Integral Equation Methods for Pricing Perpetual Bermudan Options *

Transcription

1 Journal of ppled Fnance & anng, vol., no.3,, 5-6 ISSN: (prnt verson), (onlne) Internatonal Scentfc Press, Integral Euaton Methods for Prcng Perpetual ermudan Optons * Jngtang Ma and Peng Luo bstract Ths paper develops ntegral euaton methods to the prcng problems of perpetual ermudan optons. y mathematcal dervaton, the optmal exercse boundary of perpetual ermudan optons can be determned by an ntegral-form nonlnear euaton whch can be solved by a root-fndng algorthm. Wth the computatonal value of optmal exercse, the prce of perpetual ermudan optons s wrtten by a Fredholm ntegral euaton. collocaton method s proposed to solve the Fredholm ntegral euaton and the prce of the optons s thus computed. Numercal examples are provded to show the relablty of the method, verfy the valdty of replacng the early exercse polces wth perpetual mercan optons, and explore a smplfed computatonal process usng the formulas for perpetual mercan optons. * The wor was supported n part by a grant from the project 985 and project of Southwestern Unversty of Fnance and Economcs. School of Economc Mathematcs, Southwestern Unversty of Fnance and Economcs, Chengdu (Wenjang), 63, Chna, e-mal: mjt@swufe.edu.cn School of Economc Mathematcs, Southwestern Unversty of Fnance and Economcs, Chengdu (Wenjang), 63, Chna, e-mal: ysonlp@63.com rtcle Info: Receved : February,. Revsed : prl 5, Publshed onlne : June 5,

2 5 Prcng perpetual ermudan optons JEL classfcaton numbers: G, C Keywords: Perpetual ermudan optons, perpetual mercan optons, optmal exercse boundary, collocaton methods, ntegral euaton methods Introducton Perpetual mercan optons are mercan optons wthout expry date, whch means that the optons can be exercsed at any tme n the lfetme. Perpetual ermudan optons are perpetual mercan optons that can be exercsed only on the predetermned dates. Perpetual mercan optons and the early exercse boundares have closed-form formulas (see e.g., Wlmott (998), Kwo (998) and Jang (5)). Whle there are no closed-form formulas for value and early exercse boundares for Perpetual ermudan optons. In the hstory several papers developed numercal methods to prce perpetual ermudan optons and determne the early exercse polces. oyarcheno and Levendors () developed a Wener-Hopf factorzaton method to prce e perpetual ermudan optons. Fatth () proposed terated ntegral methods to prce perpetual ermudan optons. Muro and Yamada (6) studed fnte dfference methods for prcng perpetual ermudan optons. Ln and Lang (7) nvestgated the bnomal tree methods for prcng perpetual mercan and ermudan optons. Ln (8) formulated perpetual ermudan opton prcng as a soluton of a perodc lac-scholes partal dfferental euaton and obtaned an ntegral formula for the valuaton usng contracton mappng theorem. Kay et al. (9) nvestgated the early exercse regon of perpetual ermudan optons wth two underlyng assets usng terated ntegral methods. In ths paper we propose an ntegral euaton method for prcng perpetual ermudan optons. The value of perpetual ermudan optons satsfes a Fredholm ntegral euaton wth the early exercse boundary as the parameter. The early

3 J. Ma and P. Luo 53 exercse boundary can be computed by solvng an ntegral form nonlnear euaton. Snce perpetual ermudan optons approach to perpetual mercan optons as the exercse tme step goes to zero and perpetual mercan optons have explct valuaton and closed-form early exercse polces, one may thn f t s possble to replace the early exercse polces for perpetual ermudan optons by those for perpetual mercan optons. We develop collocaton methods for solvng the Fredholm ntegral euatons. We mplement the algorthm and provde a table to verfy the valdty of replacement for the early exercse polces and nvestgate a smplfed computatonal process usng formulas for perpetual mercan optons. In the hstory for ntegral euaton methods for solvng mercan-style optons, Km (99), Huang et al. (996), Ju (998), Detemple and Tan () have studed the mplementatons of the ntegral euaton methods for prcng mercan put optons. However ther approaches for solvng the ntegral euatons are based on low-order approxmatons and the numercal uadratures are used to evaluate the EEP (Early Exercse Premum) representaton of the opton prce (see e.g., Detemple and Tan ()). Recently Ma et al. (, ) developed a hgh-order collocaton method for solvng the nonstandard ntegral euatons satsfed by the early exercse boundary. Problem statement ssume that the underlyng asset prce follows a dffuson process ds S t t rdt dw. t where r denotes the nterest rate, volatlty, W t rownan moton. Let V be the value of ermudan put optons and be the optmal exercse boundary. Then the ermudan put opton prcng problem can be formulated by (see [])

4 5 Prcng perpetual ermudan optons V( S) G( S,, T)( K ) d G( S,, T) V( ) d, S. V( ) K, S. () V( ) K S, S. where G s lac-scholes European Green s functon K s the stre prce, and S ln ( r ) T exp( r T) GS (,, T) exp{ }, T T T s ermudan exercse tme-step. Let V ( S) ( S,, T) G( S,, T)( K ) d. Then we construct a seuence { V ( S)}, such that V ( S) ( S,, T) G( S,, T) V ( ) d,,,... () s derved by Ln (8), V ( S ) can be represented as n V ( S) ( S,, T) G ( S,, T) (,, T) d,,,... (3) n where the seuence { G n ( S,, T)} satsfes n G ( S,, T) G( S,, T). () n n G ( S,, T) G( S,, T) G (,, T) d, n,3,... (5) Ln (8) also proved that the seuence { V ( S)} V( S ) on the set S,.e., unformly converges to Tang S V( S) ( S,, T) G n ( S,, T) (,, T) d. (6) n nto the above euaton and usng the second euaton n (), we obtan a nonlnear euaton for the optmal exercse boundary : K (,, T) G n (,, T) (,, T) d. (7) n

5 J. Ma and P. Luo 55 Euaton (7) wll be solved by a root-fndng algorthm secant method (see Press (99)). Euaton (), whch s a Fredholm ntegral euaton wth the computed, wll be solved by collocaton methods (see runner ()). 3 Numercal methods We frst solve euaton (7). Snce euaton (7) contans an nfnte seres n the ntegral, we need to truncate t nto a fnte sum. Denote n H(,, T) G (,, T), n n HM (,, T) G (,, T). M n When M, t s nown that can be approxmated by solvng HM H. Therefore soluton of euaton (7) K (,, T) HM (,, T) (,, T) d. Ths euaton s solved by secant method (a root-fndng algorthm, see e.g., Press (99)). Denote the numercal soluton of euaton (7) by,.e.,. Then the opton value can be obtaned by solvng V( x) G( x,, T)( K ) d G( x,, T) V( ) d, (8) wth x and V( ) K. collocaton method wll be proposed to solve euaton (8). The method s descrbed as follows. Defne a mesh: I { x h,,,, N }, h where h s predetermned mesh sze, N s the number of mesh ponts, and denote n : ( xn, xn ]. Defne a pecewse polynomal space: V { v: v, n,,, N }, ( ) m n m where m denotes m th order polynomal. Collocaton method for solvng

6 56 Prcng perpetual ermudan optons (8) s defned by where V h Vh ( x) G( x,, T)( K ) d G( x,, T) Vh ( ) d, (9) ( ) V m s the computatonal soluton,.e., Vh V, xxh { xn ch n : c cm ; n,,,, N }. Euaton (9) s referred as collocaton euaton. Now we rewrte collocaton euaton (9) nto a matrx form. For ease of exposton and actual computaton, we tae m. Defne collocaton ponts j x, j x ( x x), j,,3,,,,... N. On the global mesh, polynomal V h can be represented by h j j j V ( x) V l ( x), () where lj ( x ) are the Lagrange bass functons at ponts x, j, j,, 3,,.e., Puttng () nto (9) gves that l ( x) j x x,. j x, j x, N x j j x p p j p Vl( x) f( x,, T, K),, T) V l ( ) d, () where x X, h f ( x,, T, K) G( x,, T)( K ) d. Tang x x, j, j,, 3,,,,, N, euaton () can be rewrtten nto the form N x j (, j,,, ) [ (, j,, ) p( ) ] x p p V f x T K G x T l d V. () Ths can be further smplfed by where V = F, (3)

7 J. Ma and P. Luo 57, s a matrx, expressons are gven by: wth,,,,,, N N T V V, V, V, N T F F, F,, F, V, F,,,,,,,,, 3, are four-dmensonal row vectors. The,,,, Tl ) ( ) d,,, Tl ) ( ) d,,3,, Tl ) ( ) d,,, Tl ) ( ) d,,,, Tl ) ( ) d,,, Tl ) ( ) d,,3,, Tl ) ( ) d,,, Tl ) ( ) d, 3 wth,,, Tl ) 3 ( ) d,,, Tl ) 3 ( ) d,,3,, Tl ) 3 ( ) d,,, Tl ) 3 ( ) d,,,,,,,,,, 3,,, Tl ) ( ) d,,, Tl ) ( ) d ;,3,, Tl ) ( ) d,,, Tl ) ( ) d, x (,,, ) ( ) x x x x x x x Gx Tl d,,, Tl ) ( ) d,,3,, Tl ) ( ) d,,, Tl ) ( ) d, x x x (,,, ) ( ) x x x x x,,, Tl ) ( ) d Gx Tl d,,3,, Tl ) ( ) d,,, Tl ) ( ) d

8 58 Prcng perpetual ermudan optons, 3 x x x x x (,3,, ) 3( ) x x x,,, Tl ) 3( ) d,,, Tl ) 3( ) d, Gx Tl d,,, Tl ) 3( ) d, f ( x,,, T, K) f ( x,,, T, K) T F ( F, F, F3, F ) f ( x,3,, T, K) f ( x,,, T, K) x x x x x x,,, Tl ) ( ) d,,, Tl ) ( ) d ;,3,, Tl ) ( ) d Gx Tl d x (,,, ) ( ) x T ; V ( V, V, V3, V ). Numercal examples In ths secton, two examples are mplemented usng the method n ths paper. Numercal tests are carred out to nvestgate the valdty of replacng the early exercse polcy for perpetual ermudan optons by that for perpetual mercan optons and explore a smplfed computatonal process usng formulas for perpetual mercan optons. In the presentaton of the numercal results, we use the followng notatons: V( S ): Value of perpetual mercan optons at underlyng prce S ; V ( S ): Value of perpetual ermudan optons at underlyng prce S ; V ( S ): Value of perpetual ermudan optons wth the early exercse polcy of a perpetual mercan optons at underlyng prce S; : Early exercse boundary of perpetual mercan optons; : Early exercse boundary of Perpetual ermudan optons. Perpetual mercan optons have the followng closed-form formulas (see

9 J. Ma and P. Luo 59 e.g., Wlmott (998) and Jang (5)) rk r, r rk V( S) S r r r. () Example. Consder perpetual ermudan optons wth nterest rate r %, stre prce K=, exercse tme-step T=.5,.5,,.5, volatlty =%. Fgure shows that fact that perpetual ermudan optons converge to perpetual mercan optons as the exercse tme-step T. Table nvestgates the valdty of smplfyng the computaton of perpetual ermudan optons usng the formulas for perpetual mercan optons. Snce perpetual mercan optons have explct formulas for early exercse boundary and valuaton, t wll be mportant n practce to nvestgate f ether the computaton of early exercse boundary or the valuaton of perpetual ermudan optons can be realzed by the formulas for perpetual mercan optons. From Table, f the early exercse boundary for ermudan s replaced by that for mercan, then the computaton of euatons (7) can be avoded and the value functon of ermudan V ( S ) can be obtaned by computng euaton (). In ths case and a T =.5 (see the nd column of Table ), the value of Va ( S ) at S s Va ( ) 5.758, whle the true value of ermudan at S s V ( ) Ths means that such a replacement s acceptable. In the other case, f the early exercse polcy for ermudan s determned by solvng euaton (7) and the valuaton of ermudan s computed by formula for mercan (), then the value of ermudan at S by formula () s V ( ).839 and the true value of ermudan s V ( ).8 (see the nd column of Table ). Ths ndcates that such replacement s also acceptable. However Table tells us that t s not acceptable to use all the formulas for mercan () to compute both the early exercse boundary and value of ermudan.

10 6 Prcng perpetual ermudan optons Table : Numercal results for Example. T V ( ) a V ( ) V ( ) V ( ) V ( ) Perpetual mercan put opton V 8 6 ΔT=.5 ΔT=.5 ΔT= ΔT= S Fgure : Value of perpetual ermudan optons for Example.

11 J. Ma and P. Luo 6 Example. Consder perpetual ermudan optons wth nterest rate r %, stre prce K=, exercse tme-step T=,5,,, volatlty =%. Compared to Example., ths example consders a sgnfcantly lower nterest rate. From the numercs n the nd column of Table, the values of V ( ) 66.95, V ( ) and V ( ) are close. Hence a besdes the observatons made n Example., t s also concluded that formulas for mercan () can be used to compute both early exercse boundary and valuaton of ermudan n ths example. Table : Numercal results for Example. T V ( ) a V ( ) V ( ) V ( ) V ( )

12 6 Prcng perpetual ermudan optons Perpetual mercan put opton ΔT= V 3 ΔT=5 ΔT= ΔT= S Fgure : Value of perpetual ermudan optons for Example. 5 Concluson In ths paper we studed the ntegral euaton methods for valung perpetual ermudan optons, whch are sgnfcantly dfferent from the terated ntegral methods developed by Fattah () and Kay et al. (9). We developed collocaton methods to solve the Fredholm ntegral euatons whch characterze the value of perpetual ermudan optons. y mplementng two examples, we provded numercal tables to nvestgate a smplfed computatonal process usng formulas for perpetual mercan optons and verfy the valdty of replacng ermudan wth mercan.

13 J. Ma and P. Luo 63 cnowledgements: The frst author s grateful to Professor Matt Davson for the valuable dscussons durng the vst of Unversty of Western Ontaro n prl,. References [] S.I. oyarcheno and S.Z. Levendors, Prcng of perpetual ermudan optons, Quant. Fnan.,, (), 3-. [] H. runner, Collocaton Methods for Volterra Integral and Related Functonal Euatons, Cambrdge Unversty Press, Cambrdge,. [3] J. Detemple, The valuaton of mercan optons for a class of dffuson processes, Management Sc., 8, (), [] N. Fattah, Problems n ppled Mathematcs: nalyss of ermudan Optons, and Selected Topcs n the nalyss of Quantum Feld-Theoretcal Perturbatve Seres, PhD Thess, Unversty of Western Ontaro, London, Ontaro, Canada,. [5] J. Huang, M. Subrahmanyam and G. Yu, Prcng and hedgng mercan optons: recursve ntegraton method, Rev. Fnancal Stud., 9, (996), [6] L. Jang, Mathematcal Modelng and Methods of Opton Prcng, World Scentfc Press, Sngapore, 5. [7] N. Ju, Prcng an mercan opton by approxmatng ts early exercse boundary as a multpece exponental functon, Rev. Fnancal Stud.,, (998), [8] J. Kay, M. Davson and H. Rasmussen, The early exercse regon for ermudan optons on two underlyngs, Math. Computer Modellng, 55, (9), 8-6.

14 6 Prcng perpetual ermudan optons [9] I.J. Km, The analytc valuaton of mercan optons, Rev. Fnan. Stud., 3, (99), [] Y.K. Kwo, Mathematcal Models of Fnancal Dervatves, Sprnger-Verlag, Sngapore, 998. [] J.W. Ln, Prcng formula of perpetual ermudan optons, J. Tongj Unv. (Natural Sc.) (In Chnese), 36, (8), 3-7. [] J.W. Ln and J. Lang, Prcng of perpetual mercan and ermudan optons by bnomal tree method, Front. Math. Chna,, (7), [3] J. Ma, K. Xang and Y. Jang, n ntegral euaton method wth hgh-order collocaton mplementatons for prcng mercan put optons, Int. J. Econom. Fnan.,, (), -3. [] J. Ma and Z. Zhou, Hgh-accuracy ntegral euaton approach for prcng mercan optons wth stochastc volatlty, Int. J. Econom. Fnan., 3, (), 93-. [5] Y. Muro and T. Yamada, n explct fnte dfference approach to the prcng problems of perpetual ermudan optons, sa-pacfc Fnan. Marets, 5, (8), [6] W.H. Press,.P. Flannery, S.. Teuolsy, and W.T. Vetterlng, Secant Method, False Poston Method, and Rdders Method, 9. n Numercal Recpes n Fortran; The rt of Scentfc Computng, nd ed. Cambrdge Unversty Press, Cambrdge, pp ,. [7] M. Schwezer, On ermudan Optons, dvanced n Fnance & Stochastc. Sprnger, erln, pp ,. [8] P. Wlmott, Dervatves: The Theory and Practce of Fnancal Dervatves, John Wley & Sons Ltd, Chchester, 998.