Cap6
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Publicado: 30 de marzo de 2010
• ,;•!;:. i'-, i'1' •. ' i/
148 C H A PTE R • 6 Maningales and Martingale Rcpresentations
becau.se AQ = C,H - C,. Or. using thc replicating portfolio:
Glc + rAk/1+0,5,1 d20)
11 APricing Methodology Thus (121) can be rewritten as: CT = C,
v.f. y;
(125)
whcrc íhe A reprcscnts the operation of taking first differences.
Now, recall that the "change;1 in a product, u.r, canbe calculated using
the "product rule": . '-,.. ^y . • .•:,•./.. ..... ..
d(u.v) = du.v -r it.dv. (ll2)
Applying this lo tlic second and third terms on the right-hand side of(121)~
(123)
and(124)
whcrc we uscd íhe notation,
Aa/f = a,M - a,, A/3, = /?,.., - 0,,
and
"Another way of obfnining íhe cquations below is by simple algebra. Given ú [a,. 0,^0,^8,^-0,,^,
; note thatwc can add and subtract a^B,^, on thc right-hand side. factor out similar terms,
': ": " "
and obciin:
Regrouping.
- C, + ¿ [(
(126)
Now consider the tertns on the right-hand side of thiscxpression. The C, is the unknown of the problcm. We are, in fací, looking íor n melhod ío determine an arbitrage-free valué for this temí (hat saiisfks the pric-ing Eq. (112). The rwo other terms in thebrackcts need to be discusscd in detail.
Consider the first bracketcd term. Given the information set nt time tM, every element of this bracket will be known. Tlie D¡ , S,é are priccs ob-served in themarkcts, and thc Aa,., A/3,, is thc rcbalandng of the replicating portfolio as described by the financia! analyst. Henee, tlie firsi bracketed term has some simiiarities to the D, tenn in themartingale representa! ion (111).
The second bracketed term will be unknosvn given tiic information sel /,., because it ¡nvoJves íhe pnce changes A5, , A/?f that occur ajjcr t,, and henee maj' contain nfiMíinformation not contained in 1,1 Howcvcr, although unknown, these pnce changes are, in general, prediciapíc. Tlius \ve cannoi expect the second term to play the role of dM, ¡n the martingale...
148 C H A PTE R • 6 Maningales and Martingale Rcpresentations
becau.se AQ = C,H - C,. Or. using thc replicating portfolio:
Glc + rAk/1+0,5,1 d20)
11 APricing Methodology Thus (121) can be rewritten as: CT = C,
v.f. y;
(125)
whcrc íhe A reprcscnts the operation of taking first differences.
Now, recall that the "change;1 in a product, u.r, canbe calculated using
the "product rule": . '-,.. ^y . • .•:,•./.. ..... ..
d(u.v) = du.v -r it.dv. (ll2)
Applying this lo tlic second and third terms on the right-hand side of(121)~
(123)
and(124)
whcrc we uscd íhe notation,
Aa/f = a,M - a,, A/3, = /?,.., - 0,,
and
"Another way of obfnining íhe cquations below is by simple algebra. Given ú [a,. 0,^0,^8,^-0,,^,
; note thatwc can add and subtract a^B,^, on thc right-hand side. factor out similar terms,
': ": " "
and obciin:
Regrouping.
- C, + ¿ [(
(126)
Now consider the tertns on the right-hand side of thiscxpression. The C, is the unknown of the problcm. We are, in fací, looking íor n melhod ío determine an arbitrage-free valué for this temí (hat saiisfks the pric-ing Eq. (112). The rwo other terms in thebrackcts need to be discusscd in detail.
Consider the first bracketcd term. Given the information set nt time tM, every element of this bracket will be known. Tlie D¡ , S,é are priccs ob-served in themarkcts, and thc Aa,., A/3,, is thc rcbalandng of the replicating portfolio as described by the financia! analyst. Henee, tlie firsi bracketed term has some simiiarities to the D, tenn in themartingale representa! ion (111).
The second bracketed term will be unknosvn given tiic information sel /,., because it ¡nvoJves íhe pnce changes A5, , A/?f that occur ajjcr t,, and henee maj' contain nfiMíinformation not contained in 1,1 Howcvcr, although unknown, these pnce changes are, in general, prediciapíc. Tlius \ve cannoi expect the second term to play the role of dM, ¡n the martingale...
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