Efficiet tree methods for pricig digital barrier optios arxiv:1401.900v [q-fi.cp] 7 Ja 014 Elisa Appolloi Sapieza Uiversità di Roma MEMOTEF elisa.appolloi@uiroma1.it Abstract Adrea igori Uiversità di Roma Tor Vergata Dip.to di Matematica adrea.ligori@live.it We propose a efficiet lattice procedure which permits to obtai Europea ad America optio prices uder the Black ad Scholes model for digital optios with barrier features. Numerical results show the accuracy of the proposed method. Keywords: America optios, digital barrier optios, biomial mesh; 1 Itroductio Tree-based algorithms for optio pricig are studied sice the semial work of Cox et al. 1979) ad tur out to be very simple ad fast to be implemeted by a backward iductio. A importat characteristic which makes these procedures very appealig is that they easily iclude America-style features oce the Europea case is treated ad well setup. This makes lattice techiques widely used i the practice because although may progresses have bee doe i the developmet of exact formulas or other umerical procedures Mote Carlo, fiite differeces, etc.) for Europea optio prices, the America couterparts, that ivolve a cotrol problem, are ot so well-provided. We cosider here the Black ad Scholes model, see Black&Scholes 1973) ad Merto 1973), which is either classical ad still widely used i fiace. It meas that the uderlyig asset price process evolves as a geometric Browia motio. Here, optio prices ca be computed by usig the simple tree method due to Cox et al. 1979) CRR). However fiacial derivatives have bee becomig more ad more sophisticated ad this meas that the stadard implemetatio of the CRR biomial tree brigs to further errors i the approximatio of the Black ad Scholes prices. This is the reaso why it becomes importat to setup efficiet tree schemes, i.e. tree methods which allow oe to reduce the approximatio errors. We propose here a lattice scheme for pricig digital barrier optios. I particular a digital call optio is a optio whose payoff is equal to a fixed amout i what follows we suppose this amout is equal to 1) if the uderlyig asset at maturity is 1
greater tha a predetermied level the strike price K) or othig otherwise. Practitioers that trade these products essetially predict the directio of the market without cocerig i the specific the magitude of the movemets of the uderlyig asset price. Oe of the beefits with respect to stadard products is that the ivestmet ad the returs are fixed, so the risk ivolved ad the potetial losses are kow a priori. Digital optios ca also iclude barrier levels: they ca be activated or ullified if the uderlyig asset price process reaches certai cotractually specified levels. This more complex optio ca be used as a fiacial tool embedded i sophisticated products, such as accrual rage otes. These otes are fiacial securities that are liked for example to a foreig exchage rate ad the they pay a fixed iterest accrual if the exchage rate remais withi a specified rage ad othig otherwise, see Wystup 006) ad also Hui 1996). Europea digital optios with barrier features have a kow closed-form pricig formula both for a sigle barrier ad for double barriers that ca be easily derived from the correspodig formulas i the stadard case, for the latter oes we ca refer for istace to Reier&Rubistei 1991) i the sigle barrier case ad Ikeda&Kuitomo 199) i the double barrier case. As far as we kow, the literature of America exotic optios suggests o approximatio formulas for digital barrier optios. Our objective is the to treat the optio pricig problem related to these optios by usig lattice techiques. I particular we propose a efficiet method for the pricig of Europea digital call optios with a sigle barrier ad the, cosequetly, we also get a good method for the case of America-style optios. To this ed, we first eed to fid a explicit asymptotic expasio of the classical CRR biomial approximatio error, that is the differece betwee the price computed with the CRR tree ad the Black ad Scholes price. The we setup a algorithm such that it behaves well, i the sese that the worst cotributio i 1 the asymptotic expasio which turs out to be of order, where N is the umber of time steps of the tree) is ullified. It is well kow that the rate of covergece of the CRR tree for vailla optios o cotiuous payoff fuctios is of order 1, see for example Walsh&Walsh 00), Dieer&Dieer 004) ad Chag&Palmer 007). Moreover the results kow from the literature whe dealig with barrier optios always require the cotiuity o the payoff fuctio. For double barrier optios o a geeral class of cotiuous payoff fuctios we ca refer to Gobet 001) ad i the more specific case of call optios 1 to i&palmer 013). We recall that here the rate of covergece is of order ad this is due to the fact that the cotractual barriers do ot ecessarily coicide with the effective barriers o the tree structure. O the other had whe the payoff is assumed to be discotiuous the aalysis of the rate of covergece of the biomial algorithm is give oly for vailla optios, see Walsh&Walsh 00) ad Chag&Palmer 007). Also i this case the rate of covergece is of order 1, but ow it is caused by the positio of the discotiuity poits of the payoff fuctios o the lattice. I this framework we treat the study of the CRR biomial approximatio error i the case of digital optios with a sigle barrier ad we get a complete theoretical result that allows us to costruct a efficiet algorithm. I particular we fit the
Biomial Iterpolated attice BI) algorithm provided i Appolloi et al. 013) to this specific case. We recall that the BI procedure is based o a backward iductio o a biomial mesh with additioally suitable iterpolatios that allows oe to get very precise optio prices for call ad put double barrier optios. We adapt this procedure to the case of digital call ad put optios with a sigle barrier ad the umerical results show that we get very reliable ad accurate optio prices. The paper is orgaized as follows. I Sectio we describe the cotiuous-time model for the evolutio of the uderlyig asset price process. I Sectio 3 we propose our theoretical cotributio o the asymptotic expasio of the CRR approximatio error for Europea digital optios with barrier features. The umerical results are show i Sectio 4. The model We cosider a market model i the time iterval [0, T ] where the evolutio of the risky asset S t ) t [0,T ] is govered by the Black-Scholes stochastic differetial equatio ds t = rdt + σdb t, S 0 = > 0, 1) S t where B t ) t [0,T ] is a stadard Browia motio uder the risk eutral probability measure. The o-egative costat r is the risk-free iterest rate ad σ is the costat volatility parameter. A digital call barrier optio is a cotract whose payoff is equal to 1 if the uderlyig asset price at maturity is greater tha the strike price K, ad othig otherwise. Moreover, the payoff also depeds o whether the uderlyig stock price path ever touches certai price levels called barriers. Oce either of these barriers is breached, the status of the optio is immediately determied: either the optio comes ito existece if the barrier is a kock-ad-i type or it ceases to exist if the barrier is a kock-ad-out type. I what follows we will cosider the kock-ad-out type, the kock-ad-i type beig similar. We recall that the payoff of a digital call optio with lower barrier is give by 1 ST K1 Sif >, ) where S if = if t [0,T ] S t. The case of a digital call optio with higher barrier H is similar, as well as the case of the correspodig digital put optios with a sigle barrier. The Black ad Scholes prices of Europea optios whose payoff is equal to the oe give i ) ca be foud i Reier&Rubistei 1991). No aalytical approximatios are available for optio prices whe the payoff is as i ) i the America case. 3 The CRR biomial approximatio error Startig from the model described i Sectio, we aalyze here the error committed by usig the CRR tree method for pricig Europea digital call optios with a sigle 3
barrier. If N deotes the umber of time steps of the tree, we defie the CRR biomial approximatio error as follows Err CRR ) = price CRR ) price BS, N 3) where price CRR ) deotes the price calculated by usig the CRR tree scheme ad price BS is the Black ad Scholes price. We briefly recall here the idea of the CRR tree scheme. et us fix a iteger N. We first defie τ = T/, where T is the maturity of the optio, ad the we build a biomial tree with steps of legth τ. If we label 0, 0) the startig ode that correspods to the value S 0 =, the after i time steps i = 0, 1,..., ), the discrete process may be located at oe of the odes i, j) j = 0, 1,..., i) correspodig to the values S i,j = e j i)σ τ. 4) Hece, startig from S i,j at time ih, the process may jump at time i + 1)h to the value S i+1,j+1 or the value S i+1,j with probability p ad 1 p respectively, where p is defied as p = er τ d 5) u d ad u = e σ τ = d 1. Europea ad America prices at time 0 are the obtaied by applyig the classical backward iductio. We ow develop some argumetatios described i Chag&Palmer 007) ad i&palmer 013). We cosider the case of a digital call optio with a lower barrier, the other cases of digital optios with a sigle barrier beig similar. The idea is to fid a closed-form formula i terms of biomial coefficiets of the optio price followig Reimer&Sadma 1995) ad the use the approximatio of the biomial distributio by the ormal oe as suggested i i&palmer 013) i order to fid explicit coefficiets i the asymptotic expasio. We first stress that we eed the biomial prices of digital call optios with lower barrier of the two followig types: 1. a dow-ad-i call optio with < K;. a dow-ad-out call optio with > K. I these two cases the biomial formulas are maageable ad permits a simple treatmet. The, by usig the biomial formulas for the correspodig vailla digital call optio, it is possible to fid the asymptotic expasio of the error for a dow-adout digital call optio with < K ad a dow-ad-i digital call optio with > K. et us first itroduce two quatities, that will have a crucial role i what follows, as defied i Chag&Palmer 007) ad i&palmer 013), that we call K ad. The quatity K is set as ) K log /K) = 1 frac σ, 6) τ 4
where for every real umber x, the fractioal part of x is defied as fracx) = x x, with x idicatig the largest iteger precedig x. We observe that K is a measure of the positio of K i the log-scale i relatio to two adjacet termial stock prices. I fact if we defie j K the iteger such that S,jK 1 = u j K 1 d j K+1 < K S,jK = u j K d j K, the K = 1 if log K is the ode log S,jk 1 at maturity, K = 0 if log K lies halfway betwee the two odes log S,jk 1 ad log S,jk at maturity i.e. it is a ode from the first period before maturity) ad K = 1 if log K is the ode log S,jk at maturity. We ow describe a similar quatity correspodig to the lower barrier that we call. First, we call the effective barrier o the tree structure, that is geerally differet from the cotractual barrier. et us suppose that j is the umber of up jumps required to reach. We defie = fracl ), with l = log σ τ +. The the effective barrier ca be writte as = u j d j, where j = j + 1 1 ɛ ), with j = 1 l ad ɛ = { 0, if the effective barrier is ot a termial stock price, 1, if the effective barrier is a termial stock price. The quatity [0, 1] measures i the log-scale the positio of i relatio to two adjacet stock prices, oe of which is a ode at maturity ad the other is a ode of the first time before maturity. I the special cases i which = 0 ad = 1 we get that the effective barrier lies exactly o a ode of the tree for a more detailed discussio o this see i&palmer 013)). We ow eed to itroduce some otatios as i Reimer&Sadma 1995). We defie π d, j, j ) as the price at time 0 of a security which pays o uit at time T if the asset price at the time step is equal to S,j = u j d j ad if there exists a pair i, l) with i {0,..., } ad l {0,...i}, such that S i,l = u l d i l, ad otherwise othig, i.e. π d, j, j ) = e rt E[1 S T =S,j 1 i, l i:si,l = u l d i l ] = e rt PS T = S,j ; i, l i : S i,l ), 7) 5
where St i ) i=0,1,...,, with t i = i τ for every i = 0, 1,...,, deotes the discrete approximatio of S ih, so i particular St = ST is the discrete approximatio of S T. I order to calculate 7), we first eed to cout the umber of paths Z d, j, j ) i the biomial tree which reach the termial stock price S,j after touchig or passig through the effective barrier. The reflectio priciple see Feller 1968)) yields the umber Z d, j, j ) that for every j = 0,..., is equal to ) j, if j j, Z d, j, j ) = j j), if j < j j, 8) 0, if j > j. We ca ow prove the followig Propositio: Propositio 1 The biomial price C di digital, K, T,, ) of a dow-ad-i digital call optio with barrier < K < is equal to C di digital, K, T,, ) = e rt j i=j K ) p i 1 p) i. 9) j i Proof. et us deote with GS,j ) the payoff of a digital call optio at ode S,j, i.e. { 1, if S,j K, GS,j ) = 10) 0, if S,j < K. So the price at time 0 of a dow-ad-i digital call optio is equal to C di digital, K, T,, ) = e rt E[GST ) 1 i, l i:si,l = u l d i l ] = e rt E[GST )1 S T =S,j 1 i, l i:si,l = u l d i l ] = π d, j, j ) j=0 j=j [ K ) ] = e rt p j 1 p) j 1 j j j + )p j 1 p) j 1 j<j j j j j=j K j = e rt j=j K +1 ) p j 1 p) j, 11) j j where the last equality comes from the fact that here we suppose < K i.e. j < j K ) ad as a cosequece oe has that the cotributio due to the first sum vaishes. So the proof is complete. We ow derive the price of a dow-ad-out digital call optio with > K, that we call C do digital, K, T,, ), as the differece betwee the biomial price of the vailla digital call optio ad the biomial price of the dow-ad-i digital call optio with > K. et us start from the biomial price of the vailla digital call optio, that we deote with C digital, K, T, ). et us call with π, j) the price at time 0 of a security 6
which pays oe uit at time T if the asset price is equal to S,j = u j d j ad otherwise othig, i.e. ) π, j) = e rt E[1 S T =S,j ] = e rt p j 1 p) j. j We deote as before with GS,j ) the payoff of a digital call optio at ode S,j, see 10). So the price at time 0 of a vailla digital call optio is equal to C digital, K, T, ) = e rt E[GS T )] = e rt j=j K E[GST )1 S T =S,j ] j=0 [ ] = π, j) = e )p rt j 1 p) j. j j=j K Now from the proof of Propositio 1 we kow that the biomial price of a dowad-i digital call optio with > K i.e. j > j K ) ca be writte as 1) C di digital, K, T,, ) [ j ) = e rt p j 1 p) j + j j=j K j j= j +1 ] )p j 1 p) j, j i 13) so we ca state the followig result: Propositio The biomial price C do digital, K, T,, ) of a dow-ad-out digital call optio with barrier > K is equal to [ ) C do digital, K, T,, ) = e rt p i 1 p) i i j i= j +1 i= j +1 ] )p i 1 p) i. j i Proof. The price C do digital, K, T,, ) of a dow-ad-i digital call optio with > K is the obtaied by subtractig the price C digital, K, T, ) give i 1) to the price C di digital, K, T,, ) give i 13). We ow give the explicit coefficiets of order 1 ad 1 i the asymptotic expasio of the CRR biomial error defied i 3) for the price of digital call optios with barrier. The idea is to use the closed-form formulas of the biomial prices give i Propositio 1 ad Propositio ad the approximate them by usig emma 4.1 i i&palmer 013) o the approximatio of the biomial distributio by the ormal oe. We ca state the followig result: 7
Theorem 3 I the -period CRR biomial model, the biomial error Err CRR ) for the prices of Europea digital call optios with barrier < K is: for a dow-ad-i digital call optio: [ Err CRR ) = e rt Ã1 K + Ã ) 1 ] ) + B 1 + B K ) + B 3 K + B 4 ) ) 1 1 +O ; 3/ for a dow-ad-out digital call optio: [ Err CRR ) = e rt C 1 K + C ) 1 ] ) + D 1 + D K ) + D 3 K + D 4 ) ) 1 1 +O. 3/ The list of the costat is postpoed i Appedix A. Proof. et us cosider first the biomial price of the dow-ad-i digital call optio give i Propositio 1, that is C di digital, K, T,, ) = e rt j i=j K ) p i 1 p) i. 14) j i From equatio 5.1) i i&palmer 013) we ca write 14) as follows C di digital, K, T,, ) = e rt ) j j j 1 p K p i=0 ) p i 1 p) i. 15) i The asymptotic expasio of 15) is ow obtaied by applyig the asymptotic expasio 5.7) i i&palmer 013) of the term 1 p ) j ad the asymptotic p expasio 5.3) i i&palmer 013) of the term j j K ) i=0 p i 1 p) i. The asymptotic expasio for the dow-ad-out digital call optio is ow straightforward. I fact it ca be derived from the asymptotic expasio for the dowad-i optio ad that for the correspodig vailla optio that ca be foud i Chag&Palmer 007). i 8
Remark 4 Theorem 3 shows that the cotributio of the type 1 i the asymptotic expasio of Err CRR ) is due to the positio of barrier ad the positio of the strike with respect to the odes of the tree. I order to obtai a algorithm of order 1, we eed to set K = 0 ad = 0. It meas that i the log-scale the strike K must be positioed halfway betwee two odes at maturity i.e. it should be a ode of the peultimate period before maturity) ad the barrier must lie o a layer of odes of the tree. We ow state the followig result: Theorem 5 I the -period CRR biomial model, the biomial error Err CRR ) for the prices of Europea digital call optios with barrier > K is: for a dow-ad-out digital call optio: [ Err) = e rt Ẽ1 + Ẽ ) 1 ] ) + F 1 + F + F 3 ) 1 1 +O ; 3/ for a dow-ad-i digital call optio: [ Err) = e rt G 1 + G K + G 3 ) 1 ] ) + H 1 + H K ) + H 3 + H 4 ) ) 1 1 +O, 3/ The list of the costats i postpoed i Appedix A. Proof. By proceedig similarly to the proof of Theorem 3, we eed here to fid a asymptotic expasio of the biomial price for a dow-ad-out digital call foud i Propositio. I order to do this we apply 5.7), 5.11) ad 5.15) i i&palmer 013). The dow-ad-i case is the obtaied by cosiderig the differece of the asymptotic expasio for the vailla digital call ad the dow-ad-out digital call. Remark 6 The term K does ot appear i the expasio for the dow-ad-out optio i Theorem 5. The ituitive reaso is that i this case > K ad sice the optio stays alive if the stock price is above, ad therefore above K, the positio of K has o ifluece. 9
Remark 7 I the error expasios foud i Theorem 5 for dow-ad-i ad dowad-out digital call optios with > K it is ot possible to totally vaish the cotributio of order by settig = 0 ad K = 0. I fact there is a costat 1 term of order 1 that ca t be ullified. A possibility i order to get a algorithm of order 1 is to set ad K such that = 0 = K ad the explicitly calculate 1 the costat coefficiet that multiplies i order to subtract it to the biomial approximated price. Theorem 3 ad Theorem 5 suggest us how to set the barrier ad the strike K i the biomial tree scheme i order to get a algorithm of order 1. This theoretical result is ehaced by the umerical examples preseted i Sectio 4. Ufortuately, the extesio of the previous reasoig for digital double barrier optios is ot straightforward because o maageable closed-form formulas of the CRR biomial prices exist i this case. A possibility to deal with this issue is to use a completely differet approach. We tried to exted to discotiuous payoffs the theoretical result i Gobet 001), that studies the CRR biomial approximatio error for double barrier optios o a geeric cotiuous payoff fuctio by usig PDE techiques. We expected to obtai that the 1 cotributio of order could be explicitly writte as depedet o two differet sources: the positio of the barriers ad the positio of the discotiuity poit with respect to the odes of the tree. Curretly we are able to give just a upper boud of the CRR biomial approximatio error i this more complex case, however we address it to a future work. 4 Numerical results Theorem 3 ad Theorem 5 o the asymptotic expasio of the CRR biomial approximatio error, suggest that a algorithm of order 1 ca be obtaied if i the log-space the lower barrier lies exactly o a ode of the tree i.e. = 0) ad the strike K is positioed halfway betwee two odes at maturity i.e. K = 0). To this ed, we adapt here the Biomial Iterpolated attice itroduced i Appolloi et al. 013) BI) for pricig digital optios with a sigle barrier. The BI procedure is a efficiet algorithm for the pricig of double barrier call ad put optios that we ow briefly recall. The idea is to defie the time step t of the algorithm such that i the log-space the lower barrier ad the higher barrier H coicide exactly with two odes of the tree at maturity. The, if τ = T is the stadard time step of a CRR tree with steps, oe eeds to set where t = k = h l ) 16) kσ h l σ τ 17) 10
ad h = log H ad l = log. We recall here that x, for every x R, deotes the smallest iteger ot less tha x. The the time step t defied i 16) is obliged to take some specific values i order to match both ad H ad this implies that T / N. I order to arrive t close to time 0, oe eeds to add two further steps of legth t i order to get a fictitious time t 0 < 0 ad a time t 1 > 0 see Figure 1) so that the umber of time steps of the procedure is set as = T +. T Figure 1: Biomial Iterpolated attice mesh for double barrier optios. Sice we do ot kow a priori if the iitial price is a poit of the lattice ad i geeral it is ot), the approximatig optio price at 0, ) is provided by suitable iterpolatios i time ad i space ivolvig the prices, which are computed by the stadard backward iductio, at times t 0 ad t. To be precise we choose i t 0 ad t the two poits below ad the two above. The price at is obtaied by a agrage four poits iterpolatio i space of the prices deoted i Figure 1 with the empty circles, such prices are obtaied by a liear iterpolatio i time of the prices at the odes deoted by squares. For more details oe ca refer to Appolloi et al. 013). The, we ca easily adapt the BI algorithm to the case of digital optios with a sigle barrier. I fact we just eed to modify the choice of k defied i 17) ad the time step t defied i 16) such that the barrier is a ode of the tree i particular we set it as a ode at maturity) ad the strike K lies halfway betwee two odes at maturity i.e. it is a ode from the peultimate period). We set k = log K ad the we defie the iteger k as follows k l k = σ + 1 τ, 18) 11
so that the time step t is ow give by t = ) k l. 19) σk The umber of time steps of this adjusted procedure, that we call Adjusted BI, is set as = T +. I fact the price at time 0 is the obtaied by a backward t iductio ad by proceedig through iterpolatios i time ad i space ivolvig some specified prices at times t 0 = 0 ad t = t as for the BI algorithm. Remark 8 By usig the choice of k as i 18) so that the time step t is defied as i 19) we are able to costruct a mesh i the log-space i which the barrier is a ode at maturity ad the strike K is a ode from the peultimate period. But we stress here that i the Adjusted BI algorithm it is ot eough to build a biomial mesh betwee ad K, but we eed to exted it above K ad this is straightforward. This is a structural differece with what doe i the BI algorithm for pricig double barrier optios: i this case we just eed to setup the mesh betwee the barriers ad H. We ow preset some umerical results i order to compare the prices for sigle barrier digital optios obtaied with the stadard CRR algorithm ad those obtaied with the Adjusted BI algorithm. I particular we study dow-ad-out digital call optios i two cases: first, whe < K Sectio 4.1); secodly, whe > K Sectio 4.). The dow-ad-i case provides similar results, so we omit it. 4.1 Dow-ad-out digital call optio with < K We cosider a dow-ad-out digital call optio with lower barrier = 60, strike K = 100 ad iitial stock value equal to = 150. The other parameters are: r = 0.1, σ = 0.5 ad T = 1. I Figure we plot the Europea prices obtaied by usig the CRR biomial approximatio ad the true price calculated with the Black ad Scholes formula, i.e. C BS do digital, K, T,, r, σ) = e rt [Φd 1 ) Φd ) r σ ], 0) where d 1 ad d are defied i Appedix A ad Φ ) is the stadard ormal distributio fuctio. We observe that the biomial price oscillates widely aroud the true price ad this is due both o the positio of ad also o the positio of K with respect to the odes of the tree. It is clear that the rate of covergece of the algorithm is of order 1. 1
Figure : CRR biomial approximatio. Kock-out digital call optio with = 60, K = 100, = 150, r = 0.1, σ = 0.5 ad T = 1. I Figure 3 we plot the prices obtaied by usig the Adjusted BI algorithm. We remark here that i the x-axis we report the umber of time steps correspodig to the CRR biomial approximatio. I fact we recall that i the Adjusted BI algorithm we defie a ew umber of time steps, differet from, but havig the same order of magitude. Now there are o oscillatios ad the covergece is of order 1 : the oscillatios due ad K disappear ad the covergece is mootoe. Figure 3: Adjusted Biomial Iterpolated attice. Kock-out digital call optio with = 60, K = 100, = 150, r = 0.1, σ = 0.5 ad T = 1. 13
I Table 1 we report the prices obtaied with the CRR algorithm ad the Adjusted BI algorithm. I the first colum we write the umber of time steps of the CRR biomial approximatio. The true price is calculated by usig the Black ad Scholes formula give i 0). < K < CRR True Adjusted BI 100 0.883147 0.878791 00 0.879006 0.87873 400 0.880340 0.878667 0.878700 800 0.876786 0.878684 1600 0.878863 0.878676 300 0.877873 0.878671 Table 1: Kock-out Europea digital call optios prices with = 60, K = 100, = 150, r = 0.1, σ = 0.5 ad T = 1. 4. Dow-ad-out digital call optio with > K We ow cosider a dow-ad-out digital call optio with strike K = 60, lower barrier = 100 ad iitial stock price = 150. The other parameters are: r = 0.1, σ = 0.5 ad T = 1. I Figure 4 we plot the Europea CRR biomial prices ad the true price that is give by the followig Black ad Scholes formula C BS do digital, K, T,, r, σ) = e rt [Φd 3 ) r ] σ Φd 4 ), 1) where d 3 ad d 4 are defied i Appedix A. We observe that i this case the oscillatios are fewer tha the case < K ad this is due to the fact that the positio of the strike K has o ifluece i the error expasio. I fact the optio stays alive whe the stock price is above ad therefore above K, so the positio of K with respect to the odes of the tree has o ifluece. The the term of order 1 is oly due o the positio of, as remarked i Theorem 5, ad a costat term. However, from the umerical computatios it turs out that the costat term is of order 10 3, so it really does ot affect the error. But the covergece is still slow, i.e. the CRR algorithm has order 1. 14
Figure 4: CRR biomial approximatio. Kock-out digital call optio with K = 60, = 100, = 150, r = 0.1, σ = 0.5 ad T = 1. I Figure 5 we plot the Europea prices obtaied with the Adjusted BI algorithm ad the Black ad Scholes price give i 1). We observe that here the covergece is mootoe because the biomial mesh is costructed such that the lower barrier lies exactly o a layer of odes. The the procedure is of order 1. 15
Figure 5: Adjusted Biomial Iterpolated attice. Kock-out digital call optio with K = 60, = 100, = 150, r = 0.1, σ = 0.5 ad T = 1. I Table we report the Europea prices of a dow-ad-out digital call optio with lower barrier > K obtaied with the CRR algorithm ad the Adjusted BI algorithm. As usual, deotes the umber of time steps of the CRR biomial approximatio. The true price is calculated by usig the Black ad Scholes formula give i 1). K < < CRR True Adjusted BI 100 0.855913 0.844983 00 0.846415 0.845304 400 0.849497 0.845484 800 0.846188 0.845659 0.845571 1600 0.846107 0.845615 300 0.8465 0.845637 Table : Kock-out Europea digital call optios prices with K = 60, = 100, = 150, r = 0.1, σ = 0.5 ad T = 1. Remark 9 The theoretical proof that the rate of covergece of the Adjusted BI algorithm is of order 1 is a direct cosequece of Propositio 1 i Appolloi et al. 013), Theorem 3 ad Theorem 5. I fact, i the Adjusted BI algorithm we get the price i 0, ) by suitable iterpolatios of some selected CRR prices at times t 0 ad t, as i the stadard versio of the BI algorithm see Figure 1). But the iterpolatio rule preserves the error committed by approximatig the cotiuous-time 16
prices at the selected odes by the CRR oes. So if the CRR biomial approximatio error at these odes is of order 1, tha the Adjusted BI algorithm is still of order 1. 5 Coclusios We give here a explicit asymptotic expasio of the approximatio error related to the stadard CRR tree for pricig digital optios with a sigle barrier. The theoretical results suggest us how to set a biomial algorithm such that the worst 1 cotributio term i the error expasio, that is of order where N is the umber of time steps of the algorithm), is ullified. We get a efficiet lattice procedure ehaced by umerical examples. The extesio of the reasoig to the case of double barrier digital optios is ot straightforward. Our idea is to use a differet approach, based o PDE techiques, i order to study theoretically the CRR biomial approximatio error. We foud a upper boud of the error, but the result is still partial, so we address it to a future work. Ackowledgmets The authors thak Professor. Caramellio ad Professor A. Zaette for the useful commets ad their valuable assistace ad Professor K. Palmer for participatig i discussios. 17
Appedix A We report here the list of the costats that appear i Theorem 3 ad Theorem 5: d 11 = log K σ )T σ, T d 1 = d 11 σ T, log s d 1 = 0 K σ )T, d = d 1 σ T, σ T d 31 = log + r + 1 σ )T σ T d 41 = log + r + 1 σ )T σ T, d 3 = d 31 σ T,, d 4 = d 41 σ T, α = r 1 σ, ˆα = α + σ σ, β = σ4 4σ r + 1r σr, ˆβ = β 48σ 6, ĝ i = T ˆα d i1 + ˆβ ˆα ) T T ) + d i1 3 1 g i = T ˆα d i + ˆβ T ) + ˆα T 3 d i 1 ) 1 d i1), i = 1,, 3, 4 1 d i), i = 1,, 3, 4 ) d G 1 = e d 11 ĝ1 g 1 ), G = s 1 1 0 s 0 e d 1 ĝ g ), π π G 3 = e d 31 ĝ3 K π g 3), G 4 = e d 31 ĝ4 K π g 4), A 1 = 4 T h 1 d 1, d ), A = 4 ) T h 1 d 41, d 4 ) + x 0 e d 31 1 K, π A 3 = 4 ) T h 1 d 41, d 4 ) h 1 d 1, d )) e d 31 1 K, π ) r ) σ s 0 r + σ h i x, y) = D i ) Φx) Ke rt r σ i Φy)), σ σ for i = 0, 1,, B 1 = G 1 G + Ih 0 d 1, d ), B = B 1 G 1, B 3 = G 3 G 4 + Ih 0 d 41, d 4 ), B 4 = B 3 G 1, B 5 = G + G 3 G 4 + Ih 0 d 1, d ) Ih 0 d 41, d 4 ), B 6 = B 5 G 1, ) ) 4β + 16 3 I = α3 log T, σ ) C 1 = s 1 r σ 0 s 0 e d 1 σ T, π 18
) C = e d 31 π d 31 K d 3, ) C 3 = e d 31 π d 41 K d 4, C = 1 C C 3 ), D 1 = d π e 11 C 1 σ T 4, D = C 1 4, D 3 = D 1 + D, E 1 = 8T h d 1, d ) + C 1, E = 8T h d 41, d 4 ) + 1 3C + C 3 ), E 3 = 8T h d 1, d ) h d 41, d 4 )) C 1 + 1 3C + C 3 ), r σ s 0 e d à 1 =, π à = Ã1 4α T Φd ) B 1 = B = B 3 = B 4 = r σ, r σ [ g e d π IΦd ) r σ d )e d π, r σ e d π [d 4α T ], r [ σ e d d + 8α ] T ) + 8α T Φd ), π ], d c 1 = e 1, c = d 1 π e d 1 π, c = d3 11 + d 11 d 1 + d 1 4d 11 4 d c 3 = c e 1, π + d 11d 1 d 11) T 6σ r + T d 11 σ r, C 1 = c 1 Ã1, C = Ã, D 1 = c B 1, D = c 3 B, D 3 = B 3, D4 = B 4, e d 3 Ẽ 1 = ɛ + π r σ e d 4 π, 19
d Ẽ = e 3 + π d F 1 = e 3 g 3 d 3 π + Φd 4 )I r σ e d 4 + 4α ) Φd 4 ), π ɛ ) r σ, d F = e 3 s 0 d 3 ɛ + π e d 3 F 3 = d 3 + π + r σ Φd 4 )8α T, G 1 = Ẽ1, G = c 1, G 3 = Ẽ, r ) σ e d 4 d 4 π ɛ g 4 r σ e d 4 ɛ d 4 4ɛ α ) T, π r ) σ e d 4 d 4 π 4α T H 1 = c F 1, H = c 3, H3 = F, H4 = F 3. 0
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