The Cox-Ross-Rubinstein Option Pricing Model

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1 Fnance 400 A. Penat - G. Pennacc Te Cox-Ross-Rubnsten Opton Prcng Model Te prevous notes sowed tat te absence o arbtrage restrcts te prce o an opton n terms o ts underlyng asset. However, te no-arbtrage assumpton alone cannot determne an exact opton prce as a uncton o te underlyng asset prce. To do so, one needs to make addtonal assumptons regardng te dstrbuton o returns earned by te underlyng asset. Certan dstrbutonal assumptons can mply a complete market or te underlyng asset s rsk tat allows us to determne a unque opton prce. Te model n tese notes makes te assumpton tat te underlyng asset, ereater reerred to as a stock, takes on one o only two possble values eac perod. Wle ts may seem unrealstc, te assumpton leads to a ormula tat can accurately prce optons. Ts bnomal opton prcng tecnque s oten appled by Wall Street practtoners to numercally compute te prces o complex optons. Here, we start by consderng te prcng o a smple European opton wrtten on a non-dvdend-payng stock. In addton to assumng te absence o arbtrage opportuntes, te bnomal model assumes tat te current stock prce, S, eter moves up, by a proporton u, or down, by a proporton d, eacperod. Teprobabltyoanupmovesq, so tat te probablty o a down move s 1 q. Ts can be llustrated as us wt probablty q S % & (1) ds wt probablty 1 q Denote as one plus te rsk-ree nterest rate or te perod. Ts rsk-ree return s assumed to be te constant over tme. To avod arbtrage between te stock and te rsk-ree nvestment, we must ave d< <u. Let C equal te value o a European call opton wrtten on te stock and avng a strke prce o X. Atexpry,C = max[0,s T X]. Tus: One perod pror to expry: 1

2 C u max [0,uS X] wt probablty q C % & (2) C d max [0,dS X] wt probablty 1 q Wat s C one perod beore expry? Consder a portolo contanng sares o stock and $B o bonds. It as current value equal to S + B. Ten te value o ts portolo evolves over te perod as us + B wt probablty q S + B % & (3) ds + B wt probablty 1 q Wt two securtes (te bond and stock) and two states o te world (up or down), and B can be cosen to replcate te payo o te call opton: us + B = C u ds + B = C d (4a) (4b) Solvng or and B tat satsy tese two equatons, we ave = C u C d (u d) S (5a) B = uc d dc u (u d) (5b) Hence, a portolo o sares o stock and $B o bonds produces te same caslow as te call opton. Ts s possble because te market s complete. Tradng n te stock and bond produces payos tat span te two states. Now snce te portolo s return replcates tat o te opton, te absence o arbtrage mples 2

3 Example: IS = $50, u =2,d =.5, =1.25, X =$50,ten C = S + B (6) us = $100, ds=$25,c u =$50,C d =$0. Tereore: = 50 0 (2.5) 50 = 2 3 B = 0 25 (2.5) 1.25 = 40 3 so tat C = S + B = 2 3 (50) 40 3 = 60 3 = $20 I C< S + B, ten an arbtrage s to sort sell sares o stock, nvest $ B n bonds, and buy te call opton. Conversely, C> S + B, ten an arbtrage s to wrte te call opton, buy sares o stock, and borrow $ B. Te resultng opton prcng ormula as an nterestng mplcaton. It can be re-wrtten as C = S + B = C u C d (u d) + uc d dc u (7) (u d) R d u d = max [0,uS X]+ u R u d max [0,dS X] wc does not depend on te probablty o an up or down move o te stock, q. Tus,gven S, nvestors wll agree on te no-arbtrage value o te call opton even tey do not agree on q. Snce q determnes te stock s expected rate o return, uq + d(1 q) 1, ts does not need to be known or estmated n order to solve or te no-arbtrage value o te opton, C. However, 3

4 we do need to know u and d, te sze o movements per perod, wc determne te stock s volatlty. Butnotetattecalloptonvalue, C, doesnotdrectly depend on nvestors atttudes toward rsk. It s a relatve (to te stock) prcng ormula. Note also tat we can re-wrte C as [pc u +(1 p) C d ] (8) were p d u d. Snce 0 < p < 1, p as te propertes o a probablty. In act, ts pseudo-probablty p would equal te true probablty q nvestors were rsk-neutral, snce ten te expected return on te stock would equal : [uq + d (1 q)] S = S (9) or q = d u d = p. (10) Perapstssnotsurprsng,snceteexpresson [pc u +(1 p) C d ] does not depend on rsk-preerences, and so t must be consstent wt all possble rsk preerences, ncludng rsk-neutralty. Next, consder te opton s value wt: Two perods pror to expraton: Testockprceprocesss 4

5 u 2 S us % & S % & dus (11) ds % & d 2 S so tat te opton prce process s C uu max 0,u 2 S X % C u & C % & C du max [0,duS X] (12) % C d & C dd max 0,d 2 S X Usng te results rom our analyss wen tere was only one perod to expry, we know tat C u = pc uu +(1 p) C du C d = pc du +(1 p) C dd (13a) (13b) Wt two perods to expry, te one perod to go caslows o C u and C d can be replcated once agan by te stock and bond portolo composed o = Cu C d (u d)s B = uc d dc u (u d) o bonds. No-arbtrage mples sares o stock and C = S + B = 1 [pc u +(1 p) C d ] (14) 5

6 Substtutng n or C u and C d,weave p 2 R 2 C uu +2p (1 p) C ud +(1 p) 2 C dd = 1 R 2 p 2 max 0,u 2 S X +2p (1 p)max[0,dus X]+(1 p) 2 max 0,d 2 S X Note tat C depends only current S, X, u, d,,andtetmeuntlmaturty,2perods. Repeatng ts analyss or tree, our, ve,..., n perods pror to expry, we always obtan (15) C = S + B = 1 [pc u +(1 p) C d ] By repeated substtuton or C u, C d, C uu, C ud, C dd, C uuu, etc., we obtan te ormula: n perods pror to expraton: R n µ j=0 X p j (1 p) n j max 0,u j d n j S (16) Ts ormula can be smpled by denng a as te mnmum number o upward jumps o S or t to exceed X. Tus a s te smallest non-negatve nteger suc tat u a d n a S>X. Takng te natural logartm o bot sdes, a s te mnmum nteger >ln(x/sd n )/ln(u/d). Tereore or all j<a(te opton expres out-o-te money), wle or all j>a(te opton expres n-te-money), max 0,u j d n j S X =0, (17a) Tus, te ormula or C can be re-wrtten: R n max 0,u j d n j S X = u j d n j S X (17b) µ j=a Breakngtsupntotwoterms,weave: p j (1 p) n j u j d n j S X (18) 6

7 µ " C = S p j (1 p) n j u j d n j # R n (19) j=a µ XR n p j (1 p) n j j=a Te terms n brackets are complementary bnomal dstrbuton unctons, so tat we can wrte ts as were p 0 C = Sφ[a; n, p 0 ] XR n φ[a; n, p] (20) ³ u R p and φ[a; n, p] = te probablty tat te sum o n random varables wc equal 1 wt probablty p and 0 wt probablty 1 p wll be a. Tese ormulas mply tat C s te dscounted expected value o te call s termnal payo tat would occur n a rsk-neutral world. I we dene τ astetmeuntlmaturtyotecalloptonandσ 2 astevaranceperunt tme o te stock s rate o return (wc depends on u and d), ten by takng te lmt as te number o perods n, but te lengt o eac perod τ n 0, te Cox-Ross-Rubnsten bnomal opton prcng ormula becomes te well-known Black-Scoles-Merton opton prcng ormula 1 were z ln µ S XR τ (σ τ) σ2 τ C = SN (z) XR τ N z σ τ (21) and N ( ) s tat standard normal dstrbuton uncton. 1 In te Black-Scoles-Merton ormula, s now te rsk-ree return per unt tme rater tan te rsk-ree return or eac perod. Te relatonsp between σ and u and d wll be dscussed sortly. 7

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