itrage price process of the chooser option must be equal to the value process of the replicating portfolio. In particular,V 0=X 0= eE[ X (1+r) ] = eE[ max(C,P) (1+r) ]. 2.13. (i) Proof.Note under both actual probabilityPand risk-neutral probability eP, coin tossesω n’s are i.i.d.. So without loss of generality, we work onP. For any functiong,E n[g(S n+1, Y n+1)] =E n[g( SS S n, Y n+ SS S n)] =pg(uS n, Y n+uS n) +qg(dS n, Y n+dS n), which is a function of (S n, Y n). So (S n, Y n) 0≤n≤Nis Markov underP. (ii) Proof.Setv N(s, y) =f( y N+1 ). Thenv N(S N, Y N) =f( P S N+1 ) =V N. Supposev n+1is given, then V n= eE n[ V1+r ] = eE n[ v (S ,Y ) 1+r ] = 1 1+r [epv n+1(uS n, Y n+uS n) +eqv n+1(dS n, Y n+dS n)] =v n(S n, Y n), where v n(s, y) = ev n+1(us, y+us) +ev n+1(ds, y+ds) 1 +r . 9
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