对dS/S = r dt + sigma dz^Q积分 [推广有奖]

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schwereburg 发表于 2013-4-9 03:49:56 |AI写论文

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lns=rT+sigma*Z
S=exp(rT+sigma*Z)

对吗？心里不是很踏实

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关键词：Sigma GMA EXP 积分 sigma

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Chemist_MZ 发表于3楼查看完整内容

Sofa is correct. You miss the Ito term, which is 0.5*sigma^2 The main point here is that, for common deterministic function, the quadratic variation is zero, since they are so-called "Smooth" function, but for Brown Motion, the quadratic variation is not zero, which means you need to include the second order term when you do the stochastic differentiation. The right way to solve this SDE, ...

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shelf317 发表于 2013-4-9 04:24:42

assuming r and sigma are constant,
the solution should be
S_T = S_0 * exp( (r-1/2 sigma^2)T + sigma * W_T), where W_T is a standard brownian motion.

Check the wiki page on GBM at
http://en.wikipedia.org/wiki/Geometric_Brownian_motion

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Chemist_MZ

发表于 2013-4-9 06:34:24

Sofa is correct.

You miss the Ito term, which is 0.5*sigma^2

The main point here is that, for common deterministic function, the quadratic variation is zero, since they are so-called "Smooth" function, but for Brown Motion, the quadratic variation is not zero, which means you need to include the second order term when you do the stochastic differentiation.

The right way to solve this SDE,is doing transform y=lnS, then apply Ito formula to y with respect to S.

It's done.