The pricig of discretely sampled Asia ad lookback optios 5 The pricig of discretely sampled Asia ad lookback optios: a chage of umeraire approach Jesper Adrease This paper cosiders the pricig of discretely sampled Asia ad lookback optios with oatig ad xed strikes. I the modellig framework of Black ad Scholes (973), it is show that a chage of umeraire of the martigale measure ca be used to reduce the dimesio of these path-depedet optio pricig problems to oe i additio time. This meas that the pricig problems ca be solved by umerically solvig oedimesioal partial differetial equatios. The author demostrates how a Crak± Nicolso scheme ca be applied to the umerical solutio. Fially, the methodology is exteded to the case whe the uderlyig stock exhibits discotiuous returs, ad it is show that i this case the Asia ad lookback optio pricig problems ca be solved by umerically solvig oe-dimesioal partial itegrodifferetial equatios.. INTRODUCTION Exotic optios that have payo s that deped o the arithmetic average, the maximum, or the miimum of the uderlyig stock over a certai time period have become icreasig popular hedgig ad speculatio istrumets over recet years. Parallel to that, a growig body of literature has cosidered the pricig ad hedgig of such derivatives. Withi the Black ad Scholes (973) model, closed-form solutios have bee obtaied for lookback optio prices by Goldma, Sosi, ad Gatto (979), Goldma, Sosi, ad Shepp (979), ad Coze ad Viswaatha (99). No closed-form solutio has yet bee derived for Asia optio prices, but various trasforms ad approximatios have bee obtaied; see, for example, Gema ad Yor (993) ad Rogers ad Shi (995). The closed-form solutios for the lookback optios are based o the assumptio that the maximum is take over the whole cotiuous path of the uderlyig. But, for most traded lookback optios, the maximum is ot based o daily highs of the uderlyig over the whole life of the optio. The maximum is rather based o daily closig prices over either the whole life of the optio or oly over a discrete umber of tradig days. For such cotract speci catios, the assumptio of cotiuous observatios seems as a poor approximatio. The same goes for the average optios. The approximatios obtaied for the optios depedig o the arithmetic average are also based o the assumptio that the average is sampled over cotiuous itervals, typically the whole life of the optio. However, all traded Asia optios deped o averages sampled over a discrete ad ofte a low umber of tradig days. The cosequece is that i practice oe has to resolve to Mote Carlo simulatios i order to price these types of cotracts. Fall 998
6 Jesper Adrease I this paper we suggest a simple ad computatioally e ciet alterative to Mote Carlo simulatios for four types of path-depedet optios: the Asia optio, the average strike optio, the lookback optio with xed strike, ad the lookback optio with oatig strike. The idea is to make use of chage of umeraire techiques to obtai the optio prices as fuctios of time ad a oe-dimesioal Markovia state variable oly. The techique has previously bee applied to the pricig of lookback optios with oatig strike by Babbs (99) ad Wilmott, Dewye, ad Howiso (993), but, as idicated, this paper exteds the methodology to the pricig of three other types of path-depedet optio. Owig to the discrete observatios, the state variables ivolved here exhibit jumps at the observatio poits with probability. However, i betwee two observatio poits the state variable evolves cotiuously, so it is possible to describe the optio price as the solutio to a stadard partial di eretial equatio (PDE) i such a regio. Lettig the rst PDE geerate the termial boudary coditio of the secod, ad so forth, we obtai a sequece of PDEs that ca be solved umerically by ite-di erece techiques. I the paper we employ Crak±Nicolso schemes for the umerical solutio of the pricig problems. We could i fact also set up biomial or triomial trees for the umerical solutio, but we choose ot to for two reasos. First, the ostadard dyamics of the state variable yields problems with the stability of such trees. That is, oe has to take ureasoable small time steps i order to isure the stability of the umerical solutio. Secod, the ature of the pricig problems are similar to barrier optio pricig problems. Trees give rise to odd±eve problems for such pricig problems; see, for example, Boyle ad Lau (994). As metioed, the xed strike optio-pricig problems for both the lookback ad the Asia optio ca be coverted ito barrier optio-pricig problems. This is rather surprisig give the ature of the origial pricig problems. However, the determiig state variables that we idetify here have a `barrier' type of behavior, i the sese that if they go through a certai level, typically i the moey, their dyamics become more tractable ad it is possible to derive the risk-adjusted expectatio of the termial payo i closed form. For the oatig strike optios treated i this paper, it is also possible to apply the PDE techique to the pricig of optios with a America feature. We provide umerical examples that illustrate the speed ad the accuracy of our procedures. Our bechmark is Mote Carlo simulatios with a large umber of samples combied with a cotrol variate techique. I most cases, the ite-di erece solutios get withi pey accuracy compared to the Mote Carlo solutios i less tha oe secod of CPU time. Fially, we show how the techique ca be applied to the case whe the uderlyig exhibits discotiuous dyamics. Our model is i this case a `risk-eutralized' versio of the Merto (976) model, where the jumps are triggered by a Poisso process ad the jumps i retur are displaced logormal distributed. I this case, the sequece of PDEs is replaced by a sequece of partial itegrodi eretial equatios (PIDEs) that also ca be solved by ite-di erece techiques. The paper is orgaized as follows. The secod sectio of the paper shortly describes the modellig framework ad the mai trick applied i this paper: the chage of Volume /Number
The pricig of discretely sampled Asia ad lookback optios 7 umeraire for the martigale measure. We the have a sectio for each of the optios cosidered here, i respective order these are: the Asia ( xed) strike optio, the average strike optio, the xed strike lookback, ad the oatig strike lookback optio. Each sectio cotais umerical examples of the accuracy ad the speed of our solutio procedure. The al sectio of the paper shows how our techique also ca be applied to ocotiuous dyamics of the uderlyig stock.. THE MODEL AND CHANGE OF NUMERAIRE For simplicity we start by cosiderig the stadard Black±Scholes ecoomy with two assets: a divided payig stock ad a moey-market accout. We will later exted the model to cover the case whe the uderlyig exhibits discotiuous dyamics. We assume the existece of a equivalet martigale measure Q uder which all discouted security prices (icludig accumulated divideds) are martigales. This assumptio implies absece of arbitrage. Uder Q; the stock is assumed to evolve accordig to the stochastic di eretial equatio ds t ˆ r q dt dw t ; S t where r is the costat cotiuously compouded iterest rate, q is the costat cotiuous divided yield, is the istataeous volatility of the stock retur, ad W is a stadard Q Browia motio. If oe cosiders the pricig of currecy or commodity optios, q deotes the foreig iterest rate ad mius the proportioal cost-of-carry, respectively. The moey-market accout evolves accordig to db t B t ˆ r dt; B 0 ˆ: Suppose that a security promises a paymet of $H at time T, where H is a radom variable that ca be represeted by some well-behaved fuctioal take o the stock price up to time T. The the fair price at time t of this claim ca be represeted as F t ˆE t e r T t H ; where E t Š deotes expectatio take uder the measure Q coditioal o the iformatio at time t. Oe might also solve the security valuatio problem by applyig a chage of umeraire resultig i the alterative valuatio equatio F t ˆS t E 0 t e q T t H ; 3 S T Fall 998
8 Jesper Adrease where E 0 t Š deotes coditioal expectatio uder Q0, which is de ed by dq 0 ˆ S T dq S t e r q T t 4 o t; TŠ. By the Girsaov theorem it follows that, uder Q 0, ad so W 0 t ˆW t t; ds t S t ˆ r q dt dw 0 t : 5 Whe H depeds o the whole path of the uderlyig up to the termial date T, we should i priciple keep track of the whole path of the uderlyig up to curret time t, i order to calculate the expectatio i the valuatio equatio () or the expectatio i the alterative valuatio equatio (3). However, if we are able to come up with a Markov process x, with evolutio so that dx t ˆ t; x t dt v t; x t dw 0 t ; H S T ˆ x T for some fuctio, the it is ot eccesary to keep track of the whole path of the uderlyig. Because of the Markov property of x, the expectatio i (3) ca be evaluated by keepig track oly of the curret value of x. Hece, the de ated optio price f F=S will be a fuctio of t; x t oly ad f ca be represeted as the solutio to the oedimesioal PDE qf ˆ @f @t @f @x v @ f @x ; subject to the termial boudary coditio f T; x ˆ x. The PDE ca be solved umerically by ite-di erece techiques, which, as we will demostrate, is much faster tha solvig the expectatio by Mote Carlo methods. We will ow show that such a Markov represetatio is ideed possible for the Asia optios with xed ad oatig strikes ad for the lookback optios with xed ad oatig strikes. 3. THE ASIAN OPTION WITH FIXED STRIKE Let ad de e 0 ˆ t 0 6 t < < t 6 t ˆ T; A t ˆ X S t i 6i6 : t i6t m t ˆsupf 6 i 6 : t i 6 tg: Volume /Number
The pricig of discretely sampled Asia ad lookback optios 9 The Asia optio with xed strike promises the holder the time-t paymet A T K ; where K is some xed amoutðthe strike price. The object ow is to evaluate the time-t fair price of the optio, F t ˆS t E 0 t e q T t A T K : 6 S T Let x t ˆ A t K : 7 S t Whe we hit a observatio poit t i, the process x will jump by =. To see this, ote that, for 6 i 6, wehave x t i ˆ ˆ ˆ P i jˆ S t j K S t i P i jˆ S t j S t i K S t i P i jˆ S t j K S t i ˆ x t i : At times betwee observatios, the process x evolves cotiuously, because oly the deomiator i (7) chages as time evolves. Hece, usig Itoà 's lemma, we have 3 dx t ˆ r q x t dt x t dw 0 t X tˆti iˆ ˆ r q x t dt x t dw 0 t dm t : We see that, uder Q 0, the evolutio of x depeds oly o x itself, so x is a Markov process uder Q 0. This implies that evaluatio of the expectatio i (6) requires kowledge oly of the curret positio of x, ad we may write F t =S t ˆf t; x t. Further, we observe that if x t > 0 the x u > 0 for all u > t with probability. This implies that, for all x t > 0, f t; x t ˆ E 0 e q T t x T x t ˆ E 0 e q T t x T x t ˆ e r T t x t g t; x t : The third equality is give i the Appedix. We de e z ˆ max 0; z. X r T t e i q t i t i : t<t i 6t We de e z t ˆ lim!0 z t jj ad z t ˆ lim!0 z t jj. 3 We let A deote the idicator fuctio o the set A. 8 Fall 998
0 Jesper Adrease We ote that if x t < 0 the the process x ca oly pass through the level x ˆ 0at the future samplig poits ft i g m t <i6. Suppose x passes the level x ˆ 0 at some poit t i (i 6 ). We the have f t i ; x t i ˆ g t i ; x t i ˆ g t i ; x t i = : I the case that the level x ˆ 0 is ot passed for ay ft i g iˆ;...;, the holder of optio will receive othig. To formalize this, let us de e to be the rst passage time amog the observatio poits ft i g iˆ;...; of the level x ˆ 0 for the process x, i.e. We ca the write ˆ if ft i g iˆ;...; : x t i > 0 ˆ if ft i g iˆ;...; : x t i > = : f t; x t ˆ E 0 e r t g ;x x t 9 ˆ E 0 e r t g ;x = x t : 0 Solvig for f is the a rst passage time problem for a Markovia process. This demostrates the parallel to a `up-ad-i' barrier optio: the stock price de ated optio price, f, is the risk-adjusted expectatio of the discouted value of a payo at the rst passage time to a certai level. This problem ca be formulated as a PDE problem that ca be solved umerically, as we will show i the sectio below. Before we do so, we ote that F t KˆK 0 ˆ S t f t; A t K0 S t which gives us the possibility of solvig for more tha oe optio price oce the fuctio f is ideti ed. It should be metioed that this techique does ot eable us to solve for the America style xed strike Asia optio price. The reaso is that the state variable x does ot supply su ciet iformatio to determie the (de ated) itrisic value of the Asia optio at ay time poit before maturity of the optio. This follows from the de itio of x i equatio (7). Cosequetly, to solve the America style problem, we would have to additioally keep track of the level of the uderlyig stock, i.e. we would be back to a two-dimesioal formulatio of the xed strike Asia optio pricig problem. However, as we shall see i the ext sectio, if the strike is oatig, the America exercise problem ca be hadled usig the umeraire techique. 3. Numerical Solutio ad Numerical Results Oe ca ow set up a system of triomial or biomial trees that discretize the radom evolutio of x. Except from the observatio poits ft i g iˆ;...;, the state variable x evolves as a geometric Browia motio, so a Cox, Ross, ad Rubistei (979) biomial tree applied to the x process could be used o the regios betwee observatio poits. However, the jumps at the observatio poits are costat ad additive. This works poorly with a stadard biomial tree, which is ormally speci ed to be log-liear. So we choose ot to use this approach. ; Volume /Number
The pricig of discretely sampled Asia ad lookback optios As yet aother alterative, oe might use the fact that x t i coditioal o x t i is logormal uder Q 0. Discretizig the state space i the x dimesio will therefore make it possible to solve for the optio prices usig umerical itegratio at each poit t i ad recursig backwards to curret time. This might ot be more computatioally e ciet tha ite-di erece solutio of PDEs, sice (implicit) ite-di erece approximatio as the hardest part ivolves umerical iversio of liear tridiagoal systems ad this ca be performed i liear computer time. We therefore choose to cocetrate o the ite-di erece techiques. We ow eed to idetify the PDE system for the umerical solutio of the Asia optio price or, rather, f. This ca be doe directly i t; x. We prefer, though, to get rid of the discotiuous dyamics by itroducig We ow have y t ˆx t m t : dy t ˆ r q y t m t y t m t dw 0 t : Sice e qt f t is a Q 0 martigale, Itoà 's lemma ad the martigale represetatio theorem together imply that f is the solutio to the PDE qf ˆ @f r q y m t @f @t @y y m t @ f @y o f t; y : t i < t < t i ; y < i =; i ˆ ;;g, subject to the boudary coditios f t i ; y ˆ f t i ; y if y < i=, 3 g t i ; y i= if i = > y > i=, ad f t; y ˆ0 if y < 4 for t > t. Betwee two observatio poits, i.e. o each of the itervals t i ; t i Š, the PDE () ca be solved umerically usig a Crak±Nicolso scheme. The idea is to approximate the di eretials i () by cetral di ereces. To do this, we rewrite the PDE () as 0 ˆ q @ @t y @ @ v y @x @x f ; 5 where y ˆ r q y m t ad v y ˆ y m t : We suppress the otioal depedece of ad v o t, because m t is costat o each subiterval t i ; t i Š. If we approximate the di eretials i (5) by cetral di ereces i the poit t t; x, we get the (Crak±Nicolso) partial di erece equatio q t y y v y yy f t; y ˆ q t y y v y yy f t t; y ; 6 Fall 998
Jesper Adrease where y ad yy are di erece operators de ed by y h y ˆ h y y h y y y Š; yy h y ˆ h y y h y h y y y Š: For the iterval t i ; t i Š, we limit our state space to the discrete grid sk ; y j kˆ0;...;k; lˆ0;...;l ; with s k ˆ t i K k K t k i K ad y L l l ˆ y mi L y l max L : Here we have t ˆ t i t i =K ad y ˆ y max y mi =L. The upper boud of the grid is dictated to be the upper limit of the domai of f,soy max ˆ i =; the lower boud has to satisfy y mi <. Typically y mi ˆ ca be chose for maturities less tha oe year. By supplyig the arti cial boudary coditios yy f ˆ 0 at the boudaries y mi ad y max of the grid, we ca ow state the partial di erece equatio (6) as a sequece of matrix equatios Af s k ˆBf s k ; 7 where f is the vector f s k ˆ f s k ; y 0 ;...; f s k ; y L Š T ad A ad B are L -dimesioal tridiagoal matrices with the lth rows give by A l ˆ B l ˆ 0;...; 0; y l y v y l y ; q t v y l y ; y l y v y l y 0;...; 0; y l y v y l y ; q t v y l y ; y l for l ˆ ;...; L, ad q A 0 ˆ t A L ˆ B 0 ˆ B L ˆ ; 0; 0;...; 0 ; 0;...; 0; y L y ; q t y L ; y q t ; 0; 0;...; 0 ; y v y l y 0;...; 0; y L y ; q t y L : y ; 0;...; 0 ; ; 0;...; 0 ; Whe solvig (), subject to (3) ad (4), umerically, we start at time t. By the boudary coditios (3) ad (4), we get the values of f t. We the umerically solve back to time t by recursively solvig the matrix system (7). At time t, the umerical solutio together with the fuctio g t ; acts as termial boudary coditio for the umerical solutio o t ; t. We ow cotiue like this back to curret time, where we get the curret value of f ad thereby the optio price. Volume /Number
The pricig of discretely sampled Asia ad lookback optios 3 Note that the state space of the process y chages as time progresses. At each time, we have y 6 i = whe t i < t < t i. But the state space is costat for all t betwee two observatio poits, ad ruig backwards i time, the ew added regios have boudary coditios speci ed by the kow fuctio g ;. The fact that the matrices A ad B are tridiagoal meas that the computatioal e ort of solvig equatio (7) is of order O L. This i tur implies that the computatioal burde of the total scheme is of order O K L. If we choose K ˆ O L, this is similar to the computatioal complexity of a biomial tree as the oe of Cox, Ross, ad Rubistei (979). The solutio techique applied here is a Crak±Nicolso scheme. We refer the reader to Mitchell ad Gri ths (980) for a detailed descriptio of the properties of the Crak±Nicolso scheme, but amog its ice properties are that it is uiformly stable ad that its local precisio is of order t y, which is maximal for stadard ite-di erece schemes for partial di eretial equatios of the parabolic type. Table compares optio prices for various strikes geerated by the ite-di erece algorithm with di eret grid sizes to optio prices obtaied by Mote Carlo simulatios. For referece we also report the CPU times for geeratig the optio prices usig the two di eret techiques. We see that the ite-di erece algorithm for this optio is surprisigly accurate, ad that the prices chage very little as the grid size is chaged. The maximum relative error compared with the Mote Carlo procedure is approximately 0:4%. Here it is importat to ote that the Mote Carlo price is ot a absolute gure. It might vary a little from simulatio to simulatio; as metioed i the headig to Table, the stadard deviatio of the optio prices is approximately 3 0 3. For the reported CPU times, here ad i the followig, it should be oted that all programmig was doe i C ad the hardware used was a Hewlett-Packard 9000 Uix system. TABLE. The parameters are: r ˆ 0:05, q ˆ 0:0, ˆ 0:; T ˆ :0, t ˆ 0:0, ˆ 0, S 0 ˆ00:0, t i ˆ 0:i. MC refers to Mote Carlo solutio ad FD refers to fiite-differece solutio. The differet I's refer to the umber of time steps. We used I=0 umber of steps per jump size = i the y directio. The Mote Carlo prices are based o 0 5 simulatios with a cotrol variate techique. The stadard deviatio of the Mote Carlo estimated prices is estimated as 3 0 3. Reported CPU times are for all 9 strikes. Asia optio prices K MC FD I ˆ 500 FD I ˆ 00 FD I ˆ 50 90.0.98.99.99.98 9.5.05.05.05.05 95.0 9.7 9.7 9.7 9.7 97.5 7.67 7.66 7.66 7.66 00.0 6.4 6.3 6.3 6.3 0.5 5.0 5.00 5.00 5.00 05.0 3.96 3.95 3.95 3.95 07.5 3.08 3.07 3.07 3.07 0.0.36.35.35.36 CPU 46.0 s 0.65 s 0.06 s 0.04 s Fall 998
4 Jesper Adrease Let us brie y describe the Mote Carlo techique. We apply a cotrol variate techique to our Mote Carlo simulatios i order to decrease the umber of simulatios eeded. That is, we simulate a collectio of paths S t ;...; S t!! uder Q, ad cosider the regressio equatio e r T t A T K! X ˆ E t e r T t A T K a i S ti S 0 e r q t t i! ; where a ;...; a are costats. Note that the regressors uder the sum have zero Q mea. We ru a ordiary least-squares regressio o this ad simultaeously estimate the coe ciets fa i g ad the Q mea of the payo, i.e. the fair price of the optio. This also gives us a estimate of the stadard deviatio of the estimate of the parameters, i.e. a estimate of the stadard deviatio of the Mote Carlo optio prices. The properties of the procedure are described i detail i Davidso ad Mackio (993). Oe ca also iclude di eret powers of the stock price mius its momets as cotrol variates. We choose ot to, because the stock prices aloe give su ciet precisio for our purpose ad because the presece of additioal parameters to be estimated makes the Mote Carlo procedure more computatioally demadig. iˆ 4. THE AVERAGE STRIKE OPTION With the de itios i the previous sectio, the termial time-t payo of the average strike (put) optio ca be writte as A T S T ; where is a costat. Usig the valuatio equatio (3), we d that the time-t price of the optio is give by For t > t, de e x t by F t ˆS t E 0 t e q T t A T S T : x t ˆA t S t : Applyig the same argumet as i the previous sectio, we get, for t > t, dx t ˆ r q x t dt x t dw 0 t dm t ; x t ˆ; This is a Markov process with domai o x > 0. The object is ow to evaluate the iitialvalue problem F t =S t f t; x t ˆ E e 0 q T t x T x t : 9 8 Volume /Number
The pricig of discretely sampled Asia ad lookback optios 5 Owig to the Markovia property of x, this ca be doe by solvig a sequece of PDEs, as we shall more formally describe i the followig sectio. Suppose that we wat to evaluate a average strike optio with a America feature, i.e. the optio might be exercised at some time t i the iterval t ; TŠ with resultig payout m t A t S t : Fidig the fair price of such a cotract is a stoppig time problem, i the sese that we are supposed to d the exercise time that maximizes the value of the optio. To formalize this, let T be the set of stoppig times o the iterval t ; TŠ with respect to the ltratio geerated by the stock price. The the average strike optio with the America feature has the fair value F t ˆsup E t e r t T ˆ S t sup E 0 t T m A S e q t ˆ S t sup E 0 e q t T m x A m S x t : 0 This de es a Markovia stoppig time problem for f ˆ F=S that ca be treated i a free-boudary formulatio, as we shall illustrate i the followig sectio. Both i the America ad the Europea style case we have, for t > t, F t ˆS t f t; x t : Applyig the alterative valuatio equatio (3) to this quatity, we get, for t, F t ˆS t e q t t f t ; : It should be metioed that the above results for the average strike optio for the cotiuous observatio case have previously bee obtaied though PDE techiques by Igersoll (987) ad Wilmott, Dewye, ad Howiso (993). 4. Numerical Solutio ad Results As for the xed strike case, we itroduce ad we have y t ˆx t m t ; dy t ˆ r q y t m t dt y t m t dw 0 t : O t i < t < t i, with i >, e qt f is a Q 0 martigale ad therefore the solutio to qf ˆ @f @t r q y m t @f @y y m t @ f @y Fall 998
6 Jesper Adrease o y > i, subject to the boudary coditios f t i ; y ˆf t i ; y ; f t ; y ˆf T; y ˆ y : The America style average strike optio ca be hadled by addig the free-boudary coditio f t; y > m t y : 3 We apply a liear grid to this problem, supply the same `arti cial' boudary coditios as for the xed strike, ad agai use the Crak±Nicolso scheme as i (6). This meas that () ca be solved as a sequece of tridiagoal matrix equatios as i (7). Table gives prices geerated by the ite-di erece algorithm ad compares these quatities to umbers geerated by Mote Carlo simulatios. As for the xed strike Asia, the precisio of the ite-di erece solutio is remarkable. Eve though the grid size chages by a factor 0, the relative price chages are less tha 0:6% for all strikes. The maximum relative deviatio to the Mote Carlo solutio is about.%. But it should agai be emphasized that the Mote Carlo solutio eed ot be more accurate tha the ite-di erece solutios ad serves oly as a bechmark. The loger computer times here compared with the Asia optios is due to the fact that here a ite-di erece algorithm has to be ru for each, whereas for the Asia optios we eed oly solve oe ite-di erece grid to obtai the prices for all strikes. TABLE. The parameters are: r ˆ 0:05, q ˆ 0:0, ˆ 0:, T ˆ :0, t ˆ 0:0, ˆ 0, S 0 ˆ00:0, t i ˆ 0:i. MC refers to Mote Carlo solutio, ad FD refers to fiite-differece solutio. The differet I's refer to the umber of time steps. We used I=0 umber of steps per jump size = i the y directio. The Mote Carlo prices are based o 0 5 simulatios with a cotrol variate techique. The stadard error o the estimated Mote Carlo optio prices is approximately 3:0 0 3. Reported CPU times are for all 9 strikes. Average strike optio prices MC FD I ˆ 500 FD I ˆ 00 FD I ˆ 50 0.900 8.98 8.98 8.98 8.99 0.95 7.8 7.8 7.7 7.8 0.950 5.6 5.60 5.60 5.58 0.975 4.8 4.7 4.6 4.6.000 3.8 3.8 3.8 3.8.05.3.3.30.30.050.65.64.64.63.075.4.4.4.4.00 0.78 0.77 0.77 0.77 CPU 48.0 s.94 s 0. s 0.05 s Volume /Number
The pricig of discretely sampled Asia ad lookback optios 7 5. THE LOOKBACK OPTION WITH FIXED STRIKE For t > 0, de e S t ˆ sup S t i ; 6i6m t with the covetio S t ˆ0 for t 6 t. The xed strike lookback optio promises the time-t paymet : S t K The solutio of this pricig problem is a two-step procedure. First, we solve the optio price at time t whe S t > K. We the solve for the case S t < K by observig that i this case the optio might be viewed as a rst passage problem of S to the level K where the reward is equal to the value of the optio at S t > K. Suppose S t > K. We the have F t ˆE t e r T t S T K ˆ E t e r T t S T K De e for t > t. For 6 i 6, wehave ˆ S t E 0 t e q T t S T x t ˆS t S t e r T t K: S T x t i ˆ if x t i 6, x t i if x t i >. Elsewhere the evolutio of x is cotiuous, ad for t > t we have dx t ˆ r q x t dt x t dw 0 t x t dm t ; x t ˆ: So x is a Markov process with domai o x > 0. De e f t ˆE 0 q T t S T t e S T ˆ E 0 t e q T t x T 4 5 ˆ E 0 e q T t x T x t ; 6 where the last equality follows from the Markovia property of x. The quatity f ca be writte as f t ˆf t; x t ad ca be foud by umerically solvig the PDE related to the iitial-value problem (6). We will show how this is doe i the sectio below. Fall 998
8 Jesper Adrease This establishes the optio price at t > t for S t > K explicitly as F t ˆS t f t; x t e r T t K: 7 Suppose we are sittig at time t > t with S t < K. The rst time t i > t i 6, with S above K, we get a reward of F t i ˆS t i f t i ; x t i e r T t i K ˆ S t i f t i ; e r T t i K: 8 The secod equality is valid because, i the above, t i is the rst time S t goes above K. If a level of K or above is ot hit at ay of the samplig times t i i ˆ ;...;, the holder of the optio receives othig. Equatio (8) implies that, for t > 0 with S t < K, we may write the optio price as F t ˆE t e r t S f ; e r T K 6t where ˆ E e r t S f ; e r T K 6t S t ; 9 ˆ if ft i : S t i > Kg; iˆ;...; with the covetio if? ˆ. This shows the parallel to a up-ad-i barrier optio. Whe f is kow, F ca be foud by umerically solvig the rst passage time problem (9). We illustrate how this is doe i the sectio below. Fidig the optio price is therefore a two-step procedure. First we solve for f f u; x g for all u; x with u > max t; t. This is doe by umerically solvig a iitial-value problem from T dow to t. IfS t > K, the the optio price is give by (7). Otherwise we keep f f t i ; g 6i6 : ti>t ad solve the rst passage time problem (9). 5. Numerical Solutio ad Results The accuracy of the umerical solutio of partial di eretial equatios is geerally impoved if the variables are trasformed so that the di usio term is costat. We therefore perform a log trasformatio. Let y ˆ l x ad cosider 4 dy t ˆ r q dt dw 0 t y t dm t : Sice e qt f t is a Q 0 martigale ad y is Markovia, the solutio to the iitial-value problem (6) ca be foud as the solutio to the followig system of PDEs. O t i < t < t i i >, f solves qf ˆ @f @t r q @f @y @ f @y ; 30 4 The otatio is de ed by z ˆ mi 0; z :. Volume /Number
The pricig of discretely sampled Asia ad lookback optios 9 subject to the boudary coditios ( f t i ; y ˆ f t 9 i ; 0 if y < 0, >= f t i ; y if y > 0, >; f t ; y ˆf T; y ˆe y : 3 Now rede e y ad let y t ˆl S t =K. The rst passage time problem (9) ca be hadled by otig that, for S t < K, g F=K is the solutio to rg ˆ @g @t r q @g @y @ g @y 3 o f t; y : t i < t < t i i ˆ ;...; ; y < 0 g, subject to the boudary coditios g t i ; y ˆ g t 9 i ; y if y < 0, >= e y f t i ; 0 e r T t i if y > 0, >; g t ; y ˆ0 for y < 0: 33 The f t; i (33) should be iterpreted as fuctio of y, as i (30). This meas that we ca treat f ad g i the same grid ad simultaeously solve for f ad g; at each time step, i that respective order. At curret time t, optios of di eret strikes are geerated by F t ˆKg t; S t =K. As for the Asia optios, we apply the Crak±Nicolso scheme (6) to the umerical solutio of this problem, where we supply the `arti cial' boudary coditios 5 @ f @f @y @y ˆ @ g @g @y @y ˆ 0 at the upper ad the lower boud of the grid. We arrage the grid so that the level y ˆ 0 is o the grid ad the poits ft i g are amog the time poits of the grid. Table 3 shows optio prices geerated by ite di erece ad compares these with Mote Carlo solutios. Comparig the ite-di erece solutio o the 500 500 grid with the Mote Carlo solutio shows a maximal relative error of approximately 0:%, which is clearly withi ay reasoable demads for precisio. But the ite-di erece solutios for the two smaller grids do ot show su ciet precisio. This must be attributed to the two-step procedure ivolved here; umerical errors might be accumulated i the two steps. The coclusio is that this type of optio requires a er mesh tha the optios cosidered i the previous sectios. 6. THE LOOKBACK OPTION WITH FLOATING STRIKE With the de itios of the previous sectios the time-t payo of a oatig strike 5 These coditios are equivalet to the coditio @ f =@x ˆ @ F=@S ˆ 0. Fall 998
0 Jesper Adrease TABLE 3. The parameters are: r ˆ 0:05, q ˆ 0:0, ˆ 0:, T ˆ :0, t ˆ 0:0, ˆ 0, S 0 ˆ00:0, t i ˆ 0:i. MC refers to Mote Carlo solutio, ad FD refers to fiite-differece solutio. The differet I's refer to the umber of time steps ad also to the umber of steps i the y directio. The Mote Carlo prices are based o 0 5 simulatios with a cotrol variate techique. The stadard deviatios of the Mote Carlo prices are approximately 3:0 0 3. Reported CPU times are for all 9 strikes. Fixed strike lookback optio prices K MC FD I ˆ 500 FD I ˆ 00 FD I ˆ 50 90.0 4.4 4.39 4.7 3.8 9.5.07.06.93.47 95.0 9.78 9.77 9.64 9.8 97.5 7.57 7.56 7.43 6.96 00.0 5.48 5.47 5.34 4.87 0.5 3.53 3.5 3.39.95 05.0.75.74.6. 07.5 0.4 0.4 0.03 9.67 0.0 8.70 8.7 8.6 8.30 CPU 46.0 s 0.68 s 0.06 s 0.03 s lookback optio ca be expressed as : S T S T Of the optios cosidered i this paper, this is the easiest optio to evaluate umerically. For t > t, the fair price is give by S T F t ˆE 0 t e q T t S T ˆ E 0 t e q T t x T x t ; where x is de ed as i (5). This is also observed by Babbs (99), who treats the America style case i ways similar to what is outlied below. However, Babbs uses a biomial tree for the umerical solutio. Wilmott, Dewye, ad Howiso (993) derive the result by maipulatio of the fudametal PDE. Lettig f ˆ F=S, f solves a Markovia iitial boudary problem equivalet to (6). I the sectio below we supply the PDE with boudary coditios associated with this problem. If we wat to cosider a oatig strike lookback optio with a America feature, ote that the fair price of such a cotract ca be represeted as F t ˆsup E t e r t S S T ˆ S t sup E 0 t T e q t S S ˆ S t sup E 0 e q t x x t ; T 34 Volume /Number
The pricig of discretely sampled Asia ad lookback optios where T is the set of stoppig times o t ; TŠ adapted to the ltratio geerated by S. As i (0), this is a Markovia stoppig time problem that ca be reformulated as a freeboudary problem for f ˆ F=S. We formulate this as a PDE problem i the sectio below. I both the Europea ad the America style oatig lookback optio, we have ( F t ˆ S t e q t t f t ; if t < t, S t f t; x t if t > t. 6. Numerical Solutio ad Results As for the xed strike lookback, we choose to log-trasform the state variable ad de e y ˆ l x. We ow d that f solves the PDE qf ˆ @f @t r q @f @y @ f @y whe t i < t < t i, with i >, subject to the boudary coditios ( f t i ; y ˆ f t 9 i ; 0 y < 0, >= f t i ; y y > 0, >; f t ; y ˆf T; y ˆ e y : 35 If we cosider a America style optio, we have to add the free-boudary coditio f t; y > e y : 36 We apply the same `arti cial' boudary coditios as i the previous sectio ad agai we use the Crak±Nicolso scheme for the umerical solutio. TABLE 4. The parameters are: r ˆ 0:05, q ˆ 0:0, ˆ 0:, T ˆ :0, t ˆ 0:0, ˆ 0, S 0 ˆ00:0, t i ˆ 0:i. MC refers to Mote Carlo solutio, ad FD refers to fiite-differece solutio. The differet I's refer to the umber of time steps ad also to the umber of steps i the y directio. The Mote Carlo prices are based o 0 5 simulatios with a cotrol variate techique. The stadard deviatio of the Mote Carlo optio prices is approximately 3 0 3. Reported CPU times are for all 9 strikes. Floatig strike lookback optio prices MC FD I ˆ 500 FD I ˆ 00 FD I ˆ 50.000 0.0 0.00 9.96 9.86.05 8.7 8.6 8.3 8.4.050 6.77 6.76 6.74 6.68.075 5.5 5.50 5.47 5.43.00 4.46 4.45 4.4 4.39.5 3.59 3.58 3.56 3.53.50.88.87.85.8.75.30.9.8.5.00.83.8.8.79 CPU 46.0s.43 s 0.4 s 0.06 s Fall 998
Jesper Adrease Table 4 gives optio prices geerated usig the ite-di erece solutio ad compares these with optio prices foud by Mote Carlo simulatios. We choose oly to show prices for values of greater tha. This is because all optios with 6 are all `i the moey' with probability, because of the samplig of the maximum that we use here (we have t ˆ T). This meas that, for all <, the optio cotract has a value that equals the value of the cotract with ˆ plus S 0 e qt. Comparig the ite-di erece solutios with the Mote Carlo solutios we d that the maximal relative error is about 0:5% for the 500 500 grid, % for the 00 00 grid, ad approximately % for the 50 50 grid. This is acceptable, but the example shows that oe has to use a higher degree of precisio for the lookback tha for the Asia optios. 7. DISCONTINUOUS RETURNS OF THE UNDERLYING I this sectio we exted the model of the stock price to allow for discotiuous dyamics ad show that the techique used i the previous sectios ca also be applied to this type of stock price behavior. Uder Q, the stock is assumed to evolve accordig to the stochastic di eretial equatio ds t ˆ r q k dt dw t I t dn t ; S t 37 where r, q,, W are de ed as i Sectio, N is a Poisso process with itesity ad fi t g t>0 is a sequece of idepedet ad idetically distributed radom variables with distributio give by l I t Q N ; ad Q mea k ˆ E I t ˆ e : The processes W, I, N are assumed to be idepedet. We ote that the ecoomy is ow icomplete, i.e. there exists o perfect hedgig strategy i the stock ad the bod that replicates the payo of derivatives, ad the measure Q is ouique. However, this does ot i uece derivative pricig oce a martigale measure Q is xed as above by simply assumig the Q dyamics for the stock give by (37). Of course, the relatio betwee the objective probability measure ad the martigale measure matters if we are cosiderig portfolio ad hedgig decisios, but that is beyod the scope of this paper, so we will avoid this discussio for the remaider of the paper. De ig Q 0 as i (4), the Girsaov theorem implies that 6 ds t S t ˆ r q k dt dw 0 t I 0 t dn 0 t ; 6 Note that the Q 0 measure is uiquely related to Q, so that oce Q is xed so is Q 0. Volume /Number
The pricig of discretely sampled Asia ad lookback optios 3 where W 0 is a Q 0 Browia motio, as give i (5), fi 0 t g t>0 is a sequece of idepedet idetically distributed radom variables with distributio give by l I 0 t Q0 N ; ; ad N 0 is a Q 0 Poisso process with itesity 0 ˆ k ˆe : The processes W 0, I 0, N 0 are also idepedet uder Q 0. I this type of ecoomy, the valuatio equatios () ad (3) are still valid. 7. Path-Depedet Optios uder Jumps The tricks applied for the pricig of the optios that we cosidered i the previous sectios aturally exted to the case whe the uderlyig exhibits jumps. To see this, let us cosider the optios oe by oe. The Asia optio with xed strike has the value give by (6), ad if we de e x as i (6) we ow have dx t ˆ r q k x t dt x t dw 0 I 0 t t x t I 0 t dn0 t dm t : 38 This is clearly a Markov process with the property that if x t > 0 the x u > 0 for all u > t with probability. This implies that if x t > 0 the the de ated optio price is give by F t =S t f t ˆE 0 t x T ˆ g t; x t ; 39 where g ; is de ed as i (8). The last equality is show i the Appedix. Now, if x t < 0, the process x ca still oly pass the level x ˆ 0 at the poits ft i g iˆ;...;, which agai implies that, for x t < 0, we may write the de ated price f of the optio as the solutio to a rst passage time problem, as we did i (9). We will retur to how this is solved umerically i the sectio below. Oce f is obtaied, the optio price is give by F t ˆS t f t; x t : If, for the average strike optio cosidered i Sectio 4, we de e x as i (8), the dx t ˆ r q k x t dt x t dw 0 I 0 t t x t I 0 t dn0 t dm t ; x t ˆ; for t > t. This is a Markov process with domai o x > 0. The solutio to the de ated optio price is ow give as the solutio to the Markovia iitial-value problem f t; x t ˆ E e 0 q T t x T x t : We show how to hadle this umerically i the followig sectio. For the average strike optio with a America feature, we obtai the same type of Markovia stoppig time Fall 998
4 Jesper Adrease problem as i (0). This ca be give a free-boudary formulatio that we will cosider i the ext sectio. Give f,wehave ( F t ˆ S t f t; x t if t > t, S t e q t t f t; if t < t. The lookback optio with xed strike ca also be hadled by the techique applied i Sectio 5. The key observatios are the same. We rst ote that, for S t > K, the optio price ca be writte as i (7). De ig x ˆ S=S, Itoà 's lemma implies that, for t > t, 9 dx t ˆ r q x t dt x t dw 0 t I 0 t x t I 0 t dn0 t x t >= dm t ; 40 x t ˆ: >; This is clearly a Markov process with domai o x > 0. So f t E 0 t e q T t x T is the solutio to a Markov iitial-value problem. O the other had, if S t < K; we ca write the optio price as the solutio to a Markovia rst passage time problem as i (9), because S is still a Markovia process. So, oce f is obtaied for the poits ft i g iˆ;...;, the problem ca be hadled by umerically solvig the rst passage time problem. We will retur below to how this is doe. To summarize, agai we have a twostep procedure: if S t > K the the optio price is give by F t ˆS t f t; x t Ke r T t ; 4 otherwise the optio price is give by the solutio to a Markovia rst passage time problem like (9). Cosider ow the oatig strike lookback optio. We have see that if x ˆ S=S the x has the Markovia evolutio (40). So the Europea style optio price is give as the solutio to the Markovia iitial-value problem F t =S t f t; x t ˆ E e 0 q T t x T x t for t > t ; ad F t ˆS t e q t t f t ; for t < t : We will retur to how this ca be hadled umerically i the sectio below. The America style optio is hadled as i (34). That is, we have to solve a Markovia optimal stoppig time problem. I the followig sectio, we do this by reformulatig the problem as a free-boudary problem. 7. Numerical Solutio uder Jumps The Markovia ature of the reformulated pricig problems that we have see i the previous sectio meas that the pricig ca be doe by solvig partial itegrodi eretial equatios (PIDEs). The term `itegro' is added because the PIDEs ot oly ivolve Volume /Number
The pricig of discretely sampled Asia ad lookback optios 5 partial derivatives but also itegrals, sice the processes cosidered here have discotiuities of radom sizes at radom times. The umerical solutio of such equatios ca still be obtaied o ite grids by applyig ite-di erece techiques, but we eed to supply additioal `arti cial' boudary coditios i order to make this machiery work. This is because the itegrals i the PIDEs typically iclude terms outside the boudaries of a reasoably sized grid. We will i the followig derive the PIDEs that eed to solved umerically ad supply our choices of `arti cial' boudary coditios. I the followig we will let y mi ad y max deote the lower ad upper boudaries, respectively. These quatities are i some cases depedet o the iterval t i ; t i that we are cosiderig, but for brevity we will igore this. With y t ˆx t m t = the PIDE aalog to the PDE () for the xed strike Asia optio ca be writte as q 0 f ˆ @f @t r q k 0 E 0 I 0 f y m t @ f @y y m t @ f @y t; y m t = I 0 m t ymi 6 y m t = I 0 m t 6y max h t; y : 4 The operator E 0 I Š is de ed for ay fuctio by E 0 I I 0 ˆ d; 0 43 where is the desity for I 0 uder Q 0, ˆp exp l : The fuctio h ; is i tur de ed as h t; y ˆ 0 E 0 I f t; y m t = 0 I 0 m t y m t = I 0 m t <y y m t = mi I 0 m t >y : max The PIDE (4) is to be solved, subject to the boudary coditios (3) ad (4), o the set t; y : ti < t < t i ; y < i =; i ˆ ;...; : Before we ca solve this umerically, we eed to make a reasoable approximatio for h ;. We set f t; y ˆ0; y < y mi : As we typically will set y max ˆ i = ad y caot cross through the level i = from below, we cosequetly get the very simple approximatio h t; y ˆ0: Substitutig this ito (4) ad usig the additioal `arti cial' boudary coditio @ f =@y ˆ 0 at the lower ad the upper bouds, we ca ow umerically solve for the Asia optio price usig the ite-di erece scheme described i Adrease ad Fall 998
6 Jesper Adrease Grueewald (996). Without a ectig stabililty, the speed of the procedure might be icreased by takig a explicit approximatio for the itegral ad a implicit approximatio for the partial derivatives. For the oatig strike optio we itroduce y t ˆx t m t as i Sectio 4, ad we obtai the PIDE aalog to the PDE (), q 0 f ˆ @f @t r q k y m t @ f @y y m t @ f @y 0 E 0 I f t; y m t 0 I 0 m t ymi < h t; y ; 44 y m t I 0 m t <ymax which is valid o t; y : ti < t < t i ; y > i ; i ˆ ;...; ad has to be solved subject to the boudary coditios () ad whe the optio is America style additioally subject to the free-boudary coditio (3). For y > y max, we set f t; y ˆE e 0 q T t y T y ˆ y m t e q T t X e r T t i q t i t e q T t i6 : t i >t i the Europea case, ad for the America style optio we let For y mi, we set f t; y ˆ y : m t f t; y ˆ0 i both cases. This results i the followig approximatios for h ;. For the Europea case, we get h t; y ˆ y m t e r T t 0 l y m t y max m t X l y m t = e r T t i q t i t e q T t i6 : t i >t Whe the optio is America, we have the approximatio y h t; y ˆ m t l y m t = y max m t = y max m t : l y m t = 0 y max m t = : Substitutig these equatios ito the PIDE (44), we ca ow umerically solve the average strike optio prices usig the algorithm described i Adrease ad Grueewald (996). Cosider the lookback optio with xed strike. With y ˆ l S=S, we d that f, Volume /Number
The pricig of discretely sampled Asia ad lookback optios 7 de ed as i (6), solves the PIDE q 0 f ˆ @f @t r q k @f @y @ f @y o the set E 0 I 0 f t; y l I 0 ymi <y l I 0 <y max h t; y 45 t; y : ti < t < t i ; i ˆ ;...; ; subject to the boudary coditios (3). Let ( f t; y ˆ f t; y mi if y < y mi ; e r T t y if y > y max : The last coditio is obtaied by takig the discouted coditioal Q 0 expectatio of x T as if there were o jumps i x at the observatio poits ft i g iˆ;...;. The fuctio h ; is the approximated by y h t; y ˆe r T t y ymax 0 ymi y f t; y mi : The PIDE (45) ca ow be solved umerically o a grid. This gives us the solutio for the optio price whe S t > K. If this is ot the case, we proceed by otig that, for S t < K with the de itios y ˆ l S=K ad g ˆ F=K, the PIDE equivalet to the PDE (3) is r g ˆ @g @t r q k @g @y @ g @y o E I g t; y l I ymi <y l I <y max h t; y 46 t; y : ti < t < t i ; i ˆ ;...; ; subject to the boudary coditios (33). The operator E I Š is de ed as i (43), with the modi catio that the Q 0 desity is ow replaced by the Q desity. By aalogy with the previous cases, the fuctio h ; is de ed as h t; y ˆE I g t; y l I y l I <ymi y l I >ymax : For y < y mi we set g t; y ˆ0; ad for y > y max we let g t; y t ˆ E t e r t t m t g t m t ; y t m t ˆ E t e r t t m t e y t m t f t m t ; e r T t ˆ e q t m t t y t f t m t ; e r T t : Fall 998
8 Jesper Adrease From this, we obtai h t; y ˆ 0 e q t m t y t y ymax y e r T t ymax : Usig this we ca umerically solve the PIDE (46) with the ite-di erece machiery. Fially, let us cosider the lookback optio with oatig strike. De ig y ˆ l S=S ad f ˆ F=S, we d that f is the solutio to (45) o t; y : ti < t < t i ; i ˆ ;...; ; subject to the boudary coditios (35) ad if the optio is America style also subject to the free-boudary coditio (36). What is left is to supply a approximatio of h ; for this optio. For y < y mi, we set f t; y ˆf t; y mi for both the America ad Europea style cases. For y > y max ; we set f t; y ˆe r T t y e q T t for the Europea case. This correspods to the discouted Q 0 expected termial payo if we igore the fact that the optio could go out of the moey ad the (possible) jumps at the observatio poits ft i g iˆ;...;. For the America style optio, we set f t; y ˆe y for y > y max. I doig so, we get the followig approximatio for h whe the optio is Europea: y h t; y ˆe r T t y ymax y 0 e q T t ymax 0 ymi y f t; y mi : For the America style optio, we get the approximatio y h t; y ˆe y ymax y 0 ymax 0 ymi y f t; y mi : With this, we ca umerically solve for the price of the lookback optio with oatig strike. Volume /Number
The pricig of discretely sampled Asia ad lookback optios 9 8. CONCLUSION This paper has described a approach to the umerical pricig of discretely observed path-depedet optios that is highly competitive i terms of accuracy ad speed compared with Mote Carlo simulatios. We have illustrated this by umerical examples for four types of path-depedet optio. A secod advatage of this pricig techique compared with Mote Carlo techiques is the ability to price the oatig strike America style optios. This caot be doe by stadard Mote Carlo methods. I the Black±Scholes ad the jump framework, the techique applies to most types of Europea optios o the average ad the maximum (or miimum). Amog the types of optio that have ot bee cosidered i this paper but ca be priced usig our approach are combiatios of maximum, miimum, ad average ad digital optios o the average ad/or the maximum. APPENDIX Derivatio of the Equatios (8) ad (39) Let x be de ed as i (7) ad let be a determiistic fuctio. Usig Itoà 's lemma ad (38), we obtai d t x t Š ˆ 0 t x t r q k t x t dt x t t dw 0 t I 0 t t x t I 0 t dn0 t t dm t : Isertig t ˆe r q t ; we obtai I 0 t d t x t Š ˆ x t t dw 0 t t x t I 0 t dn0 t kdt We kow that the process s x u u dw 0 u I 0 u I 0 u dn0 u kdu t s>t t dm t : 47 is a Q 0 martigale, so itegratig (47) ad takig the Q 0 expectatio yields e r q T E 0 t x T ˆ e r q t x t T e r q u dm u ˆ e r q x t X e r q t i: i : t<t i >t t Fall 998
30 Jesper Adrease Fially, we obtai e q T t E 0 t x T ˆ e r T t x t X e r T t i q t i t : i : t<t i 6t For ˆ 0 we have equatio (8), ad for geeral we have equatio (39). REFERENCES Adrease, J., ad Grueewald, B. (996). America optio pricig i the jump-di usio model. Workig paper, Aarhus Uiversity, Demark. Babbs, S. (99). Biomial valuatio of lookback optios. Workig paper, Midlad Motagu Capital Markets, Lodo. Black, F., ad Scholes, M. (973). The pricig of optios ad corporate liabilities. Joural of Political Ecoomy, 8, 637±654. Boyle, P., ad Lau, S., (994). Bumpig up agaist the barrier with the biomial method. Joural of Derivatives,, 6±4. Coze, A., ad Viswaatha, R. (99). Path-depedet optios: The case of lookback optios. Joural of Fiace, 46, 893±907. Cox, J., Ross, S., ad Rubistei M. (979). Optio pricig: A simpli ed approach. Joural of Fiacial Ecoomics, 7, 9±63. Davidso, R., ad Mackio, J. (993). Estimatio ad Ifereces i Ecoometrics. Oxford Uiversity Press. Gema, H., ad Yor, M. (993). Bessel processes, Asia optios ad perpetuities. Mathematical Fiace, 34, 349±375. Goldma, M., Sosi, H., ad Gatto, M. A. (979). Path-depedet optios: Buy at the low, sell at the high. Joural of Fiace, 34, ±8. Goldma, M., Sosi, H., ad Shepp, L. (979). O cotiget claims that isure ex-post optimal stock market timig. Joural of Fiace, 34, 40±44. Igersoll, J. (987). Theory of Fiacial Decisio Makig. Rowma ad Little eld. Merto, R. (976). Optio pricig whe the uderlyig stock returs are discotiuous. Joural of Fiacial Ecoomics, 5, 5±44. Mitchell, A., ad Gri ths, D. (980). The Fiite Di erece Method i Partial Di eretial Equatios. Wiley. Rogers, L., ad Shi, Z. (995). The value of a Asia optio. Joural of Applied Probability, 3, 077±088. Wilmott, P., Dewye, J., ad Howiso, S. (993). Optio Pricig: Mathematical Models ad Computatio. Oxford Fiacial Press. Volume /Number