m******2
发帖数: 564 | 1 还是有个问题啊
就是那种Mean-Reversion的过程怎么去drift
比如
u=(A-Wt)h
这种过程还能用Girsanov去掉随时因Wt变化的drift吗？
Shreve的证明很形式化，但是没有触及到这种本质问题。
比如一个股票几乎永远在10-14元波动，它的期权就一定和同方差的另一支股票一样吗
？Girsanov定理的也适用这种吗？
至少按照probability乘法法则，这种是不适用的，因为dX之间不independent |
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M****i
发帖数: 58 | 2 You are welcome.
Generally speaking, the martingale convergence theorem can not be applied
here because I(t) is only a semimartingale, not a martingale:
I(t)=g(t)*M(t),
where
g(t)=exp(-\int_0^t c(r)dr)
M(t)=\int_0^t c(s)*exp(\int_0^s c(r)dr) dB(s).
Note that g(t) is of finite variation and only M(t) could be a martingale.
This can also be verified by my previous example
c(t)=1/(1+t). In this case I(t)=B(t)/(1+t), which is not a martingale, but
it converges to 0 a.s. by iterated logarithm.
For yo... 阅读全帖 |
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L*******t
发帖数: 2385 | 3 Thanks, 所以是GARCH Option Pricing Model。这个在学界似乎讨论的比较多，发在JF
，RFS，JBF，JFE上的文章页蛮多的。我个人比较喜欢Chorro, Guegan, Ielpo 2012的
文章，因为比较喜欢Levy过程。。。只是CGI2012没有closed form只有MC数值解。
讨论一下GARCH OPM的好处吧，个人觉得离散时间不用受到连续时间Girsanov的限制，
因为在连续时间Girsanov需要innovation的volatility保持不变来维持absolute
continuous。而且离散时间至少在只需要对日终价格进行定价的情况下是一个比较好的
选择。Model时变系数比较方便，最后如果GARCH Option Pricing Model的log return
的似然函数能写出来的话，不用解有closed form就能很方便的calibration。
似乎也有讨论GARCH Diffusion的文章。蛮有意思的。
问个问题，如果我实在没有closed form了，series expansion解算不算？实现起来也
很快的,... 阅读全帖 |
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w**********5
发帖数: 1741 | 4 In probability theory, the Girsanov theorem (named after Igor Vladimirovich
Girsanov) describes how the dynamics of stochastic processes change when the
original measure is changed to an equivalent probability measure.[1]:607
The theorem is especially important in the theory of financial mathematics
as it tells how to convert from the physical measure which describes the
probability that an underlying instrument (such as a share price or interest
rate) will take a particular value or values to t... 阅读全帖 |
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Q***5
发帖数: 994 | 5 Here is one way to prove it, if we are allowed to use some other results
about Brownian motion:
Since B.M. is homogeneous, we only have to prove the conclusion at t = 0.
Proof by contradiction.
Let G(h) = (B(h)-B(0))/h
If the conclusion does not hold, then there a>0 and N>0, such that
prob(G(h)>-N)>0 for any 0
which means that with Prob>0 the Brownian motion B never goes below the line
x=-Nt
on the interval (0,a).
By Girsanov's Thm, W_t = B_t +Nt is a B.M. under an equivalent probability,
an |
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e***x
发帖数: 13 | 6 I use Girsanov to make Xt/b to be a Brownian motion under measure Q, and use
Doob's OST to find the Laplace transform of T, the rest of problem is
trivial. |
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r*****t
发帖数: 286 | 7 ☆─────────────────────────────────────☆
litaihei (李太黑) 于 (Sat Apr 14 12:39:53 2007) 提到:
X(t) = a t + b W(t) , where W(t) is standard Brownian motion.
T = Min(t, X(t) = x ) where x > 0;
What is E(T)
What is Var(T)
☆─────────────────────────────────────☆
expix (yun) 于 (Sat Apr 14 14:04:45 2007) 提到:
I use Girsanov to make Xt/b to be a Brownian motion under measure Q, and use
Doob's OST to find the Laplace transform of T, the rest of problem is
trivial.
☆───────────────────────────────── |
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s*******s
发帖数: 11 | 8 我在国内的时候用过这本书。姜在数理金融界以PDE著名，书内有很多观点都很新；涉
及到了常见产品(主要是Equity Derivative)的定价模型。但我觉得有点不足，是没有
把PDE和Martingale的观点结合起来。另一点就是没有涉及更Exotic的产品定价和对冲
模型。
模型定价领域，自然是要把PDE的理论和方法发展到及至。事实正是如此，在姜的这本
著作中，PDE演化成了一个无比强大的数学工具，无论是简单的欧式期权，还是复杂的
美式期权，亚式期权，和路径
都是用risk neutral方法，虽然有的期权定价也得到了PDE模型，但并没有直接用PDE方
法求解。相对而言，risk neutral方法虽然应用更容易，但需要理解martingale，
change of measure，Girsanov t
你能做到最好，最顶尖，你就成功了。）
高。上海师范大学数理信息学院院长张寄州教授认为该书是“国内外第一本用偏微分方
程方法来建立金融衍生物的定价模型的专著”。 |
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m*******s
发帖数: 758 | 9 Fokker-Planck equation for diffusion process.
不需要Girsanov transforma, change of measure
前面贴的Ito写的Kolmogorov的贡献 早在SDE形式出现之前就提出了
一般Markov process的 Kolmogovov forward/backword equation ,
i.e. FPE.
造成一个
为0
解SDE和用risk neutral解对应问题的关系？ |
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s***n
发帖数: 76 | 10 最近电面了几个公司把问的题目说出来
公司a:
1 what is the assumption for BS model
2 stl concepts
3 difference between forward and option
4 忘了
公司b:
1 pde ode, green function
2 why expectation result is almost as same as pde solution
3 root solver
4 optimization method
5 numerical recipes
6 girsanov theorem
公司c:
1 how to detect cyclic link list
2 hash table for translation a dictionary
3 忘了 |
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r****t
发帖数: 10904 | 11 "So the calculated prices should be the same under both measures. "
why? Can u explain more about this?
to me, after Girsanov the sigma of the BM changed, isn't it?
the
the |
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q********u
发帖数: 53 | 12 Exchange option.
exp(-rt)max(S-P,0)=Pexp(-rt)max(S/P-1,0)
Then std BS with adjust term Pexp(-rt).
Using Girsanov Theorem, the likelihood will be canceled by the expansion of
P. |
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m*********g
发帖数: 646 | 13 RE这个。
我觉得就是Girsanov.在转换成 equivalent risk-neutral measure时，已经充分考虑
了risk premium.
|
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c*********g
发帖数: 154 | 14 我们知道，No arb <=> 对任何numeraire都存在一个probability measure（equivalent to physical probability），使得市场上所有的tradable assets作用这个numeraire之后，在这个probability measure下都是一个martingale。
根据这个fundamental theory，我们可以将bank account作为这个tradable asset，将stock做为这个numeraire，然后将ito rule作用到d(Bt/St)上，并令这个东东的drift为零（martingale），就可以推出这个结论了。当然，需要有Girsanov Theorem做为前提保证。
这个结论非常好用的哦，大家记住吧，哈哈 |
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p*****k
发帖数: 318 | 15 wushine, this seems str8forward by change of measure
(Girsanov) and taking into account the R-N derivative.
not sure which book is the best reference, but Joshi
definitely talked about this exact topic when he
discussed the barrier options |
|
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z****i
发帖数: 406 | 17 d(Xt) = 1/2*(ln2)^2 *Xt dt + ln2*Xt dBt
= ln2*Xt (1/2*ln2 dt + dBt)
Let Wt = 1/2 ln2 *t + Bt, we want Wt to be a standard Brownian motion under
the new probability measure Q. (suppose the original measure is P).
Then by Girsanov,
dQ/dP = exp(-(1/2*ln2)^2*t/2- 1/2*ln2 Bt), which is a martingale under P.
(this is kind of working it out backwards..)
Let me know if this is not correct. |
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d*e
发帖数: 843 | 18 Good point, it's not a martingale, and thanks a lot for all your
thoughts on this question. I'm not very familiar with Girsanov Theorem
though.
A second thought: $I(t)$ is still a supermartingale if we choose $c(s)$
non-negative and satisfyingall the conditions such that $I(t)$
converge in $L^2$ to 0.
Now that $I(t)$ is $L^2$-bounded implies that $I(t)$ is uniformly
integrable. By the MCT, a uniformly integrable supermartingale can be
closed and so $I(t)$ has an almost sure limit. ... 阅读全帖 |
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z****i
发帖数: 406 | 19 话说art rones, 因为他的reading list, 我买了本teaching yourself c++ in 21
days， 然后发现那本书其实不浅。
他每次打电话来就问，你知不知道Ito lemma是用来干嘛的阿， Girsanov 又是什么阿
。
问我，你申过GS的工作没有，我说只申过实习，他说，哦，那没事， 然后他终于把我
简历递到了GS。 结果是GS的小秘联系我去面试，Art满腔委屈的说，因为他们在他们的
数据库里找到了我联系方式，就把他Art给抛弃了。 当然我面试挂了，所以对他其实没
有什么影响 |
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q**********a
发帖数: 4 | 20
geometric
change measure, Girsanov theorem
E_tail(Y) = E(YZ)
Here Z(t) = EXP(integral(sigma*dt)-1/2*integral((sigma^2)*dt))=
((S(t)/S(0))*EXP(integral(rdt)),
Y(t) = log (S(t))
dY = rdt + sigma*dw (under risk neutral)
and under new measure
d_w_tail = dw - sigma*dt
E(YZ) can be written as
E(log(S)*S/S_0*EXP(integral(rdt)) = E_tail(Y)
E(log(S)*S) = E_tail(Y)*discount*S_0
under tail measure
dY = (r + sigma^2)dt + sigma*dw_tail
Y(t) = log(S_0) + (r +sigma^2)t + N(0, sigma^2*t)
E(log(S)*S) = S_0*di... 阅读全帖 |
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M****i
发帖数: 58 | 21 Assume that dS(t)=rS(t)dt+\sigma S(t)dW(t) under risk neutral measure,
then the price process of your claim is
V(t)=S(t)(lnS(t)+(\sigma^2/2+r)(T-t)).
Key points:
1) Formula for conditional expectation under change of measure;
2) Girsanov theorem.
Proof: Let Z(t)=\sigma W(t)-(\sigma^2)t/2,
P'=Z(T)P on F(T), W' is P' Brownian motion.
By risk neutral pricing formula,
V(t)=E[exp(-r(T-t))S(T)lnS(T) | F(t)]
=S(0)exp(rt)E[Z(T)(\sigmaW(T)-(\sigma^2)T/2)+lnS(0)+rT | F(t)]
=S(t)E'[\sigmaW(T)-(\sigma^2)T/2... 阅读全帖 |
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M****i
发帖数: 58 | 22 For 17, I found B(t,T)=exp(-R(t)(T-t)+(T-t)^3/6),
where R(t) is the interest rate process which is assumed to be a BM under
risk neutral measure. It seems that this question is a special case of the
bond price in Hull-White model, so the solution is in fact direct. Otherwise
, risk neutral pricing formula + Girsanov give the same result, but seems
less direct. |
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l*******l
发帖数: 248 | 23 这ID，哈哈，我都想叫Girsanov了lol |
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s****r
发帖数: 2386 | 24 Anyone knows Girsanov died from rock/mountain climbing, with a stoch
calculus book in his bag? |
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s****r
发帖数: 2386 | 25 lol, Shreve's academic grandpa got his PhD roughly when Girsanov died |
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l****o
发帖数: 2909 | 26 check the property of a special form of girsanov's SDE's solution. |
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L**********u
发帖数: 194 | 27 我更晕，你在装糊涂还是真糊涂。
这可是很重要的东西。
Girsanov定理种filtration是任意的都可以，
但是martingale 表示定理中filtration必须要是由W（t）生成的。 |
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N*******D
发帖数: 4 | 28 Under what measure do you make your argument?
In real measure, a stock with downward jump must be compensated with higher
expected return for the downward jump risk, so the probability of (ST > K )
are not necessarily smaller.
In the tradition risk neutral measure, things must be more complicated with
jumps, since Girsanov theorem did not mention jump. I don't know any
theories about jump yet, but I feel the risk neutral measure with jumps
might be different.
jump |
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S*********g
发帖数: 5298 | 29 哈哈，第一个问题是ito，第二个是girsanov theorem
是吧？
他跑单帮的，跟谁都问这俩问题，都说投GS，给reading list
不过他那个readling list还挺基本的，看看没坏处。 |
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d********t
发帖数: 9628 | 30 我太土了，本来以为还知道得挺多的，连girsanov都不懂，这下更加绝望了。 |
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Q***5
发帖数: 994 | 31 数学上， change of numeraire 不过是Girsanov 定理及那个不起眼的Lemma5.2.2 的
简单推论。Shreve单开一章讲这一技巧，已经挺对得起它了。
另外， section 9.3.6 GK formula 难道不是讲 FX option吗？
个人觉得 change of numeraire 最重要的应用是forward LIBOR model。 |
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r**m
发帖数: 7 | 32 Let w_t be the brownian motion under the original risk-neutral measure Q
_0, and \tilde(w)_t be the brownian motion under Q_s(with s(t) as
numeraire). Change the probability measure from Q_0 to Q_s using Girsanov
theorem with exp(-1/2\sigma^2t+\sigma w_t) as the Radon-Nikodym
derivative.
We have \tilde(w_t) = w_t -\sigma t. Therefore, the original drift rt
becomes (r+sigma^2)t. |
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g******r
发帖数: 29 | 34
觉得是用 Girsanov 定理
若 Z_t 是一个 P 上面的 exponential martingale
考虑 dQ / dP = Z_t
而且有 Z_t in P ~ 1/Z_t in Q
E_p[ f ( Z_t ) ] = E_q[ f(Z_t) * 1/Z_t ] = E_p[ Z_t f( 1/Z_t )] |
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m******2
发帖数: 564 | 35 听说Shreve80年代搞了一个简单的Probability Ratio相乘的证明不够严谨
严谨的证明必须用Martigale 方法，请问哪里有详细的Martingale方法证明啊？
我特别想知道这两种方法的比较，后者严谨在哪里？我怎么觉得后者在脱了裤子放屁？ |
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h*****u
发帖数: 204 | 36
Shreve的书证明了一维的啊 用的是martingale的方法啊 |
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k*******d
发帖数: 1340 | 38 有个猎头就喜欢问你知道Girsanov Theorem吗？知道Ito formula干啥的吗？
也就他会问，其他的都不会问 |
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f********k
发帖数: 136 | 39 It's very important. Actually, the Risk Neutral Pricing is essentially based
on the idea of "change of measure" and Girsanov Theorem, which changes the
physical measure to risk neutral measure.
If you are going to learn derivative pricing stuffs, you will inevitably
need to deal a lot of "measure theory" related topics. For example, for
interest or commodity products, you need to understand what's "forward
measure". For FX products, a very popular technique is "change of numeria",
which is again... 阅读全帖 |
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f********k
发帖数: 136 | 40 It's very important. Actually, the Risk Neutral Pricing is essentially based
on the idea of "change of measure" and Girsanov Theorem, which changes the
physical measure to risk neutral measure.
If you are going to learn derivative pricing stuffs, you will inevitably
need to deal a lot of "measure theory" related topics. For example, for
interest or commodity products, you need to understand what's "forward
measure". For FX products, a very popular technique is "change of numeria",
which is again... 阅读全帖 |
|
h*****2
发帖数: 16 | 41 True... importance sampling is an application of Girsanov theorem |
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