Properties of American Derivative Securities

Transcription

1 Capter 6 Propertes of Amercan Dervatve Securtes 6.1 Te propertes Defnton 6.1 An Amercan dervatve securty s a sequence of non-negatve random varables fg k g n k= suc tat eac G k s F k -measurable. Te owner of an Amercan dervatve securty can exercse at any tme k, and f e does, e receves te payment G k. (a) Te value V k of te securty at tme k s kf V k =max(1 + r) IE[(1 + r) ; G jf k ] were te maxmum s over all stoppng tmes satsfyng k almost surely. (b) Te dscounted value process f(1 + r) ;k V k g n k= s te smallest supermartngale wc satsfes V k G k 8k almost surely. (c) Any stoppng tme wc satsfes s an optmal exercse tme. In partcular s an optmal exercse tme. (d) Te edgng portfolo s gven by V [(1 + r) ; G ] 4 =mnfk V k = G k g k (! 1 :::! k )= V k+1(! 1 :::! k H) ; V k+1 (! 1 :::! k T) k = 1 ::: n; 1: S k+1 (! 1 :::! k H) ; S k+1 (! 1 :::! k T) 85

2 86 (e) Suppose for some k and!, we ave V k (!) =G k (!). Ten te owner of te dervatve securty sould exercse t. If e does not, ten te seller of te securty can mmedately consume and stll mantan te edge. V k (!) ; 1 fie[v k+1 jf k ](!) 1+r 6.2 Proofs of te Propertes Let fg k g n k= be a sequence of non-negatve random varables suc tat eac G k s F k -measurable. Defne T k to be te set of all stoppng tmes satsfyng k n almost surely. Defne also Lemma 2.18 V k G k for every k. Proof: Take 2 T k to be te constant k. V k 4 =(1+r) k max f IE [(1 + r) ; G jf k ] : Lemma 2.19 Te process f(1 + r) ;k V k g n k= s a supermartngale. Proof: Let attan te maxmum n te defnton of V k+1,.e., (1 + r) ;(k+1) V k+1 (1 + r) ; G jf k+1 : Because s also n T k, we ave fie[(1 + r) ;(k+1) V k+1 jf k ] fie[(1 + r) ; G jf k+1 ]jf k [(1 + r) ; G jf k ] max IE f [(1 + r) ; G jf k ] = (1 + r) ;k V k : Lemma 2.2 If fy k g n k= s anoter process satsfyng Y k G k k = 1 ::: n a.s., and f(1 + r) ;k Y k g n k= s a supermartngale, ten Y k V k k = 1 ::: n a.s.

3 CHAPTER 6. Propertes of Amercan Dervatve Securtes 87 Proof: Te optonal samplng teorem for te supermartngale f(1 + r) ;k Y k g n k= mples Terefore, fie[(1 + r) ; Y jf k ] (1 + r) ;k Y k 8 2 T k : V k = (1 + r) k max f IE[(1 + r) ; G jf k ] = Y k : (1 + r) k max f IE[(1 + r) ; Y jf k ] (1 + r) ;k (1 + r) k Y k Lemma 2.21 Defne C k = V k ; 1 fie[v k+1 jf k ] 1+r = (1 + r) k n(1 + r) ;k V k ; f IE[(1 + r) ;(k+1) V k+1 jf k ] o : Snce f(1 + r) ;k V k g n k= s a supermartngale, C k must be non-negatve almost surely. Defne k (! 1 :::! k )= V k+1(! 1 :::! k H) ; V k+1 (! 1 :::! k T) S k+1 (! 1 :::! k H) ; S k+1 (! 1 :::! k T) : Set X = V and defne recursvely X k+1 = k S k+1 +(1+r)(X k ; C k ; k S k ): Ten X k = V k 8k: Proof: We proceed by nducton on k. Te nducton ypotess s tat X k = V k for some k 2f 1 ::: n; 1g,.e., for eac fxed (! 1 :::! k ) we ave We need to sow tat X k (! 1 :::! k )=V k (! 1 :::! k ): X k+1 (! 1 :::! k H)=V k+1 (! 1 :::! k H) X k+1 (! 1 :::! k T)=V k+1 (! 1 :::! k T): We prove te frst equalty; te proof of te second s smlar. Note frst tat V k (! 1 :::! k ) ; C k (! 1 :::! k ) = = 1 fie[v k+1 jf k ](! 1 :::! k ) 1+r 1 1+r (~pv k+1(! 1 :::! k H)+ ~qv k+1 (! 1 :::! k T)) :

4 88 Snce (! 1 :::! k ) wll be fxed for te rest of te proof, we wll suppress tese symbols. For example, te last equaton can be wrtten smply as We compute V k ; C k = 1 1+r (~pv k+1(h)+ ~qv k+1 (T )) : X k+1 (H) = k S k+1 (H) + (1 + r)(x k ; C k ; k S k ) = V k+1(h) ; V k+1 (T ) S k+1 (H) ; S k+1 (T ) (S k+1(h) ; (1 + r)s k ) +(1 + r)(v k ; C k ) = V k+1(h) ; V k+1 (T ) (u ; d)s k (us k ; (1 + r)s k ) +~pv k+1 (H)+ ~qv k+1 (T ) = (V k+1 (H) ; V k+1 (T ))~q +~pv k+1 (H)+ ~qv k+1 (T ) = V k+1 (H): 6.3 Compound European Dervatve Securtes In order to derve te optmal stoppng tme for an Amercan dervatve securty, t wll be useful to study compound European dervatve securtes, wc are also nterestng n ter own rgt. A compound European dervatve securty conssts of n +1dfferent smple European dervatve securtes (wt te same underlyng stock) exprng at tmes 1 ::: n; te securty tat expres at tme j as payoff C j. Tus a compound European dervatve securty s specfed by te process fc j g n, were eac j= C j s F j -measurable,.e., te process fc j g n j= s adapted to te fltraton ff k g n. k= Hedgng a sort poston (one payment). Here s ow we can edge a sort poston n te j t European dervatve securty. Te value of European dervatve securty j at tme k s gven by V (j) k =(1+r) kf IE[(1 + r) ;j C j jf k ] k= ::: j and te edgng portfolo for tat securty s gven by (j) k (! 1 :::! k )= V (j) k+1 (! 1 :::! k H) ; V (j) k+1 (! 1 :::! k T) k = ::: j; 1: S (j) k+1 (! 1 :::! k H) ; S (j) k+1 (! 1 :::! k T) Tus, startng wt wealt V (j), and usng te portfolo ( (j) ::: (j) j;1 ), we can ensure tat at tme j we ave wealt C j. Hedgng a sort poston (all payments). P Superpose te edges for te ndvdual payments. In n oter words, start wt wealt V = j= V (j). At eac tme k 2f 1 ::: n; 1g, frst make te payment C k and ten use te portfolo k = k (k+1) + k (k+2) + :::+ k (n)

5 CHAPTER 6. Propertes of Amercan Dervatve Securtes 89 correspondng to all future payments. At te fnal tme n, after makng te fnal payment C n,we wll ave exactly zero wealt. Suppose you own a compound European dervatve securtyfc j g n j=. Compute V = nx j= V (j) 2 4 n X j= (1 + r) ;j C j 3 5 and te edgng portfolo s f k g n;1 k=. You can borrow V and consume t mmedately. Ts leaves you wt wealt X = ;V. In eac perod k, receve te payment C k and ten use te portfolo ; k. At te fnal tme n, after recevng te last payment C n, your wealt wll reac zero,.e., you wll no longer ave a debt. 6.4 Optmal Exercse of Amercan Dervatve Securty In ts secton we derve te optmal exercse tme for te owner of an Amercan dervatve securty. Let fg k g n k= be an Amercan dervatve securty. Let be te stoppng tme te owner plans to use. (We assume tat eac G k s non-negatve, so we may assume wtout loss of generalty tat te owner stops at expraton tme n f not before). Usng te stoppng tme, n perod j te owner wll receve te payment C j = I f =jg G j : In oter words, once e cooses a stoppng tme, te owner as effectvely converted te Amercan dervatve securty nto a compound European dervatve securty, wose value s V ( ) 2 4 n X j= (1 + r) ;j C j 3 5 j= 2 3 = IE f X 4 n (1 + r) ;j I f =jg G j 5 [(1 + r) ; G ]: Te owner of te Amercan dervatve securty can borrow ts amount of money mmedately, f e cooses, and nvest n te market so as to exaclty pay off s debt as te payments fc j g n are j= receved. Tus, s optmal beavor s to use a stoppng tme wc maxmzes V ( ). Lemma 4.22 V ( ) s maxmzed by te stoppng tme = mnfk V k = G k g: Proof: Recall te defnton 4 V =maxie f [(1 + r) ; G ] = max V ( ) 2T 2T

6 9 Let be a stoppng tme wc maxmzes V ( ),.e., V = IE f (1 + r) ; G : Because f(1 + r) ;k V k g n k= s a supermartngale, we ave from te optonal samplng teorem and te nequalty V k G k, te followng: V IE f (1 + r) ; V jf (1 + r) ; V f IE (1 + r) ; G = V : Terefore, and V (1 + r) ; V V = G a.s. = IE f (1 + r) ; G We ave just sown tat f attans te maxmum n te formula ten But we ave defned V = max 2T f IE [(1 + r) ; G ] (4.1) V = G a.s. = mnfk V k = G k g and so we must ave n almost surely. Te optonal samplng teorem mples (1 + r) ; G = (1 + r) ; V Takng expectatons on bot sdes, we obtan fie (1 + r) ; G f IE (1 + r) ; V jf (1 + r) ; G jf f IE (1 + r) ; G : = V : It follows tat also attans te maxmum n (4.1), and s terefore an optmal exercse tme for te Amercan dervatve securty.