求教HULL 里面关于B-S 一道题目

以下是引用qdzhang在2007-9-7 5:02:00的发言：

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.

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).

FYI, go to the appendix of that chapter

BTW, risk-neutral valuation is NOT to solve PDE ...

以下是引用qdzhang在2007-9-8 4:04:00的发言：

BTW, risk-neutral valuation is NOT to solve PDE ...

yeah, you're talking

以下是引用qdzhang在2007-9-6 1:49:00的发言：
题目说的很清楚，用risk neutral valuation to value the derivative.
Risk-neutral valuation 通常分三布：
1. Assume that the expected return from the underlying asset is the risk-free interest rate, i (ie, assume mu = r)
2. Calculate the expected payoff from the derivative.
3. Discount the expected payoff at the risk-free interest rate.
需要注意的是，risk-neutral valuation只是一个工具来解BS方程。它的结构不仅在risk-neutral世界里适用，在实际世界里也适用。

以下是引用qdzhang在2007-9-8 4:04:00的发言：
FYI, go to the appendix of that chapter

BTW, risk-neutral valuation is NOT to solve PDE ...

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.

以下是引用jgrsun在2007-9-7 12:47:00的发言：

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).

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。

以下是引用jgrsun在2007-9-8 21:35:00的发言：

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.

Thanks man. You pointed out what I didn't explain clearly.

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？