse or risk-seeking. Looks like you can't, or you have risk-fourth something to show us.

Admit now you don't really understand what you're talking about, okay? Better than being pointed out in an interview, as that would be a slap in your face.

By the way, partial differential equation, pal, not partial-derivative equation.

The BS PDE is already risk-neutral, that's why you don't see the real world drift term.

Now you tell me you use risk neutral valuation, which is a very important tool, to solve that PDE, zzz....

题目是JOHN HULL 《Options, futures and other derivative securities, 5th ed》第12章 B-S MODEL里面的习题，跟原题有些不一样。

确实不用B-S MODEL，

第一步，应该还好，PAYOFF 刚刚服从一个期望是U-Q^2/2, 方差是O/T^0.5的标准正态分布；

第二步关键还是用 risk-neutral ， 用risk-free rate 取代U， 再配上时间因素，E^-rT,结果就是E^-rT*（r--Q^2/2).

yeah, you're talking

看了appendix, 问题如下: the question is to calculate the price at the time 0 , if i ues the method as the appendix show, the result as i mentioned, i think it just the expect return of the prize.BTW, this is the return rate of the prize, just the rate, not the true value of this derivative.

是你自己改的题目吗，说法完全不严谨

应该是这样，假设现在的时间为t，到期时间为T，到期的payoff为1 / (T - t) * log(S_T / S_t)，求t = 0时候的价格f_0

f_t = e^(-r*(T - t)) * E_Q [1 / (T - t) * log(S_T / S_t)]，这个叫做discounted expectation of payoff under the risk neutral measure Q

代进去t = 0

f_0 = e^(-r*T) * E_Q [1 / T * log(S_T / S_0)] = e^(-r * T) / T * ( E_Q [log(S_T)] - log(S_0) )

E_Q[log(S_T)] = log(S_0) + (r - 1 / 2 * sigma ^ 2) * T，因为你假设S_t是GBM

代进去，f_0 = e^(-r * T) * (r - 1 / 2 * sigma ^ 2)

题目是书上的，我们用的那本教材是《fundamental of options, futures》,和《Options, futures and other derivative securities, 5th ed》一样，不过比它要简单一些，基本上内容是一样的。这道题目是课后习题，12章里面第12题。

Okay, I guess that's why we have the word "fundamentals" in the title.

Anyway, I still believe Hull should have worded it in a more careful way.

我有一点不是很清楚，通过上面的方法求出的f_0 = e^(-r * T) * (r - 1 / 2 * sigma ^ 2)，它的两个分式，前面代表的是discounted expectation ， 后面的一个分式表示的是a dollar amount of this derivative payoff, 也就是说其只是个增长率，而非其价格本身，为什么两个乘起来就能代表derivative 在0时刻的价格呢？

他题目本身规定的Payoff就是个underlying的增长率。

错

e^(-r * T)只是discount factor，numeraire在t = 0的价格

(r - 1 / 2 * sigma ^ 2)是expected payoff under risk neutral measure

搞不懂a dollar amount equal to 1/T * log(S_T / S_0)什么意思吗，你在上学吗，干吗不问老师或者TA，你付了学费了，别浪费了

假如你买了这么一个东西，t = 0，T = 1，S_0 = 100，假如到期的时候S_T = 200，payoff是log(2) = 0.69，你收钱，假如S_T = 50，payoff是log(0.5) = -0.69，你付钱，所以你payoff是个random variable depending on S_T

under the risk neutral measure，expected payoff = (r - 1 / 2 * sigma ^ 2)

这个东西其实有个正式的名称，叫log contract

我想知道，为什么不严谨，还有其他的可能性吗

没有其他可能性了，我漏看了要求B里面最后的at time zero，还是你后来翻了书加上去的，要是你开始就写了，那就是我漏掉了