Chulhan Bae and Doobae Jun

Transcription

1 Korean J. Mah No. pp hps://doi.org/0.568/kjm ANALYIC SOLUIONS FOR AMERICAN PARIAL ARRIER OPIONS Y EXPONENIAL ARRIERS Chulhan ae and Doobae Jun Absrac. his paper concerns barrier opion of American ype where he underlying price is moniored during only par of he opion s life. Analyic valuaion formulas of he American parial barrier opions are obained by approximaion mehod. his approximaion mehod is based on barrier opions along wih exponenial early exercise policies. his resul is an exension of Jun and Ku [0] where he exercise policies are consan.. Inroducion American opions are widely raded in he over couner marke because American ype opions give heir holders an addiional privilege of early exercise. For hese reasons he valuaion of he American opion price has been very imporan issue in financial economics. rennan and Schwarz [] and Parkinson [5] have proposed a numerical approach o value American opions. hey have approached o a soluion of he lack-scholes parial differenial equaion using finie differences. Cox e al. [4] used he binomial model o reduce he size of errors. Liang e al. [3] found convergence rae of he binomial model. Received June 07. Revised June Acceped June Mahemaics Subjec Classificaion: 65C50 6P0. Key words and phrases: American opion parial barrier opion exponenial barrier. his work was suppored by asic Science Research Program hrough he NRF of Korea funded by he Minisry of Science 05RDAA Corresponding auhor. c he Kangwon-Kyungki Mahemaical Sociey 07. his is an Open Access aricle disribued under he erms of he Creaive commons Aribuion Non-Commercial License hp://creaivecommons.org/licenses/by -nc/3.0/ which permis unresriced non-commercial use disribuion and reproducion in any medium provided he original work is properly cied.

2 30 Chulhan ae and Doobae Jun hey have modeled an American pu opion as a variaional inequaliy. hese numerical mehods are very flexible and easy o implemen. However even afer employing conrol variae or convergence exrapolaion hey require a very long ime. Many people have ried o reduce he ime consuming ask. Sullivan [6] approximaed he opion value funcion hrough Chebyshev polynomials and applied a Gaussian quadraure inegraion scheme a each discree exercise dae. In order o find he value of American pu opion Longsaff and Schwarz [4] used he Mone Carlo simulaion mehod. In he Mone Carlo simulaion mehod opimal sopping is significan problem. hey resolved he problem by comparing he condiional expeced value of coninuing wih he value of immediae exercise if he opion is currenly in he money. Kim e al. [] inroduced a simple ieraive mehod o deermine he opimal exercise boundary for American opion. hey allowed us o compue he values of American opions and heir Greeks quickly and accuraely. Anoher flow of he American opion pricing is o deermine lower and upper bounds for American opion value. Kim [] Jacka [8] and Carr e al. [3] obained an analyic inegral-form soluion of American opions where he formulas represen he early premium of American opion as inegral. roadie and Deemple [] provided an upper bound on he value of American opions using a lower bound for he early exercise boundary. Ju [9] approximaed American opion price using early exercise boundary as a muli-piece exponenial funcion. Approximaion mehod of American opions is proposed by Ingersoll [7]. He approximaed exercise policy employing a simple class of funcion and chose a fines policy in ha class adoping sandard opimizaion echnique. his mehod is simple and speedy. Concreely he discussed American pu using wo ypes approximae consan barrier and exponenial barrier. arrier opions have been exensively raded over he couner marke since 967. hese opions are acivaed or expired when he price of he underlying asse crosses a barrier level during he opion s life. Heynen and Ka [6] sudied parial barrier opions where he underlying price is moniored during only par of he opion s lifeime. If barrier is observed from a fixed dae afer iniial saring dae unil expiraion i is called forward saring barrier opion. If barrier is disappeared a

3 Analyic soluions for American parial barrier opions 3 a designaed dae before he expiry dae i is named early ending barrier opion. Heynen and Ka [6] deermined pricing formulas for parial barrier opions in erms of bivariae normal disribuion funcions. For branch of he American barrier opion problem Gao e al. [5] alered sandard American opion o an American barrier opion employing approximaion echnique. Ingersoll [7] presened American up and in pu price by an approximaion mehod based on barrier opions using consan and exponenial exercise policies. Jun and Ku [0] approximaed analyic valuaion formulas of he American parial barrier opion based on barrier opions along wih consan early exercise. his paper concern valuing he American parial barrier opion where barrier is observed jus from fixed dae o expiry dae. his paper exend approximaion mehod derived by Jun and Ku [0] o American parial barrier opion as exponenial exercise policy. his mehod includes he case of consan early exercise policy. Secion proposes he approximaion of American barrier opion based on exponenial exercise policies. Secion 3 provides he valuaion of American parial barrier opion using exponenial exercise policies.. Approximaion of American barrier opion using exponenial barriers Le r be he risk-free ineres rae q be a dividend rae and > 0 be a consan. We assume he price of he underlying asse S follows a geomeric rownian moion = S 0 expµ W where µ = r q and W is a sandard rownian moion under he risk-neural probabiliy P. In his secion we consider he parial barrier opion of American ype as exponenial early exercise policies. American opion holders gain more benefi han European opion on early exercise. An American up-and-in pu opion can be exercised before he expiraion ime when i is in he money bu only afer he sock price rises above he knock-in barrier. We deal wih he up-and-in pu where he barrier appear a a specified ime sricly afer he opion s iniiaion. If he underlying asse price never his he up-barrier over he ime period beween and expiraion paymen of opion is zero. Oherwise if he asse

4 3 Chulhan ae and Doobae Jun price reaches he up-barrier beween and expiraion his opion can early exercise. In order o obain he approximaion o value American parial barrier opion using barrier derivaives under exercise policies Ingergsoll [7] used following digials: le DS ; A be he value a ime of receiving one dollar a ime if and only in he even A occurs and DSS ; A be he value a ime of receiving one share of sock a ime if and only if he even A occurs. he D is said o be a digial or binary opion and he DS is said o be a digial share. he quaniy ES K τ ; A denoes he value a ime of paymen X K τ a he firs ime τ ha he sock price S his he barrier K τ provided he even A occurs before ime where X is a srike price. he E is said o be a firs-ouch digial. We consider he class of exercise policies K e is a se of exponenial funcions whose elemens are in he form of K = K 0 e δ wih consan K 0 and δ 0. Since opions wih exponenial barrier have analyical soluions under lack-scholes condiions an exponenial barrier is a naural choice. Consider an American up-and-in pu expiring wih srike price X. Le us denoe by up barrier and by K he opimal exercise policy. Le τ Y denoe he firs ime he sock price is equal o Y and τ Y Y denoe he firs ime afer τ Y ha he sock price is equal o Y. Le E = { < τ < τ K > S < X} be he even of exercise a mauriy under he opimal policy and E = { < τ τ K < } be he even of early exercise under he opimal policy. hen he value of he up-and-in pu can be wrien as UIP = X DS ; E DSS ; E ES K ; E he barrier approximaion for his pu akes he maximum value wihin a class of resriced policies. For example for exponenial exercise policies K UIP UIP exp = max k K e [X DS ; E 3 DSS ; E 3 ES K ; E 4 ] where E 3 = { < τ < τ K > S < X} E 4 = { < τ τ K < } and τ K is he firs ime he sock price his he exponenial policy barrier K afer hiing he barrier. he values for hese digials are

5 Analyic soluions for American parial barrier opions 33 given by [ µ { DS ; E 3 =e r N h K µ K δ { N K h K X N h X N } h K }] [ µ { DSS ; E 3 = e q N h K µ { K K N h X N N h X h K }] } [ q p qp δ K q p ES K ; E 4 =X K N g K q p qp ] N g K K [ q p q p δ K q p K K N g K q p q p ] N g K S K where N is he sandard normal disribuion funcion ln z µ h z = ln z µ δ h z = g z = ln z q δ q p µ = r q µ = r q p = µ δ q = p r δ. h i and g are he same as h i g excep µ = r q and p q in replacemen of µ p q for i = separaely.

6 34 Chulhan ae and Doobae Jun 3. Valuaion of American parial barrier opion using exponenial exercise policies Now we presen he valuaion of American parial barrier opion using exponenial exercise polices. Le X = ln S 0 and E m be he expecaion operaor under m-measure. hen X is a rownian moion wih drif µ. Le k = k 0 δ for k 0 δ 0 are consan. Define τ b and τ bk by sopping imes for his process defined as he firs ime ha X = b > X 0 afer ime and he firs ime afer τ b ha X = k < b respecively. Lemma 3.. For x k he probabiliy ha he process X reaches b afer ime and hen his k before expiraion and X is greaer han x is P τ bk X > x X 0 = 0 µ = exp b k 0 exp δ µ δ k 0 b G x exp k 0 G x where G x = N b µ δ G x = N b µ δ k 0 b x µ k 0 x µ. Proof. If b > k hen { X b τ bk } = { X b τ k }. P τ bk X > x X 0 = 0 = P X < b τ bk X > x X 0 = 0 = b P X b τ k X > x X 0 = 0 π e b x µ P τ bk X > x X = x dx e x µ P τ k X > x X = x dx. π

7 Analyic soluions for American parial barrier opions 35 Firs we calculae P τ bk { X > x X = x. In order o reflec a pah a τ b we define X b X if τ = u X if > τ u P τ bk X > x X = x µ = exp b x P τ k X > x X = b x. hen we reflec his pah before is firs ouch a k 0 again. P τ bk X > x X = x µ µ δ = exp b x exp k 0 b x δ k0 b x x δ N µ. hus b π e x µ P τ bk X > x X = x dx µ = exp b k 0 δ u µ δ k 0 b N and b k 0 b x µ e x µ P τ k X > x X = x dx π µ δ b µ δ = exp k 0 N k 0 x µ. Lemma 3.. he probabiliy ha he process X reaches b afer ime and hen falls below k before ime is P τ bk X 0 = 0 µ = exp b k 0 exp δ k 0 b G k µ δ bµ exp k 0 G k exp G 3 k G 4 k

8 36 Chulhan ae and Doobae Jun where G 3 x = N b µ G 4 x = N b µ x b µ x µ. Proof. Since {τ bk X k } = {τ b X k } P τ bk X 0 = 0 = P τ bk X > k X 0 = 0 P τ b X k X 0 = 0 When X > b he even {τ b X k } = {X k } and P τ b X k X 0 = 0 b = e x µ P τ b X k X = x dx π b b = b e x µ P X k X = x dx π e x µ µ exp π N b x k0 b x µ δ µ dx e x µ k0 x N µ δ µ dx π uµ u µ = exp N N u µ l 0 u µ δ. l 0 µ δ Lemma 3.3. For k x b he probabiliy ha he process X reaches b afer ime and hen does no fall below k before expiraion and is value a ime is less han x is

9 Analyic soluions for American parial barrier opions 37 P τ b < τ bk > X x X 0 = 0 µ = exp b k 0 exp δ k 0 u [G x G k ] µ δ exp k 0 [G x G l ] bµ exp [G 3 x G 3 k ] G 4 x G 4 k Proof. P τ b < τ bk > X x X 0 = 0 = P τ b < X x X 0 = 0 P τ b < τ bk X x X 0 = 0 = P τ b < X x X 0 = 0 P τ bk X x X 0 = 0 = P τ b < X x X 0 = 0 P τ bk X 0 = 0 P τ bk X > x X 0 = 0 P τ b < X x X 0 = 0 can be calculaed wih k = l 0 δ = x. he second and hird probabiliies are calculaed by Lemma 3. and Lemma 3.. heorem 3.4. he value of a digial opion and a digial share a ime for he even E 8 = {τ K < } are DS ; E 8 =e r K e r K µ K µ δ δ K N h 4 h N h 4 K h e r µ N h 3 h e r N h 3 h K K

10 38 Chulhan ae and Doobae Jun DSS ; E 8 = e q K e q K µ K µ δ δ K N h 4 h N h 4 K h e q µ N h 3 h e q N h 3 h K K Proof. Apply Lemma 3.3 wih b = ln k = K ln = ln K 0e δ and x = ln X o derive he risk-neural probabiliy of early exercise. We calculae each erm of P τ K. P τ K µ = K K µ δ K δ N h 4 h K N h 4 K h µ N h 3 h N h 3 h K K hus he value of digial opion a ime is obained. DS ; E 8 = e r P τ K Lemma 3.5. If he sock does no pay dividends he value of a firs ouch digial for he even E 8 = {τ K < } is

11 Analyic soluions for American parial barrier opions 39 ES K ; E 8 = X K τ K τ [ K K r K r δ N h4 δ N h4 S h K h K r N h3 h N h3 h K K ] where h i is he same as h i excep µ = r in replacemen of µ for i = 3 4. Proof. he firs-ouch digial pays X K τ a ime τ Kτ. his money can be used o purchase X Kτ K τ shares of he sock a ime. Since he shares do no pay dividends i is worh X K K S a mauriy i.e. ES K ; E 8 = X K τ K τ DSS K ; E 8 where DSS K ; E 8 is he value when q = 0 in heorem 3.4. Le K denoe he opimal exercise policy. We denoe he E 5 = {τ < τ K > S < X} be he even of exercise a mauriy under he opimal policy and E 6 = {τ K < } be he even of early exercise under he opimal policy. hen he value of his parial up-and-in pu is wrien as P UIP = X DS ; E 5 DSS ; E 5 ES K ; E 6 For he barrier approximaion of his opion we consider a class of all exponenial exercise policies. Le E 7 = {τ < τ K > S < X} be he even of exercise a mauriy under an exponenial policy K and E 8 = {τ K < } be he even of early exercise under policy K. hen we can approximae he opion price as P UIP P UIP exp = max[x DS ; E 7 DSS ; E 7 ES K ; E 8 ]. k K e

12 40 Chulhan ae and Doobae Jun If he se of policies considered conains all coninuous funcions hen he resuling pu value will be exac. Since he se K e is he se of all exponenial funcions hen he resuling value will be an approximaion providing a lower bound o he pu price. We firs presen he digial opions in case of barrier greaer han srike price for an American parial barrier opion. heorem 3.6. For X he values of a digial opion and a digial share a ime < for he even E 7 = {τ < τ K > S X} are DS ; E 7 [ = e r K DSS ; E 7 K µ [ = e q K where δ F K F K K X µ δ F K F X K µ F 3 F 4 X K F 3 X K µ F 4 K δ ] F K F K K X µ δ F K F X K µ F 3 F 4 X K F 3 X F 4 K ] F x y = N h 4 x h y

13 Analyic soluions for American parial barrier opions 4 and F x y = N h 4 x h y F 3 x y = N h 3 x h y F 4 x y = N h 4 x h y h 3 z = ln z µ h 4 z = ln z µ δ. F i x y and h j z are he same as F i x y h j z excep µ = r q in replacemen of µ for i = 3 4 j = 3 4 separaely. Proof. Apply Lemma 3.3 wih leing b = ln k = K ln = ln K 0e δ and x = ln X o derive he risk-neural probabiliy of exercise a mauriy. hen P τ < τ K > S X µ K = K K δ N h 4 h K X µ δ K N h 4 h X N h 4 µ N h 3 h N h 4 h K N h 3 h N h 3 h X K h X X K N h 3 h K

14 4 Chulhan ae and Doobae Jun hus DS ; E 7 = e r P τ < τ K > S X is obained as desired. he digial share DSS ; E 7 can be valued by changing µ o µ = r q and replacing he discoun facor e r o e q. heorem 3.7. he value of he firs-ouch digial for he even E 8 = {τ K < } is ES K ; E 8 [ q p qp δ K K q p =X H S K K τ K q p qp δ K K K q p H K τ K q p qp q p H H K τ K K τ S ] K [ q p K q p δ K q p K H S K K τ K q p q p δ K K K q p H K τ K q p q p q p H H K τ K K τ S ] K where H x y = N g x g y H x y = N g 3 x g y and

15 Analyic soluions for American parial barrier opions 43 ln z q δ q p ln z g z = g z = g 3 z = ln z q g 3 z = ln z q. q δ q p H i x y is he same as H i x y excep r δ in replacemen of r for i =. Proof. y Lemma 3.5 we noe ha ES K ; E 8 = X Kτ K τ DSS ; E 8 for 0 < < and < τ <. i.e. ES K ; E 8 = X K τ K τ K τ [ K r K δ N h4 S h K K r δ N h4 h K r N h3 h N h3 [ K r K h K ] K δ N h4 S h K K r δ N h4 h K r N h3 h N h3 h K K ]. When he sock price pays dividends he asse price follow he coninuous diffusion process d = r q d dw. In order o calculae he firs erm wih consan paymen X we se V = S q p

16 44 Chulhan ae and Doobae Jun where hen by Io s lemma p = µ and q = p r. dv = rv d q pv dw We may apply Lemma 3.5 o he process V since does no conain he dividend erm. he barriers for V corresponding o and K are q p and K q p. Furhermore he volailiy is replaced by q p. hen X K τ DSS K ; A 8 is X V Kτ p q [ q p r K q p q p q p N h4 q p V K q p V h V K q p q p δ q p h q p K q p r δ K q p V N h4 V q p V r q p q p q p q p N h3 h V V V K q p ] V V N h3 h q p K q p [ q p qp δ K K q p = X H S K K τ K q p qp δ K K K q p H K τ K q p qp q p H H K τ K K τ S ]. K For he second erm wih exponenial paymen K τ we noe when a paymen grows exponenially a rae δ discouning he paymen a he ineres rae r is equivalen o discouning a consan paymen a he rae r δ herefore se V = S q p where

17 hen by Io s lemma Analyic soluions for American parial barrier opions 45 p = µ δ and q = p r δ. dv = r δv d q p V dw. References [] M. roadie and J. Deemple American opion valuaion: new bounds approximaions and a comparison of exising Review of Financial udies [] M. J. rennan and E. S. Schwarz Savings bonds reracable bonds and callable bonds Journal of Financial Economics [3] P. Carr R. Jarrow and R. Myneni Alernaive characerizaion of American pu opions Mahemaical Finance [4] J. C. Cox S. A. Ross and M. Rubinsein Opion pricing:a simplified approach Journal of financial Economics [5]. Gao J. Huang and M. Subrahmanyyam he valuaion of American barrier opions using he decomposiion echnique Journal of Economic Dynamics and Conrol [6] R. C. Heynen and H. M. Ka Parial barrier opions he Journal of Financial Engineering [7] J. Ingersoll Approximaing American opions and oher financial conracs using barrier derivaives Journal of Compuaional Finance [8] S. D. Jacka Opimal sopping and he American pu Mahemaical Finance [9] N. Ju Pricing and American Opion by Approximaing Is Early Exercise oundary as a Mulipiece Exponenial Funcion he Review of Financial udies [0] D. Jun and H. Ku Valuaion of American parial barrier opions Review of Derivaives Research [] I. J. Kim he analyic valuaion of American opions Review of financial sudies [] J. Kim. G. Jang and K.. Kim A simple ieraive mehod for he valuaion of American opions Journal of Quaniaive Finance [3] J. Liang. Hu L. Jiang and. ian On he rae of convergence of he binomial ree scheme for American opions Numerische Mahemaik [4] F. A. Longsaff and E. S. Schwarz Valuing American opions by simulaion:a simple leas-squares approach Review of Financial sudies [5] M. Parkinson Opion pricing:he American pu he Journal of usiness

18 46 Chulhan ae and Doobae Jun [6] M. A. Sullivan Valuing American pu opions using Gaussian quadraure. he Review of Financial udies Chulhan ae NICE P&I Inc Seoul 0738 Republic of Korea baechulhan9@gmail.com Doobae Jun Deparmen of Mahemaics and Research Insiue of Naural Science Gyeongsang Naional Universiy Jinju 588 Republic of Korea dbjun@gnu.ac.kr