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求教求用Girsanov定理求解此文章1220页（10）式第二行中的期望值,困惑困惑~~~~~睡不着觉，期待牛人帮帮忙~~

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关键词：困惑困惑 anov 牛人帮 期望值 RSA 期望值 文章

不要买那个文章，我发错了，网上有下载的

我再把我的问题描述下：Klein, P.C. “Pricing Black-Scholes Options with Correlated
Credit Risk.” Journal of Banking and Finance, 20 (1996), pp.
1211-1229.
这篇文章中的公式（10）中第二行期望用GIRSANOV定理如何求解？

没找到哪里可以下

http://www.sciencedirect.com/science/article/pii/0378426695000526在这个网站

aliceww2012 发表于 2013-5-20 17:44
我再把我的问题描述下：Klein, P.C. “Pricing Black-Scholes Options with Correlated
Credit Risk.” Jo ...

OK, I briefly go through the paper.

I give an example:

suppose you want to solve E*(ST|ST>K,VT>D).

The thing you want to do is to write it into E*(ST)E(?)(I{ST>K,VT>D}) where I{} is an indicator function taking 1 only when ST>K and VT>D.

I think your question is want to know how to use Girsanov to change from risk neutral measure (*) to the new unknown measure (?)

To do this, you have to construct a likelihood ratio to "eliminate" ST.

E*(ST|ST>K,VT>D)=E*(ST) E*(I{ST>K,VT>D}*ST/E*(ST)), if you take ST/E*(ST) as a likelihood ratio, you can change the measure from * to ?

that is E*(ST) E*(I{ST>K,VT>D}*ST/E*(ST))=E*(ST) E(?)(I{ST>K,VT>D})

your next task is to find out the drift of S and V in the new measure using Girsanov.

Note that ST/E*(ST)=St*exp((r-0.5*sigma^2)(T-t)+sigma*W(T-t))/(St*exp(r(T-t))

so ST/E*(ST)=exp(-0.5*sigma^2*(T-t)+sigma*W(T-t))
=exp(-∫(t to T)0.5(-sigma)^2ds -∫(t to T) (-sigma)dW(s))

we know that ST/E*(ST) is the likelihood ratio

dQ(?)/dQ(*)=exp(-∫(t to T)0.5(-sigma)^2ds -∫(t to T) (-sigma)dW(s))

what the Girsanov theorem says is that if dW is a brownian motion under Q* measure, dW+theta*dt is also a brownian motion under Q(?) measure. The two measures are linked with the likelihood ratio:

dQ(?)/dQ(*)=exp(-∫(t to T)0.5(theta)^2ds -∫(t to T) (theta)dW(s))

So you compare the two dQ(?)/dQ(*) and find out that theta=-sigma (sigma here means the volatility of S)

this gives you the result that under Q(?) measure the drift of dS and dV are r+sigma_S^2 and r+sigma_V*sigma_S.

You now have the joint distribution of S and V under the new measure.

The remaining task is just to integrate the bivariate joint lognormal on the area ST>K and VT>D, its easy.

明白你的意思，但是我要求的是E*(STVT|ST>K,VT>D).这样的话用你的方法，GIRSANOV定理不能直接套，套不出来啊

是不是要分两步？

aliceww2012 发表于 2013-5-21 17:21
是不是要分两步？

Absolutely.

In that case you need to change measure twice. First take out S. Second take out V. It's more complicated but it's doable, not a big deal. Actually, S and V are symmetric, you can easily get the drift of dS and dV after you change measure twice.
Best,

You are a genius.I have done it by your help.Thanks a lot.