b***k
发帖数: 2673 | 1 ☆─────────────────────────────────────☆
yayadoudou (yayadou) 于 (Wed Jun 18 22:17:54 2008) 提到:
算一个option value, option is perpetual and written on a stock,
stock current price $80, if the stock price exceeds $100,
the owner gets $100 and the option is closed; otherwise, continue
make assumption of any model or interest rate;
provide a reasonable soln for the option value。
do not know how to solve it, anyone has good ideas?
Thanks!
☆─────────────────────────────────────☆
Jadeson (Jadeson) | d*******n
发帖数: 524 | 2 Have no idea what you guys were talking about here.
So I just derived it myself.
Here I use the lowercase s to represent the logarithm of stock price:
s = ln(S)
and use v to represent the value of the derivative.
then under GBM assumption for the stock price:
s = (r-1/2sigma^2)t + sigma*W
which is a biased Brownian Motion.
(Note: this W is a BM only in the risk-neutral probability space,
not the real probability space. All the following BMs,
expectations and Martingales are in the risk-neutral s | d*******n
发帖数: 524 | 3 看来一堆人的讨论，觉得蛮搞笑的，没有几个搞清楚了问题是什么的。。。。。。
如果你知道risk-neutral pricing的话，这个问题就是最基本的随机微积分的作业题，
求一下E[100*exp(-r*tau)]，什么高深讲义都不要，就是最最基本的几个理论。
【在 b***k 的大作中提到】
: ☆─────────────────────────────────────☆
: yayadoudou (yayadou) 于 (Wed Jun 18 22:17:54 2008) 提到:
: 算一个option value, option is perpetual and written on a stock,
: stock current price $80, if the stock price exceeds $100,
: the owner gets $100 and the option is closed; otherwise, continue
: make assumption of any model or interest rate;
: provide a reasonable soln for the option value。
: do not know how to solve it, anyone has good ideas?
: Thanks!
: ☆─────────────────────────────────────☆
| t**********a
发帖数: 166 | 4 Your answer seems amusing too, how can you say others not knowing the
problem?
The big question is: why are you assuming GBM, anywhere in the problem
mentioned that? Is the answer going to be different if it is not GBM?
Actually the answer is model-independent ...
【在 d*******n 的大作中提到】
: 看来一堆人的讨论，觉得蛮搞笑的，没有几个搞清楚了问题是什么的。。。。。。
: 如果你知道risk-neutral pricing的话，这个问题就是最基本的随机微积分的作业题，
: 求一下E[100*exp(-r*tau)]，什么高深讲义都不要，就是最最基本的几个理论。
| J*****n
发帖数: 4859 | 5
你是不是觉得你随机积分很牛B阿？
你又搞清楚了什么呢？
【在 d*******n 的大作中提到】
: 看来一堆人的讨论，觉得蛮搞笑的，没有几个搞清楚了问题是什么的。。。。。。
: 如果你知道risk-neutral pricing的话，这个问题就是最基本的随机微积分的作业题，
: 求一下E[100*exp(-r*tau)]，什么高深讲义都不要，就是最最基本的几个理论。
| Q***5
发帖数: 994 | 6 Just wonder, can you prove this? I think this is also one of your earlier
questions: Jadeson's argument only show that this option's price can not be
greater than 80, but can we prove that it can not be less than 80 either?
Consider the following weird law: between 0 and some time T>0, S is GBM. For
any t>T, S(t)=S(T) (i.e., sigma drop to 0 at T) -- also we assume r=0 to
eliminate arbitrage possibility.
It seems that, for this kind of law of nature, the option is the same as an
option with matur | t**********a
发帖数: 166 | 7 As I said, you need r=0 and allowing infinite shorting. Then for the lower
bound, you can short 1 share, this should allow you buy the option. You get
paid from the option to clear the short position on underline.
be
For
an
【在 Q***5 的大作中提到】
: Just wonder, can you prove this? I think this is also one of your earlier
: questions: Jadeson's argument only show that this option's price can not be
: greater than 80, but can we prove that it can not be less than 80 either?
: Consider the following weird law: between 0 and some time T>0, S is GBM. For
: any t>T, S(t)=S(T) (i.e., sigma drop to 0 at T) -- also we assume r=0 to
: eliminate arbitrage possibility.
: It seems that, for this kind of law of nature, the option is the same as an
: option with matur
| Q***5
发帖数: 994 | 8 Assuming r=0 is fine, not quite sure about infinite shorting: that means you
can borrow and keep a stock forever -- that is an arbitrage by itself: you
can sell the stock, enjoy the proceeds and don't have to worry about
returning the 'borrowed' stock.
【 from tradingGamma: 】
As I said, you need r=0 and allowing infinite shorting. Then for the lower
bound, you can short 1 share, this should allow you buy the option. You get
paid from the option to clear the short position on underline. | m******c
发帖数: 55 | 9 哈哈，这道题目的争论太好玩了，大家都是牛人.
首先是永久期权, 所以stock一定会超过100块, 在time t.
所以option writer 在t 时间 一定要付出100块.
那在t时间的100块的current 价格是多少呢. 在这道题目没有给出任何其他条件的情况
下, current $80的股票在t时候已经是100块了, 所以t 时间的100块是current的80块.
我同意Jadeson 的答案，思路简单明了. 一道面试问题，用得着那么复杂吗 | Q***5
发帖数: 994 | 10 I agree with you, as an interview question, the interviewer is likely
expecting a simple argument as made by Jadeson.
But, it is quants' duty to make things unnecessarily complicated, for the
sake of job security if nothing else. If you make your model so simple, that
even your CEO understands, then you should always prepare for that
undesireble freedom he will grant you at any time. Just Kidding.
Jadeson's argument makes sense -- but as tradingGamma and I argued, that
argument only shows that t
【在 m******c 的大作中提到】
: 哈哈，这道题目的争论太好玩了，大家都是牛人.
: 首先是永久期权, 所以stock一定会超过100块, 在time t.
: 所以option writer 在t 时间 一定要付出100块.
: 那在t时间的100块的current 价格是多少呢. 在这道题目没有给出任何其他条件的情况
: 下, current $80的股票在t时候已经是100块了, 所以t 时间的100块是current的80块.
: 我同意Jadeson 的答案，思路简单明了. 一道面试问题，用得着那么复杂吗
| | | s*******d
发帖数: 64 | 11 discounted stock price should be a martingale under the risk neutral measure
due to a no arbitrage argument.
Then it is a martingale bounded by 0 and 100e(-r*tau) under Q conditioning
on tau.
Use OST the Q(S_tau=100)=80/(0+100e(-r*tau))=0.8e(r*tau) conditioning on tau
Then the value of the options is
E^Q(e(-r*tau)E^Q(payoff|tau))=E^Q(e(-r*tau)*100*0.8e(r*tau))=80
Then a trial to set it free from GBW model | i********y
发帖数: 346 | 12 The result is surely independent on r(Q measure) or sigma(P measure). Using
Q measure, value of option= E(100exp(-rt)), Using P measure, the value is E(
100exp(-sigma*t)) . Both lead to 80
that
exactly
to
【在 Q***5 的大作中提到】
: I agree with you, as an interview question, the interviewer is likely
: expecting a simple argument as made by Jadeson.
: But, it is quants' duty to make things unnecessarily complicated, for the
: sake of job security if nothing else. If you make your model so simple, that
: even your CEO understands, then you should always prepare for that
: undesireble freedom he will grant you at any time. Just Kidding.
: Jadeson's argument makes sense -- but as tradingGamma and I argued, that
: argument only shows that t
| p******i
发帖数: 1358 | | Q***5
发帖数: 994 | 14 That is given.
The problem now is: whether the result is independent of the BS model
assumption. That's, whether the conclusion is still true if you do not
assume that the stock price S is geometric B.M. with determinstic sigma.
Jadeson's argument does not rely on the BS model assumption, but it seems
that it only shows that the option price does not exceed 80, it is not very
clear from that argument why the option price should be exactly 80.
Using
E(
【在 i********y 的大作中提到】
: The result is surely independent on r(Q measure) or sigma(P measure). Using
: Q measure, value of option= E(100exp(-rt)), Using P measure, the value is E(
: 100exp(-sigma*t)) . Both lead to 80
:
: that
: exactly
: to
| i********y
发帖数: 346 | 15 I believe it has nothing to do with BS model. Let's change the problem that
the investor can get $90 if the stock price reaches $100. Then the answer
should be 80*90/100=$72, meaning the value of the option now is $72. And one
can still hedge by shorting a stock at $80. When the stock reaches $100, he
receives $90. Then he uses the received $90 plus another $10 from the
proceeds of early shorting to by the stock and hence closes out the position
. Note here "another $10 from the proceeds of earl
【在 Q***5 的大作中提到】
: That is given.
: The problem now is: whether the result is independent of the BS model
: assumption. That's, whether the conclusion is still true if you do not
: assume that the stock price S is geometric B.M. with determinstic sigma.
: Jadeson's argument does not rely on the BS model assumption, but it seems
: that it only shows that the option price does not exceed 80, it is not very
: clear from that argument why the option price should be exactly 80.
:
: Using
: E(
| Q***5
发帖数: 994 | 16 I don't follow your logic in:
It is true (at least in BS model) that under the risk neutral measure, no
arbitrage assumption implies that the AVERAGE growth rate of the stock is
the same as the risk free rate -- but that is on average, it does not mean
that every time the stock grows from 80 to 100, your $10 (actually -$10)
proceeds from the short position discouted to $8 at time 0.
For the example you give, you can make a simple argument as Jadeson's: If
you sell a call option for $72 today, y | Q***5
发帖数: 994 | 17 Consider the following (wired) world:
Interest rate 0;
A stock is currently 80. The next day, it will take either 120 or 40 (both
with prob>0). After that, on the n-th day, it will be either 120 (1.05)^n or
40(0.95)^n. (with prob>0 for both cases)
Such a world is arbitrage free, because you can make S(n) an martingale.
In this world, that option price (which can be calculated using risk neutral
measure) is no greater than 80*100/120: I can sell the option for 80*100/
120, buy 100/120 unit of the | Q***5
发帖数: 994 | 18 I think your argument makes sense. (How can I forget to use OST!!)
So, I guess the following more general conditions will gurantee the exact
option price of 80:
Existence of equivalent risk neutral measure;
Continous path;
Conditions for OST to hold.
measure
tau
【在 s*******d 的大作中提到】
: discounted stock price should be a martingale under the risk neutral measure
: due to a no arbitrage argument.
: Then it is a martingale bounded by 0 and 100e(-r*tau) under Q conditioning
: on tau.
: Use OST the Q(S_tau=100)=80/(0+100e(-r*tau))=0.8e(r*tau) conditioning on tau
: Then the value of the options is
: E^Q(e(-r*tau)E^Q(payoff|tau))=E^Q(e(-r*tau)*100*0.8e(r*tau))=80
: Then a trial to set it free from GBW model
| l******g
发帖数: 81 | 19 hehe
其实还有个隐含的条件的，就是price是连续的
如果tick size 是 30, 就不是80了，：）
【在 p******i 的大作中提到】
: 好多牛人，好多牛逼的解答
|
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