Monday, February 23, 2015

stochastic calculus - Dumb question: is risk-neutral pricing taking conditional expectation?



Dumb question: is risk-neutral pricing taking conditional expectation?



In trying to recall intuition for risk-neutral pricing, I think I read that we should price derivatives risk-neutrally because the risk is already incorporated in the stock or something. I also think I remember NNT saying something about how certain information is irrelevant in the expected price of oil if the information is public.



This made me think of risk-neutral pricing in terms of conditional expectation.


Simple I guess but it wasn't discussed in classes since conditional expectation and Radon-Nikodym was taught after one period model.


From what I recall of the one period model:


(Ω,F,P)=((u,d),2Ω,real world))

Bonds: {Bt}
B0=1,B1=1+R
Stocks: {St}
S0(0,)
P(S1(u)=S0u)=pu>0
P(S1(d)=S0d)=pd=1pu
European call option: X X(u)=S1(u)K
X(d)=0
Price process: {Π(X,t)}


where t=0,1,u>1+R>d>0.


It can be shown that


Π(X,0)=11+REQ[X]=11+R(quX(u)+qdX(d))


where qu,qd are the risk-neutral probabilities under Q, equivalent to P


Also, I think σ(S1)=σ(X)={,Ω,{u},{d}}


Dumb question rephrased:




ZL1(Ω,F,P) s.t. EQ[X]=EP[X|Z]?



Well, the left hand side is a constant while the right hand side a random variable so I'm not sure that that would make sense


How about



ZL1(Ω,F,P) s.t. EQ[X]=EQ[EP[X|Z]]?






  • For (2),




  • One thing I did:




    1. EQ[X] is constant and thus Zmeasurable  ZL1(Ω,F,P)




    2. zEQ[X]dP=zXdP  z  σ(Z)







E[EQ[X]1z]=E[X1z]  z  σ(Z)

EQ[X]E[1z]=E[X1z]  z  σ(Z)
EQ[X]P(z)=E[X1z]  z  σ(Z)
P(z)=E[X1z]EQ[X]  z  σ(Z)


There doesn't seem to be such a Z.



  • Another thing I did: Well, I did think of Radon-Nikodym (duh)


EQ[X]=EP[XdQdP]

.



I guess Z=dQdP otherwise not sure how that's relevant but I guess since Q(z)=zdQdPdP zσ(Z)2Ω

,


dQdP is a version of E[dQdP|Z]




  • Gee how informative. Well, I think σ(Z) can be only either {,Ω}, in which case the Z is any (almost surely?) constant random variable or 2Ω=σ(X)=σ(S1), in which case qu=1u which would make sense iff qu is degenerate, which I guess violates equivalence assumption.




  • For (3),





I guess Z=S1? I'm not sure what that says. I was kinda expecting (lol) that real world probabilities E[1A] and risk-neutral probabilities E[1A|B]=E[1A1B]E[1B] would be like, respectively, prior P(A) and posterior P(A|B) probabilities.




Edit: dQdP=qupu1u+qdpd1d

?


Based on Section 4.5 of Etheridge's A Course in Financial Calculus, I guess


dQdP=(qupu)u(qdpd)1u


This avoids indicator functions in favour of exponents as in the binomial theorem, binomial model or binomial distribution.



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