Diener and Diener and Walsh follow as special cases. In addition, by making. smooth, as numerically observed by Tian. Moreover, we propose the center

Transcription

1 Smooth Covergece i the Biomial Model Lo-Bi Chag ad Ke Palmer Departmet of Mathematics, Natioal Taiwa Uiversity Abstract Various authors have studied the covergece of the biomial optio price to the Black-Scholes price as the umber of periods teds to ifiity. I this article, we cosider the biomial model with a additioal parameter λ ad show that i the case of a Europea call optio the biomial price coverges to the Black-Scholes price at the rate /. Results of Dieer ad Dieer ad Walsh follow as special cases. I additio, by makig special choices for λ, we ca achieve smooth covergece. I particular, we are able to prove that the covergece i Tia s flexible biomial model is smooth, as umerically observed by Tia. Moreover, we propose the ceter biomial model which also exhibits smooth covergece for Europea call optios but has the advatage that it gives faster smooth covergece tha the flexible biomial model for digital call optios. Key words Biomial model, Black-Scholes model, optio pricig, smooth covergece JEL Classificatio G3 Mathematics Subject Classificatio(000) 6P05 This author was supported by NSC grat 93-8-M

2 Itroductio I this paper, we study the rate ad smoothess of the covergece of the Europea call optio price give by the biomial model to the Black- Scholes price as the umber of periods teds to ifiity. We assume the stock price evolves as i the Black-Scholes model ad use the followig otatios: S 0 as the iitial stock price, K as the strike price, r as the cotiuously compouded iterest rate, σ as the volatility, T as the time to maturity, ad C BS = S 0 Φ(d ) Ke rt Φ(d ) as the Black-Scholes call optio price at t = 0, where Φ(.) is the stadard ormal distributio fuctio. I the biomial model, we assume the curret stock price S either rises to Su or falls to Sd at the ed of the ext period, ad if is the umber of periods, we use the otatio C() for the -period biomial model call optio price ad set t = T. The first proofs that the biomial optio price coverges to the Black- Scholes price as the umber of time periods teds to ifiity were give by Cox, Ross ad Rubistei (979) ad Redlema ad Bartter (979). The, a more geeral proof was provided by Hsia (983). However, oe of these authors studied the rate or smoothess of the covergece. Hesto ad Zhou (000) studied the rate of covergece. They work with the Cox, Ross, ad Rubistei (CRR) model, where u = e σ t, d = e σ t, ad show that the error, the differece betwee the biomial ad Black- ( ) Scholes prices, is O for a geeral class of optios.

3 3 For Europea call optios, the covergece is faster. Leise ad Reimer (996) proved that for the CRR model, the Jarrow ad Rudd (983) model ad the Tia (993) model, the error for Europea call optios is O ( ), where Jarrow ad Rudd take u = e σ t+(r σ ) t, d = e σ t+(r σ ) t, ad Tia takes u = MV (V + + V + V 3), d = MV (V + V + V 3) with M = e r t ad V = e σ t. However, ote that Leise ad Reimer did ot give a explicit formula for the coefficiet of. Fracie ad Marc Dieer (004) prove the followig result for the CRR model, which we state i our otatio: I the -period CRR biomial model, if S 0 =, the biomial price at t = 0 of the Europea call with strike price K ad maturity T = satisfies C() = C BS + e d / where = frac d )r r. 4σ π A σ ( ) ( ) + O [ ] log(/k)+ log d log(u/d) ad A = σ (6 + d + d ) + 4(d So the coefficiet of / i the error depeds o the quatity, which i geeral oscillates betwee ad as teds to, ad accouts for the well-kow oscillatio of the biomial call price aroud the Black-Scholes price. We will say more about below i coectio with smooth covergece. Walsh (003) (see also Walsh ad Walsh (00)) cosiders a geeral class of optios but he chooses u ad d differetly. I the otatio of this paper, he proves the followig result for Europea call optios:

4 4 σ t+r t I the -period biomial model with eve, let u = e ad d = e σ t+r t. If S 0 is the iitial stock price, the the biomial price at time t = 0 of a Europea call with strike price K ad maturity T satisfies C() = C BS + S 0e d / 4σ πt A σ T e rt ( ) ( ) + O, where [ ] log(s0 /K) + log d = frac log(u/d) ad A is a costat. Note that Walsh does ot give a explicit formula for A. Actually the e rt term before ( ) should ot be there, as ca easily be see from Walsh s paper. I this paper, we take u = e σ t+λ σ t ad d = e σ t+λ σ t, where λ is a geeral bouded sequece. (It is equivalet to cosiderig all possible choices of u = u ad d = d such that u /d = e σ t ad log(u d ) is bouded.) Note that λ = 0 gives the CRR model, λ = r/σ Walsh s model ad λ = r/σ / the Jarrow ad Rudd model. The reaso we cosider this more geeral model is that it allows us, as i Tia (999), to achieve smooth covergece (defied below) by suitable choice of λ. We prove the followig theorem. Mai Theorem: For the -period biomial model, where u = e σ t+λσ t, d = e σ t+λσ t, () with λ a arbitrary bouded fuctio of, if the iitial stock price is S 0 ad the strike price is K ad maturity is T, the

5 5 () the price of a digital call optio satisfies ad C d () = e rt Φ(d ) + e rt e d / π () the price of a Europea call optio satisfies where C() = C BS + S 0e d 4σ πt [ d + B ] ( ) + o A σ T ( ) [ ] log(s0 /K) + log d = frac, log(u/d) + o ( ), B = d3 + d d + d 4d + ( d d d ) T (r λσ )+ T d 4 6σ σ (r λσ ), A = σ T (6 + d + d ) + 4T (d d )(r λσ ) T (r λσ ). Takig S 0 = T = ad λ = 0 i (), we obtai the result of Fracie ad Marc Dieer. Takig r = λ/σ i (), we obtai the result of Walsh (without the e rt term metioed above). The secod focus of this paper is smooth covergece, which we defie as follows. Defiitio: If the -period biomial model price C() of a optio satisfies e() = C() C BS = f() m ( ) + o m as, where f() is bouded ad does ot ted to zero, we say the rate of covergece is of order / m. If f() ca be take as costat, we say the covergece is smooth.

6 6 Whe the covergece is smooth of order / m, we fid that the error ratio r() = e() e() ρ = m as. I this case ot oly is the covergece mootoe, we ca also use extrapolatio to accelerate the covergece. I fact, if we defie the sequece fid that Ĉ() C BS = e() ρ r() ρ C BS tha the origial sequece. ρc() C() Ĉ() = ρ, we so that Ĉ() coverges faster to I geeral, the i the Mai Theorem is a oscillatory fuctio of so that smooth covergece does ot usually hold. If j is the iteger such that S j = S 0 u j d j+ < K S 0 u j d j = S j, the = log(s j/k) log(s j /S j ). So measures the positio of K o the log scale i relatio to the two adjacet termial stock prices. I fact, if K = S j = 0 if K = S j S j ad approaches as K approaches S j. So covergece i the CRR model or Walsh s model is almost always osmooth because the positio of K oscillates betwee the two adjacet stock prices so that oscillates betwee ad. However the covergece is smooth i special cases such as i Fracie ad Marc Dieer (999) for at the moey Europea calls, sice for such optios K always falls o a termial stock price if is eve ad is positioed at the geometric average of the two middle stock prices if is odd.

7 7 Leise ad Reimer (996) also study smooth covergece. Without provig it they observe that the smoothess of covergece seems to deped o the relative positio of the strike price withi the stock price tree. They costruct a biomial model whose u ad d are chose so that the strike price is positioed at the ceter of the stock price odes at maturity. They observed smooth covergece i umerical simulatios but did ot prove it. Hesto ad Zhou (000) describe two approaches to achieve smooth covergece. First they replace the biomial optio prices immediately prior to the termial period by the Black-Scholes values, computig the remaiig biomial prices as usual. Their secod approach is to smooth the osmooth payoff fuctio g(x). They also observe smooth covergece i umerical simulatios without provig it. I his flexible biomial model Tia (999) adjusts u ad d i the CRR model so that the exercise price K coicides exactly with a stock price at a termial ode. Suppose u 0 = e σ t, ad d 0 = /u 0. The Tia fids the j 0 so that S 0 u j0 0 d j0+ 0 < K S 0 u j0 0 d j0 0, ad takes u = e σ t+λσ t, d = e σ t+λσ t, choosig λ so that K = S 0 u j0 d j0. We ote that Tia observed from umerical simulatios that his model exhibited smooth covergece but did ot prove it. The origial motivatio for this paper was to prove smooth covergece i the flexible biomial model. It follows from our Mai Theorem.

8 8 Corollary : For the -period flexible biomial model, (a) the price of a digital call optio with strike price K satisfies C d () = e rt Φ(d ) + e rt e d π + O ( ) so that the flexible biomial price C d () coverges to the Black-Scholes price e rt Φ(d ) smoothly with order / ; (b) the price of a Europea call optio with strike price K satisfies C() = C BS + S 0e d 4σ πt A + o ( ), where A = σ T (6 + d + d ) + 4T (d d )r T r so that the flexible biomial price C d () coverges to the Black-Scholes price smoothly with order /. I this paper, we develop a ew biomial model called the ceter biomial model. The iterestig feature of this model is that we get covergece of order / for the digital call optio whereas for the flexible model the covergece was oly of order /. I this model, we defie u, d ad j 0 as Tia does but ow we choose λ so that K is the geometric average of S 0 u j0 d j0 ad S 0 u j0 d j0+. The ituitive reaso for the better covergece for the digital optio is that the payoff fuctio for a digital optio has a jump at the strike price if it coicides with a termial stock price but has o jump if the strike price is situated betwee stock prices. Our result for the ceter biomial model is a corollary of the Mai Theorem.

9 9 Corollary : For the -period ceter biomial model, (a) the price of a digital call optio with strike price K satisfies C d () = e rt Φ(d ) + e rt e d / π A + o ( ) where A = σ (d 3 + d d + d 4d ) + 4σ T ( d d d )r + T d r 4σ so that the ceter biomial digital call optio price C d () coverges to the Black-Scholes price e rt Φ(d ) smoothly with order /; (b) the price of a Europea call optio with strike price K satisfies C() = C BS + S 0e d 4σ πt A + o ( ), where A = σ T ( 6 + d + d ) + 4T (d d )r T r so that the ceter biomial price C() coverges to the Black-Scholes price smoothly with order /. I this paper, we have studied the covergece of the biomial model oly for Europea optios. The covergece for America optios has also bee discussed i the literature. Ami ad Khaa (994) first gave a proof of the covergece of the biomial method for America optios. Jiag ad Dai (999) ad Qia, Xu, Jiag ad Bia (005) employed the theory of viscosity solutios to show uiform covergece. Lamberto (998) showed that for America puts i the CRR model the covergece order is betwee / 3 ad /. Leise (998) showed that for America puts the CRR model coverges with order / ad the models of Jarrow ad Rudd (983) ad Tia (993) coverge with order betwee / ad /. However there appear to be o rigorous results cocerig smooth covergece.

10 0 Fially, we summarize the cotets of this paper. I Sectio, we give the proof of the Mai Theorem ad the two corollaries. Sectio 3 is the coclusio ad Sectio 4 is a Appedix i which we prove a Lemma eeded for the proof of the Mai Theorem. Proof of Mai Theorem ad Corollaries I this sectio, we prove the Mai Theorem ad Corollaries as stated i the Itroductio. First from Pliska (999, pp.0-), we have that i the biomial model with periods ad parameters u, d, r, whe the curret stock price is S 0, the price of a Europea call optio with maturity T ad strike price K is give by C() = S 0 k=j ( ) ˆp k ˆq k Ke rt k k=j ( ) p k q k, k provided that 0 < p <, where p = er t d u d, q = p, ˆp = pue r t, ˆq = ˆp ad j = [γ], γ = log(k/s 0) log d log(u/d) ([x] = mi{m N : m x}). Note the biomial price of a digital call optio with maturity T ad strike K is C d () = e rt k=j ( ) p k q k, k a compoet of the biomial Europea call price, ad its Black-Scholes price is e rt Φ(d ), a compoet of the Europea call price SΦ(d ) Ke rt Φ(d ). Thus, we first prove part () of our mai theorem, ad the prove part ().

11 Our fudametal tool is the followig Lemma, which is a extesio of a result of Uspesky (937, p.0) o approximatig the biomial distributio by the ormal distributio. The proof of the Lemma is i the Appedix. Lemma : Provided that p = p / as ad 0 j = j + for sufficietly large, k=j ( ) k p k q k = ξ π ξ e u du + q p 6 πpq (( ξ )e ξ ( ξ )e ξ ) + π (ξ e ξ (ξ ) ξ e ξ (ξ )) + o ( ) where ξ = j p / q + / ad ξ pq =. pq Proof of Mai Theorem: Throughout the proof we assume is so large that 0 < p < ad 0 γ +, which implies that 0 j = [γ] +. Proof of (): First we expad the risk eutral probability p i powers of t / up to order 3. From the defiitio () ad usig Taylor expasio, p = er t d u d = / + α t / + β t 3/ + o( t 3/ ) () where α = r (λ + /)σ σ, β = σ4 (4λ + ) 4σ r + (r λσ ). (3) 48σ Next we estimate ξ = j p / = pq ( j + p + ). (4) pq

12 Because j = γ + frac( γ), γ + + α t / = d ad usig equatio (), j + p + = j α t / + β t 3/ + o( t / ) (5) = + d + βt t / + o( t / ). Also, sice pq = p( p) = 4 α t+o( t 3/ ), we get, usig the biomial series theorem, Hece, from (4), usig (5) ad (6), pq = + α t + O( t 3/ ). (6) ξ = d + + δ ( ) + o, where δ = T (α d + β T ). (7) Next we examie the terms i C d () i Lemma oe by oe. Let I ξ ξ First we estimate I = ξ I = ξ e u du = I I = e u du = d ξ e u du e u du. With f(x) = x e u du + ξ By Taylor expasio, for some η betwee ξ ad d, f( ξ ) = e d / ( ξ d ) d e d / where f (η) = e η [ (7) that f( ξ ) = e d / + δ d d ξ d e u e u du. du, we write e u du = πφ(d ) + f( ξ ). ( ξ d ) + f (η) ( ξ d ) 3, 3! + η e η is bouded. The it follows from equatio ] + o ( ), ad so I = πφ(d ) + e d [ + δ d ] + o ( ). (8)

13 3 Next, we estimate I = ξ, ξ = q + / pq ξ for 0. The, whe 0, I e u du. Sice p / ad q / as as. Hece we ca fid 0 such that ξ e u du = e ξ = e q + / pq = o( ). (9) ( Next from equatio (6), ) pq = + α T +O 3/ ad from equatio (), p q = α ( ) T + O 3/ so that we have q p = 4α T pq ( ) + O. Next ote that ( ξ)e ξ ( ξ)e ξ ( d )e d as. Hece q p 6 ) (( ξ πpq )e ξ ( ξ)e ξ = 4α T 6 π ( ( d )e d ) ( ) + o. (0) Next, usig ξ d ad ξ as, we have π (ξ e ξ (ξ ) ξ e ξ (ξ )) = d e d / (d ) π Usig (8)-() i Lemma, we obtai +o ( ). () e rt C d () = ( ) k=j k p k q k = Φ(d ) [ + e d / d { π + δ + ( α T 3 d ) } ( d ) ] ( ) + o. () Multiplyig by e rt ad replacig α ad δ by their defiitios, we obtai part () of the Mai Theorem.

14 4 Proof of (): As i the proof of (), we assume that is so large that 0 < p < ad 0 γ +. The, usig Lemma with p, q, ξ, ξ replaced by their hatted versios, we obtai the relatio k=j ( k )ˆpk ˆq k = ˆξ π ˆξ e u du + ˆq ˆp 6 πˆpˆq {( ˆξ )e ˆξ ( ˆξ )e ˆξ } + π (ˆξ e ˆξ (ˆξ ) ˆξ e ˆξ (ˆξ )) + o ( ), where ˆξ = j ˆp / ˆpˆq ad ˆξ = ˆq + /. Now we estimate ˆp. From ˆpˆq defiitio () ad usig Taylor expasio ˆp = u ude r t u d = + ˆα t/ + ˆβ t 3/ + o( t 3/ ), where ˆα = r (λ /)σ σ σ, ˆβ 4 (4λ ) 4σ r (r λσ ) =. (3) 48σ Next by replacig p, q, α ad β i the derivatio of equatio (7) by the same quatities with hats o ad usig the fact that here γ + + ˆα t / = d, we fid that ˆξ = d + + ˆδ ( ) + o, where ˆδ = T (ˆα d + ˆβ T ), with ˆα ad ˆβ as i equatio (3). Proceedig as we did to get (), but replacig p, q, ξ, ξ, α, β, δ by their hatted versios ad d by d, we obtai the relatio ( k=j k )ˆpk ˆq k = Φ(d ) [ { + e d / + ˆδ d ( π + ˆα T 3 d ) } ( d ) ] ( ) + o. (4)

15 5 Now multiplyig (4) by S 0, () by Ke rt, subtractig ad usig the well-kow fact that S 0 e d / = Ke rt e d /, we get where B is C() = S 0 Φ(d ) Ke rt Φ(d ) + S 0e d ( ) B π + o, ˆδ δ σ T + ( d ) T (8(ˆα α) σ) σ T (d + d ) (8α d ). Simplifyig we obtai part () of our Mai Theorem. Next, we prove the two corollaries. Proof of Corollary : For the flexible biomial model, λ = log( K S 0 ) (j 0 )σ t σ, j 0 = [ γ], γ = log( K S 0 ) + σ t t σ t with [ γ] = mi{m N : m γ}. Note that λ = ( γ j0) σ T so that σ T < λ 0. Thus λ = λ 0 as ad hece is certaily a bouded sequece. Next we observe that γ = log(k/s 0) log d log(u/d) = log(k/s 0) + σ t λσ t σ t = j 0 so that = frac( γ) = frac( j 0 ) =. Usig this ad also the fact that λ = λ 0, we obtai Corollary from our Mai Theorem. Proof of Corollary : For the ceter biomial model, λ = log( K S 0 ) (j 0 )σ t σ, j 0 = [ γ], γ = log( K S 0 ) + σ t t σ t with [ γ] = mi{m N : m γ}. Note λ = ( γ j0)+ σ T so that σ T < λ σ T. Thus λ = λ 0 as ad hece is certaily a bouded sequece. Next γ = log(k/s 0) log d log(u/d) = log(k/s 0) + σ t λσ t σ t = j 0 /

16 6 so that = frac( γ) = frac((/) j 0 ) = 0. The we apply our Mai Theorem to obtai Corollary. 3 Coclusio I this paper we studied the covergece of the biomial price of a optio to the Black-Scholes price as the umber of periods teds to ifiity. We cosidered a geeral biomial model with a additioal parameter λ ad showed that the error i the -period biomial digital call optio price is of order / ad that i the Europea call optio price it is of order /. However the asymptotic relatio we derived showed that, i geeral, the covergece is ot smooth. Nevertheless oe choice of λ yields Tia s flexible biomial model ad our asymptotic relatio eables us to verify that covergece i this model is ideed smooth, as observed by Tia. Aother choice yields what we call the ceter biomial model which also exhibits smooth covergece but has the additioal feature that the covergece is of order / for both the Europea call optio ad digital call optio whereas i the flexible biomial model the covergece is oly of order / for the digital call optio. 4 Appedix: Proof of Lemma Our aim here is to prove Lemma i Sectio. First by Uspesky (937, p.), we write k=j ( ) p k q k = J(ξ ) J(ξ ) (5) k

17 7 where ξ = j p / B where, ξ = q+/ B, with B = pq ad J(ξ) = π π 0 R si(ξ B ϕ χ) si ϕ dϕ, χ = arg (pe iϕ + q) pϕ, R = pe iϕ + q. As i sectio 6 of Uspesky (937, p.4), usig equatio (5) o Uspesky (937, p.5), we ca write J(ξ) = J (ξ) + o ( ), where J (ξ) = τ π 0 R si(ξ B ϕ χ) si ϕ dϕ, with τ = 3B /4. Before we go o, we first list equatios (7), (8) ad (9) o Uspesky (937, p.3,4) as follows: R e Bϕ < 6 B ϕ 4 e Bϕ, (6) χ = 6 B (p q)ϕ 3 + Nϕ 5, where N < B p q ( pqτ ) 4. (7) By extedig the argumet i Uspesky, we ca prove the followig more refied estimate for R. We omit the details of the proof. R e Bϕ e 4 B( 6 pq)ϕ4 < e Bϕ B ϕ 6. (8) Now, we partitio J (ξ) as follows: J (ξ) = I 0 + I + I + I 3 + I 4 + I 5 + I 6,

18 8 where I 0 = π I = τ π 0 I = π I 3 = τ π 0 I 4 = τ π 0 τ 0 e Bϕ si(ξ B ϕ χ) ϕ ( R si ϕ e Bϕ ϕ dϕ, ) (si(ξ B ϕ χ) si(ξ B ϕ) ) dϕ, ( ) τ 0 R si ϕ ϕ ϕ si(ξ B ϕ)dϕ, ( ϕ R e Bϕ e 4 B( pq)ϕ4) 6 si(ξ B ϕ)dϕ, ( ) ϕ e Bϕ e 4 B( 6 pq)ϕ4 si(ξ B ϕ)dϕ, I 5 = τ ϕ π 0 (R e Bϕ ) si(ξ B ϕ)dϕ, I 6 = τ π 0 e Bϕ ϕ si(ξ B ϕ)dϕ. Note Uspesky writes J (ξ) = I 0 + Î + Î, where Î = π Î = τ π 0 τ 0 R ( ϕ si ϕ ϕ ) si(ξ B ϕ χ)dϕ, ( R e Bϕ) si(ξ B ϕ χ)dϕ. We have used a more refied decompositio i order to obtai a more precise result by usig (8). First, followig sectio 9 ad sectio 0 i Chapter VII i Uspesky (937), we obtai I 0 = ξ π 0 ( ) e u q p du + 6 πpq ( ξ )e ξ + o (9) uiformly i ξ. The, by usig (6), (7) ad the Lauret expasio si ϕ = ϕ + ϕ + O(ϕ3 ), we ca show that Next we write I = o ( ) ( ), I = o ad I 5 = o ( ). (0) I 6 = B τ e x x si(ξx)dx. 4B π 0

19 9 Because as, B τ = 3B /4 teds to, ad Bτ e x x si(ξx) dx = o(), we get I 6 = 4B π Next, by (8), we have I 3 π 0 0 e x x si(ξx)dx+o e Bϕ B ϕ 5 dϕ = πb ( ) = 0 4πB ( ) π ξe ξ +o. e u / u 5 du = o () ( ). () Next, by usig the fact that 0 e x x x / for x < 0, we fid that I 4 = π τ 0 ( ) ϕ e Bϕ 4 B 6 pq ϕ 4 si(ξ ( ) B ϕ)dϕ + o. The, as i the treatmet of I 6, we get I 4 = ( ) π ( 6pq) 4B π ξ(3 ξ )e ξ + o. From (9), (0), (), () ad the last equatio, we deduce that J(ξ) = J (ξ) + o ( ) = ξ u e π du + q p 0 6 πpq ( ξ )e ξ + π uiformly with respect to ξ. The, as required, J(ξ ) J(ξ ) = π ξ ξ So usig (5), we get the Lemma. e u du + q p 6 πpq π ξe ξ (ξ ) + o ( [( ξ )e ξ ( ξ )e ξ ] + π [ξ e ξ (ξ ) ξ e ξ (ξ )] + o ( ). ) Refereces. Ami, K.; A. Khaa: Covergece of America optio values from discrete to cotiuous-time fiacial models, Mathematical Fiace 4, (994)

20 0. Cox, J., Ross, S.A., Rubistei, M.: Optio pricig: a simplified approach, Joural of Fiacial Ecoomics 7, 9-63 (979) 3. Dieer, F., Dieer, M.: Asymptotics of the biomial formula for optio pricig, (999) 4. Dieer, F., Dieer, M.: Asymptotics of the price oscillatios of a Europea call optio i a tree model, Mathematical Fiace 4, 7-93 (004) 5. Hesto, S., Zhou, G.: O the rate of covergece of discrete-time cotiget claims, Mathematical Fiace 0, (000) 6. Hsia, C.-C.: O biomial optio pricig, The Joural of Fiacial Research 6, 4-50 (983) 7. Jarrow, R., Rudd A.: Optio Pricig, Homewood, Illiois (983) 8. Jiag, L., Dai, M.: Covergece of biomial tree method for America optios, i Proceedigs of the Coferece o Partial Differetial Equatios ad their Applicatio, Sigapore: World Scietific, 06-8 (999) 9. Lamberto, D.: Error estimates for the biomial approximatio of America put optios, The Aals of Applied Probability 8, (998) 0. Leise, D. P. J.: Pricig the America put optio: a detailed covergece aalysis for biomial models, Joural of Ecoomic Dyamics ad Cotrol, (998). Leise, D. P. J., Reimer, M.: Biomial models for optio valuatio-examiig ad improvig covergece, Applied Mathematical Fiace 3, (996). Pliska, S.R.: Itroductio to Mathematical Fiace: Discrete Time Models, Oxford: Blackwell (997) 3. Qia, X., Xu, C., Jiag, L., Bia, B.: Covergece of biomial tree method for America optios i a jump-diffusio model, SIAM Joural of Numerical Aalysis 4, (005)

21 4. Redlema, R., B. Bartter: Two state optio pricig. The Joural of Fiace 34, (979) 5. Tia, Y.S.: A modified lattice approach to optio pricig, The Joural of Futures Markets 3, (993) 6. Tia, Y.S.: A flexible biomial optio pricig model, The Joural of Futures Markets 9, (999) 7. Uspesky, J. V.: Itroductio to Mathematical Probability, McGraw-Hill (937) 8. Walsh, J.B.: The rate of covergece of the biomial tree scheme, Fiace ad Stochastics 7, (003) 9. Walsh, J.B., Walsh, O.D.: Embeddig ad the covergece of the biomial ad triomial tree schemes, i Numerical Methods ad Stochastics, Fields Istitute Commuicatios Vol. 34, Providece: Amer. Math. Soc., 0- (00)