s***e
发帖数: 267 | 1 Try to write some notes while learning some new stuff. Here is one way to
derive BS without stochastic
calculus:
http://shede.wordpress.com/2010/02/21/bs/ |
t********t
发帖数: 1264 | 2 By the arbitrage theorem....
其实你这样推导并不是不需stochastic calculus，而是所有需要的地方都省略在上面
这句话里了 |
s***e
发帖数: 267 | 3 My understanding is that arbitrage theorem only utilizes the duality theorem
in linear programming, and it does not involve stochastic calculus...
【在 t********t 的大作中提到】
: By the arbitrage theorem....
: 其实你这样推导并不是不需stochastic calculus，而是所有需要的地方都省略在上面
: 这句话里了
|
m******n
发帖数: 354 | 4 Yes, your approach does not need stochastic calculus.
But you posit that the arbitrage price must be some expectation price.
This assumption is only satisfied under the martingale measure. To
understand that, one needs stochastic calculus.
You just hide that. |
t********t
发帖数: 1264 | 5 then you have to show that the option is a riskless asset, with the help of
dynamic hedging, in order to apply the arbitrage theorem. This part is not
trivial. Actually it is the most important part in BS derivation. |
s***e
发帖数: 267 | 6 Do you mean the statement that
C = E[exp(-rt) * (s_t - K)_+] ?
It seems to me that:
1) The option value is a deterministic function of the s_t, the stock price
at expiration. In particular, its value is (s_t - K)_+;
2) Thus the risk-neutral value of the option will be its expectation, since
s_t is random (after adjusting interests), so we have the above formula.
I still don't see where you need stochastic calculus for the derivation.
From (1) to (2) it seems to me that the logic is a simple expe
【在 t********t 的大作中提到】
: then you have to show that the option is a riskless asset, with the help of
: dynamic hedging, in order to apply the arbitrage theorem. This part is not
: trivial. Actually it is the most important part in BS derivation.
|
s***e
发帖数: 267 | 7 Well, I don't know stochastic calculus (haven't gotten there yet), but I
feel that I understood the logic behind the derivation. To make it risk-
neutral, you need to have the value of option match that expectation (since
for any given s_t, its value is determined) wrt s_t's distribution. Not sure
which part I got wrong.
【在 m******n 的大作中提到】
: Yes, your approach does not need stochastic calculus.
: But you posit that the arbitrage price must be some expectation price.
: This assumption is only satisfied under the martingale measure. To
: understand that, one needs stochastic calculus.
: You just hide that.
|
S*********g
发帖数: 5298 | 8 How do you define risk neutral value?
How do you proof there is such a risk neutral value?
How do you proof the risk neutral value should equal to the price?
price
since
logic,
solution
【在 s***e 的大作中提到】
: Do you mean the statement that
: C = E[exp(-rt) * (s_t - K)_+] ?
: It seems to me that:
: 1) The option value is a deterministic function of the s_t, the stock price
: at expiration. In particular, its value is (s_t - K)_+;
: 2) Thus the risk-neutral value of the option will be its expectation, since
: s_t is random (after adjusting interests), so we have the above formula.
: I still don't see where you need stochastic calculus for the derivation.
: From (1) to (2) it seems to me that the logic is a simple expe
|
s***e
发帖数: 267 | 9 Well, I guess you are saying similar to what was said earlier by others, I
need to construct the delta-neutral approach to prove that if the call price
does not follow that equation, then there is arbitrage.
However, I still don't think I need that approach. Consider the following
simplest example. If you toss a FAIR coin and it is head, you get $2, and if
it is tail, you lose $1. What is the option price to play one game?
Well, it is the expected value, which is $0.5. Why? otherwise, by law of
【在 S*********g 的大作中提到】
: How do you define risk neutral value?
: How do you proof there is such a risk neutral value?
: How do you proof the risk neutral value should equal to the price?
:
: price
: since
: logic,
: solution
|
S*********g
发帖数: 5298 | 10 This is NOT a good example because you cannot hedge this gamble.
Your argument in this case has nothing to do with risk neutral/delta hedge.
Let me ask you this question.
What is the price of a call option with strik price 1 if the stock has 0.5
probability to be 2 and 0.5 probability to be 0.5? Assume risk-free rate is
zero.
What is the price if it has 0.7 probability to be 2 and .3 probability to be
0.5?
price
if
you
,
【在 s***e 的大作中提到】
: Well, I guess you are saying similar to what was said earlier by others, I
: need to construct the delta-neutral approach to prove that if the call price
: does not follow that equation, then there is arbitrage.
: However, I still don't think I need that approach. Consider the following
: simplest example. If you toss a FAIR coin and it is head, you get $2, and if
: it is tail, you lose $1. What is the option price to play one game?
: Well, it is the expected value, which is $0.5. Why? otherwise, by law of
|
|
|
s***e
发帖数: 267 | 11 Ok, let me try using the expectation rule.
The first case, price of the option should be 0.5*(2-1) + 0.5*0=0.5;
for the secon case, its price should be 0.7*(2-1)+0.3*0=0.7.
is
be
【在 S*********g 的大作中提到】
: This is NOT a good example because you cannot hedge this gamble.
: Your argument in this case has nothing to do with risk neutral/delta hedge.
: Let me ask you this question.
: What is the price of a call option with strik price 1 if the stock has 0.5
: probability to be 2 and 0.5 probability to be 0.5? Assume risk-free rate is
: zero.
: What is the price if it has 0.7 probability to be 2 and .3 probability to be
: 0.5?
:
: price
|
S*********g
发帖数: 5298 | 12 hehe, you did not use risk neutral and your answer is not correct.
Please read page 60-63 of http://stat-www.berkeley.edu/users/evans/shreve.pdf
(Chapter 3)
【在 s***e 的大作中提到】
: Ok, let me try using the expectation rule.
: The first case, price of the option should be 0.5*(2-1) + 0.5*0=0.5;
: for the secon case, its price should be 0.7*(2-1)+0.3*0=0.7.
:
: is
: be
|
T*******t
发帖数: 9274 | 13 你现在很活跃啊
【在 S*********g 的大作中提到】
: hehe, you did not use risk neutral and your answer is not correct.
: Please read page 60-63 of http://stat-www.berkeley.edu/users/evans/shreve.pdf
: (Chapter 3)
|
S*********g
发帖数: 5298 | 14 晚上不玩游戏了，闲的慌，呵呵
【在 T*******t 的大作中提到】
: 你现在很活跃啊
|
T*******t
发帖数: 9274 | 15 不做超弦，也来花街混了？
【在 S*********g 的大作中提到】
: 晚上不玩游戏了，闲的慌，呵呵
|
S*********g
发帖数: 5298 | 16 算是在这个行业里混饭吃了。
不过我们不在街上，在乡下
【在 T*******t 的大作中提到】
: 不做超弦，也来花街混了？
|
u********e
发帖数: 263 | 17 超弦兄在哪个乡下啊
【在 S*********g 的大作中提到】
: 算是在这个行业里混饭吃了。
: 不过我们不在街上，在乡下
|
S*********g
发帖数: 5298 | 18 DC附近的乡下
【在 u********e 的大作中提到】
: 超弦兄在哪个乡下啊
|
T*******t
发帖数: 9274 | 19 二房？
【在 S*********g 的大作中提到】
: DC附近的乡下
|
S*********g
发帖数: 5298 | 20 不是不是。
上去逛mall迷路的时候看到过二房的大楼。
我在一小hedge fund混饭吃。
【在 T*******t 的大作中提到】
: 二房？
|
|
|
T*******t
发帖数: 9274 | 21 wa....DC附近，肯定是做政治盘的
【在 S*********g 的大作中提到】
: 不是不是。
: 上去逛mall迷路的时候看到过二房的大楼。
: 我在一小hedge fund混饭吃。
|
S*********g
发帖数: 5298 | 22 哈哈，这个想法不错，以后可以试试
上次tiger woods一讲话，电视里trading floor里的人都不动了，volume噌就下来了
obama一讲话，指数就跌下来了，他一讲完，指数就弹回去了。
【在 T*******t 的大作中提到】
: wa....DC附近，肯定是做政治盘的
|
u********e
发帖数: 263 | 23 DC 附近的hf，有巴马罩着，肯定没问题啊。
【在 S*********g 的大作中提到】
: 不是不是。
: 上去逛mall迷路的时候看到过二房的大楼。
: 我在一小hedge fund混饭吃。
|
a**n
发帖数: 3801 | 24 概念上完全错了啊
嘿嘿
压根没理解risk neutral measure啥意思
price
if
you
,
【在 s***e 的大作中提到】
: Well, I guess you are saying similar to what was said earlier by others, I
: need to construct the delta-neutral approach to prove that if the call price
: does not follow that equation, then there is arbitrage.
: However, I still don't think I need that approach. Consider the following
: simplest example. If you toss a FAIR coin and it is head, you get $2, and if
: it is tail, you lose $1. What is the option price to play one game?
: Well, it is the expected value, which is $0.5. Why? otherwise, by law of
|
x**y
发帖数: 10012 | 25 我对risk neutral measuer也是一知半解的
请指点
【在 a**n 的大作中提到】
: 概念上完全错了啊
: 嘿嘿
: 压根没理解risk neutral measure啥意思
:
: price
: if
: you
: ,
|
S*********g
发帖数: 5298 | 26 最简单的例子看我前面给的那个link里的第3章
【在 x**y 的大作中提到】
: 我对risk neutral measuer也是一知半解的
: 请指点
|
x**y
发帖数: 10012 | 27 你发了好多
【在 S*********g 的大作中提到】
: 最简单的例子看我前面给的那个link里的第3章
|
a**n
发帖数: 3801 | 28 你现在咋理解的啊
【在 x**y 的大作中提到】
: 你发了好多
|
s***e
发帖数: 267 | 29 I guess what the "meaning" of risk neutral is that you can hedge without
loss, for any result. The coin example
if you play once then you cannot guarantee to win, you only win if playing
multiple times.
Not sure whether such "risk neutral" has any practical meaning in real world
. In my opinion, as long as you can
win as you play more, then you make money...
【在 a**n 的大作中提到】
: 概念上完全错了啊
: 嘿嘿
: 压根没理解risk neutral measure啥意思
:
: price
: if
: you
: ,
|
a**n
发帖数: 3801 | 30 看来arbitrage啥意思都没搞懂。。。
arbitrage跟expectation都没关系，别说大数定理了
world
【在 s***e 的大作中提到】
: I guess what the "meaning" of risk neutral is that you can hedge without
: loss, for any result. The coin example
: if you play once then you cannot guarantee to win, you only win if playing
: multiple times.
: Not sure whether such "risk neutral" has any practical meaning in real world
: . In my opinion, as long as you can
: win as you play more, then you make money...
|
|
|
s***e
发帖数: 267 | 31 Thanks a lot. Don't know this book as pdf version, hehe.
【在 S*********g 的大作中提到】
: hehe, you did not use risk neutral and your answer is not correct.
: Please read page 60-63 of http://stat-www.berkeley.edu/users/evans/shreve.pdf
: (Chapter 3)
|
S*********g
发帖数: 5298 | 32 15087
【在 x**y 的大作中提到】
: 你发了好多
|
s***e
发帖数: 267 | 33 我现在的理解就是不论结果如何，总能gain profit，我并没有说它和概率有关。
只是如果概率知道，算expectation就可以构造实际可以稳定盈利的系统，这个应该算
是更广义更有实际价值的吧。
【在 a**n 的大作中提到】
: 看来arbitrage啥意思都没搞懂。。。
: arbitrage跟expectation都没关系，别说大数定理了
:
: world
|
m******n
发帖数: 354 | 34 Okay, let's talk about the very first thing:
The strong law of large numbers does not work for capital market at all!
Consider a forward: with the same underlying stock price process, the arbitrage(-free) price is not the expectation price(under real probabilty)!
But the arbitrage(-free) price should be the enforced market price! So we
want to change it to some expectation, so that we can calculate it using
stochastic tools generally.
To this end, we do change of measure (here: the probability m |
t********t
发帖数: 1264 | 35 That's the loophole in your derivation. Although you happen to get the same
result, but the logic is not right. LLN is irrelavent for option pricing.
In the coin tossing example you made, you can only be guaranteed to break
even on average. But in every single bet, you may win or lose and are not
risk-free at all. Nobody (who is risk-averse) is gonna bet with you if he
can only expect to earn the risk-free interest rate. It is where the risk
preference of the investor will play a role.
In option
【在 s***e 的大作中提到】
: Well, I guess you are saying similar to what was said earlier by others, I
: need to construct the delta-neutral approach to prove that if the call price
: does not follow that equation, then there is arbitrage.
: However, I still don't think I need that approach. Consider the following
: simplest example. If you toss a FAIR coin and it is head, you get $2, and if
: it is tail, you lose $1. What is the option price to play one game?
: Well, it is the expected value, which is $0.5. Why? otherwise, by law of
|
s***e
发帖数: 267 | 36 Thanks, very helpful. I think I got it.
arbitrage(-free) price is not the expectation price(under real probabilty)!
measure,
,
【在 m******n 的大作中提到】
: Okay, let's talk about the very first thing:
: The strong law of large numbers does not work for capital market at all!
: Consider a forward: with the same underlying stock price process, the arbitrage(-free) price is not the expectation price(under real probabilty)!
: But the arbitrage(-free) price should be the enforced market price! So we
: want to change it to some expectation, so that we can calculate it using
: stochastic tools generally.
: To this end, we do change of measure (here: the probability m
|
s***e
发帖数: 267 | 37 Got it. I think I start from the very beginning I was thinking the buy-side
thing and equal "arbitrage" with "gain profit", which is wrong. So no
arbitrage means you can hedge risk-free no matter what results.
On the other hand, because the no-arbitrage way to price derivatives, I felt
that there is systematic ways/holes to make profit by utilizing it. For
example, in the SuperString's example, the two options have the same price.
Then why don't you buy one and sell the other if that is the cas
【在 t********t 的大作中提到】
: That's the loophole in your derivation. Although you happen to get the same
: result, but the logic is not right. LLN is irrelavent for option pricing.
: In the coin tossing example you made, you can only be guaranteed to break
: even on average. But in every single bet, you may win or lose and are not
: risk-free at all. Nobody (who is risk-averse) is gonna bet with you if he
: can only expect to earn the risk-free interest rate. It is where the risk
: preference of the investor will play a role.
: In option
|
t********t
发帖数: 1264 | 38 In superstring's example, of course you can build such portfolio and expect
to win in the long run. But remember this is not Risk-Free Arbitrage. If you
regard this as a bet between two and you are the one side, then the other
side is not the option seller, but the stock market. What you earn is from
the stock market, not from the seller. In some degree, the sell side just
tranfer the risk from the stock market to the buyer, by selling the option
and delta hedging it.
If you as well as many can
【在 s***e 的大作中提到】
: Got it. I think I start from the very beginning I was thinking the buy-side
: thing and equal "arbitrage" with "gain profit", which is wrong. So no
: arbitrage means you can hedge risk-free no matter what results.
: On the other hand, because the no-arbitrage way to price derivatives, I felt
: that there is systematic ways/holes to make profit by utilizing it. For
: example, in the SuperString's example, the two options have the same price.
: Then why don't you buy one and sell the other if that is the cas
|
p****u
发帖数: 2596 | 39 看不懂的人飘过:(
【在 s***e 的大作中提到】
: Try to write some notes while learning some new stuff. Here is one way to
: derive BS without stochastic
: calculus:
: http://shede.wordpress.com/2010/02/21/bs/
|
s***e
发帖数: 267 | 40 I think you assume the option seller is sell option for hedge, not for
speculation. In real world most option
traders trade to speculate. And in SuperString's example, I don't see how
the stock can correct itself.
Now it seems there is this no-arbitrage price, and also there is this "fair
price" if we define it by the
expected value of the derivative. If you can find something cheaper than
fair price (or vice versa), then you
can consistently make profit by doing the opposite (no hedge). I guess
【在 t********t 的大作中提到】
: In superstring's example, of course you can build such portfolio and expect
: to win in the long run. But remember this is not Risk-Free Arbitrage. If you
: regard this as a bet between two and you are the one side, then the other
: side is not the option seller, but the stock market. What you earn is from
: the stock market, not from the seller. In some degree, the sell side just
: tranfer the risk from the stock market to the buyer, by selling the option
: and delta hedging it.
: If you as well as many can
|
|
|
s***e
发帖数: 267 | 41 看不懂你也比我强，呵呵，因为这个推倒错的离谱，已经被n多人指出来了。搞了半天
最基本的概念都没有弄清楚，不过还
是很惊奇，怎么这么巧呢，呵呵
【在 p****u 的大作中提到】
: 看不懂的人飘过:(
|
S*********g
发帖数: 5298 | 42 You are wrong again.
The mistake in your derivation is in the formula mu=r-sigma^2/2.
mu is independent of r and sigma in real world.
Your comment on hedger vs speculator does not make sense either.
fair
this
【在 s***e 的大作中提到】
: I think you assume the option seller is sell option for hedge, not for
: speculation. In real world most option
: traders trade to speculate. And in SuperString's example, I don't see how
: the stock can correct itself.
: Now it seems there is this no-arbitrage price, and also there is this "fair
: price" if we define it by the
: expected value of the derivative. If you can find something cheaper than
: fair price (or vice versa), then you
: can consistently make profit by doing the opposite (no hedge). I guess
|
s***e
发帖数: 267 | 43 Yes the mu part in the derivation is wrong, now I totally agree with on that
. But did you see how I got the
result? It is derived by using the "fair price" principle I mentioned. I am
just trying to see that this may not be
a co-incidence, as I did not arbitrarily pick up a value.
As for hedge vs speculation, I think I know what they means as I have
extensive experience in trading options.
【在 S*********g 的大作中提到】
: You are wrong again.
: The mistake in your derivation is in the formula mu=r-sigma^2/2.
: mu is independent of r and sigma in real world.
: Your comment on hedger vs speculator does not make sense either.
:
: fair
: this
|
T*******t
发帖数: 9274 | 44 shede真可怜啊
【在 S*********g 的大作中提到】
: You are wrong again.
: The mistake in your derivation is in the formula mu=r-sigma^2/2.
: mu is independent of r and sigma in real world.
: Your comment on hedger vs speculator does not make sense either.
:
: fair
: this
|
Q***5
发帖数: 994 | 45 It's kind of - * - = +. There are 2 problems in your derivation: 1. under
real world prob, the expected return of stock does not equal risk free rate.
2. to calculate option price, the risk neutral prob should be used, not the
real world measure.
But when you made these 2 mistakes simultaneously, something wonderful
happens: you accidently used the risk neutral measure (although you thought
it is real world measure), and when you take expectation w.r.t this "real
world measure", you essentially
【在 s***e 的大作中提到】
: Yes the mu part in the derivation is wrong, now I totally agree with on that
: . But did you see how I got the
: result? It is derived by using the "fair price" principle I mentioned. I am
: just trying to see that this may not be
: a co-incidence, as I did not arbitrarily pick up a value.
: As for hedge vs speculation, I think I know what they means as I have
: extensive experience in trading options.
|
S*********g
发帖数: 5298 | 46 Good luck on your option tradings.
that
am
【在 s***e 的大作中提到】
: Yes the mu part in the derivation is wrong, now I totally agree with on that
: . But did you see how I got the
: result? It is derived by using the "fair price" principle I mentioned. I am
: just trying to see that this may not be
: a co-incidence, as I did not arbitrarily pick up a value.
: As for hedge vs speculation, I think I know what they means as I have
: extensive experience in trading options.
|
s***e
发帖数: 267 | 47 I think your explanation is correct. Indeed I learned a lot after the
previous discussions. Thanks for the encouragement!
rate.
the
thought
【在 Q***5 的大作中提到】
: It's kind of - * - = +. There are 2 problems in your derivation: 1. under
: real world prob, the expected return of stock does not equal risk free rate.
: 2. to calculate option price, the risk neutral prob should be used, not the
: real world measure.
: But when you made these 2 mistakes simultaneously, something wonderful
: happens: you accidently used the risk neutral measure (although you thought
: it is real world measure), and when you take expectation w.r.t this "real
: world measure", you essentially
|
t********t
发帖数: 1264 | 48 建议你还是把risk-neutral pricing看看想明白了再讨论speculation的问题，要不讨
论起来很麻烦
我前面说的意思是，只讲理想状态，就是在买方卖方模型中，sell-side不跟市场赌，
他是个卖服务的介质，一边卖option给别人，另一边从市场买卖股票做delta-hedge从
而把风险从一端完全传导给另一端。所以如果你从它卖的option里发现统计意义上的获
利机会，你不是从sell-side获利，而是通过它这个介质从市场获利，sell side完全保
持中性，只赚价差，不参与你跟市场的赌博。而统计意义上获利机会的存在会在长期影
响股票供需直至机会消失。通过如下方法影响供需，你build a portfolio with long
option A and short option B, the sell side needs to long stock A and short
stock B to hedge the opposite position. Driven by such supply and demand,
stock A price up an
【在 s***e 的大作中提到】
: I think your explanation is correct. Indeed I learned a lot after the
: previous discussions. Thanks for the encouragement!
:
: rate.
: the
: thought
|
