The Direct Control and Penalty Methods for American Put Options

Transcription

1 The Drect Control and Penalty Methods for Amercan Put Optons by Ama Peprah Asare A thess presented to the Unversty of Waterloo n fulfllment of the thess requrement for the degree of Master of Mathematcs n Computatonal Mathematcs Supervsors: Prof. Peter Forsyth and Prof. George Labahn Waterloo, Ontaro, Canada, 2013 c Ama Peprah Asare 2013

2 I hereby declare that I am the sole author of ths thess. Ths s a true copy of the thess, ncludng any requred fnal revsons, as accepted by my examners. I understand that my thess may be made electroncally avalable to the publc.

3 Abstract Prcng Amercan optons gves rse to nonlnear Hamlton-Jacob-Bellman (HJB) Partal Dfferental Equatons (PDEs). These HJB PDEs can be solved usng a penalty or a drect control approach. In ths essay, we explore the characterstcs of these two methods the drect control method and penalty method, under the Amercan put opton prcng doman. We also examne a thrd method, a combnaton of the best propertes of the above mentoned methods. Included are algorthms and numercal results to support our fndngs. In all cases, we use a postve coeffcent dscretzaton of the PDE to ensure the numercal scheme converges to the vscosty soluton. v

4 Acknowledgements It s my pleasure to thank all the people who have made ths work possble. My heartfelt apprecaton goes to my supervsors, Peter Forsyth and George Labahn for ther useful comments, gudance, ntellectual and fnancal support, through the wrtng of ths work. Especally for ther mpeccable attenton to detal, whch has made ths essay much better than t would have been otherwse, I am deeply grateful. To Yuyng L, I say thank you for the correctons and all the Computatonal Fnance classes. I am also partcularly apprecatve of the support of Justn Wan, the current Drector of the Centre of Computatonal Mathematcs, Unversty of Waterloo, for the helpful dscussons and great advce. In the same ven, I wll lke to thank Anthea Dunne, the Admnstratve Offcer for the Computatonal Mathematcs program, for the awesome get-togethers and her resourcefulness. For the ntellectual and emotonal support from all the wonderful classmates I have had n the Computatonal Mathematcs program, I am very thankful. To the professors I have had the pleasure of studyng under at the Unversty of Waterloo, I want to say thanks for the educatonal and lfe lessons I have had. I wll surely put them nto practce. To my famly, by blood and by choce all over the world, and to all the knd people I have met whose actons and words have gven me encouragement to carry ths work through, I want to say thank you. v

5 Table of Contents Lst of Tables Lst of Fgures v x 1 Introducton Overvew: The Drect Control and Penalty Methods Objectves Model The Amercan Opton Constrant and the LCP Drect Control Formulaton Penalty Formulaton Dscretzaton Drect Control Penalty Method Solutons to the Dscrete Equatons Drect Control method Penalty Method The Polcy Iteraton Algorthm v

6 4.4 Effects of the Drect Control Scalng Factor and Penalty Parameter on the Soluton to the Dscrete Equatons Scalng Factor Penalty Parameter Numercal Results Standard Amercan put opton Convergence propertes Performance Tme-dependent Amercan constrant Performance Concluson Proposal Approach Numercal Results Convergence Performance Summary 36 References 38 v

7 Lst of Tables 5.1 Data for Amercan put Grd/tme step data for convergence study, Amercan put. On each grd refnement, new fne grds are nserted between each two coarse grd nodes, and the tme step control parameter s halved Convergence study, opton value at S=100. tol = 10 6, ε = 1/Ω = 10 6 τ Convergence study, opton value at S=100. tol = 10 6, ε = 1/Ω = 10 2 τ Convergence study, opton value at S=100. tol = 10 6, ε = 1/Ω = 1 τ Parameters for drect control and penalty methods, teratons study Maxmum teratons study wth constant tme steps Iteratons study. Varable tme steps. On grd refnement, tme step control parameter s halved, and new fne grds are nserted between each two coarse grd nodes. Intally, dnorm = 0.2, τ = Iteratons per step wth strke, K = 100, N = 801. Varable tme steps used Parameters: drect control and penalty methods, tme-dependent Amercan constrant study Tme-dependent Amercan constrant study wth constant tme steps. Other parameters n Table Tme-dependent Amercan constrant study wth varable tme steps. On grd refnement, tme step control parameter s halved, and new fne grds are nserted between each two coarse grd nodes. Intally, dnorm = 0.2, τ = Other parameters n Table Data for Amercan put v

8 6.2 Grd/tme step data for convergence study of the hybrd method, Amercan put. On each grd refnement, new fne grds are nserted between each two coarse grd nodes, and the tme step control parameter s halved Hybrd Method. Convergence study, opton value at S=100. ε = 1/Ω = 10 6 τ Hybrd Method. Convergence study, opton value at S=100. ε = 1/Ω = 10 2 τ Hybrd Method. Convergence study, opton value at S=100. ε = 1/Ω = 1 τ Parameters for drect control and penalty methods, teratons study Grd/tme step data for hybrd method, Amercan put opton. On each grd refnement, new fne grds are nserted between each two coarse grd nodes, and the tme step control parameter s halved Comparson of teratons and elapsed tme (seconds) for the drect control and hybrd methods for a standard and tme-dependent Amercan constrant. Varable tme steps used x

9 Lst of Fgures 2.1 Payoff dagram for an Amercan put opton, llustratng the payoff, V, opton value, V x

10 Chapter 1 Introducton A fnancal opton s a contract that enttles ts holder to buy or sell an asset under certan condtons at a future tme. Ths asset, usually a stock, s referred to as an underlyng asset or smply an underlyng. Optons may be classfed based on the rghts a holder has and/or when they can be exercsed. For example, a put opton s a contract that gves the holder the rght but not the oblgaton to sell an underlyng at a specfc prce, known as a strke prce. The wrter, the buyer of the opton, s oblged to buy the underlyng at the specfed strke prce. A call opton, on the other hand, s a contract that gves a holder the rght but not the oblgaton to buy an underlyng at a specfed strke prce. The wrter, ths tme the seller, s oblged to sell the underlyng at the specfed strke prce. An Amercan opton allows ts holder to exercse anytme before or up to the specfed expraton date. A European opton, however, allows ts holder to exercse an opton only on the expraton or maturty date. The proft an opton holder stands to make from exercsng an opton s called a payoff. Ths essay dscusses the methods used to prce the far value of an Amercan put opton at ntal tme. Valuaton of fnancal optons s of both theoretcal nterest and practcal relevance. Consequently, consderable work has been done to fnd new approaches to solve for the prcng of optons, n addton to contnuously mprove upon exstng ones. Exstng approaches nclude Monte Carlo smulatons [11], lattce methods [17], and partal dfferental equaton (PDE) approaches [24]. For an underlyng asset, Monte Carlo smulatons generate several possble random prce paths forward n tme. The value of an opton s calculated by computng the present day value of the average of the payoffs calculated for each smulated path. Ths approach s advantageous when we want to handle multple underlyng assets. However, t cannot be 1

11 easly used to handle an early exercse constrant. It also has the dsadvantage of slow convergence to a soluton for problems t can handle. For prcng an opton usng a lattce method, the duraton from the ntal tme to the opton s maturty date s splt nto dscrete ntervals (tme dscretzaton) known as tme steps, and possble stock prces are calculated at each dscrete tme step, over a lattce. Then workng backwards from the expry date, the opton prces at each pont on the lattce are calculated untl ntal tme. The opton prce calculated for ntal tme s the value of the opton. It may seem that wth ths method, the entre lattce wll need to be stored n memory. But ths s not the case, as calculatng the prce of an underlyng depends only on the prce of the underlyng n the prevous tme step, and the opton values, on the underlyng prce and opton values at the next tme step. Ths approach s easy to mplement for smple cases such as prcng a European opton, but may become complex for more complcated cases. The number of computatons and approxmaton errors grow exponentally wth ncreasng number of tme steps, and as a result, the lattce method can become computatonally expensve and numercally unstable unless the number of tme steps s restrcted [24]. The prce of an opton over tme may be posed as a partal dfferental equaton (PDE). The PDE approach may be used for many dfferent types of optons.to get the prce of an opton gven a PDE, the latter needs to be dscretzed,.e. converted nto a set of dscrete algebrac equatons. There are several technques that may be used to dscretze a PDE. These nclude fnte dfference, fnte volume, and fnte element methods. Fnte dfference methods are more popular, manly because they are relatvely easy to mplement, and allow for easy calculaton of hedgng parameters, whch are essental for rsk-managng fnancal dervatves [5]. After dscretzaton, the equatons are solved teratvely over dscrete tme ntervals n order to arrve at the opton prce. Our work n ths essay focuses on solvng for the prce of an Amercan opton usng a PDE approach wth fnte dfference dscretzaton. Under ths approach, we wll use two dfferent means to represent the opton prcng problem, called the drect control and penalty methods. 1.1 Overvew: The Drect Control and Penalty Methods There are many models that have been used to represent the behavor of stock prces n the market today. The most popular of these s the Black-Scholes model and the methods under dscusson n ths essay wll employ the famous Black-Scholes PDE, frst developed 2

12 n 1973 by Black and Scholes n The Prcng of Optons and Corporate Labltes, to assst n solvng the Amercan opton prcng problem. The Black-Scholes PDE cannot be solved as t s, but can be used n ts closed form to solve for the prce of a European opton [24]. Such a closed-form soluton generally does not exst for an Amercan opton. Instead, we use numercal methods, n ths case, fnte dfference methods, to approxmate the prce of an Amercan-type opton. An Amercan opton s more dffcult to prce than ts European counterpart, manly because of ts early exercse feature. The early exercse constrant s an mportant factor to be taken nto account when prcng Amercan optons. For an Amercan call payng no dvdends, the opton value s the same as that of a European call on the same underlyng wth the same maturty date [22]. Hence the optmal exercse polcy here s to not exercse. However, for a put opton, also payng no dvdends, t may be optmal to exercse before expry tme and dong so wll affect the value of the opton. Therefore the optmal exercse polcy must be known beforehand and taken nto consderaton when prcng Amercan optons. To prce an Amercan opton usng the PDE approach, we pose the prcng problem as a dfferental lnear complementarty problem 1 (LCP). Ths LCP s formulated as a Hamlton- Jacob-Bellman (HJB) PDE, dscretzed and solved usng fully mplct or Crank-Ncolson tme steppng schemes, whch are numercal schemes used to approxmate partal dervatves. We wll explore two methods for formulatng the LCP, a drect control method and a penalty method. The drect control method was orgnally proposed n [9] n the context of HJB PDEs and adapted to the case of an early exercse opton n [15, 21]. Usng a drect control method, the HJB PDE may be posed as an optmal control problem, and solved drectly. Ths s done by dscretzng the optmal control problem, and solvng the resultng non-lnear equatons teratvely. The drect control method has the advantage of solvng the exact dscretzed LCP, wth second order convergence wthn a specfed convergence tolerance. The penalty method has been studed by many authors [10, 27, 5] and found to be a very effcent way to approxmate the soluton of an Amercan opton prcng LCP. In ths approach, a penalty term s ntroduced n the LCP n order to enforce the early exercse constrant. The resultng problem s dscretzed and solved teratvely. Ths method can be used on any type of dscretzaton for one or multdmensonal problems, and on unstructured meshes. Quadratc convergence s observed for a reasonable penalty value as tme steps and mesh sze tend to zero, usng a varable tme step selector as proposed n [10], wthn a specfed convergence tolerance. The addton of the penalty term, however, 1 In general, gven x, y R, a lnear complementarty problem s a problem of the form x 0, y 0; x > 0 = y = 0, y > 0 = x = 0 and y = ax + b for some a, b R. 3

13 ntroduces a penalty error n the solutons of the dscretzed PDEs and as a result, the Amercan constrant s only satsfed approxmately [10]. Earler work done on the drect control and penalty methods have shown that the two are very compettve n terms of effcency when solvng for the prce of an opton. The queston we wll lke to answer wth ths work s whch of the two s more effcent and why, for the problems under consderaton n ths essay. 1.2 Objectves In ths essay, we dscuss two methods used to solve the problem of the valuaton of Amercan optons, namely the drect control and penalty methods, both of whch fall under the PDE approach, wth fnte dfference dscretzaton. We present an teratve algorthm, whch can be modfed to work wth ether drect control and penalty approaches. Solvng the Amercan opton prcng problem wth these methods usng our teratve algorthm, we observe the convergence rates assocated wth the methods, the maxmum number of nonlnear teratons requred by each method to solve the problem, and the performance of the methods on a tme-dependent Amercan constrant. The man results of ths essay are: The drect control method can be modfed to make t more effcent than t already s. The penalty method s superor to the drect control method n convergng to a soluton. The essay s arranged as follows: Chapter 2 presents the Amercan opton prcng problem mathematcally, and defnes models and formulas, ncludng drect control and penalty forms of the problem, that wll be used n the rest of the essay. Followng ths, we dscretze the equatons from Chapter 2 n Chapter 3, then descrbe an algorthm whch wll be used to solve the dscretzed equatons n Chapter 4. Chapter 5 presents some numercal results. We show how the drect control method can be made more effcent n Chapter 6, and summarze our work n Chapter 7. 4

14 Chapter 2 Model In ths chapter, we present the mathematcal models and formulae whch wll lay a foundaton for the rest of the work n ths essay. Let S be the prce of an underlyng rsky asset whch we assume follows a stochastc process n tme t, ds = µsdt + σsdz, S (0, ) (2.1) where dz s the ncrement of a Wener process, µ s the drft, and σ s the volatlty. We defne V (S, t) as the prce of an opton at t when the market prce s S, for some functon V : (0, ) [0, T ] R, where T s the expry date of the contract. Applyng Ito s Lemma [22] to equaton (2.1), leads to the Black-Scholes equaton [22], σ 2 2 S2 V SS + rsv S + V t = rv. (2.2) where r s the rsk-free rate of return. Let τ = T t be the tme n the backward drecton. Then V can be redefned n terms of S and τ as V (S, τ), and equaton (2.2) can be rewrtten as σ 2 2 S2 V SS + rsv S V τ = rv. (2.3) Equaton (2.3) s the PDE whch wll ad us n solvng for the prce of an Amercan put opton. We prefer to use backward tme, τ, when solvng for the prce of an Amercan opton, because then we can use as an ntal condton to equaton (2.3), the payoff, V = max(k S, 0), 5

15 whch occurs at tme T or τ = 0. K s the strke prce. At τ = 0, V (S, 0) = V. 2.1 The Amercan Opton Constrant and the LCP To prevent arbtrage (rsk-free gan) for an Amercan opton, we enforce the constrant V V. If V < V, arbtrage can be easly realzed by buyng the opton for V, exercsng t by sellng the underlyng for K, and then repurchasng the underlyng n the market for S, makng a rsk-free proft of K V S. So wth the constrant V V enforced, where the opton value V > V, t s more proftable for the holder to sell than to exercse. There, V satsfes the Black-Scholes equaton (2.3). Where V = V, the opton should be exercsed. At those ponts, V s not governed by the Black-Scholes equaton (2.3). These constrants combned satsfy the nequalty σ 2 2 S2 V SS + rsv S V τ rv 0. (2.4) Fgure 2.1 shows the payoff, V for an underlyng, and ts opton value, V, before expry. We see n Fgure 2.1 that the constrant V V s enforced. So untl t s proftable to exercse, we have When t s proftable to exercse we have, V V, (2.5) σ 2 2 S2 V SS + rsv S V τ rv = 0. (2.6) V = V, (2.7) σ 2 2 S2 V SS + rsv S V τ rv 0. (2.8) Puttng equatons (2.5), (2.6), (2.7) and (2.8) together, we arrve at ths partal dfferental lnear complementarty problem (LCP): V τ LV 0, (V V ) 0, (2.9) and [(V V = 0) or (V τ LV = 0)], 6

16 Payoff dagram for an Amercan put opton, and the opton value pror to expry, as functons of S 100 Payoff Value 90 Opton Value Value ,000 S, underlyng prce Fgure 2.1: Payoff dagram for an Amercan put opton, llustratng the payoff, V, opton value, V. where L s a lnear dfferental operator and ( σ 2 LV 2 S2 V SS + rsv S rv The LCP n (2.9) can also be formulated as the HJB equaton ). (2.10) mn [V τ LV, V V ] = 0. (2.11) Intal and Boundary Condtons The Black-Scholes equaton has nfntely many solutons, so we mpose some ntal and boundary condtons to ensure we arrve at the vscosty soluton. For the Amercan put opton, our ntal condton, as mentoned earler, s V (S, 0) = max (K S, 0). (2.12) For the boundary condtons, as S, the prce of the underlyng approaches zero at any tme τ, the opton s lkely to be exercsed and so the opton prce, V approaches K. However, as S grows larger, the opton s unlkely to be exercsed and so V loses value and approaches zero. Therefore, we have the followng boundary condtons: 7

17 Left-boundary condton: lmv (S, τ) = K. (2.13) S 0 Rght-boundary condton: lm V (S, τ) = 0. (2.14) S Now, wth HJB equaton (2.11) defned, and ntal and boundary condtons specfed, we can defne drect control and penalty forms of equaton (2.11). 2.2 Drect Control Formulaton In control form, the HJB equaton (2.11) can be wrtten as max [ϕ (V V ) (1 ϕ) (V τ LV )] = 0. (2.15) ϕ {0,1} where ϕ s a control parameter. Equaton (2.15), may be solved drectly. Hence the name drect control. A value of ϕ = 1 mples early exercse s optmal. That of ϕ = 0 mples t s optmal to hold the opton. It s natural to scale a drect control formulaton (2.15) of the HJB PDE (2.11). We do so to ensure that all the varables to be compared n equaton (2.15) are of the same unts. In addton, the use of an approprate scalng factor allows for more accuracy n determnng the optmal control, ϕ on a fnte precson arthmetc machne [13]. Usng Ω as a scalng factor, equaton (2.11) becomes mn [V τ LV, Ω(V V )] = 0, Ω > 0, (2.16) and then s re-wrtten n a non-lnear drect control form as max [Ωϕ (V V ) (1 ϕ) (V τ LV )] = 0. (2.17) ϕ {0,1} 2.3 Penalty Formulaton In penalzed form, the HJB equaton (2.11) may be wrtten non-lnearly as lm ε 0 mn[v τ LV, V V + ε(v τ LV )] = 0. (2.18) 8

18 The penalty form (2.18) of the LCP (2.9) s attaned by replacng t wth the non-lnear PDE ( ) σ 2 V τ 2 S2 V SS + rsv S rv = max (V V, 0). (2.19) ε Addng the penalty term, max(v V,0) to equaton (2.19) ensures that the soluton satsfes ε (V V ) δ for 0 < δ 1, as the postve penalty parameter ε 0. Essentally, where V V 0, equaton (2.19) resembles the Black-Scholes equaton (2.3). However, where 0 < V V < δ, the Black-Scholes nequalty s satsfed, assurng the early exercse rule s not volated [5]. Wth a control ϕ, equaton (2.18) can be wrtten as [ ( )] V lm V τ LV max ϕ V = 0. (2.20) ε 0 ϕ {0,1} ε In (2.20), a value of ϕ = 1, mples early exercse s optmal, and ϕ = 0 mples holdng the opton s optmal. Typcally we dscretze equaton (2.18). If we denote the dscretzaton parameter by h, then t s common to choose ε = O (h), whch means that the penalty method s formally only frst order convergent. 9

19 Chapter 3 Dscretzaton In ths chapter, we transform equatons (2.17) and (2.18) nto a set of dscrete non-lnear algebrac equatons, usng fnte dfference approxmatons to the partal dervatves. Let S = { S n 1, S n 2,..., S n max }, 0 S S max, (3.1) be a fnte grd of ponts to approxmate the underlyng, τ n = n τ, 0 τ n N τ = T (3.2) be a set of dscrete tmes, and V n = { V n 1, V n 2,..., V n max } (3.3) be the opton prce, where V n = V (S, τ n ). (3.4) Then we can approxmate the partal dervatves n equaton (2.3) by ( σ 2 S 2 V SS 2 (V τ ) n V n+1 ) n V n τ ( ) σ 2 S 2 2 ( V n +1 V n S +1 S ) S +1 S 1 2 ( ) V n V 1 n S S 1 (3.5) (3.6) (rv ) n rv n (3.7) 10

20 For the case of rsv s, there are three ways to dscretze. We can use ether forward, backward or central dfferencng to yeld ( ) V (rsv s ) n n rs +1 V n, forward dfference (3.8) S +1 S ( ) V or (rsv s ) n n rs V 1 n, backward dfference (3.9) S S ( 1 ) V or (rsv s ) n n rs +1 V 1 n, central dfference. (3.10) S +1 S 1 Central dfferencng gves the hghest accuracy among the three. But t may not always be approprate for dscretzng equaton (2.3), for reasons whch we dscuss n the next paragraph. Substtutng equatons (3.5), (3.6), (3.7) and one of (3.8), (3.9), (3.10), dependng on the choce of dfferencng nto equaton (2.3) and rearrangng, we arrve at wth V n+1 = V n (1 (α + β + r) τ) + V 1 τα n + V+1 τβ n, (3.11) or α = α central = β = β central = σ 2 S 2 (S S 1 ) (S +1 S 1 ) rs S +1 S 1 (3.12) σ 2 S 2 (S +1 S ) (S +1 S 1 ) rs S +1 S 1 (3.13) α = α forward = σ 2 S 2 (S S 1 ) (S +1 S 1 ) (3.14) or β = β forward = α = α backward = β = β backward = σ 2 S 2 (S +1 S ) (S +1 S 1 ) + rs S +1 S 1 (3.15) σ 2 S 2 (S S 1 ) (S +1 S 1 ) rs S S 1 (3.16) σ 2 S 2 (S +1 S ) (S +1 S 1 ). (3.17) We select α and β such that the coeffcents of V n+1, V n, V 1 n are always non-negatve (postve coeffcent dscretzaton). We satsfy ths constrant by choosng central dfferencng 11

21 whenever possble, provded α and β are postve. Naturally, t would be advantageous to use central dfferencng always, snce t gves second order accuracy, whle forward or backward dfferencng gves only frst order accuracy. However we cannot choose central dfferencng always, because α and β may not always be postve when central dfferencng s used. So at each node, we select a dfferencng method, usng Algorthm 3.0.1, consderng at each pont, f we wll obtan a postve coeffcent or not. One of the central, forward or backward dfferences wll always have α and β postve. The PDE can converge to many dfferent solutons. Usng a postve coeffcent dscretzaton ensures that the soluton converges to the vscosty soluton [10]. Algorthm Postve Coeffcent Selecton Algorthm f α central 0 and β central 0 then α = α central β = β central else f α forward α = α forward β = β forward else α = α backward β = β backward end f 0 and β forward 0 then Wth the terms we have just dscretzed, we proceed to dscretze the drect control (2.17) and penalty (2.18) forms of equaton (2.11) Drect Control Let L h be the dscrete form of the L operator and θ ndcate the tme steppng scheme to be used. Then the dscretzaton of equaton (2.17) gves ( V n+1 ) (1 ϕ n+1 ) τ θlh V n+1 V n+1 τ + Ωϕ n+1 V n+1 = (1 ϕ n+1 +(1 ϕ n+1 = V τ ; 12 ) V n V τ + Ωϕn+1 )(1 θ)(l h V n ); < max (3.18) = max

22 wth { ( V {ϕ n+1 } arg max Ωϕ(V V n+1 n+1 V n ) (1 ϕ) ϕ {0,1} τ )} θ(l h V n+1 ) (1 θ)(l h V n ). (3.19) We use θ = { 1/2, for a Crank-Ncolson tme steppng scheme 1, for a fully mplct tme steppng scheme. Fully mplct schemes arse from the use of backward dfferences, and Crank-Ncolson schemes from central dfferences [24]. Wth second order accuracy, the Crank-Ncolson scheme s the most logcal choce for dscretzaton. However, prevous work has shown that solutons of these dscretzed forms usng a Crank-Ncolson scheme show spurous oscllatons. However, by usng a Rannacher modfcaton [20], we are able to observe smoothness n the soluton. The Rannacher modfcaton conssts of usng, nstead of a Crank-Ncolson scheme throughout the duraton of the problem, a fully mplct scheme n the frst two tme steps. The varable θ enables us to easly swtch between schemes. Let M be a trdagonal matrx wth entres [MV n ] = α V 1 n (α + β + r) V n + β V+1. n (3.20) Then LV from equaton (2.10) can be represented n dscretzed form as equaton (3.20). Usng ths substtuton, the matrx form of equaton (3.18) can be wrtten as [ ( ) ] [ ] I I (1 ϕ n+1 ) τ θm + Ωϕ n+1 I V n+1 = (1 ϕ n+1 ) + (1 θ) M V n +Ωϕ n+1 V τ (3.21) and equaton (3.19) rewrtten as ϕ n+1 = { 1, f (V V n+1 ) Ω > [ MV n ( I θm) (V n+1 V n ) ] τ. (3.22) 0, otherwse I s an max max dentty matrx and V s a vector of sze max. 13

23 3.0.2 Penalty Method Let L h be the dscrete form of the L operator and θ ndcate the tme steppng scheme to be used. Then the dscretzaton of equaton (2.18) gves V τ n+1 θlh V n+1 + ϕn+1 ε V n+1 wth V n+1 τ {ϕ n+1 = V n τ + ϕn+1 ε V + (1 θ)(l h V n ); < max (3.23) = V τ ; where ε represents the penalty parameter. Agan = max. { ϕ ( ) } } arg max V ϕ {0,1} ε V n+1, (3.24) θ = { 1/2, Crank-Ncolson 1, fully mplct Usng M from equaton (3.20), equaton (3.23) can be smplfed and wrtten n matrx form as [ ] [ ] I ϕn+1 I θm + τ ε I V n+1 = + (1 θ)m V n + ϕn+1 τ ε V (3.25) and equaton (3.24) as ϕ n+1 = { 1, f (V V n+1 ) > 0 0, otherwse I s an max max dentty matrx and V s a vector of sze max... (3.26) 14

24 Chapter 4 Solutons to the Dscrete Equatons There s no analytcal formula for the exact soluton to the LCP n (2.9). We use the penalty and drect control formulatons of the HJB equaton (2.11) to numercally approxmate the soluton of the LCP n (2.9) usng a polcy teraton algorthm. The polcy teraton algorthm used to solve dscretzed (2.11) starts wth the payoff, V, as an ntal state, and at each tme step, repeatedly generates a guess of the optmal control polcy and uses that guess n a lnear system to solve for the opton prce untl convergence. Wth the excepton of the frst tme step whch uses V as ts ntal state, all other polcy teratons used wthn the lfetme of the opton use as an ntal state, the state from the prevous teraton. Formally put, polcy teraton solves non-lnear algebrac equatons of the form A (P ) U = C(P ), wth P = arg max{ A(P )U + C(P )}. P {0,1} A s a matrx of sze max max, and U, C are vectors of sze max. P s a set of controls. In the next sectons, we defne A, C, and P precsely for the drect control and penalty methods, and llustrate the polcy teraton algorthm. 15

25 4.1 Drect Control method Assumng we are at the kth teratve step of the polcy teraton algorthm, durng the nth dscrete tme step, we have for < max, [ ( A P k) ] [( ) ] I V = (1 P k+1 ) τ θm V + ΩP k+1 V. (4.1) ( [ ] ) I C(P k ) = (1 P k+1 ) + (1 θ) M V n + ΩP k+1 V. (4.2) τ P k+1 = { 1, f ( V V k+1 0, otherwse ) Ω > ([ MV k] (( I τ θm) ( V k+1 V k)) ).(4.3) At = max, [ A k V ] = V max τ ; Ck = V max τ. (4.4) I s an max max dentty matrx, and M s as defned n (3.20). Ω s the scalng factor. 4.2 Penalty Method Assumng we are at the kth teratve step of the polcy teraton algorthm, durng the nth dscrete tme step, we have for < max, At = max, [ A ( P k ) V ] = C(P k ) = P k+1 = [( ) ] I τ θm V + P k+1 V. (4.5) ε [( ) ] I + (1 θ)m V n + P k+1 V. (4.6) τ ε { 1, f ( ) V V k+1 > 0. (4.7) 0, otherwse [ A k V ] = V max τ ; Ck = V max τ. (4.8) I s an max max dentty matrx, and M s as defned n (3.20). ε s the penalty parameter. 16

26 4.3 The Polcy Iteraton Algorthm The followng polcy teraton algorthm, Algorthm s used for both drect control and penalty method mplementatons, substtutng the values for A(P k ) and C(P k ) from equatons (4.1) and (4.2) for a drect control method, or (4.5) and (4.6) for a penalty method. The polcy teraton algorthm s used wthn each dscrete tme step, untl the LCP s solved. Algorthm Polcy teraton k = 0 V k = V. V s the soluton vector from the prevous tme step, or the payoff for the frst tme step. V s of sze max. whle (k 0) do P k arg max P {0,1} Solve A(P k )V k+1 = C(P k ) f k > 0 and max break end f k = k + 1 end whle { A(P )V k + C(P )} V k+1 V k max [ scale, V k+1 ] < tol then The term scale used n Algorthm s to ensure that unrealstc levels of accuracy are not enforced. Typcally, for optons valued n dollars, we use scale = 1. If the dfference between tme steps s constant throughout the soluton of the problem, we say we have used a polcy teraton algorthm wth constant tme steps. Otherwse, we have used varable tme steps, where the dfference between tme steps changes. In ths essay, whenever we use varable tme steps, we use the varable tme step selector presented n [10]. Usng ths selector, we take small tme steps where the soluton, V, changes rapdly over the last two tme steps, and larger tme steps where the change between solutons s not great. 17

27 4.4 Effects of the Drect Control Scalng Factor and Penalty Parameter on the Soluton to the Dscrete Equatons Scalng Factor Dfferent choces of the scalng factor, Ω, wll lead to dfferent drect control forms, all convergng to the same soluton [14]. The sze of Ω affects how fast the polcy teraton algorthm, Algorthm converges to a soluton n fnte precson arthmetc. The speed referred to here s measured n terms of the teratons per step, whch s the number of teratve steps used by the polcy teraton algorthm wthn each dscrete tme step. Although Ω should only affect the speed of the algorthm for convergence to a soluton, for suffcently small Ω, the soluton does not converge, as a result of usng floatng pont arthmetc [15]. Ideally, Ω = 1 C τ, (4.9) wth and C < 1 τ C > ( tolerance δ 2δ tolerance, (4.10) ) ( ) ( S) 2 mn. (4.11) 2θS 2σ2 δ 10 6 (double precson). tolerance s the desred level of accuracy. θ refers to ether a fully mplct or Crank-Ncolson tme steppng scheme [15]. The scalng factor, Ω, should have nverse unts of tme, so that the same unts are beng compared n equaton (2.17) Penalty Parameter The number teratons per step used by Algorthm when solvng (2.18) s nsenstve to the sze of the penalty parameter, ε, unless the algorthm does not converge due to floatng pont error. However, a poor choce n penalty parameter,.e. too large a value, wll result n poor convergence as the number of cells used to approxmate the grd S 18

28 approaches nfnty and tme steps are refned. Ths s because the ntroducton of the penalty parameter, ε(v τ LV ), n equaton (2.18) also ntroduces a penalzaton error, whch wll be present at any fnte grd sze. Choosng too small a value can also ntroduce round-off error, whch causes the penalty method to not converge. Let ε = C τ, where C s a dmensonless constant. Then usng the bounds for C as presented n equatons (4.10) and (4.11), the consstency error due to the penalty parameter, ε(v τ LV ), s small compared to the dscretzaton error at reasonable grd szes and tme steps [15]. Usng these bounds, convergence can be expected despte nexact arthmetc effects on the soluton of equaton (2.18). 19

29 Chapter 5 Numercal Results Ths chapter outlnes the numercal results obtaned for prcng Amercan put optons usng both drect control and penalty methods usng the polcy teraton algorthm, Algorthm The convergence propertes of both methods are nvestgated, and the effects of the scalng factor and penalty parameter as dscussed n the prevous chapter are demonstrated numercally. The results also nclude a comparson of the total teratons, maxmum number of teratons out of all the teratons used wthn each tme step, and the tme used n solvng for the prce of standard and tme-dependent Amercan put optons, usng the drect control and penalty methods. Ths s to ascertan whch of the methods s more effcent. The tests are carred out on MATLAB R2012b on an Intel Core 7 machne wth processor speed 3.1 GHz and 16GB of RAM. 5.1 Standard Amercan put opton A standard Amercan put opton s an Amercan put opton wth payoff max (K S, 0), where K s the strke prce and S s the prce of the underlyng Convergence propertes Usng the data n Table 5.1 and a non-unformly spaced grd wth propertes n Table 5.2, we run Algorthm for both methods, observng the dfferences n the opton prce attaned for the two methods at S = 100, as the non-unform grd S s refned, and ther convergence ratos. Refnement of the grd s acheved by nsertng new fne grds between 20

30 each two coarse grd nodes. At each level of refnement, the tme step control parameter s halved. We also observe the average number of teratons each method uses per tme step, for each level of refnement. The term convergence rato or smply put, rato, gven a specfc level of refnement, ndcates the rato between (a) the change n soluton between the prevous level of refnement and the one before t, and (b) the change n soluton at the current level of refnement and the one before t. Value refers to the opton prce realzed. Itn/Step refers to the average number of teratons used per tme step. Rsk free rate, r Volatlty, σ Maturty tme, T Strke, K tol Smax Table 5.1: Data for Amercan put. Level of Refnement S Nodes Tme steps Table 5.2: Grd/tme step data for convergence study, Amercan put. On each grd refnement, new fne grds are nserted between each two coarse grd nodes, and the tme step control parameter s halved. We can see from Tables 5.3, 5.4, and 5.5 that the convergence ratos for the penalty method decrease as the penalty parameter, ε s ncreased. The decreasng convergence rato can be explaned by the penalzaton error, caused by the term ε(v T LV ) n the penalty formulaton of the LCP n equaton (2.18). As ε 0, the penalty form of the LCP s consstent wth the LCP tself. But as ε ncreases, the penalzaton error ncreases, thus affectng the soluton and the order of convergence. Mathematcally, the scalng factor does not affect the soluton to the opton prcng problem. As dscussed n the prevous chapter, we see from Tables 5.3, 5.4, and 5.5 that the 21

31 Level Penalty Method Drect Control Method Value Rato Itn/Step Value Rato Itn/Step Table 5.3: Convergence study, opton value at S=100. tol = 10 6, ε = 1/Ω = 10 6 τ. Level Penalty Method Drect Control Method Value Rato Itn/Step Value Rato Itn/Step Table 5.4: Convergence study, opton value at S=100. tol = 10 6, ε = 1/Ω = 10 2 τ. Level Penalty Method Drect Control Method Value Rato Itn/Step Value Rato Itn/Step Table 5.5: Convergence study, opton value at S=100. tol = 10 6, ε = 1/Ω = 1 τ. sze of the scalng factor affects how fast the polcy teraton algorthm, Algorthm converges to a soluton, n terms of the number of teratons used per tme step. 22

32 We also see from Tables 5.3, 5.4, and 5.5 that the teratons per tme step are generally lower for the drect control method, for a good choce of the scalng parameter. Ths ndcates that the drect control method may be more effcent. We study the performance n detal n the next subsecton Performance Here, we observe the teratons per step used for both drect control and penalty methods n detal. We study the maxmum and total number of teratons used by these methods to solve the Amercan opton prcng problem at dfferent levels of refnement. Both constant and varable tme steps are used on a non-unform grd, densely populated n the neghborhood of K, for 7 levels of refnement. Data used to carry out these tests s gven n Table 5.6. M here refers to the total number of tme steps, and N, the number of spatal nodes. Max Itns refers to the maxmum number of teratons used wthn a tme step out of all the teratons per tme step. Total Itns refers to the total number of teratons used. Elapsed Tme s measured n seconds and s the total tme used to solve for the prce of an opton at a gven level of refnement. A note of cauton to the reader, concernng the elapsed tme, s to not rely too heavly on the tme used as a means to judge the performance of both methods, snce the tme used s dependent on the algorthm used, as well as the actual computer program used to run the algorthm, assumng both programs are run on the same machne. Concernng the algorthms, t s more expensve to compute the control for the drect control method (equaton (4.3)), than t s for the penalty method (equaton (4.7)). However, t s less expensve to solve the lnear system wthn the polcy teraton algorthm for the drect control method, snce the matrx n (4.2) can be created so that t s more sparse than the matrx n (4.6). The elapsed tme therefore can be used as a good measure of comparson between tests for the same method, but not so relable a measure for tests between dfferent methods, as the methods are mplemented wth dfferent computer programs and wth dfferent controls and lnear systems whch requre dfferent computatonal tmes to solve. The value of the total teratons used s a better measure to be used for comparsons of the performances of both methods. We observe from Tables 5.7 and 5.8 that the number of maxmum teratons, total teratons, and tme taken for the drect control and penalty methods ncrease wth ncreasng 23

33 r σ T Smax tol ε or 1/Ω K τ 100 Table 5.6: Parameters for drect control and penalty methods, teratons study. Drect Control Method Max Itns Total Itns Elapsed Tme M, N = M, N = M, N = M, N = M, N = M, N = M, N = Penalty Method Max Itns Total Itns Elapsed Tme M, N = M, N = M, N = M, N = M, N = M, N = M, N = Table 5.7: Maxmum teratons study wth constant tme steps. N. Ths suggests that the speed of convergence to a soluton depends on N, the number of spatal nodes. The dependence of the speed of convergence on N can be seen n the total number of teratons, as they roughly ncrease by a factor of two, as the grd s refned and N s ncreased approxmately by a factor of two. The tme used for both methods does not roughly double, lke N and the total number of teratons do, for the followng reason: For a partcular method, a dfferent sparse lnear system s solved for each teraton, dependng on the control used (equaton (4.3) or (4.7)). The tme requred to solve each sparse lnear system depends on the number of nonzero elements, and so may vastly dffer across teratons and tme steps. As a result, the total tme used may not be proportonal to the total number of teratons. It can also be seen that when both constant and varable tme steps are used, the penalty method starts off wth a lower number of total teratons, but as grd and tme step sze are 24

34 Drect Control Method Max Itns Total Itns Elapsed Tme M = 44, N = M = 86, N = M = 170, N = M = 337, N = M = 671, N = M = 1337, N = M = 2669, N = Penalty Method Max Itns Total Itns Elapsed Tme M = 44, N = M = 86, N = M = 170, N = M = 337, N = M = 671, N = M = 1337, N = M = 2669, N = Table 5.8: Iteratons study. Varable tme steps. On grd refnement, tme step control parameter s halved, and new fne grds are nserted between each two coarse grd nodes. Intally, dnorm = 0.2, τ = refned, the number of teratons used grow at a faster rate than that of the drect control method, resultng n a hgher number of total teratons than the drect control method. Even though the penalty method uses more teratons than the drect control method for constant tme steps, t uses less tme. Ths behavor does not occur when varable tme steps are used. The maxmum teratons used s a key element n fndng out what happens to the total number of teratons n Tables 5.7 and 5.8 as the grd sze ncreases. Usng the drect control method, changng the data n Table 5.6, and the grd, by modfyng ts sze and nter-node dstances, numercal experments reveal that the drect control method uses ts hghest number of teratons n ts frst tme step. The penalty method does ths at latter tme steps, close to the end of the lfetme of the opton. The free boundary s the dvdng pont between the regon to exercse and the regon to hold the opton. Both methods move one node per teraton [21] untl the free boundary s reached. However, the drect control method always starts to move from the exercse boundary, the pont on the grd where the strke prce s, whle the penalty method starts 25

35 a lttle closer to the optmal soluton for that tme step. Ths dfference n startng ponts s due to the set of controls used (equatons (4.3) and (4.7)). As a result, the drect control method always uses more teratons n the frst tme step to arrve at the same ntermedate soluton as the penalty method. After that tme step, both methods behave n a smlar manner, arrvng at the same ntermedate soluton at the end of each tme step, wth the drect control method sometmes usng one less teraton than the penalty method. Ths explans why the penalty method uses a lesser number of maxmum teratons than the drect control method, but may end up wth a hgher number of total teratons. Table 5.9 compares the number of teratons per tme step, usng data from Table 5.6, on the same grd used to carry out the tests n ths subsecton. Tme step # Drect control Penalty Table 5.9: Iteratons per step wth strke, K = 100, N = 801. Varable tme steps used. Clearly, from Tables 5.7 and 5.8, the use of varable tme steps s more advantageous n comparson to that of constant tme steps. It requres a smaller number of teratons and uses less tme. For ths problem, usng varable tme steps, we can say that the drect control method s more effcent than the penalty method. 5.2 Tme-dependent Amercan constrant The numercal examples so far have used a fxed value for the strke prce K, for the Amercan constrant. We refer to ths as a standard Amercan constrant. A tme-dependent Amercan constrant s one n whch the strke prce, K, s a functon of tme. For ths constrant, we change Algorthm slghtly by usng wthn our whle loop, values of K(τ) between K (0) = 100 and K (T ) = 200, dependng on the tme left untl the end of the algorthm. Gven τ and T, we approxmate K (τ) usng lnear nterpolaton. The next set of tests, usng the above constrant, smulate the practcal problem of prcng a convertble bond wth accrued nterest effects [1, 12]. We test the performance of the drect control and penalty methods for the smulatons. 26

36 5.2.1 Performance The non-unform grd from the prevous secton s modfed for the tests n ths secton, by makng t densely populated n the neghborhood of the possble strke prce values, 100 K (τ) 200. Data used to carry out these tests s gven n Table M refers to the total number of tme steps and N, the number of spatal nodes. Max Itns refers to the maxmum number of teratons used wthn a tme step out of all the teratons per tme step. Total Itns refers to the total number of teratons used. Elapsed tme s measured n seconds and s the total tme used to solve for the prce of an opton at a specfed level of refnement. r σ T Smax tol ε=1/ω K(τ) τ K(0) + τ T (K (T ) K (0)) Table 5.10: Parameters: drect control and penalty methods, tme-dependent Amercan constrant study. We see from Tables 5.11 and 5.12 that usng constant tme steps, the penalty method uses a lower number of total teratons and tme than the drect control method. For ths constrant the drect control and penalty methods do not always move one node per teraton untl convergence. There s a pont after whch they both move one node per teraton untl the free boundary s reached. We call the nodes from that pont the path to the free boundary. Before that pont, the drect control mostly moves one node per teraton. The penalty method may move one node per teraton, or even more, especally for large grd szes. As a result, the penalty method s more lkely to use a lesser number of total teratons than the drect control method does. We realze through the numercal experments conducted on ths problem that the penalty method s able to converge to a soluton, much faster than the drect control method. It uses a lesser amount of total teratons and tme. 5.3 Concluson For a standard Amercan put opton, the drect control method solves the dscretzed HJB equaton usng less teratons and tme than the drect control method, especally when varable tme steps are used. 27

37 Drect Control Method Max Itns Total Itns Elapsed Tme M, N = M, N = M, N = M, N = M, N = M, N = M, N = Penalty Method Max Itns Total Itns Elapsed Tme M, N = M, N = M, N = M, N = M, N = M, N = M, N = Table 5.11: Tme-dependent Amercan constrant study wth constant tme steps. Other parameters n Table The penalty method outperforms the drect control method, generally usng lesser teratons and tme when both constant and varable tme steps are used for solvng for the opton prce of a smulated convertble bond. In [16], a sem-smooth Newton method for the underlyng contnuous varatonal nequalty n a sutable functon space s analyzed and superlnear convergence shown. Resnger and Wtte n [21] propose that drect control method n some sense s nherently dscrete, and that the penalty method can be seen as a dscretzaton of the teraton n [16] and so has a well-defned lmt for vanshng grd sze. Ths sheds some lght on why the penalty method seems to converge faster than the drect control method, especally notceable n the case of the tme-dependent constrant, as the grd s refned. Overall, the numercal performances of the methods on both problems are very smlar for small grd szes, but the dstncton between ther performances grows clearer as grd data and tme step are refned. The drect control method has the advantage of solvng the exact dscretzed LCP n equaton 2.9, whle the penalty method solves an approxmaton of the LCP, due to the addton of a penalty parameter. Whle ths can be a dsadvantage for the penalty method, the choce of a good penalty parameter can make the error ntroduced 28

38 Drect Control Method Max Itns Total Itns Elapsed Tme M = 31, N = M = 68, N = M = 141, N = M = 286, N = M = 575, N = M = 1154, N = M = 2310, N = Penalty Method Max Itns Total Itns Elapsed Tme M = 31, N = M = 68, N = M = 141, N = M = 286, N = M = 575, N = M = 1154, N = M = 2310, N = Table 5.12: Tme-dependent Amercan constrant study wth varable tme steps. On grd refnement, tme step control parameter s halved, and new fne grds are nserted between each two coarse grd nodes. Intally, dnorm = 0.2, τ = Other parameters n Table by the addton of the penalty term smaller than the error ntroduced by dscretzaton for typcal grd szes [15], so that the penalzaton error may have less mpact on the soluton. A good penalty parameter also causes the error due to round-off to be neglgble. Usng a good penalty parameter, the penalty method has the advantage of beng able to locate the free boundary faster than the drect control method does. Wth that beng sad, the author beleves the use of the penalty method wth a good penalty parameter, should be the preferred method. 29

39 Chapter 6 Proposal In ths chapter, we propose an alternatve way of usng the drect control method to solve for the prce of an Amercan opton, whch makes t slghtly more effcent than t already s. Ths approach s a combnaton of the drect control and penalty methods and so we wll call t a hybrd approach. The hybrd approach s also proposed n [21], where the smlartes of the two methods are explored. 6.1 Approach It has been mentoned earler n ths essay that when solvng for the opton prce of a standard Amercan put opton, the penalty method, for the frst teraton wthn the frst tme step, arrves at a soluton closer to the optmal soluton for that tme step, than the drect control method, due to the dfferent controls used (equatons (4.3) and (4.7)). Ths gves t a good start, causng t to use less teratons n the frst tme step than the drect control method. After the frst tme step, both methods behave n almost the same manner. For the hybrd method, we propose that n that frst teratve step of the frst tme step, we use the control for the penalty method and n the followng tme steps untl the free boundary s found, we use the control for the drect control method. In dong so, we take advantage of the penalty method s good startng pont. The hybrd method should show an mprovement over the drect control method, but the mprovement s not expected to be very great, snce the only modfcaton s only made n the frst tme step of the drect control method. 30