Joural of Mathematical Fiace, 016, 6, 810-841 http://www.scirp.org/joural/jmf ISSN Olie: 16-44 ISSN Prit: 16-434 Pricig Asia Optios: A Compariso of Numerical ad Simulatio Approaches Twety Years Later Akos Horvath 1, Peter Medvegyev 1 Departmet of Fiace, Uiversity of Viea, Viea, Austria Departmet of Mathematics, Corvius Uiversity of Budapest, Budapest, Hugary How to cite this paper: Horvath, A. ad Medvegyev, P. (016) Pricig Asia Optios: A Compariso of Numerical ad Simulatio Approaches Twety Years Later. Joural of Mathematical Fiace, 6, 810-841. http://dx.doi.org/10.436/jmf.016.65056 Received: August 9, 016 Accepted: November 15, 016 Published: November 18, 016 Copyright 016 by authors ad Scietific Research Publishig Ic. This work is licesed uder the Creative Commos Attributio Iteratioal Licese (CC BY 4.0). http://creativecommos.org/liceses/by/4.0/ Ope Access Abstract The calculatio of the Asia optio value has posed a great challege to fiacial mathematicias as well as practitioers for the last two decades. Sice there exists o aalytical valuatio formula to date, oe has to resort to other methods to price this commoly used derivative product. Oe possibility is the usage of simulatio approaches, which however are especially iefficiet for Asia optios, due to their depedece o the etire stock price trajectory. Aother alterative is resortig to semi-aalytical methods, based o the iversio of the optio price s Laplace trasform, which however are proe to severe umerical difficulties. I this paper, we seek aswer to the questio whether it is possible to improve o the efficiecy of the semi-aalytical approach, implemetig ad comparig differet umerical algorithms, so that they could be applied i real-life situatios. We look ito whether today s superior computer eviromet has chaged the relative stregth of umerical ad simulatio approaches with regards to Asia optio pricig. Based o a comprehesive aalysis of speed ad reliability, we fid that the Laplace trasform iversio method ca be further ehaced, pushig dow the prior critical value from 0.01 to 0.005 ad the calculatio time from 0-30 secods to 3-4 secods. This reders the umerical approach readily applicable for practitioers; however, we coclude that the simulatio approach is a more efficiet optio whe σ T < 0.01. Keywords Asia Optios, Laplace Trasform Iversio, Mote Carlo Simulatio 1. Itroductio Asia optios are popular hedgig istrumets i the hads of fiacial risk maagers, owig to their special payout structure ad cost-effectiveess. With regard to exercise DOI: 10.436/jmf.016.65056 November 18, 016
behavior, these optios are Europea, i the sese that they ca be exercised oly at a pre-agreed (maturity) date; however, as opposed to (plai) vailla products, Asia optios have a exotic payout at maturity ( 0, T ) ( ) [ ] f T, S T = Avg S K +, (1) 0,T S stads for the time average of the uderlyig asset s prices from time 0 to time T (i.e. durig the lifetime of the optio) 1. The above equatio implies that Asia optios are path-depedet; moreover, sice their value depeds o the average of all uderlyig prices durig the optio s lifetime, they may be cosidered as the quitessece of path-depedet derivatives. The exotic ature of this optio type has importat fiacial ad mathematical implicatios for us. From a fiacial (or practical) perspective, the stabilizig effect of averagig i Equatio (1) makes Asia optios a cheaper, yet more reliable alterative to their vailla couterparts for fiacial risk maagers. By takig the average of market prices for a give time period, the adverse effects of potetial market maipulatio ad price jumps are miimized, which reders these optios a cost-effective hedgig istrumet, especially i market eviromets where either the volume is low (e.g. i the corporate bod or commodity markets), or the volatility is high (e.g. i the currecy ad iterest rate markets). As a result, accordig to a survey icludig 00 derivative-usig o-fiacial U.S. firms coducted by Bodar et al. [1], Asia optios are the most popular exotic payout optios chose by o-fiacial firms for risk maagemet. From a mathematical (or theoretical) perspective, the valuatio of Asia optios is particularly challegig eve i the most simple pricig models, which is partly due to path-depedece ad partly because the probability distributio of averages is usually ukow. The average itself is defied as either the arithmetic or the geometric mea of the uderlyig asset s prices; the former beig more commo o the market, while the latter havig more coveiet mathematical properties. I the classical Black-Scholes framework, the value of the geometric-average Asia optio ca be determied just as easily (for the closed-form formula refer to Turbull et al. []) as i the vailla case, sice the product of the logormally distributed asset prices follows logormal distributio as well. I cotrast, the valuatio of the arithmetic-average Asia optio poses a far greater challege tha its geometric couterpart, ad a closed-form formula has ot bee derived to date. Nevertheless, i their work Gema ad Yor [3] derive a aalytical expressio for the Laplace trasform of the so-called ormalized Asia optio price, makig it possible to determie the optio value by meas of umerical (iversio) methods. At the same time, Gema ad Eydelad [4] fid that these methods are itractable for small values of σ T. Aother approach to pricig arithmetic-average Asia optios is usig Mote Carlo simulatio, which is a far more robust, yet computatioally demadig method. Em- 1 Note that the maturity date ad the ed of the averagig period eed ot coicide. I practice, however, they usually do, ad thus o distictio is made i this article. where K stads for the pre-agreed strike price ad Avg[ ] 811
ployig the geometric-average Asia optio price as cotrol variate, Fu et al. [5] make a compariso betwee the two approaches (i.e. betwee umerical iversio methods ad cotrol variate Mote Carlo methods), ad fid that difficulties with the Laplace trasform arise whe σ T is lower tha 0.01. Below this value, the iversio algorithms employed fail to coverge, ad oe has to resort to the simulatio approach or make approximatios (e.g. by iterpolatio). This is a especially uwelcome result, as σ T is ofte smaller tha this critical value uder usual market circumstaces, thus rederig the umerical approach impractical. I this paper, we seek aswer to two importat questios aturally arisig from the results above. Cosiderig the developmet i computatioal power, 1) Is it possible to fid a more efficiet algorithm tha the Euler method employed by Fu et al. [5]? If so, how does the critical value of covergece chage for the umerical approach? ) What is a optimal daily read-replicatio umber combiatio for the Mote Carlo method, with regard to both discretizatio ad simulatio error? How has the relatio of the differet approaches chaged i the last te years? I Sectio, we itroduce the basic priciples of Asia optio (ad i geeral, fiacial derivative) pricig, makig a clear distictio betwee the differet valuatio approaches. The, i Sectio 3, we proceed with a empirical aalysis, comparig the performace of pricig algorithms i MATLAB. Fially, we make recommedatios o the best valuatio practices, ad coclude.. Valuatio Methodology I order to price Asia optios, it is ecessary to agree o the specific risk-eutral framework used, which is the Black-Scholes model i this paper. I this sectio, first we briefly recapitulate the assumptios ad results of this model, the we itroduce the risk-eutral valuatio logic ad make a distictio betwee the mai pricig methods. Fially, we apply these to the particular valuatio problem of Asia optios, as well as discuss what computatioal difficulties arise with the differet algorithms..1. The Black-Scholes Model I their origial model, Black ad Scholes [6] make the followig assumptios about the behavior of asset prices, iterest rates ad fiacial markets: the risk-free iterest rate is costat over time (as opposed to models i which iterest rates are determiistic or stochastic processes); the stock price (of ay firm) follows geometric Browia motio with a variace rate proportioal to the square of the stock price ; the optio is a Europea-exercise optio; the stock pays o divideds (or other iterim cash flows); there are o trasactio costs of tradig or borrowig; there are o limitatios to tradig (icludig the oes cocerig short-sellig or the tradig of fractioal quatities); 81
there are o limitatios to takig out a loa or makig a deposit at the risk-free iterest rate. Although the last few assumptios are rather techical ad oly esure that the fuctios ad processes are well-behaved, the first three assumptios form the core of the model ad defie very specific stock price ad iterest rate behavior. There are effectively three fiacial products i this setup: a risk-free bod B, a stock S, whose price follows geometric Browia motio, ad a Europea-exercise derivative product f, whose price is a real fuctio of the stock s price, which, if looked at as processes, have the respective price dyamics db t Brdt µ + σ ds t S t dt S t dw t f f 1 f f d f ( t, S( t) ) + µ S( t) + σ S ( t) dt+ σs( t) d W( t), t S S S where W stads for the stadard Wieer process. It follows (from Itô s lemma) that the logarithm of S is a ormally distributed variate havig the dyamics 1 dx( t) dl( St ) = µ σ dt+ σd Wt. () Based o the assumptios above, usig a replicatio method called dyamic delta hedgig, Black ad Scholes [6] derive the reowed partial differetial equatio (PDE) for the derivative price 1 rf t r S t S t, f f f t S S σ = + + which is a otable result, sice i may cases it ca be solved by meas of stadard PDE methods... Risk-Neutral Valuatio Havig chose the specific price dyamics of asset prices, i theory we ca calculate the expected value of the optio s payout at maturity, which, if discouted at the proper rate of retur, yields the optio s value. Alas, this rate of retur is ukow i most cases, moreover, it teds to chage durig the derivative product s lifetime, rederig the above pricig logic uusable. Fortuately, there exists a elegat fiacial mathematical trick called risk-eutral valuatio, which makes it possible to discout the expected value of the optio s payout at the (kow) risk-free iterest rate. I what follows we briefly recapitulate the teets of this logic, the we show how it is applied i practice. Notatio.1. Let Ω,, { }, t be a filtered probability space, where { } t 0 t t 0 is the atural filtratio geerated by Wieer process W. Defiitio.. A π( t) ( β( t), γ ( t) ) = process pair is called ivestmet strategy if (3) 813
both processes are predictable with respect to t. Defiitio.3. A ivestmet strategy is self-fiacig (SF) if t : β γ B t d t + S t d t = 0 holds. Notatio.4. Let V π be the value process of the portfolio (determied by a give π ivestmet strategy) which cotais β ( t) bods ad ( t) γ stocks at ay give V time. Accordigly, let V = π π be the discouted value process of the portfolio. B It ca be show that give the right model coditios (i.e. o arbitrage, market completeess), the dyamics of the stock price could be expressed as σ ds t = rs t dt+ S t d W t, with respect to a uique probability measure. It follows that the discouted value process V π of ay portfolio determied by a give π self-fiacig ivestmet strategy will be a martigale, implyig ( ) π s π E V t = V s s t. Applyig the martigale represetatio theorem, it ca be show that for ay Europea-exercise derivative there exists a correspodig self-fiacig ivestmet strategy * π replicatig the value process of a particular derivative. I preparatio for the fial step, let us defie a risk-eutral stock otig that the distributio of r r + r ds t rs t dt σ S t d W t, (4) ( r) S with respect to measure is the same as the distributio of S with respect to measure, which is summarized i the followig theorem. Theorem.5. (Risk-Neutral Valuatio) The preset value of a Europea-exercise derivative equals to the expectatio of the derivative price with respect to probability measure discouted at the risk-free iterest rate. By assumig that the stock price has risk-eutral dyamics as defied i Equatio (4), this expectatio ca be calculated with regards to probability measure. ( ) rt ( ) rt ( r ) f 0, S 0 = e E f ts, t e E f ts, = t (5) Havig described the fudametal logic of risk-eutral valuatio, let us ow tur to itroducig the differet pricig methods, which apply the above priciples i practice...1. Aalytical Methods Aalytical methods aim at derivig a closed-form expressio for the price of the particular derivative. This could be attempted by either solvig Equatio (3) with the appropriate boudary coditios give by the payout of the derivative at expiry, or calculatig the risk-eutral expected value of the derivative at expiry, usig the Risk-Neutral Valuatio priciple outlied i Sectio.. Whichever method we choose, the existece of a closed-form expressio is ot guarateed, depedig o the payout structure of the derivative. Hece, the valuatio 814
problem becomes purely mathematical ad eve though i some cases the derivatio of a pricig formula is extremely challegig, fiacial mathematicias put substatial effort ito it. The reaso for this is that a closed-form expressio for a particular fiacial derivative s price yields a easy-to-calculate value, rederig aalytical methods computatioally superior to other (e.g. umerical or simulatio) approaches. Noetheless, apart from a few otable examples (e.g. vailla optios or geometric-average Asia optios), it is ot possible to derive a pricig formula. For istace, there has ot bee oe foud yet i the case of arithmetic-average Asia optios.... Numerical Methods Although a closed-form solutio ofte proves to be elusive, sometimes it is possible to derive a complex but determiistic (semi-aalytical) pricig formula. For istace, we may arrive at a complex (sigle/double/triple) itegral form, which ca be approximated by meas of umerical methods (e.g. quadratures). As we explai i Sectio.3, this is the case for arithmetic-average Asia optios, where oly the Laplace trasform of the optio price could be determied i closed-form ad the optio price itself has to be calculated usig umerical iversio techiques...3. Simulatio Methods As a alterative to aalytical ad umerical methods, we ca tur to simulatio techiques kow as Mote Carlo methods. There exist umerous approaches but the basic priciple is the same (see Kalos ad Whitlock [7]): 1) Usig radom umber geeratig algorithms, geerate outcomes for a radom variate with kow distributio. ) Applyig the Law of Large Numbers, estimate the ukow expected value with the average of the geerated outcomes. 3) Whe applicable, icrease the sample size, so as to costrai the estimate ito a desired cofidece iterval. Mote Carlo methods are computatioally itesive but easy to uderstad ad flexible algorithms i the sese that they are compatible with a broad family of (physical, mathematical or fiacial) stochastic models. From the perspective of fiacial derivative valuatio, the flexibility of Mote Carlo simulatio allows for complex stock price movemets (e.g. jump diffusio) ad derivative payouts (e.g. path-depedecy or early exercise), which are otherwise extremely hard to hadle by aalytical methods. Without goig ito further details o the specific algorithms used i Sectio 3, we must poit out the otoriously slow covergece of simulatio methods. As stated earlier, these techiques rely heavily o the Law of Large Numbers, estimatig the expected value by the average of simulated outcomes, which implies that the stadard 0.5 deviatio of the estimate will be proportioal to. I practice this meas that for every additioal decimal digit precisio, we have to make a hudred times greater simulatio, which meas that the traditioal Mote Carlo method becomes computatioally itractable after the first 4-6 decimal digits of precisio improvemet. For the above metioed reasos, it is ecessary to employ advaced simulatio 815
techiques so as to reduce the variace of the estimate. Without aimig at completeess, we list a few variace reductio techiques here (for a detailed descriptio refer to Kleije et al. [8]): commo radom umbers (icludig atithetic variates); cotrol variates; coditioig ad stratified samplig; importace samplig ad splittig; quasi (ot fully radom) Mote Carlo simulatio. Depedig o the statistical properties of the estimated radom variate, some techiques may be more suitable to a certai simulatio problem tha others. Based o the fidigs of Kema ad Vorst [9], the method of cotrol variates looks the most promisig i the case of Asia optio valuatio. We discuss this i more detail i the ext sectio, where we look ito the properties of differet simulatio techiques, ad explai how they could be efficietly implemeted i optio pricig..3. Pricig Asia Optios After this brief itroductio to the valuatio framework, let us itroduce a umerical valuatio method based o the work of Gema ad Yor [3], the we describe the simulatio methods used later i Sectio 3. I the case of arithmetic-average Asia optios, which we from ow o refer to as Asia optios for the sake of simplicity, the payout i Equatio (1) is determied by the arithmetic mea of the uderlyig product s prices. I the cotiuous case 1 ( r ) ( r) T AT Avg[ 0, ] S = S t d t, T T 0 T (6) ( r) where S is the risk-eutral uderlyig product process defied earlier. All we eed to do ow is apply the risk eutral valuatio priciple, as described i Equatio (5), ad calculate the call optio price: + + rt ( r) rt AT C = e E ( Avg[ 0, ] ) e. T S K = E K (7) T Oce the call optio price is calculated, we may use the put-call parity, which is valid for all Europea-exercise optios, to arrive at the put optio price. Lemma 1 (Put-Call Parity) rt AT P = e E K T + AT rt AT = e E K E K (8) T T rt = C F + e K where F is the Asia forward price, as defied i the ext lemma. Lemma (Asia Forward Price) The forward price of the arithmetic-average of + 816
stock prices for a give time iterval [ 0,T ] is give by h( ν + 1) rt AT rt S F E = T T σ ( ν + 1) where 0 4 e 1 e e, (9) σ r ν r = 1. σ σ Therefore, i theory we are able to price Asia optios. However, solvig the simple-lookig pricig formula i Equatio (7) hides several pitfalls hautig fiacial mathematicias for decades. The fudametal problem is caused by the ukow probability distributio of AT. More specifically, the sum of logormally distributed radom variates is ot logormal, which makes the expected value i Equatio (7) hard to calculate. Eve though based o a result of Mitchell [10], who show that the sum of logormals is close to logormal, Turbull et al. [] derive a approximatio for the optio value, this could oly be cosidered as a cotrol tool rather tha a soud pricig formula by market practitioers ad risk maagers..3.1. The Laplace Trasform Method The breakthrough ca be credited to Gema ad Yor [3], based o whose semial paper a series of authors, icludig Carr ad Schröder [11] as well as Dufrese [1], coduct a excitig discussio over the validity ad applicability of their approach. Before statig the mai result, let us do a crucial trasformatio to the expressio i Equatio (7): where + rt AT C = e E K T + rt S ( 0) 4 ( ν ) = e E A ( σ T 4) K (10) T σ rt S ( 0) 4 ( ν ) σ T KTσ = e C,, T σ 4 4S( 0) { } + ν ( ) ( ν ) T ( ν + ) A T exp t W t d t (11) 0 ( ν ) C TK, E A T K, (1) the latter beig referred to as the ormalized price of the Asia optio. Followig [11], i the rest of this paper let us deote σ T KTσ h ad q. 4 4S 0 The revolutioary fidig of Gema ad Yor [3] is that there exists a aalytical ν formula for the Laplace trasform of C hq, with respect to h. 817
Theorem.6 (Gema-Yor) The Laplace trasform of the Asia optio s ormalized price, defied i Equatio (1), with respect to h is ( ν { ) }( λ ν) 0 ( ν ) λh C hq,, q, = e C hq, dh 1 β ( q) ( β) 1 u β α + 1 = exp u ( 1 u) d u, λ α + 1 Γ 0 q (13) where for every γ ν γ ν γ λ + ν α + β,,, { ( ν )} λ > max 0, + 1. Usig this result, it is possible to determie the optio s price by first calculatig the above Laplace trasform, the ivertig it to yield the ormalized optio price ad fially substitutig ito Equatio (10). However, as described i Cohe [13], the problem of ivertig the Laplace trasform is a otrivial oe (apart from a few special cases), ad oe ofte has to resort to umerical iversio techiques. There have bee various techiques iveted, based o either of the followig two iversio theorems. Theorem.7 (Theorem) If the Laplace trasform of a cotiuously differetiable fuctio f exists, the { }( t) f t 1 1 ( π) γ + it λt = λ lim e d, i λ λ (14) T γ it where γ is a real umber so that has o sigularities o or right from γ o the complex plae. Theorem.8 (Post-Widder Theorem) If the Laplace trasform of a cotiuously differetiable fuctio f exists, the where { ( λ )}( t) f t 1 1 = lim,! t t is the -th derivative of the particular Laplace trasform. Although i cotrast to the Bromwich formula the Post-Widder formula is restraied to the real lie ad thus looks easier to hadle, its speed of covergece is otoriously slow ad the techiques based o them are proe to computatioal difficulties, as we ca see i Fu et al. [5]. Hece, the two algorithms that we use are based o the Bromwich theorem, servig as a check for each other. I what follows, we preset these algorithms described i Abate ad Whitt [14]. The Euler Algorithm Abate ad Whitt [15] call this algorithm the Euler algorithm, sice they employ Euler summatio to accelerate covergece. Without goig ito further details, their basic idea is to look at the Bromwich itegral i Equatio (14), as though it was a Fouri- As the Laplace trasform is uique, i the sese that if { f} = { g} about oe approach leadig to a differet result tha aother., the f = g, we eed ot worry 818
er-series, by substitutig for λ = a + iu : { }( t) 1 1 ( π) γ + it λt λ = lim e λ dλ i T γ it 1 T ( γ + iu lim e ) t ( γ iu) d T T = + π γ t e iut = e ( γ iu) d. u π + Their fial result is the algorithm which also Equatio [5] used i their calculatios, beig apparetly a very (if ot the most) efficiet umerical algorithm i ivertig the Laplace trasform derived by Gema ad Yor [3]. Algorithm.9 (Euler) Let us defie ad A A e A e s t Re + a t t t t k = 1 k ( 1) k ( π) A+ i k ak ( t) Re. t The, the iverse Laplace trasform could be approximated by the Euler series m ˆ m,,,, k = 0 k 1 m { ( λ )}( t) f( t Am) = s+ k( t) where A, m ad are chose parameters of the algorithm, for which Abate ad Whitt [15] recommed A = 18.4, m = 11 ad = 15 (icreasig as ecessary). The Talbot Algorithm Similarly to the Euler algorithm, this algorithm is also based o the Bromwich theorem. The idea of Talbot, described i Cohe [13], is to mitigate the disturbig effects of oscillatio as we move alog the itegratio cotour parallel to the imagiary axis by deformig (i.e. alterig) the itegratio cotour to a more appropriate oe (let us call it Γ ), which ecloses all sigularities of i the complex plae. This idea ca be formally summarized as σ t 1{ ( λ 1 it t ce c t )}( t) = γ λ λ lim e d e ( c ) d, i + λ λ = T γ it i σ + λ λ Γ ( π) ( π) which is applied by Abate ad Whitt [14] i the followig umerical iversio algorithm. Algorithm.10 (Talbot) Let us defie M ( π) k δ0, δ k ( cot ( kπ M) + i) 5 5 ad 1 e δ 0, 1 π 1 cot π cot π e δk + +. γ γk i k M k M i k M The, the iverse Laplace trasform could be approximated by the series u 819
ˆ δk, =, t M 1 1 { ( λ) }( t) f ( t M ) Re γk 5t k = 0 where M is a chose parameter of the algorithm. Implemetatio By meas of the above algorithms, the Laplace trasform ( ν ) { C }(, q, ) λ λ ν could ow be umerically iverted with respect to λ to obtai ( ν ) f h C hq,. We use MATLAB for the implemetatio of the itroduced algorithms. The codes are preseted i Sectios A.1 ad A. of the Appedix. For the sake of clarity, we strive to preserve the otatio preseted earlier i this paper ad isert as may commets ito the code as possible. Without goig ito uecessary techicalities, let us poit out a couple of practical features ad ehacemets i the code: Both algorithms are based o substitutig back ito the Laplace trasform at differet loci o the complex plae. Most MATLAB fuctios ca take complex argumets, so this should ot be a setback. Oe importat hit to bear i mid though is the use of 1i istead of i i the code, idicatig that we refer to the imagiary umber ad ot a variable. A crucial step i calculatig the Laplace trasform preseted i Equatio (13) is complex itegratio o the ( 0,1 ) iterval. Sice solvig this problem is ot i the scope of our research, we use the built-i itegral fuctio for this purpose, which is i tur based o the proprietary global adaptive quadrature of MathWorks. I additio, at this poit we feel obliged to emphasize the cosiderable achievemet of Gema ad Yor [3] i providig a itegral o a closed iterval, as the hadlig of improper itegrals would have bee a umerically much less tractable problem. Based o our experiece, durig the implemetatio of the algorithms we have to work with quatities of very differet magitudes. More specifically, the itegral i 500 Equatio (13) is usually of magitude 10 100 to 10, the precedig fractio beig large eough to compesate. This poses a big challege because computers use floatig-poit umbers, ad thus multiplicatio ad divisio of such differet magitude umbers could lead to the complete loss of precisio. The measures take to mitigate this problem are 1) Takig the (atural) logarithm of the whole expressio i Equatio (13) so as to expoetially reduce the magitude of variables. Although this step does ot ecessarily icrease precisio, it defiitely ears space i terms of magitude. ) Explicitly istructig MATLAB to use the Symbolic Math Toolbox, which allows the user to choose a arbitrary level of precisio for calculatios. With regard to our particular problem, we cosider a precisio of 5 sigificat digits sufficiet. Havig uderstood the umerical approach based o the Laplace trasform of the Asia optio s ormalized price, we are ready to price Asia optios with two algo- 80
rithms. But before proceedig to the empirical aalysis cocerig the precisio ad robustess of these algorithms, let us review the simulatio approach, which will serve as a ultimate check ad basis of compariso for the Laplace trasform method..3.. The Mote Carlo Method Keepig i mid the basics of the Mote Carlo method from the ed of Sectio., we kow that this is a highly-flexible, robust approach, suitable to various frameworks ad product types. Flexibility, however, has a price takig the form of variace i the case of simulatio methods, which meas that the simulated price bears some ucertaity ad fluctuates aroud the optio s true (aalytical) value. Strivig to cut this variace to a miimum level, we study i details how the variace reductio techiques of atithetic variates ad cotrol variates work i the case of fiacial derivative valuatio. Basic Method As the ame suggests, this is the simplest form of Mote Carlo method, takig the arithmetic-mea as a ubiased estimator of the expected value: which has a variace of where X f ( ω ) i= 1 X i ˆ X i E X =, i= 1 ( X) ( ˆ X i D D E( X) ) = D =, i= 1 i = i are IID radom variates represetig the i th outcome i our simulatio. The radom fuctio f, whose expected value we wish to estimate is the discouted payout fuctio of the optio as suggested by the priciple i Equatio (5): f rt ( ω) Payout ( ω) e, which, i the particular case of Asia call optios, takes the form of havig a expected value rt ( r ( ω) ) [ 0, T ] ( ( ω) ) f = e Avg S K, rt ( r ( ( ω) ) ) [ 0, T ] ( ( ω) ) + E f = e E Avg S K, which is precisely the price of the optio give i Equatio (7). As we ca see, it is this risk-eutral expected value that we estimate through simulatio of the uderlyig product s risk-eutral trajectory which will be costructed as where ( r { ) t ( ω) j } j 0, m ( T 365) ( r { S ) ( ω) } t t [ T ] 0,, j S S( 0) exp Xt ( ω), i i=0 + j 0, m ( T 365) 81
X t i ( ω) 0 if i = 0 σ r ( ti ti 1) + σ ( W( ti) W( ti 1) ) if i > 0 is the risk-eutral log-retur applyig Equatio () ad m idicates the umber of daily reads (i.e. the mesh of the equidistat time grid). We must poit out that i the case of path-depedet derivatives simulatio methods have two sources of estimatio error: 1) The geerated trajectories (outcomes) deped o the radom evet ω, ad thus the calculated estimator is also a radom variate, fluctuatig aroud the expected value that we look for. This simulatio error could be reduced by icreasig, which is the umber of trajectories (outcomes) i our simulatio. ) The geerated trajectories are discretized, which meas they are oly approximatios of the true, cotiuous-time trajectories required for the calculatio of the path- depedet derivative payout. I other words, we substitute the itegral with a sum i the payout of the Asia optio ad o matter how great ad thus how accurate our estimator is, we still face some discretizatio error. This error could be reduced by icreasig m, which is the umber of fragmets we break up each day of iterval [ 0,T ] ito. Method of Atithetic Variates Keepig the logic ad weak poits of the basic simulatio method i mid, let us see how we could improve o our estimate of the expected value. First of all, let us try ad reduce the variace of the estimator. A suitable techique is the method of atithetic variates, described i Kleije et al. [8]. The pricipal idea here is usig each geerated radom umber twice, creatig two trajectories from each log-retur sequece, where oe is egatively correlated (atithetic) to the other. Although the way the atithetic trajectories are created is greatly product specific, it is usually a good idea to fid a step i the simulatio algorithm whe we deal with symmetrically distributed radom variates, ad the simply mirror them about their expected value. I our particular case of log-retur sequeces, we simply take the egative of each log-retur ad thus create a mirrored trajectory. Callig them sample A ad sample B, let us create a ew estimator ˆ ˆ ( a) ˆ E X A + E( XB) Xi + Xi Eati ( X) =, ( a) where X i ad X i are the payouts based o the i th origial ad atithetic trajectory, respectively. This estimator has a variace of ( a) D ( X ) + Cov ( X, X ), i= 1 ( a ) ( ˆ Xi + X i ati ) = i 1 = ( a) ( a) D ( X) + D ( X ) + Cov X, X D E X D = (15) 8
where we use the symmetric ature of the radom walk. The beefit of usig this method is twofold: 1) It is apparet from the above results that as log as the origial ad atithetic payouts have a egative covariace, the variace of the estimator will be reduced compared to that of the basic method, while the estimator remais ubiased. As we demostrate i Sectio 3, thaks to the low correlatio of the origial ad the mirrored trajectories, the variace reductio achieved by this relatively simple method is substatial. ) Sice we use every radom umber twice, we oly eed to geerate / outcomes istead of the origial oes, which meas that the method of atithetic variates is (almost) twice as efficiet as the basic method. Method of Cotrol Variates As demostrated through the above example of atithetic variates, it is possible to keep the estimator ubiased, while sigificatly reducig its variace. As it turs out, it is possible to further improve o our simulatio techique by usig cotrol variates istead of atithetic variates (for further details refer to Kleije et al. [8]). This method is based o the logic that if we add a appropriate (i.e. highly correlated, zero expectatio) expressio to our origial estimator, the variace reductio could be achieved while the estimator remais ubiased. Let us deote the cotrol variate with Y, while preservig our origial variate X. I this case, the estimator is ˆ ˆ ˆ X i + cyi Ecotrol X E X + ce ( Y ) =, where c is a properly chose 3 costat, X i ad Y i are the i th outcomes of the origial ad cotrol variates i our simulatio, respectively. It is easy to see that the variace will be ( ˆ cotrol ) i i = i= 1 D E X D X + cy i= 1 + + (, ) D X c D Y ccov X Y = Kema ad Vorst [9] show that i the case of the arithmetic-average Asia optio price, the use of the geometric-average Asia optio price ca be used as a effective cotrol variate. I a succeedig research, Fu ad Mada [5] compare the discrete ad the cotiuous geometric-average Asia optio price s performace as cotrol variate with iterestig results. They fid that eve though the appropriate formula for a discrete estimator would be the discrete average formula, by usig the cotiuous average formula a greater variace reductio ca be achieved. I other words, by usig a biased (i.e. o-zero expectatio) cotrol variate, we ot oly reduce the variace of our origial estimator, but also add a appropriate bias to it, which compesates for the above metioed discretizatio error iheret i the basic simulatio method. For this Cov ( X, Y 3 * ) It ca be easily show that the optimal value of the costat is c = D ( Y), which miimizes the variace of the estimator.. (16) 83
reaso, we also use this discretizatio adjusted cotrol variate i our simulatios. It is crucial to ote the advatages of this method: 1) It reduces the simulatio error of our origial estimator itroduced i the basic method, eve more so tha the method of atithetic variates 4. ) It effectively elimiates some of the discretizatio error iheret i the simulatio techique by usig a discretizatio adjusted cotrol variate..3.3. Implemetatio Before proceedig with the empirical aalysis of our valuatio framework i the ext sectio, let us make a few otes with regard to the practical implemetatio of the simulatio methods discussed above. Firstly, we perform the simulatio followig the algorithms preseted i this sectio. Although the true variaces ad covariaces are ukow, they could be estimated from the sample. Secodly, the greatest challege durig the simulatio is to avoid ruig out of memory. Thaks to the capacity of moder computers, this is usually ot a issue whe simulatig oly the edpoits of trajectories as i the case of Europea-exercise vailla optios. However, whe the etire trajectory eeds to be geerated ad stored i the memory (e.g. i the case of path-depedet derivatives), we ru out of resources alarmigly fast. Without goig ito further techical details (e.g. issues with memory pagig ), what we do to avoid this evetuality is break each trajectory ito small eough sectios that the geerated log-retur matrix fits ito the memory. The, we roll over the trajectory, calculatig the (geometric) mea step by step, efficietly reusig the available memory space agai ad agai as show i Figure A5 Memory of the Appedix. 3. Empirical Aalysis After the itroductio of the differet approaches to Asia optio pricig, ow we take a more practical perspective ad compare the efficiecy ad computatioal characteristics of the methods described i Sectio.3 by aalyzig outputs from MATLAB. The computatios are performed uder Widows 7 operatio system by meas of a Itel Core i7. GHz CPU 8 GB RAM hardware. First, we examie how the two semi-aalytical (i.e. Euler ad Talbot) algorithms fare compared to each other, the we cotrast simulatio outputs for differet readig frequecies ad sample sizes. The, we compare the semi-aalytical ad the simulatio approaches ad make suggestios o their appropriate usage i real life situatios. 3.1. Numerical Results At first glace the semi-aalytical algorithms preseted i the previous sectio might look awkward but i effect they are very efficiet for certai parameter rages of the Asia call optio price C( SKrTσ,,,, ). As Fu ad Mada [5] poit out, difficulties σ T t < 0.01. Our calculatios cofirm that the critical [ ] seem to begi whe 4 Based o our experiece, the correlatio betwee the arithmetic-average Asia optio price ad the geometric-average Asia optio price is very high (above 0.9), which idicates that a very sigificat variace reductio could be achieved. For further details, refer to Sectio 3. 84
parameter is ideed σ T, which fudametally govers how far away the uderlyig product s price ca get from its curret state durig the lifetime of the optio, ad directly determies the locus h at which the ormalized optio price C ) ( hq, ), defied ( ν i Equatio (1), must be calculated. For the sake of simplicity, let us fix σ at oe 5 ad observe what effect chages i T (i.e. time to maturity) have o a ATM ( S = K = 1 ) Asia call optio price s calculatio efficiecy. As hited i Sectio.3.1, the mai parameters of the Euler ad Talbot algorithms, drivig computatioal precisio, are ad M, respectively. Leavig the other parameters at their default values 6 as suggested by Abate ad Whitt [15], the way we measure the efficiecy of each algorithm is iteratively icreasig the precisio parameters util the last four decimal digits of the optio price is stabilized. The results are show i Figure 1, where we depict i log-log charts the times ad parameter levels that each algorithm required to reach the desired precisio. From these charts, we may coclude the followig: The Euler algorithm is more robust tha the Talbot algorithm with respect to both calculatio time ad parameter levels. I geeral, settig = 10 produces a Asia optio value of at least four sigificat digit precisio. The Talbot algorithm is more reliable tha the Euler algorithm i the sese that if m is set sufficietly high, the the precise Asia optio value ca be calculated eve for small values of σ T (give eough computatio time). I cotrast, the Euler algorithm fails to coverge whe σ T < 0.005, o matter how high is set. Whe σ T falls below a sufficietly low threshold, we ca see a rapid icrease i both calculatio times ad parameter levels. However, whe σ T remais outside the dager zoe, the semi-aalytical method proves to be extremely efficiet ad solve the iversio problem uder - 3 secods. By repeatig the above calculatio test with differet combiatio of optio parameters, we coclude that it is oly the value of σ T which determies the speed of the (a) Figure 1. Compariso of the Euler ad Talbot aalytical algorithms. 5 Naturally, σ = 1 is a urealistic sceario, but sice it is the value of σ T that drives the Asia optio price give i Equatio (10), it is a coveiet choice, makig it easier for us to capture the effects of chages i T. 6 Fu et al. [5] does a extesive research o tweakig the Euler algorithm s other two parameters (i.e. A ad m), cocludig that the default settig (i.e. A = 18.4 ad m = 11 ) suggested by Abate ad Whitt [15] is ideed optimal. (b) 85
Laplace trasform iversio approach. A importat questio to be aswered though is why the chose umerical algorithms have covergece issues whe σ T is small. Cosiderig that this is a pheomeo that also other authors, such as Fu et al. [5] ad Craddock et al. [16] cofirm, it seems sesible to assume that as σ T 0, iversio complicatios arise due to the iheret umerical characteristics of the Laplace trasform calculated by Gema ad Yor [3], o matter which iversio algorithm we choose. Ideed, observig the charts preseted i Sectio A.3 of the Appedix, we fid that as σ T 0, the Laplace trasform gets umerically out of had : I the case of the Euler algorithm, the Laplace trasform oscillates ad quickly teds to zero as k (i.e. the locus i the complex plae) chages, which limits the effect of the precisio parameter, there beig o reaso to use > 0 i the calculatios. The problem is that as σ T 0, the Laplace trasform rapidly dimiishes to orders smaller tha 10, makig the embedded umerical itegratio, which is out- 10 lied i Equatio (13), icreasigly hard to calculate with the desired precisio. I the case of the Talbot algorithm, the Laplace trasform shows some oscillatig ature agai as k (i.e. the locus i the complex plae) chages but remais zero outside a give rage. The problem here is ot that the trasform dimiishes to magitudes which are hard to hadle umerically but that this specific locatio of oscillatio is ukow, moreover, it shifts alog the k axis towards ifiity as σ T gets smaller. It follows that i order to reach the desired precisio, we have to set k large eough that this locatio of iterest is icluded i the fial summatio withi the algorithm. This etails the usage of a ever larger summatio, or i other words the value of precisio parameter M (ad thus the calculatio time) eeds to be icreased as σ T 0. Although with the Euler algorithm we quickly ru ito a dead-ed street, sice the embedded itegral caot be calculated umerically to a arbitrary precisio, the deformatio of the Bromwich-cotour i case of the Talbot algorithm seems to leave some space for improvemet. If we could (either umerically or aalytically) pipoit the locatio of oscillatio without havig to calculate all the zeros before, we could sigificatly improve o the algorithm s performace. Moreover, we cosider it a achievemet that we could sigificatly reduce calculatio times compared to the oes produced by Fu et al. [5] as show i Figure A6 i the Appedix. The former calculatio time of 30-40 secods have bee reduced to a mere 3-4 secods. How much of this achievemet could be attributed to better implemetatio ad how much to better equipmet, though, is hard to judge. 3.. Simulatio Results I the previous sectio we ivestigate how the two umerical algorithms perform for differet combiatios of optio parameters. Before makig a compariso with the simulatio approach, let us look ito the latter o its ow ad examie the three Mote Carlo algorithms itroduced i Sectio.3.. As explaied earlier, i the case of pathdepedet derivatives (such as Asia optios) simulatio techiques have two gover- 86
ig parameters:, which is the umber of trajectories simulated, m, which is the umber of fragmets each day is partitioed ito, where determies the simulatio error, whereas m determies the discretizatio error. Strivig to isolate the effects of these parameters as much as possible, first we examie what level of m leads to a satisfactory approximatio of the cotiuous Asia optio price, ad oly the will we tur to adjustig (lowerig) i order to attai a estimate with desirable precisio. Just as i the case of umerical methods, we costrai our aalysis to ATM ( S = K = 1 ) Asia call optios ad σ = 1. The beefit of this restrictio is twofold. O the oe had, simulatio results will be easily comparable to the respective umerical results. O the other had, the relatioship betwee T ad the estimatio error will serve as a good proxy for the relatioship betwee σ T ad the estimatio error. However, before proceedig with the fie-tuig of the above parameters ad measurig the performace of the differet simulatio algorithms, let us have a word o what level of error we cosider satisfactory. I order to make a later compariso betwee simulatio ad umerical methods possible, we aim at the same (i.e. four-decimal) valuatio precisio. To achieve this, we eed to set the simulatio parameters so that the cofidece iterval 7 aroud the estimated optio value is sufficietly small to esure the desired precisio. Sice the exact probability distributio of the Asia optio price is ukow, let us suffice with Chebyshev s iequality ad give a approximatio for the cofidece iterval by 1 P ( E ( X ) X k σ ), k which implies that by choosig k = 4, we ca be about 95% certai that the true Asia optio value falls betwee Eˆ ( X) 4 ˆ σ ad Eˆ ( X) + 4 ˆ σ. I other words, the estimated stadard deviatio has to be of order 10 so that the desired four-decimal pre- 5 cisio is satisfied. 3..1. Discretizatio Error Keepig i mid the fidigs about simulatio precisio, let us determie what level of m esures a good approximatio of the uderlyig product s cotiuous trajectory. If we set = 100000, which is relatively large, we expect to receive results with low simulatio error. It follows that by chagig m, we should be able to discer chages i discretizatio error, oly slightly disturbed by ay residual simulatio error i the output. Applyig this logic, we ru the simulatio for differet maturities ad gathered the output i Sectio A.4 of the Appedix, displayig the absolute error (i.e. the absolute differece of the simulated optio value ad the true optio value). From these charts, we ca make the followig observatios: The cotrol variate method has lower (discretizatio) error tha the other two methods by about oe order of magitude. 7 I order to defie a cofidece iterval, we should also pick a suitable sigificace level. Let this be 95%. 87
There is a rapid decrease i discretizatio error whe usig m = 10 istead of m = 1. However, the discretizatio error reductio is ot sigificat whe m is further icreased to m = 100. Iterestigly, the atithetic variate method does ot reduce but rather amplifies discretizatio error. O a secod thought, this pheomeo is ot surprisig at all, as i the case of the atithetic variate method each geerated radom umber is used twice, so as to cut dow o simulatio error ad time. However, this method also etails that ay discretizatio error i a give simulated trajectory is emphasized (duplicated) as there are less idepedet trajectories used i the simulatio process. By takig a closer look at the absolute errors for differet levels of m, we might otice patters, such as peaks or troughs, especially i case of the first two methods. This is ot a coicidece, as we use the same radom umber sequeces for each valuatio, which meas that loger simulatios (i.e. where T is greater) use the same radom umbers as their shorter couterparts, ad thus ay discretizatio error will also be preserved. This slowly decayig ature of the estimatio error is very useful, makig it easier for us to idetify those simulatio methods which effectively reduce discretizatio error. For istace, it is very hard to discer patters i the chart of the cotrol variate method, which idicates that this method does exactly what it professes, amely that it reduces discretizatio error iheret i the simulatio of the arithmetic-average by offsettig it with the oe iheret i the simulatio of the geometric-average. From the above results, we ca coclude that it is best to choose the cotrol variate method, as it effectively reduces discretizatio error. Also, it is uecessary to set m greater tha 10, as creatig more refied trajectories sigificatly icreases simulatio time but does ot attai a proportioate discretizatio error reductio. Hece, i the rest of our work we set m = 10. 3... Simulatio Error From a perspective of discretizatio error reductio, the method of cotrol variates is clearly the best performig oe. Now, let us make a compariso betwee simulatio methods based o the simulatio error of their outputs, measured by the stadard deviatio of the estimate. The estimated stadard errors produced by each simulatio method (usig a give umber of trajectories) are show o a log-log scale i Figure, from which we draw the followig ifereces: The atithetic method achieves oly a slight (about 15% - 0%) reductio i simulatio variace compared to the basic method. The cotrol variate method achieves a substatial (about 90% - 95%) reductio i simulatio variace compared to the basic method. The variace reductio is so efficiet that as few as = 10000 trajectories could be sufficiet to make a accurate estimate (especially whe σ T is relatively small). 88
The stadard error from regular ad atithetic methods grows oly at a rate ~ x 0.55, whereas the cotrol variate method s stadard error grows at a faster rate ~ x 1.05. It looks as though the sigificat discretizatio error reductio observed earlier has a price, ad hece the stadard error of the cotrol variate method catches up with the stadard error of other methods as σ T icreases. The questio aturally arises whether we quatify the variace reductio by the atithetic ad the cotrol variate methods. For this, we eed to estimate the correlatio of atithetic variates ad cotrol variates, respectively. As we ca see i Figure 3, these correlatios are ot costats but they deped o the value of T (i.e. the legth of the simulated trajectories). We kow from Equatios (15) ad (16) that ( ˆ ati ) D E X ad (for a variace-optimal ( a) ( a) D ( X ) + Cov ( X, X ) D ( X) ( 1 + ρ ( X, X )) * c ) Figure. Compariso of the differet simulatio methods ( m = 10 ). Figure 3. Correlatio of proxy variates at variace reductio methods. 89
( ˆ cotrol ) D E X = + + (, ) D X c D Y ccov X Y D ( X) Cov D ( a) provided D ( X) D X ad D ( X) D ( Y). Comparig these simulatio errors with the variace of the basic method, which is ( X, Y ) ( Y) D( X) 1 ρ ( XY, ) ( X) ( ˆ D D E( X) ) =, we ca quatify the variace reductio. I the case of the atithetic method, ρ ( 0.4, 0.) esures a 0% - 40% variace reductio, which is ideed a 10% - 0% stadard error reductio. I cotrast, ρ 1 i the case of the cotrol variate method results i a close to 100% stadard error reductio, just as experieced earlier. Also, by lookig at the above calculatios, where D ( X ) is multiplied by ( 1 ρ ) + i the a- tithetic ad by ( 1 ρ ), i the cotrol variate case, we uderstad why the simulatio error of the latter grows faster as ρ 0. From the results preseted we may coclude that the cotrol variate method effectively reduces both discretizatio error ad simulatio error eve for m = 10. Usig this method, the estimatio error remais below the desired chose as per Table 1. 3.3. Compariso of Valuatio Approaches 5 10 if is appropriately Havig compared the differet umeric ad simulatio methods separately ad havig idetified the optimal parameter values for each, let us compare the two approaches i the search for a aswer to our secod research questio. Namely, give today s computatioal capacity what method (algorithm) should we use for pricig Asia optios? We have see that the Euler ad Talbot algorithms are powerful tools for pricig Asia optios, though beig rather sesitive to low values of σ T. Also, we have cocluded that the cotrol variate method is able to determie the price of Asia optios very accurately by efficietly reducig both discretizatio ad simulatio variace. The oly questio remaiig is how fast these algorithms are, compared to oe aother. Could we set up a preferece order, maybe eve as a fuctio of σ T? To aswer this questio, let us have a look at Figure 4, where we ca see what time it takes for the Euler algorithm to calculate the Asia optio price to the desired four-decimal precisio; Table 1. Optimal parametrizatio of the cotrol variate method ( m = 10 ). σ T <0.05 10000 <0.05 100000 <0.1 1000000 830
Figure 4. Compariso of calculatio times usig differet methods ( m = 10 ). the cotrol variate algorithm to ru a simulatio for a give, showig each curve oly util the desired precisio ca be maitaied. Let us ote that while the time required by umeric algorithms is fudametally a fuctio of σ T, simulatio times deped basically o T (i.e. how log trajectories eed to be geerated). This implies that for each σ, the relative positio of timigs chages, as we may observe the differeces betwee Figure 4 ad Figure A4 (i Sectio A.4 of the Appedix), calculated for σ = 0% ad σ = 10%, respectively 8. Based o the figures, we claim that the cotrol variate algorithm geerally beats umerical algorithms i terms of calculatio time for usual values of σ ad T. However, we should also icorporate aother factor i our aalysis before makig a coclusio, amely calculatio precisio. As we observe i Figure ad Table 1, eve whe usig the cotrol variate method, the stadard error of the estimate does ot satisfy our desired level of precisio whe σ T exceeds a give limit, idicatig that it might be worth to switch to umeric algorithms uder such coditios. Based o our fidigs, we may set up the followig decisio rules to optimize the Asia optio pricig methodology. 4. Coclusio By maitaiig a practical approach to the valuatio problem i this paper, we pay special attetio to the implemetatio of the umeric algorithms ad to the specific computatioal issues that arose i MATLAB. I Sectio 3, we test the two semi-aalytical (umerical) methods ad the three simulatio methods for differet optio parameter settigs, strivig to aswer the questios how the efficiecy of the differet algorithms as well as their relative performace have chaged due to the techological developmet of the last decade. 8 Uder usual market coditios, the stock price volatility lies betwee these two values, ad thus Figure 4 ad Figure A4 show the extremities for the σ parameter. 831
Poitig out the importace of σ T i the Asia optio pricig problem i Sectio 3.1, we maage to further improve o the reliability ad efficiecy of the umerical iversio algorithms, pushig the critical value of σ T dow to 0.005 with 3-4 secods calculatio time (i compariso to the critical value of 0.01 with 30-50 secods calculatio time, as show i the work of Fu et al. [5]). Also, based o the work of Abate ad Whitt [14], we successfully employ the Talbot algorithm as a viable alterative to the Euler algorithm i Laplace trasform iversio. Although the latter proved to be more robust, the Talbot algorithm showed superior computatioal efficiecy (i.e. a valuatio time of scarcely two secods) for greater values of σ T. Based o the simulatio results i Sectio 3. we cofirm that the method of usig the geometric-average Asia optio price as cotrol variate is a extremely efficiet oe, rederig traditioal Mote Carlo methods useless i the particular valuatio problem of arithmetic-average Asia optios. Ideed, whereas for the latter methods the geeratio of 100 000 trajectories is ot sufficiet to price the optio with the desired (fourdigit) accuracy, the cotrol variate method requires oly 10,000 trajectories to fulfill our requiremets, takig fractios of a secod to compute the Asia optio price. Hece, we ca experiece a iterestig dichotomy betwee umerical ad simulatio methods: oe approach teds to excel whe the other shows iferior performace ad vice versa, depedig o the value of σ T as show i Table. It follows that the two approaches complemet each other, so that there is always a applicable algorithm which is able to price a Asia optio withi 5 secods with a precisio of four decimal digits (i our perceptio, this speed ad precisio is acceptable uder most market circumstaces). Regardig the compariso of the umerical ad simulatio approaches let us make two subtle but crucial remarks. Firstly, we caot emphasize eough the geometricaverage Asia optio price s importace i this particular valuatio problem. Had it ot bee for this cotrol variate s fortuate availability, the simulatio approach would have performed far worse tha i this special case. Thus, before wavig aside the Laplace trasform iversio methods for their otorious sesitivity to small values of σ T, we should remember that i valuatio problems where there is o such cotrol variate available these umerical algorithms could serve as a valuable pricig tool. Secodly, let us poit out the Laplace trasform method s weakess, amely that it is limited by the assumptios of the Black-Scholes model. Presumably the strogest of these teets is the costat volatility assumptio, the relaxatio of which could fudametally alter the Laplace trasform formula derived by Gema ad Yor [3]. Also, if Table. Optimal method selectio rules for pricig Asia optios. σ T optimal method <0.05 MC with cotrol variate ( m = 10, = 10k ) <0.1 Euler algorithm ( = 10 ) >0.1 Talbot algorithm ( M = 10 ) 83
the assumptio of geometric Browia motio is relaxed (e.g. log-returs are modeled by the more geeral Lévy processes), the validity of the umerical results is questioable. I cotrast, the flexibility of simulatio methods makes it possible to adjust the origial valuatio framework to a wider rage of assumptios, thus eablig market participats to use more realistic stock processes for Asia optio pricig. Refereces [1] Bodar, G.M., Hayt, G.S. ad Marsto, R.C. (1998) Wharto Survey of Fiacial Risk Maagemet by U.S. No-Fiacial Firms. Fiacial Maagemet, 7, 70-91. http://dx.doi.org/10.1314/jcf.1998.04 [] Turbull, S.M., McLea, S. ad Wakema, L.M. (1991) A Quick Algorithm for Pricig Europea Average Optios. Joural of Fiacial ad Quatitative Aalysis, 6, 377-389. http://dx.doi.org/10.1314/jcf.1998.04 [3] Gema, H. ad Yor, M. (1993) Asia Optios, Bessel Processes ad Perpetuities. Mathematical Fiace, 3, 349-375. http://dx.doi.org/10.1314/jcf.1998.04 [4] Gema, H. ad Eydelad, A. (1995) Domio Effect: Ivertig the Laplace Trasform. Risk, 8, 65-67. [5] Fu, M.C., Mada, D.B. ad Wag, T. (1999) Pricig Cotiuous Asia Optios: A Compariso of Mote Carlo ad Laplace Trasform Iversio Methods. Joural of Computatioal Fiace,, 49-74. http://dx.doi.org/10.1314/jcf.1998.04 [6] Black, F. ad Scholes, M. (1973) The Pricig of Optios ad Corporate Liabilities. Joural of Political Ecoomy, 81, 637-654. http://dx.doi.org/10.1086/6006 [7] Kalos, M.H. ad Whitlock, P.A. (008) Mote Carlo Methods. d Editio, Joh Wiley & Sos, Ic., Hoboke. http://dx.doi.org/10.100/978357661 [8] Kleije, J.P.C., Ridder, A.A.N. ad Rubistei, R.Y. (010) Variace Reductio Techiques i Mote Carlo Methods. CetER Discussio Paper Series, No. 010-117. [9] Kema, A.G.Z. ad Vorst, A.C.F. (1990) A Pricig Method for Optios Based o Average Asset Values. Joural of Bakig & Fiace, 14, 113-19. http://dx.doi.org/10.100/978357661 [10] Mitchell, R.L. (1968) Permaece of the Log-Normal Distributio. Joural of the Optical Society of America, 58, 167-17. http://dx.doi.org/10.1364/josa.58.00167 [11] Carr, P. ad Schröder, M. (004) Bessel Processes, the Itegral of Geometric Browia Motio, ad Asia Optios. Theory of Probability & Its Applicatios, 48, 400-45. http://dx.doi.org/10.1137/s0040585x97980543 [1] Dufrese, D. (004) Bessel Processes ad a Fuctioal of Browia Motio. Ceter for Actuarial Studies, Departmet of Ecoomics, Uiversity of Melboure, Parkville. http://hdl.hadle.et/11343/3434 [13] Cohe, A.M. (007) Numerical Methods for Laplace Trasform Iversio. Spriger Sciece & Busiess Media, Berli. [14] Abate, J. ad Whitt, W. (006) A Uified Framework for Numerically Ivertig Laplace Trasforms. INFORMS Joural o Computig, 18, 408-41. http://dx.doi.org/10.187/ijoc.1050.0137 [15] Abate, J. ad Whitt, W. (1995) Numerical Iversio of Laplace Trasforms of Probability Distributios. ORSA Joural o Computig, 7, 36-43. http://dx.doi.org/10.187/ijoc.1050.0137 833
[16] Craddock, M., Heath, D. ad Plate, E. (000) Numerical Iversio of Laplace Trasforms: a Survey of Techiques with Applicatios to Derivative Pricig. Joural of Computatioal Fiace, 4, 57-8. http://dx.doi.org/10.1314/jcf.000.055 834
Appedix A.1. Euler Algorithm 1 fuctio x = Euler(, P, S, K, T, sigma) %% Recommeded parameters by Abate-Whitt (1995). 3 m = 11; 4 A = sym(18.4); 5 6 %% Parameters of the ormalized optio price. 7 h = sym((t*sigmaˆ)/4); 8 q = sym((k*t*sigmaˆ)/(4*s)); 9 u = sym( *log(p)/(sigmaˆ*t) 1); 10 11 %% Lambda calculatio (Euler algorithm). 1 arg = sym(zeros(( + m) + 1, 1)); 13 for k = 0:( + m) 14 arg(1 + k) = (A + *k*pi*1i)/(*h); 15 ed 16 17 if sum(sigle(abs(arg)) < max(0, *(sigle(u) + 1))) = 0 18 x = 999; % A easy to idetify umeric error output. 19 retur 0 ed 1 try 3 %% Substitutio ito the Laplace trasform. 4 mu = sqrt(*arg + uˆ); 5 alpha = (mu + u)/; 6 beta = (mu u)/; 7 8 itegrad = @(alpha, beta, q, u) u.ˆ(double(beta) ).* 9 (1 u).ˆ(double(alpha) + 1).*exp( u/(*double(q))); 30 a = sym(itegral(@(u)itegrad(alpha, beta, q, u), 0, 1, 31 'ArrayValued', true, 'AbsTol', 0, 'RelTol', 1)); 3 a = log(a) + log(*q)*(1-beta) log(.*arg) 33 log(alpha + 1) mfu('lgamma', double(beta)); 34 a = real(exp(a)); 35 36 %% Summatio (Euler algorithm). 835
37 s = (exp(a/)/h).*(( 1).ˆ(0:( + m))).*a; 38 s(1) = s(1)/; 39 for k = 1:(+m) 40 s(1 + k) = s(1 + k) + s(k); 41 ed 4 43 x = sym(0); 44 for k = 0:m 45 x = x + choosek(m, k)*ˆ( m)*s(1 + ( + k)); 46 ed 47 48 x = x*p*(s*4)/(t*sigmaˆ); 49 x = double(x); 50 51 catch 5 x = double( 999); 53 ed 54 55 ed A.. Talbot Algorithm 1 fuctio x = Talbot(M, P, S, K, T, sigma) %% Parameters of the ormalized optio price 3 h = sym((t*sigmaˆ)/4); 4 q = sym((k*t*sigmaˆ)/(4*s)); 5 u = sym( *log(p)/(sigmaˆ*t) 1); 6 7 %% Lambda calculatio (Talbot algorithm). 8 d = sym(zeros(m,1)); 9 d(1) = *M/5; 10 for k = 1:(M 1) 11 d(k + 1) = *k*pi/5 * (cot(k*pi/m) + 1i); 1 ed 13 14 g = sym(zeros(m,1)); 15 g(1) = 1/ * exp(d(1)); 16 for k = 1:(M 1) 17 g(k + 1) = (1 + 1i*(k*pi/M)*(1 + (cot(k*pi/m))ˆ) 836
18 1i*cot(k*pi/M))*exp(d(k+1)); 19 ed 0 1 arg = d/h; 3 if sum(sigle(abs(arg)) < max(0, *(sigle(u) + 1))) = 0 4 x = 999; % A easy to idetify umeric error output. 5 retur 6 ed 7 8 try 9 %% Substitutio ito the Laplace trasform. 30 mu = sqrt(*arg + uˆ); 31 alpha = (mu + u)/; 3 beta = (mu u)/; 33 34 itegrad = @(alpha, beta, q, u) u.ˆ(double(beta) ).* 35 (1 u).ˆ(double(alpha) + 1).*exp( u/(*double(q))); 36 a = sym(itegral(@(u)itegrad(alpha,beta,q,u), 0, 1, 37 'ArrayValued', true, 'AbsTol', 0, 'RelTol', 100)); 38 a = log(a) + log(*q) * (1-beta) log(.*arg) 39 log(alpha + 1) mfu ('lgamma', double(beta)) + log(g); 40 a = real(exp(a)); 41 4 %% Summatio (Talbot algorithm). 43 x = sym(0); 44 for k = 0:(M 1) 45 x = x + a(k + 1); 46 ed 47 48 x = x*/(5*h)*p*(s*4)/(t*sigmaˆ); 49 x = double(x); 50 51 catch 5 x = double( 999); 53 ed 54 55 ed 837
A.3. Behavior of the Laplace Trasform (a) (b) Figure A1. Magitude of the Laplace trasform at k usig the Euler algorithm ( = 0). (c) 838
(a) (b) Figure A. Magitude of the Laplace trasform at k usig the Talbot algorithm (M = 50). (c) 839
A.4. Simulatio Approach Figure A3. Absolute error of algorithms for differet levels of m ( = 100 000). 840
Figure A4. Compariso of calculatio times usig differet methods (m = 10). Physical Memory Usage History Figure A5. Optimized memory maagemet of simulatio algorithms. Figure A6. Speed improvemet achieved usig differet methods. 841