It suffices to compare them at T.
Consider P t T.PT= Ck-9-0-1
at time account kértttl
tim e -1 is )=k t’
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The value ofthe money
and ke” ‘t-
ke- r’t- erttt
)>0 t. d k
Thenwehave p+
similar to the above argument,
ckérttt’ -5+51Pt F71DP
I.F-Ca+ DéH%-
Efm = CT riotert- t)
Consider two wealth processes t-ke-rct-tl-Fount-fzt-T.pt
+ke”‘t erCT-
= CsgKY+☐er”→”+K. Eat= PT-1ST-1☐
erct-TD-lk-S-5-sz-DEHT-TD-hen-Q-r-E.at
‘ This F71DP
completes the proof ,
which implies ,
Weonlyfocuson the calloptioncase.
Ee case , 2k) = @ ST -214-1–219–151
Se-, 2k)= 2G-
cis suppose the exercise time ofpcs, Ka)
then it follows fromthe c k- so-1 that
convexity of
PT 1St, K) = Ckx-5+51sick, -8-7-1
+ a-A) ( Ka- STY ⇐ APels,ki)+ l1-d)Pt(Ska)
Therefore ,
Ptls, Kit. Pres,Kx)= Ka- Sz- hCki- so
Ptl S, Ka) s Xptls, ki)
1- A Pecs, KD + ( 1-DR Cska)
completes the
Can be proved in the same way.
+(ta)(Ka- S) proof .
Define T-ct.XH-e-rtfltxi-7-ffcis.is/e-tsds. It follows from Ito ‘s Gamla that
DFH. ✗+7 = – re-rtfctxi-df-e-rtdflt.lt) + cet,Xt)e-Holt
= e-rt{1ftHiXD-1 -2MtXt)f××HX+)
+ ultraf- }
rf-H.xtl-ctxi-ydl-sctxi-lf.lt/X+1dWt
Then we have the
Moreover, Fit ✗+7 is
drift ofthe above
differential equals 0 conclusion .
which yields
is easyto seethat a martingale.
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