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Binomial Tree??

Binomial Tree??# Economics - 经济

c*e2099-11-16 08:11

1 楼

If it is an American option with dividend payment, Black-sholes model cannot
be used. hence, binomial tree should be used. But if the dividend payment is
small relative to the share price, let us say, 30cents for 11$, can I still
simply use Black-sholes model to calcualte the option price?

z*e2099-11-16 08:11

2 楼

Black-scholes only applies to framework of European options.
Divident payment, if considered as a continuous cash flow
with a fixed percentage return of stock price,
can be easily included in B-S formula. In the case of constant
divident payment, a way to do it is to consider the stock
price S consists of 2 parts: D and S' where D is the
known cash flow of dividents, and S' is the "real" stochastic
price. By subtracting the present value of D from S, and
applying B-S formula or binomial trees

【在 c***e 的大作中提到】

: If it is an American option with dividend payment, Black-sholes model cannot
: be used. hence, binomial tree should be used. But if the dividend payment is
: small relative to the share price, let us say, 30cents for 11$, can I still
: simply use Black-sholes model to calcualte the option price?

q*i2099-11-16 08:11

3 楼

it is probably a little bit more complicated than that. Let's assume
dividends are paid at known interval of known amount, irregardless
of stock price.
just prior to the dividend payout date, the option holder has to
evaluate her option: to excersize the option and get the dividend
now, forgoing all future upside and risk lossing the stock value; or
keep the option, forgoing current dividend payment but retaining
future upside.
So best way is probably go back wards: find the late dividend
date b

【在 z***e 的大作中提到】

:
: Black-scholes only applies to framework of European options.
: Divident payment, if considered as a continuous cash flow
: with a fixed percentage return of stock price,
: can be easily included in B-S formula. In the case of constant
: divident payment, a way to do it is to consider the stock
: price S consists of 2 parts: D and S' where D is the
: known cash flow of dividents, and S' is the "real" stochastic
: price. By subtracting the present value of D from S, and
: applying B-S formula or binomial trees

z*e2099-11-16 08:11

4 楼

Yes let us assume that.
Use the binomial tree method, assume risk-neutral world,
construct the tree, then calculate price backward. The nightmare
is, when you construct the tree, the branches do not reconnect
as the size of the tree grows exponentially. One way to overcome
this problem, at least theoreticaly, is to consider the stock as
two things: a bond D that pays a known cash flow and a stock S'=S-D
that follows a stochastic process. The fact that option holder
does not get divident afte

【在 q**i 的大作中提到】

: it is probably a little bit more complicated than that. Let's assume
: dividends are paid at known interval of known amount, irregardless
: of stock price.
: just prior to the dividend payout date, the option holder has to
: evaluate her option: to excersize the option and get the dividend
: now, forgoing all future upside and risk lossing the stock value; or
: keep the option, forgoing current dividend payment but retaining
: future upside.
: So best way is probably go back wards: find the late dividend
: date b

q*i2099-11-16 08:11

5 楼

maybe I'm naive, but once you exercize the option, some branches of the
tree will be truncated. the reason you can get a beautiful solution
on european options is the symmetrics of the tree. I don't know, and
I doubt if that beauty can be preserved if the tree is truncated.
maybe I should dig out my copy of Hull's book (which I remember
as most dealing with continuous time).
I still think the difficulty is mutual excusivity of cash flow
from the bond and exercising of the option. this problem i

【在 z***e 的大作中提到】

:
: Yes let us assume that.
: Use the binomial tree method, assume risk-neutral world,
: construct the tree, then calculate price backward. The nightmare
: is, when you construct the tree, the branches do not reconnect
: as the size of the tree grows exponentially. One way to overcome
: this problem, at least theoreticaly, is to consider the stock as
: two things: a bond D that pays a known cash flow and a stock S'=S-D
: that follows a stochastic process. The fact that option holder
: does not get divident afte

z*e2099-11-16 08:11

6 楼

The way I see it, the binomial tree works as long as a
discrete model. Once you set up the tree that only contains
possible future prices, you can start calculating derivative
prices backward: be it european, american, asian, lookback etc.
The only difference is how the intrinsic values are determined
in each of the instruments.
Computional complexity plays a key role in practice and you
need to make sure that the branches "reconnect" rather than
grow exponentially. In european or american op