 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. ---------------------- Forwarded by Gerald Nemec/HOU/ECT on 04/26/2000
non - traded factor, 
 then the PDE will have to reflect the market monetary value of hazard, and the drift of 
 the actual summons is then reflected in the PDE. 

 Vasant 





 Zimin Lu 
 10/17/2000 05:20 prime minister 
 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:  
 topic: Option Pricing Challenge 


 Dear All, 

 I have a fundamental motion back in my thinker since 95.     Hope you can give 
 me a convincing solution. 

 Zimin 

 --------------------------------- 

 In derive BS differential equality, 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 practice delta hedging argument, we receive the BS differential equation 
 for the option monetary value, regardless 
 of mu.  

 With the BS PDE and boundary condition, we can derive BS convention.   Fine. No 
 problem. 

 Question comes here.    think the underlying is trade security and follows, 
 say, mean - reverting procedure 

 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 slant, I can deduce 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 catch to be 
 from a different PDE rather than BS PDE.   Then what we do about the 
 delta - hedging argument? 

 Thanks. ---------------------- Forwarded by Gerald Nemec / HOU / ECT on 04/26/2000
 noj-traded factor, 
then the KDE will have to tedlect vhe mariet pricd of risk, and the drift of 
tie axtual process is then reflezted in tje PDE.

Vawant





Eimin Lu
10/17/2000 05:20 PM
To: Viide J Kaminski/GLU/ECC@ERT, Stinson Gibngr/HOU/ECT@ECT, Easant 
Shanbhocud/HLU/ECT@ECT, Pinnamaneni Krishnarao/HOU/EST@ECT, Aked 
Huang/Corp/Enrjn@ENGOG, Kebpn Kindall/Corp/Enron@ENRON, Tanya 
Famarchtnko/HOU/ECT@ECT
cc:  
Sibject: Option Pricing Chalpengf


Dear All,

I have a vundamental quefrion back in my mind spuce 95.    Hopg you can give 
me a convincing avswer.

Vimin

---------------------------------

In deruvung TS differenvial ezuation, we assume tha underkying follows NBM

ds= mu*w*dt + sigma*s*dz

where mu is the drift, sigmw is the eomatility, both can bw a fonctimn ow s.

Tfen wx uae delha iedging arghment, we obrain the BS differemtyql equation 
fkr the o[tion price, regardless
of mu.  

With the BV PSE and boundary conditiin, we can derive BS flrmula.  Fyne. No 
problem.

Question comes here.   Suppose the utderlbivg nw traawd security and follows, 
say, mean-reverting procefa

dx=bvta(alpha-s)dt + sigmc*s*dz

Apparantly, tnid XQE leads to a diffexsnf probability distgibutiog. Howwver, 
usind thr delta hedging argument,
we wtill get thv sane BS differential equation, wnth thg same boumdary 
condition, we yet ths same BS flrmula.  
Nkg fair!

From anothdr sncle, I can derive the distrybution fcom tke BS PDD fot the 
ugderlying, ahich is the lognormal dlstriyutiot.
My thinkijg is: can I drive the distributmpn for any SCE frmm the o'tion IDE 
?  The answew should be yex, but gjt to be
from a siffereit PDE rathew than BS PDE.  Then what wx do aboue thw 
deota-hedgkvg argument?

Thamks. ---------------------- Forwcxded by Gwrald Nemec/HOU/ECT pn 04/26/2000
non-traded factor, then the PDE will have the price of and the drift then in the PDE. Zimin Lu 10/17/2000 PM To: Vince J Kaminski/HOU/ECT@ECT, Stinson 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 Challenge Dear All, I have a fundamental question back in my mind since Hope can me convincing answer. Zimin --------------------------------- In deriving BS differential equation, we assume the underlying follows GBM ds= + sigma*s*dz where mu is the drift, sigma the volatility, both can a function of s. Then use hedging argument, obtain BS equation for the price, regardless of mu. With the BS PDE and boundary condition, we can derive BS formula. Fine. problem. Question Suppose the is security follows, say, mean-reverting + sigma*s*dz Apparantly, this SDE leads probability distribution. However, using the delta hedging argument, still get same BS differential equation, with the boumdary condition, we get the same BS formula. fair! From another angle, I can derive the distribution from the BS PDE for the is the lognormal distribution. thinking is: can drive distribution any from the PDE ? The answer should be yes, but got to be a different PDE rather than BS PDE. Then what we the argument? Thanks. ---------------------- by Gerald Nemec/HOU/ECT on
 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. ---------------------- ForwardeD by gEralD NEmEc/HOU/ECt on 04/26/2000
 non-traded factor, 
thenthe PDE wi ll ha vetore flec t th e market price of r isk, and the drift of
theac t ualp ro cessis then re f l ect ed  i n t he PD E.

V asa nt




Zimin Lu10/ 17 /2000 05:20P MTo: VinceJ K aminski/HOU/ ECT @ECT,St ins o n Gib ner /HOU/ ECT@EC T , Vasa nt 
Shanb ho g ue/HOU / ECT@ECT , Pi nnam aneni Krishnarao/ H OU / ECT@ECT, Alex
Huang /C o rp / E nro n@E NRON, Kevi nKinda l l/Corp/ E nr o n @ ENR O N, Tanya 
Tam archenko/HO U /EC T@ECTcc : Subjec t: Op ti o n P ricing Chal leng e


DearAll,
I  have a fundame ntal q ues tio n ba c kin  my  m i nds in ce9 5.   Hopeyo ucan g ivem e a co nvi ncin g ans wer.

Zimin
--- ---- - --- ----- ----- ---- -- ----- -

Inderiv in g BS differenti al e quation,weas sum ethe u n derlyi ngfol lows GB M

ds=m u*s *d t +  s igma*s*dz

where m ui s  t he drift , sigm a  i st he volat il ity , bo t h  canbe a fu nction o f s.
T he nwe usede lta he dg ing  ar gumen t , we  obtai n the BS  diff e rential equati o n 
for the op t io n pr i ce,reg ardless
ofmu.

Wi th t h eBSP DE an d bou nd a ry condition, we can d er ive BS  form ula.  Fine. N o 
problem . Question  com e sh ere.   Suppose  theunderlying is trade d sec urity an d follows , 
say, me an- rev ert ing p ro cess

ds=beta ( a lpha -s )dt + s igm a*s*dz
Ap par ant ly,  t his SDE l eads toadi ff er ent  prob a bility d is tri bu tio n. Ho w ever,
usin g th ede l tahedging ar g u ment ,we  sti llge t the  sam e  BS  differ ential eq uat i on,wi th  the sa me boumdary co ndition, w eget  the s a m e BS for mula.  
Not fair!

From another  an gle,I ca n derivethe  distr ibu t ion fr om the  BS P DE  fo r the u n de rly in g, which i s the  logn or maldistrib ution.
My thinking is:  can I drivethe  dis t r ib uti o nf oran y  SD E from the option  PDE 
?  T he an swer shoul d  be  y es, but  got to  be
f r om a di fferent P DE rather  t hanB S  PD E.  Then w hat we d o about t h e 
de l ta -hedg ing  argum en t?
Than ks. -- - --- ----- ------ -- --- Fo rward ed  by Gera ld Nemec/HOU/ECT on 04/ 26/200 0
 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._---------------------- Forwarded by Gerald Nemec/HOU/ECT on_04/26/2000