关于伊藤过程和几何布朗运动的疑问

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hull的《期权、期货及其他衍生产品（7e)》一书中提到两个很类似的描述变化量的随机过程：
伊藤过程，dx=a(x,t)dt+b(x,t)dz              （1），
几何布朗运动的离散形式，dS/S=udt+gdz                     （2）,
其中dz是维纳过程,a(x,t)是预期漂移率，b(x,t)是方差率，u是预期收益率，g是股票价格波动率；
由第二个式子：dS=Sudt+Sgdz                                （3）;
（3）与（1）对比有：
a(x,t)=Su,这与作者定义的期望漂移等于股价乘以期望收益率相符；
但b(x,t)=Sg就无法理解了，左边是方差，右边是股价乘以标准差，这怎么会相等呢？望高手解答下，谢谢了！

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关键词：布朗运动 股票价格波动 期望收益率 hull 股票价格 伊藤 疑问

Ito process is a more general process. Hull does not say that the two process are the same.

In particular, Geometric Brownian Motion is a specific form of Ito process. If you let X=S, you will find that, uS is a(Xt,t),sigmaS is b(Xt,t).  a(Xt,t) and b(Xt,t) can be any adapted process, whatever you want. If you plug uS, and sigmaS into it you get the GBM.

I don't think Hull give some meaning of b(Xt,t). So I don't know where you see that b standards for variance. If you really want some answer, I will say that b^2 is used to describe the variance term, not b.

"g是股票价格波动率"
我觉得g应该理解为收益率的瞬时波动率

为什么你会认为左边是方差呢？

Chemist_MZ 发表于 2012-10-24 02:42
Ito process is a more general process. Hull does not say that the two process are the same.

In pa ...

不好意思是我打错了，b^2是方差率。
几何布朗运动和伊藤过程的确是不同的，只是我之前用这两个模型预测很短的时间内股价增量分布时得到相似但不同的解，所以感到很疑惑。具体来说：
由GBM：令S为现在股价，ΔS/S服从期望值为uΔt,方差为(g^2)*Δt的正态分布。即ΔS服从期望值为SuΔt，方差为((Sg)^2)*Δt的正态分布，注意到该方差是受S影响的；
由伊藤过程：令x为股价，则Δx服从期望为a(x,t)Δt，方差为（b^2)*Δt的正态分布，当时我认为这里的方差不受S影响，与GBM的不一致所以挺困惑的，不过经您提醒，伊藤过程的b是x和t的函数，所以该方差还是受股价x的影响的，
但更极端的我想，如果考虑a和b为常数的情形，即由伊藤过程改为广义维纳过程，那么此时期望为aΔt，a=Su,这与作者定义的期望漂移等于股价乘以期望收益率相符，但方差为（b^2)*Δt,就不受股价影响了（如7版的例12-2），此时就和GBM不同了。不知道我不是把概念搞混了还是有其他的解释？谢谢了！

Chemist_MZ 发表于 2012-10-24 02:42
Ito process is a more general process. Hull does not say that the two process are the same.

In pa ...

我注意到中译本的书将广义维纳过程中的b^2称为变量的“方差率”，而GBM中的g称为股价的波动率，是不是说前者本身考虑了股价的影响（比如已经乘以了股价），而后者只是变化的比率（即只是一个百分数，未乘以股价）？这样好像就解释的通了。

黑目shadow 发表于 2012-10-24 23:19
我注意到中译本的书将广义维纳过程中的b^2称为变量的“方差率”，而GBM中的g称为股价的波动率，是不是说前 ...

Very nice observation. I don't know whether I have caught your point, but I try to explain it.

I may not answer your question directly. Instead, I try to clarify some concepts so that you may have a clear picture and solve your puzzle yourself.

1. You mentioned that you try to simulate the distribution of ΔS/S in a very short time interval. However, you cannot do it directly. Since we know that Brownian motion is a kind of "rough" function whose quadratic variation is not zero. If you simulate with random number directly, you actually ignore its quadratic term. You can solve dS=uSdt+gSdW, with some elementary stochastic calculus knowledge. And the solution is actually lnSt=lnS0+(u-1/2*g^2)t+g*Wt which means
ΔS/S=dlnS=(u-1/2*g^2)dt+g*dW. However, if you think ΔS/S~N(uΔt,(g^2)*Δt), this only holds when time interval is very large which means the stock price does not move continuously. Since we know that the Brownian motion is a continuous function, there may cause some problems when you simulate the stock price discretely. The fundamental difference is its mean.

2. Why we call dS=uSdt+gSdW a GBM, because the solution is in exponential form: that is St=S0*exp(...), which means that the volatility is the volatility of stock return not the stock price itself and the stock price is log normally distributed (such process prevent the stock price take a negative value)

3.If you apply the BM with a drift(generalized wiener process) to model the stock process
dS=adt+bdW, which means the stock price is normally distributed, you can integral it directly and find that S=at+bW. This is an early way to modeling stock process. You mentioned that "a" is a constant, but later you say a=Su. There is an obvious paradox here. S is stochastic therefore "a" is surly not a constant, which is contradictory to your previous assumption.

4. There are many ways to modeling stock processes, you can not always dig an intuitive meaning on each process. We just know that what parameters are responsible for mean and what are for variance. We don't say they are variance or they are mean. Just like we call b a variance or a volatility term, does not mean itself is variance or volatility. Or whether it is the volatility of the price or return. We just know that if b is bigger, the stock price will fluctuate more and so does the return.

If you want more detail or more specific question, add my qq.

感谢。。。。。。。。。。。。。

学习了

sg为dz（服从几何布朗运动，均值为0，方差为1）的系数，由于每个dt是相互独立的，所以可以求得ds的方差为dz的sg倍