c**********e
发帖数: 2007 | 1 I have a question on the deduction of BS PDE. Please help!
The price of the stock is S.
The price of the derivative is V.
The portfolio is P = V - (partial V)/(partial S) * S.
Then dP = dV - (partial V)/(partial S) * dS. Why is this true? Why is (
partial V)/(partial S) treated as constant?
Thanks a lot! |
f*******g
发帖数: 377 | 2 (partial V)/(partial S) is delta
it is previsible
it was pre-determined before the new moves dS & dV (and thus dP)
so should be treated as constant
【在 c**********e 的大作中提到】
: I have a question on the deduction of BS PDE. Please help!
: The price of the stock is S.
: The price of the derivative is V.
: The portfolio is P = V - (partial V)/(partial S) * S.
: Then dP = dV - (partial V)/(partial S) * dS. Why is this true? Why is (
: partial V)/(partial S) treated as constant?
: Thanks a lot!
|
c**********e
发帖数: 2007 | 3 I always feel the proof is not robust, but found no alternative.
【在 f*******g 的大作中提到】
: (partial V)/(partial S) is delta
: it is previsible
: it was pre-determined before the new moves dS & dV (and thus dP)
: so should be treated as constant
|
r***n
发帖数: 6 | 4 self-financing
【在 c**********e 的大作中提到】
: I always feel the proof is not robust, but found no alternative.
|
n******m
发帖数: 169 | 5 I had the same question as OP before, here is the answer I came up with:
In fact, dc/dS is not constant. the argument says this portfolio should
evolve as cash. but it is not a fix amount of cash, it should be m(t)$1,
where m(t) is not constant either. m(t)=v-dc/ds S.
the evolution of the cash looks like:
d(m(t)$1)=d(m(t))$1+m(t)d($1)=d(m(t))$1+rm(t)$1
because d(m(t))$1+d(dc/ds)S=0, so you eventually have dP=rP (BS) |
t**********a
发帖数: 166 | 6 check the definition of self-financing and derivation in Bjork's book |
c**********e
发帖数: 2007 | 7 The same problem happens to the deduction of PDE for exchange option.
From c_t - (\partial c_t)/(\partial S_1t) S_1t - (\partial c_t)/(\partial S_
2t) S_2t = 0, the following is obtained
d c_t - (\partial c_t)/(\partial S_1t) d S_1t - (\partial c_t)/(\partial S_
2t) d S_2t = 0.
No explanation is given!!! I do not feel it straight-forward. |
y**********0
发帖数: 425 | 8 i think it is in the first step to derivate the BS formula, when the writer
want to hedge against risk, so the delta is in fact a constant a, and after
several steps, it comes to the a equals delta.
thus the derivative of the portfolio is dP = dV - (partial V)/(partial S) *
dS. It is after several steps that this formula come out, not the first, or
originally. You have to check it. |
n******m
发帖数: 169 | 9 The same happen here
d(dc/dS1)S1+d(dc/dS2)S2=0
So, it was omitted.
To see why this is true, think about you have a linear combination of S1 and
S2, which combine to a fix total value: a(t) S1 +b(t) S2 =Total
Then d(a)S1+d(b)S2=d(total)=0.
As people pointed out above, this portfolio is self financing, because by
calculation, we see d( a(t)S1(t)+b(t)S2(t) )= a(t)dS1(t)+b(t)dS2(t).
S_
S_
【在 c**********e 的大作中提到】
: The same problem happens to the deduction of PDE for exchange option.
: From c_t - (\partial c_t)/(\partial S_1t) S_1t - (\partial c_t)/(\partial S_
: 2t) S_2t = 0, the following is obtained
: d c_t - (\partial c_t)/(\partial S_1t) d S_1t - (\partial c_t)/(\partial S_
: 2t) d S_2t = 0.
: No explanation is given!!! I do not feel it straight-forward.
|
L**********u
发帖数: 194 | 10 It is self-financing hypothesis for BS PDE.
【在 c**********e 的大作中提到】
: I have a question on the deduction of BS PDE. Please help!
: The price of the stock is S.
: The price of the derivative is V.
: The portfolio is P = V - (partial V)/(partial S) * S.
: Then dP = dV - (partial V)/(partial S) * dS. Why is this true? Why is (
: partial V)/(partial S) treated as constant?
: Thanks a lot!
|
r**a
发帖数: 536 | 11 See the book "Arbitrage theory in Continuous Time" Chapter 6.
【在 c**********e 的大作中提到】
: I have a question on the deduction of BS PDE. Please help!
: The price of the stock is S.
: The price of the derivative is V.
: The portfolio is P = V - (partial V)/(partial S) * S.
: Then dP = dV - (partial V)/(partial S) * dS. Why is this true? Why is (
: partial V)/(partial S) treated as constant?
: Thanks a lot!
|
s*******0
发帖数: 3461 | 12 V= a*S+b*B
if self financing, then dv=ads+bdb
bdb did not related to s so dv/ds=a constant
if it is true? |
r**a
发帖数: 536 | 13 a并不是常数。看Shreve 324页。或者arbitrage theory in continuous time chapter
6.
【在 s*******0 的大作中提到】
: V= a*S+b*B
: if self financing, then dv=ads+bdb
: bdb did not related to s so dv/ds=a constant
: if it is true?
|
