Option pricing and numéraires

Transcription

1 Opton prcng and numérares Daro Trevsan Unverstà degl Stud d Psa San Mnato - 15 September 2016

2 Overvew 1 What s a numerare? 2 Arrow-Debreu model Change of numerare change of measure 3 Contnuous tme Self-fnancng portfolos Prcng va change of numerare Exchange rates Swap optons

3 What s a numérare? A numérare (or numerare s a chosen standard by whch value s computed. Example (currences ar numerares We may compute values w.r.t to USD 1$ or EUR 1 e or JPY (1 Y. Of course, others mght prefer use commodtes: 1 OZ of gold could be a numerare. Clearly, once we choose a numerare e.g. 1 USD, we determne the value of other assets:

4 The change of numerare problem Problem In theory, does t really matter whch numerare we choose? Of course, n practce there are reasons to prefer gold to other commodtes e.g. corn, lve cattle, or one currency wth respect to another (poltcal reasons... But ntutvely there should be no theoretcal reason (at least at the scale of nvestors to measure value n gold or USD (there used to be also the gold standard Can we deduce/explot ths fact n our fnancal models? The strong underlyng prncple wll be always absence of arbtrage opportuntes.

5 Back to the Arrow-Debreu model Recall (from Tuesday the smple model wth N 1 securtes (.e. bonds, stocks or dervatves a = (a 1, a 2,..., a N can be held long or short by any nvestor. Two tmes: t = 0 and a fxed future t = 1. At t = 0 we have the observed spot prces of the N securtes p = (p 1, p 2,..., p N = (p N =1 R N At t = 1, the market attans one state among M possble scenaros s {1,..., M}. If s s attaned dvdends (prces of the securtes at t = 1 D s = ( D s 1, D s 2,..., D s N R N

6 Theorem No arbtrage exstence of postve weghts (π s s=1,...,m such that M M p = D s π s,.e. p = D s π s for every {1,..., N}. s=1 s=1 Next we assume that there s a rsk-free securty a 1 (e.g. a bond such that n any scenaro s we have D s 1 = 1 a 1 s our numerare Defne R and ˆπ (the nterest rate and rsk-neutral probablty by the relaton 1 + R = 1 π s M, ˆπ s = M. s=1 πs r=1 πr Then p 1 = 1 [ ] 1 + R, p D = Eˆπ = 1 + R M D s π s. =1

7 Change of numerare What happens f nstead of a 1 we fx another asset, e.g. a 2 as numerare? Assume D s 2 > 0 for every state of the market. the value of the asset a, measured n unts of a 2, at t = 1 s v s := Ds D2 s, f the market s n the state s. We rewrte the value at t = 0 of a as M M p = D s D s ( π s = D s D2 s 2 π s s=1 s=1 hence f we measure the value n unts of a 2, snce M ( p 2 = D r 2 π r we have p p 2 = M s=1 r=1 v s (D2π s s M r=1 (Dr 2 πr = E [ 2 v s ], where E 2 s a dfferent probablty measure than Eˆπ.

8 Change of numerare change of probablty We found an nstance of the general mechansms : Passng from the numerare a to b corresponds to a change of probablty, from πs a (D s := aπ s M r=1 (Dr aπ r to π b s = (D s bπ s M r=1 ( D r b π r. The value of the asset a (w.r.t. the numerare b at t = 0 s gven by the expectaton w.r.t. π b of the values at tme t = 1 [ ] p = E b D s p b Db s = M s=1 D s Db s πs b.

9 Contnuous-tme (Itô models Let us model the market wth a probablty space (Ω, A, P, a fltraton (F t t [0,T ] Itô processes S t = (St =1,...,N A portfolo H t = ( H 1 t,..., H N t has value V t := H t S t = N Ht St =1 A numerare s a strctly postve Itô process D t. The prces actualzed wth respect to D become S t D t.

10 Change of numerare and self-fnancng strateges Proposton The self-fnancng condton s nvarant wth respect to any chosen numerare,.e. N dv t = Ht dst, t (0, T f and only f d ( Vt D t = =1 N =1 ( Ht S d t, t (0, T D t

11 We use Itô formula for product d(vg = GdV + VdG + dgdv wth G = 1/D. Snce we have ( d H S = Hd ( S d(vg = GHd S ( ( + H S dg + dg d H S = GHd S ( + H S dg + dghd S = H ( GS + SdG + dgds = H ( d SG

12 Change of probablty Theorem Let P 0 be a probablty (equvalent to P such that every St St 0, {1,..., N} s a martngale, and also Consder the new probablty D t St 0. Then each s a P D -martngale (as also D t D t = 1. P D = 1 DT P 0. D 0 ST 0 S t D t {1,..., N}

13 Proof E 0 [ F t] the condtonal expectaton w.r.t. P 0 and E D [ F t] the condtonal expectaton w.r.t. P D. S t S 0 t s a P 0 martngale S t S 0 t [ = E 0 S T F S T 0 t ]. We have a formula for condtonal expectaton w.r.t. dfferent probabltes: Theorem E D [X A] = E 0 [XfI A ] E 0 [fi A ] = E 0 [Xf A] E 0 [f A] For any (P D ntegrable random varable X, wth f = 1 DT. D 0 ST 0 E D [X F t] = E 0 [Xf F t] E 0 [f F t], dpd wth f = P 0 = 1 DT. D 0 ST 0 [ ] E D S T F t = D T [ E 0 S ] T D T F D T D 0 S T 0 t [ E 0 1 D 0 D 0 E ] = D 0 DT F S T 0 t [ 0 S ] T F S T 0 t [ ] = E 0 DT F S T 0 t S t S 0 t D t S t 0 = S t D t.

14 Prcng va change of numerare A consequence of the prevous theorem s the possblty to compute prces w.r.t. P D nstead of P 0. Corollary The value at tme t [0, T ] of a asset X can be computed as [ ] [ ] V t = St 0 E 0 X F ST 0 t = D te D X F t. D T Also the self-fnancng hedgng strategy can be computed w.r.t. D t. If D t = S t for some, ths has the advantage of reducng the number of parameters by one. Let us consder some examples of applcatons.

15 A remark Assume that an Itô process S t > 0 s n the form (ds t = S t(µ tdt + σ tdw t, hence the quadratc varaton s d[s] t = dsds = St 2 σt 2 dt. The process S 1 t we have s an Itô process, wth and by Itô formula wth f (x = 1 x, f (x = 1 x 2, f (x = +2 1 x 3, ds 1 = f (SdS f (Sd [S] = 1 S 2 ds + 1 S 3 S2 σ 2 dt = 1 (S t(µ tdt + σ tdw t + S 1 St 2 t σt 2 dt (( = S 1 µ t + σt 2 dt σ tdw t

16 Segel paradox on exchange rates Assume that we have two dfferent currences, 1 and 2, and correspondng (determstc bonds wth nterest rates r 1, r 2. B 1 t = e r 1t, B 2 t = e r 2 t B s expressed n currency {1, 2} a stochastc exchange rate R t dr t = R t(µdt + σdw t so that a unt of currency 2 equals R t unts of currency 1 In currency 1, the market s gven by two assets: (B 1 t, B 2 t R t. In currency 2, the market s gven by two assets: (B 1 t R 1 t, B 2 t.

17 In the equvalent martngale measure P 0 (rsk neutral measure we have B 2 t R t B 1 t = e (r 2 r 1 t R t must be a martngale. By Ito formula (wth respect to the measure P 0 ( ( ( d Bt 2 R t/bt 1 = e (r 2 r 1 t R t (r 2 r 1 dt + dr dr = R (r 1 r 2 dt + σdwt 0. Also wth respect to P 0, the equaton for the nverse exchange rate s ( d R 1 (( = R 1 (r 1 r 2 + σ 2 dt σdwt 0 there s an extra term σ 2 whch gves no symmetry. The choce of one numerare transfers all the rsk to the others assets. Choose currency 2 as a numerare,.e. D t = B 2 t R t. Then n the probablty P D, we have that Bt 1 Bt 2Rt s a P D -martngale. ths leads to the natural equaton ( d R 1 ( = R 1 t (r 1 r 2 dt σdwt D t

18 Swap optons Consder a market wth three assets S 0, S 1, S 2, ( dst 0 = S 0 rdt, dst 1 = S 1 µ 1 dt + σ 1 dwt 1, dst 2 = St 2 ( µ 2 dt + σ 2 dw 2 t wth Wt 1, Wt 2 ndependent. We want to prce the swap opton that gves the possblty to the holder to exchange S 2 T wth S 1 T wthout addtonal costs. Its value at tme T s ( (ST 1 ST 2 + S 1 = T S 2 T + 1 ST 2 Idea: swap opton s a call opton wth strke prce 1, f numerare s S 2.

19 ( S 1 ( d = S1 (... dt σ S 2 S 2 2 dwt 2 + σ 1 dwt 1 = S1 S 2 (... dt + σ1 2 + σ 1 dw 1 σ2 t σ 2 dwt 2 2 σ1 2 + σ2 2 σ 21 + σ22 dw ( = S1 (... dt + S 2 Where W s a Brownan moton w.r.t. P. If we use the rsk-neutral probablty P 2, correspondng to the numerare S 2, ( S 1 d = ( σ S1 d1 2 + σ22 db S 2 S 2 where B s a P 2 -Brownan moton and [ ] [ (S V t = St 2 E 2 1 ( T ST 2 + F t = St 2 E 2 S 1 T S 2 T ( = St 2 C t, T, S1 t, 1, 0, σ 2 St σ2 2 S 2 T + ] 1 F t where C (t, T, x, K, r, σ Black-Scholes formula for prce of call opton.