Ch 4. Binomial Tree Model
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1 Ch 4. Binomial Tree Model I. One-Period Binomial Tree II. CRR Binomial Tree Model III. Esimaion and Calibraion of µ and σ IV. Dividends and Opion Pricing V. Inroducion of Combinaorial Mehod Appendix A. Binominal Tree Model for Jump-Diffusion Processes This chaper is devoed o inroduce he binomial ree model, which is also known as a kind of laice model. The laice models, such as he binomial ree model inroduced in his chaper or he finie difference mehod inroduced in he nex chaper, are popular numerical mehods for opion pricing, paricularly for pricing American-syle derivaives. They are also flexible since only nominal changes of he payoff funcion are needed for dealing wih complex, nonsandard opion payoff funcion. I. One-Period Binomial Tree Figure ( = 1 18 = =.5 (i Consrucing a porfolio: long shares and shor 1 call 4-1
2 Figure c (ii Deciding he value of : if 1 = 18 porfolio is riskless 4 = 1 =.5 (iii Hence, a =.5, he value of he porfolio is.5 1 = 18.5 = 4.5 c = 4.5 e r.5, where =.5 and r =.1 c =.633 The emergence of he risk free ineres rae r is due o he no-arbirage argumen. 18 Figure 4-3 c u S c = c c = u d c d = cu c d S u S d T risk free rae r cu c d S u S d S c = (S u cu c d S u S d c ue rt c = cu c d ( cu c d u c ue rt c = e rt ( cu c d ert (cu c du + ( c u c = e rt ( (cu c de rt c uu+c d u+uc c u+c d d c d d ( ert d = e rt ( (ert dc u = e rt ( ert d and 1 ert d + (c d (ert dc d c u + (1 ert d c d are like binomial probabiliies, so hey are denoed as p and 1 p. 4-
3 = e rt (p c u + (1 p c d For differen opions, he above equaion remains valid, bu differen payoffs c u and c d should be considered. I is worh noing ha p is no he probabiliy for c u (or for he upward movemen of S. However, p can be regarded as he probabiliy for c u (or for he upward movemen of S in he risk neural world. This is because in he real world, if he expeced sock reurn is µ, S e µt = S u q + S d (1 q q = eµt d u d. Similarly, in he risk neural world, since he expeced reurns for all securiies are he same and equal o r, S e rt = S u p + S d (1 p p = ert d u d. Therefore, p and 1 p are ermed as risk neural probabiliies in he binomial ree framwork. The opion pricing equaion c = e rt (p c u + (1 p c d in he binomial ree model is consisen wih he RNVR because boh he expeced growh rae of he underlying asse and he discoun rae of he opion payoff are he risk free rae. If he upward and dowanward probabiliies in he real world are considered, i is difficul o idenify a proper discoun rae o discoun he expeced payoff of opions, i.e., c = e?t (q c u + (1 q c d. In pracice, for securiies wih more risk or uncerainy, i should apply higher discoun raes o fuure expeced payoffs. Furhermore, i is well-known ha opions are riskier han heir underlying asses due o he high-leverage characerisic for invesing in opions. So, i is no suied o use he expeced reurn for he underlying asse, µ, o discoun he expeced payoff of he opion. If we reconsider he above numerical example of he one-period binomial ree and assume µ = 16%, hen we can derive he probabiliy q in he real world o be.741. Nex by equalizing e?t (q c u + (1 q c d o be.633, which is he rue opion value in he numerical example, we can derive he discoun rae for he opion o be 4.58%. In fac, he proper discoun raes for expeced payoffs of opions depend no only on he expeced reurns (µ and volailiies (σ of underlying asses, bu also on he differen K and T of opions. As a consequence, i is almos impossible o derive heoreical opion prices direcly in he real world. 4-3
4 II. CRR Binomial Tree Model Lognormal propery If X is lognormally disribued, i.e., ln X follows a normal disribuion wih mean = E[ln X] and variance = var(ln X, hen E[X] = e E[ln X]+ 1 var(ln X, and var(x = e E[ln X]+var(ln X (e var(ln X 1. T N( + (µ σ T, σ T E[S T ] = S e µt, and var(s T = Se µt (e σt 1 + N( + (µ σ, σ E[S + ] = S e µ var(s + = S e µ (e σ 1 S (1 + µ (1 + σ 1 (because e x 1 + x when x = S σ + S (µ (σ S σ (because he erm wih is relaively oo small (In he nex paragraph, we will use his propery o derive he value of he parameers u, d, and p in he CRR binomial ree framework. Deriving u, d, and p in he CRR (Cox, Ross, and Rubinsein (1979 binomial ree model, which is he mos common and famous binomial ree model. Figure 4-4 p S u S 1 p S d (i Maching mean: p S u + (1 p S d = E[S + ] = S e r p = er d 4-4
5 (ii Maching variance: var(s + = E[S+ ] E[S + ] σ = p u + (1 p d [p u + (1 p d] (boh sides (1/S = p u + (1 p d p u p (1 p u d (1 p d = u (p p + [(1 p (1 p ] d p (1 p u d = u p (1 p + (1 p [1 (1 p] d p (1 p u d = p (1 p [u u d + d ] = p (1 p (u d p (1 p = p p = er d er d e r +d ( = er u u d e r d+d e r + d e r d ( = er (+ d u d e r ( = er (u+d u d e r ( σ = e r (u + d u d e r Define d = 1 u, σ = e r (u + 1 u u 1 u er u + 1 u = σ +1+e r e r = e r σ + e r + e r e r (1 r, e r (1 + r, and r σ σ + u (σ + u + 1 = u = σ +± (σ + 4 = σ +± σ 4 +4σ +4 4 (σ 4 = σ + 1 ± σ Since is far larger han for a small, and σ is relaively smaller han σ we can ignore he firs erm σ 1 ± σ e +σ (because u > 1 u e σ 4-5
6 Implemenaion of he CRR binomial ree model: Figure 4-5 n Sn (, d S(1, S u S(, S u Si (, j d i j j Sn (, j d n j j S(, S S(,1 S S(1,1 S d S(, S d i j j 1. Si (, j d, for jinand T/ n n j j. cn (, j payoff for d a mauriy r 3. ci (, j e [ pci ( 1, j (1 p ci ( 1, j1] (backward inducion i j j 4. For American opions, ci (, j max( ci (, j, exercise value for d 5. c(, is he opion price oday Snn (, n d Problems of he CRR model: 1 p = er d = er e σ e σ e σ is no necessary in [, 1] unless is small enough. The approximaions used o derive var(s + as well as u and d are vaild only when is very small. 4-6
7 Anoher binomial ree model: by considering he logarihmic sock price space and he consrain of p = 1/, we can derive p = 1 σ, u = e(r +σ σ (r, d = e σ. Figure 4-6 p ln u 1 p ln d + N(lnS + (r σ, σ E[ + ] = + (r σ, var( + = σ Maching mean and variance of + : { p( + ln u + (1 p( + ln d = + (r σ (1 p( + ln u + (1 p( + ln d ( + (r σ = σ ( In his model, p is fixed o be.5, and only u and d are lef as unknowns. From Eq. (1, we can derive.5 ln u +.5 ln d = (r σ define D = ln d and µ = (r σ ln u = µ D Replace ln u in Eq. (.5( + µ D +.5( + D ( + µ = σ ( + µ D + ( + D ( + µ = σ D µd + µ σ = D = µ± 4µ 4(µ σ = µ ± σ = ln d ln u = µ σ { Because u > d u = e µ+σ σ (r = e +σ d = e µ σ σ (r = e σ σ ln u = (r ln d 4-7
8 Advanages of his alernaive binomial ree model: 1 here is no approximaion; p mainains o be a posiive number beween and 1; 3 he convergence rae is generally beer han he CRR model for pricing plain vanilla opions. Disadvanages of his alernaive binomial ree model is due o S ud S : 1 Since here is no sock price layer coinciding wih he specified barrier price, i is less effecien o price he family of barrier opions. Since here is no sock price layer equal o S, i is impossible o apply a quick mehod, known as he exended ree model, o esimae some Greek leers, like, Γ, and Θ. Trinomial Tree u = e σ 3, d = 1, p u d = (r σ + 1, p 1σ 6 m =, 3 p u = (r σ + 1 (mach (i mean of S 1σ 6 +, (ii variance of S +, (iii p i = 1, (iv d = 1, (v p u m =, where (iv and (v are arbirarily imposed consrains
9 III. Esimaion and Calibraion of µ and σ E[ln(S T /S ] = (µ σ T ln(e[s T /S ] because assume T N((µ σ T, σ T KT Assume S 1 = S e η (T = 1 year (η: coninuonsly componuding reurn per annum E[S T /S ] = e KT E[S T ] = S e KT (S 1 is sochasic η follows a disribuion (E[S T ] is a known number K is consan η = 1 S N(µ σ, σ K = µ = expeced growh rae according o µ σ is he expeced value of he coninuously compounding reurn per annum he lognormal propery of S T on page 4-5 i. Considering wo rading days, and + 1, i. Considering wo rading days, and + 1, we can derive S +1 = S e η d and η d = +1 S hen E[S +1 /S ] = e K d and K d = ln E[S +1 /S ] ii. Calculaing he average of +1 S for n days, ii. Calculaing he average of S+1 S for n days and he resul is he esimaion of µ d σ d. aking he naural log, we can esimae K d = µ d 1 S1 n (ln S + S ln S Sn n 1 ln( 1 n ( S1 S + S S Sn S n 1 = 1 n (ln R 1 + ln R + + ln R n = ln( 1 n (R 1 + R + + R n = 1 n (ln(r 1 R R n = ln(r 1 R R n 1 n The geomeric average of daily reurns esimae he coninuously compounding reurn µ d σ d The sandard deviaion of he series of daily 1 The arihemic average of daily reurns esimaes he daily expeced growh rae µ d The sandard deviaion of he series of S+1 S S, ln S S 1,..., ln S Sn n 1 generaes he esimaion is NOT he esimaion of σ d of σ d Noe ha he esimaed resuls based on he daily daa, i.e., µ d and σ d, need o be annualized o derive he corresponding annual resuls. 4-9
10 Implied volailiy (he calibraion ( 校準 of σ For any opion pricing funcion c(s, K, T, r, σ, S is he sock price oday, K and T can be found in he opion conrac, and r is he risk-free rae corresponding o he ime o mauriy T. As for σ, i is commonly esimaed based on he hisorical sock prices. However, he marke price of an opion reflecs he consensus of he forward-looking σ of marke paricipans and may no equal he heoreical opion value based on he hisorical σ. Through equalizing c(s, K, T, r, σ and he marke opion price, i is possible o calibrae σ from he forward-looking viewpoin. The value of σ saisfying f(σ c(s, K, T, r, σ marke opion price = is called he implied volailiy. Here wo roo-finding algorihms are inroduced o solve for he implied volailiy. Bisecion Mehod Firs find [a n, b n ] such ha f(a n f(b n <. The ieraive wo seps o find [a n+1, b n+1 ] are as follows. (i Calcuae x n = a n + bn an (ii If f(a n f(x n < a n+1 = a n, b n+1 = x n else a n+1 = x n, b n+1 = b n Newon s Mehod x n+1 = x n f(xn f (x n Based on he firs-order Taylor series: f(x = f(x n + f (x n (x x n (Find he roo of f(x, i.e., solve f(x =. f(x n = f (x n (x x n x = x n+1 = x n f(xn f (x n Quadraical convergence: According o he second-order Taylor series, f(x = f(x n + f (x n (x x n + f (ξ (x x n, where ξ is beween x n and x. Solve f(x = : x x n + f(xn f (x = f (ξ n f (x (x x n n x x n+1 = f (ξ f (x (x x n n (ppose f (ξ f (x is bounded by a finie number M, i.e., f (ξ n f (x M <. n x x n+1 M x x n (The error is smaller han he produc of a finie number and he square of he error of he n-h ieraion. 4-1
11 IV. Dividends and Opion Pricing This secion inroduces how o modify he opion pricing models if he dividend yield q or known cash dividends D a he pre-specified ime poin are considered: Table 4-1 Three Differen Models for Dividend Paymens ppose he ime poin oday is Model 1: Dividend yield Model : Known cash dividends a as a percenage of Black-Scholes Model Binomial Tree Model European European American or = = no available Figure 4-8 Figure 4-8 Model 3: Known cash dividends a or Figure 4-1 Figure 4-1 Model 1: dividend yield q I is known ha he disribuions of S T are he same under Figure 4-7 { S + paying dividend yield q S e qt + no dividend paymen 4-11
12 In he Black-Scholes formula, simply replace S wih S e qt c = S e qt N(d 1 K e rt N(d where d 1 = ln(s/k+(r q+ σ σ T T, d = d 1 σ T As a maer of fac, he original dπ = ( f + 1 f S σ S d r π d new dπ = dπ + q ( f S S d = r π d (where q ( f S S d is he dividend recived by he porfolio holder f + f S (r qs + 1 f S σ S = rf (where (r qs replaces rs of he original PDE I is worh noing ha in he above PDE, when we consider he dividend yield q, i does no mean o replace r wih r q. In fac, for he underlying asse S, he expeced growh rae is from r o become r q, bu he discoun rae for he derivaive f is sill r because he righ-hand side of he above PDE remains rf. For he binomial ree model Mehod 1: replace S wih S e qt + no dividend paymen Mehod : change he risk neural probabiliy p p S u + (1 p S d = S e (r q p = e(r q d u d However, during he backward inducion phase, we sill use r o discoun he expeced opion value a he nex ime poin, i.e., f = e r [p f u + (1 p f d ] (For European opions, boh above mehods generae correc resuls, bu for American opions, only he second mehod can generae correc resuls. 4-1
13 Model : known cash dividends as a percenage δ of he sock price a he ime poin (In pracice, i is rare for companies o disribue cash dividends in his way. For he Black-Scholes formula, i is unavailable o deal wih his problem. For he binomial ree model, i is simple o deal wih his problem (see Figure 4-8. Figure (1 3 (1 (1 (1 S (1 S (1 (1 S (1 (1 (1 3 (1 4 (1 Model 3: known cash dividends D a ime poin For he Black-Scholes formula, replace S wih S De r. For he binomial ree model, Mehod 1: replace S wih S De r. However, his mehod only works for European opions, bu i canno be applied o pricing American opions. { S + paying D dollars a (This is because he disribuions of S τ (τ are he same under S De r + no dividend paymen Mehod : deduc he known dividend D from all sock prices on he divided paymen dae. This mehod canno work because i makes he ree o be non-recombined. Figure 4-9 D S S D D 4-13
14 Mehod 3: his mehod can mainain he recombined feaure of he binomial ree Figure S S S r( De r( 1 + De De r( De r ( 3 Define S = S De r( = S De r Sep 1: build S -ree following he radiional way, bu i should be noed ha in heory σ = σ S /(S De r(, where σ is he volailiy for he sochasic par of he sock price and σ is he oal volailiy of he sock price. (For a differen volailiy σ, we should derive new u, d, and p in heory. In pracice, however, i is rare o calculae σ. The firs reason is ha he difference beween σ and σ is minor. The second reason is ha since σ is an esimaed value based on he hisorical daa, performing he above adjusmen will no necessarily lead o a beer esimaion for σ in he forward-looking sense. Sep : for i <, replace S i wih S i + De r( i (Since S = S De r( = S De r in Sep 1, S -ree is he same as he ree in Mehod 1 afer, which formulaes he sock price process afer he dividend paymen appropriaely. However, he sock prices before are no consisen wih he sock price process before he dividend paymen, i.e., S S and E[S i ] E[S i ] for i <. So, he adjusmen in Sep should be performed. 4-14
15 Currency opion Replace he dividend yield q wih he foreign risk-free rae r f (Because he underlying asse S is a dollar of he foreign currency in domesic dollars, and holding he foreign currency can earn he foreign risk-free rae, a foreign currency is analogous o a sock paying a known dividend yield. c = Se r f T N(d 1 Ke rt N(d where d 1 = ln(s/k+(r r f + σ T σ T, d = d 1 σ T Fuures opions Call holders: have he righ o ener a long posiion fuures wih he deliver price o be F T and receive cash F T K if F T K, where F T is he laes selemen fuures price before T (usually F T is he closing price on he dae T and K is he srike price in fuures opions. Pu holders: have he righ o ener a shor posiion of fuures wih he deliver price o be F T and receive cash K F T if F T K. Since he fuures is worh zero when i is iniiaed (in he above cases, i is a T, i can be concluded ha he payoff of fuures opion is similar o ha of exercising plain vanilla opions, and he only difference is o replace S T wih F T. This observaion helps us o develop he pricing formula for fuures opions following he same way o develop he opion formula for plain vanilla opions. 4-15
16 Black-Scholes formula for fuures opions: ppose he fuures price follows he geomeric Brownian moion: df F c(f, is he call on fuures, and according o he Iô s lemma = µd + σdz dc = ( c + F c µ F + 1 c F σ F d + c F σf dz Consruc porfolio π = c + ( F c F = c (he iniial value of fuures is dπ = dc + ( F c c df = ( + 1 c F σ F d = rπd = r( cd c + 1 c F σ F = rc (Comparing wih he PDE for sock opions, hese is no such erm F c (r qf. So, seing q = r and S = F in he Black-Scholes formula o derive he formula for fuures opions. c = e rt [F N(d 1 KN(d ], where d 1 = ln(f/k+ σ T σ T Binomial Tree Because q = r, p = e(r q d Anoher way o derive p Figure 4-11 Fuures (a F 1 c value( c = 1 d and d = d 1 σ T (bu we sill use r as he discoun rae payoff T ( ( Fu F cu ( Fd F cd = c = [ (F u F c u ]e rt (he iniial value of he fuures is c = e rt (p c u + (1 p c d, where p = 1 d c c F u Fd = u d The Black-Scholes model as well as he binomial ree model are versaile models: Trea sock index, currency, and fuures like a share of sock paying a dividend yield q For sock index opions: q = average dividend yield on he index over he opion life For currency opions: q = r f For fuures opions: q = r 4-16
17 V. Inroducion of Combinaorial Mehod Combinaorics ( 組合數學 is a branch of pure mahemaics concerning he sudy of discree (and usually finie objecs. Aspecs of combinaorics include couning he objecs saisfying cerain crieria, deciding when he crieria can be me, or finding larges, smalles, or opimal objecs. Combinaorial mehod for European opions Based on he binomial ree framework, applying he combinaorial mehod is far faser han he backward inducion procedure. In fac, i is no necessary o build he binomial ree in he combinaorial mehod, which is anoher advanage of he combinaorial mehod because is can save memory space for compuer programming. Figure 4-1 S n S u d n1 S u d M M S u d M M M n j j M 1 n1 S u d n S u d K European opion value = e rt n j= T is pariioned ino n ime seps ( n j ( n p n j (1 p j max(s u n j d j K,, where j also denoed as C n j, is he combinaion of j from n, u = e σ, d = e σ, and p = e (r q d., For he binomial ree model, is complexiy is O(n, whereas for he above combinaorial mehod, he complexiy is O(n. The difference of required compuaional ime is subsanial for a large number of n, e.g., n >
18 Appendix A. Binominal Tree Model for Jump-Diffusion Processes The conen in his appendix belongs o he advanced conen. Binomial ree model for he jump-diffusion process in Amin (1993: Consider Meron s (1976 lognormal jump-diffusion process as follows. ds S = (r q λk Y d + σdz + (Y 1dq, where K Y = E[Y 1] and ln Y N(µ J, σj. By he Iô s lemma, d = (r q 1 σ λk Y d + σdz + ln Y dq = αd + σdz + ln Y dq The laice model proposed by Amin (1993 Figure 4-13 i i ln ln S S ln ln S i S i i i ln S ln S i i i i i n T 4-18
19 Backward inducion: Figure 4-14 M J M J Define he branching probabiliies as follows. prob( + = α + σ = (1 λ P u + λ N(k P (k = prob( + = α σ = (1 λ P d + λ N(k, prob( + = α + kσ = λ N(k, for M J k M J where N(k = N((α + (k + 1σ µ J /σ J N((α + (k 1σ µ J /σ J if k ±M J, M J in( 3σ J σ, N(M J = 1 N((α + (M J 1σ µ J /σ J, and N( M J = N((α (M J 1σ µ J σ J. (Noe ha λ N(k can fully capure he effec of he jump componen ln Y dq of d. 4-19
20 Deermine P u and P d by maching he mean and variance of he remaining diffusion process, d = αd + σdz P u S u + P d S d = S e α P u S u + P d S d (P u S u + P d S d = S σ, P u + P d = 1 where u = e α +σ and d = e α σ. Analogous wih he CRR binomial ree model, we can infer P u = eα d. However if we consider he probabiliies of (1 λ P u and (1 λ P d for he upward and downward branches, respecively, when he jump componen is inroduced. The mean and variance generaed by he probabiliies of (1 λ P u and (1 λ P d are no more S e α and S σ. (1 λ P u S u + (1 λ P d S d = (1 λ S e α (1 λ P u S u + (1 λ P d S d [(1 λ P u S u + P d S d] (1 λ S σ, (1 λ P u + (1 λ P d = (1 λ For he variance equaion, i can be rewrien as (1 λ [P u S u + P d S d (1 λ (S e α ] (1 λ S σ, if is small such ha (1 λ 1. So, he sandard deviaion become (1 λ S σ. By comparing wih he mean, (1 λ S e α, one can find ha he deviaion of he mean from is rue value is more serious han he deviaion of he sandard deviaion from is rue value because (1 λ is closer o 1 han (1 λ when is small. To offse he above undesirable effecs, we should solve P u and P d in he following sysem of equaions. P u S u + P d S d = Seα 1 λ P u S u + P d S d (P u S u + P d S d = S σ 1 λ P u + P d = 1 In Amin (1993, only he adjusmen for he mean is considered, and hus he probabiliy P u can be derived as P u = e α 1 λ d. 4-
21 Opion value of a node can be expressed as Opion value of = P (k (opion value for + = + α + kσ M J k M J If he adjusmen for he variance is also considered, one can adjus he grid size of he mulinomial ree o achieve his goal. By defining σ σ = 1 λ, u = e α +σ, d = e α σ, one can derive he following ree srucure and he corresponding branching probabiliies. Figure 4-15 * M J * * * * * M J prob( + = α + σ = (1 λ P P u + λ N(k (k = prob( + = α σ = (1 λ Pd + λ N(k, prob( + = α + kσ = λ N(k, for M J k M J where N(k = N((α +(k+ 1σ µ J /σ J N((α +(k 1σ µ J /σ J if k ±M J, M J in( 3σ J σ, N(M J = 1 N((α +(M J 1σ µ J /σ J, N( M J = N((α (M J 1 σ µ J /σ J, and P u = e α 1 λ d u d and P d = 1 P u. 4-1
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