re : option pricing challenge zimin ,
to generalize your initial comment , for any process ds = mu ( s , t ) * s * dt +
sigma ( s , t ) * s * dz ,
the delta - hedging argument leads to the black - scholes pde .
this is true for any arbitrary functions mu and sigma , and so includes gbm ,
mean reversion , and others .
there is no problem with this , because in the risk - neutral world , which is
what you enter if you can hedge ,
the drift of the " actual " process is irrelevant .
i believe your concern is that you would like to see a different option price
for mean reversion process . this can only happen if the asset is not
hedgeable , and so the actual dynamics then need to be factored into the
option pricing . if you assume that the underlying is a non - traded factor ,
then the pde will have to reflect the market price of risk , and the drift of
the actual process is then reflected in the pde .
vasant
zimin lu
10 / 17 / 2000 05 : 20 pm
to : vince j kaminski / hou / ect @ ect , stinson gibner / hou / ect @ ect , vasant
shanbhogue / hou / ect @ ect , pinnamaneni krishnarao / hou / ect @ ect , alex
huang / corp / enron @ enron , kevin kindall / corp / enron @ enron , tanya
tamarchenko / hou / ect @ ect
cc :
subject : option pricing challenge
dear all ,
i have a fundamental question back in my mind since 95 . hope you can give
me a convincing answer .
zimin
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
in deriving bs differential equation , we assume the underlying follows gbm
ds = mu * s * dt + sigma * s * dz
where mu is the drift , sigma is the volatility , both can be a function of s .
then we use delta hedging argument , we obtain the bs differential equation
for the option price , regardless
of mu .
with the bs pde and boundary condition , we can derive bs formula . fine . no
problem .
question comes here . suppose the underlying is traded security and follows ,
say , mean - reverting process
ds = beta ( alpha - s ) dt + sigma * s * dz
apparantly , this sde leads to a different probability distribution . however ,
using the delta hedging argument ,
we still get the same bs differential equation , with the same boumdary
condition , we get the same bs formula .
not fair !
from another angle , i can derive the distribution from the bs pde for the
underlying , which is the lognormal distribution .
my thinking is : can i drive the distribution for any sde from the option pde
? the answer should be yes , but got to be
from a different pde rather than bs pde . then what we do about the
delta - hedging argument ?
thanks .
re : option pricing challenge
zimin ,
to generalize your initial comment , for any process ds = mu ( s , t ) * s * dt +
sigma ( s , t ) * s * dz ,
the delta - hedging argument leads to the black - scholes pde .
this is true for any arbitrary functions mu and sigma , and so includes gbm ,
mean reversion , and others .
there is no problem with this , because in the risk - neutral world , which is
what you enter if you can hedge ,
the drift of the " actual " process is irrelevant .
i believe your concern is that you would like to see a different option price
for mean reversion process . this can only happen if the asset is not
hedgeable , and so the actual dynamics then need to be factored into the
option pricing . if you assume that the underlying is a non - traded factor ,
then the pde will have to reflect the market price of risk , and the drift of
the actual process is then reflected in the pde .
vasant
zimin lu
10 / 17 / 2000 05 : 20 pm
to : vince j kaminski / hou / ect @ ect , stinson gibner / hou / ect @ ect , vasant
shanbhogue / hou / ect @ ect , pinnamaneni krishnarao / hou / ect @ ect , alex
huang / corp / enron @ enron , kevin kindall / corp / enron @ enron , tanya
tamarchenko / hou / ect @ ect
cc :
subject : option pricing challenge
dear all ,
i have a fundamental question back in my mind since 95 . hope you can give
me a convincing answer .
zimin
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
in deriving bs differential equation , we assume the underlying follows gbm
ds = mu * s * dt + sigma * s * dz
where mu is the drift , sigma is the volatility , both can be a function of s .
then we use delta hedging argument , we obtain the bs differential equation
for the option price , regardless
of mu .
with the bs pde and boundary condition , we can derive bs formula . fine . no
problem .
question comes here . suppose the underlying is traded security and follows ,
say , mean - reverting process
ds = beta ( alpha - s ) dt + sigma * s * dz
apparantly , this sde leads to a different probability distribution . however ,
using the delta hedging argument ,
we still get the same bs differential equation , with the same boumdary
condition , we get the same bs formula .
not fair !
from another angle , i can derive the distribution from the bs pde for the
underlying , which is the lognormal distribution .
my thinking is : can i drive the distribution for any sde from the option pde
? the answer should be yes , but got to be
from a different pde rather than bs pde . then what we do about the
delta - hedging argument ?
thanks .