Computational Finance

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1 Department of Mathematcs at Unversty of Calforna, San Dego Computatonal Fnance Dfferental Equaton Technques [Lectures 8-10] Mchael Holst February 27, 2017 Contents 1 Modelng Fnancal Optons wth the Black-Scholes Equaton (Lecture 8) 2 11 Dfferental Equaton Models of Fnancal Optons 2 12 Basc Ideas and Varables for Modelng Standard Optons 3 13 The Black-Merton-Scholes Model of Standard Optons 5 14 Dervaton of the Black-Scholes Equaton Usng Ito s Lemma 6 15 Reformulaton of Black-Scholes for Numercal Soluton 7 2 Fnte Dfference Methods for the Black-Scholes Equatons (Lectures 9-10) 9 21 Fnte Dfference Methods: Basc Formulaton and Propertes 9 22 Explct and Implct Methods for Parabolc Equatons Stablty, Consstency, Convergence Amercan Optons as a Free Boundary Problem Software Packages for usng Fnte Dfference Methods 17 3 Advanced Dfferental Equaton Technques (Lecture 10) Fnte Element Methods for the Black-Scholes Equatons Exotc Optons and the Black-Scholes Framework Nonlnear Optons Models Numercal Methods for Exotc Optons 17 References 17 ccomucsdedu/~mholst/teachng/ 1

2 1 Modelng Fnancal Optons wth the Black-Scholes Equaton (Lecture 8) Recall that by an opton, we mean the rght to buy (a call opton) or sell (a put opton) one unt of a potentally rsky underlyng asset (typcally a stock) wthn an agreed upon fxed perod T at an agreed upon fxed prce K per unt By rght, we mean we have the opton, but we we do not an oblgaton to carry out the transacton Snce an opton (and related fnancal nstruments) are bult and prced based on an underlyng asset, they are refered to as a partcular type of (fnancal) dervatve An opton s smply a contracton between a seller (or wrter), and a buyer (or holder); the holder purchases the opton from the wrter, payng what s called a premum In ths way, optons are effectvely a fnancal nstrument for bettng on a stock prce ether rsng or fallng What value should the seller charge for the premum? What premum should a buyer be wllng to pay? In ths secton, we wll develop dfferental equaton models of fnancal optons to help us answer these questons An outlne of ths secton s as follows In Secton 11, we wll look at the basc deas behnd modelng optons wth dfferental equatons, based on the Black-Merton-Scholes model of the market We wll derve the Black- Scholes equaton usng Ito s Lemma, and then examne how the Black-Scholes equaton can be used to model standard optons as well as exotc optons We wll also look at some nonlnear optons models that go beyond the basc Black-Scholes equaton Snce the Black-Scholes equaton cannot be solved n closed form except for the smplest stuatons, n Secton 2 we wll develop basc some basc algorthms for solvng the Black-Scholes equatons numercally usng a computer Ths wll requre that we understand a lttle bt more about the Black-Scholes equatons, and about how to develop and use numercal methods for equatons of ths type Our focus n Secton 2 wll be on fnte dfference methods, ncludng ther basc formulaton and ther propertes Some key concerns are stablty, consstency, and convergence, and ths wll lead us to look at two dfferent types of methods, namely explct and mplct methods One of our key concerns wll be how to treat boundary condtons; whle the Black-Scholes model of European optons ft ncely nto the standard formulaton of fnte dfference methods, one of the dffcult features of usng the Black-Scholes model for Amercan optons s the appearance of a free boundary We wll examne some technques for handlng free boundares numercally, and look at some of the avalable software tools such as MATLAB Most of our dscusson of fnancal optons, and ther modelng usng dfferental equatons, follows [4, 2] The materal on ordnary and partal dfferental equatons, and numercal methods for ther soluton, may be found n [1, 3] 11 Dfferental Equaton Models of Fnancal Optons We now develop some basc deas for modelng fnancal optons usng dfferental equatons Let us frst make some defntons for contnung our prelmnary dscusson above: opton: rght to buy or sell one unt of underlyng asset wthn fxed tme at fxed cost underlyng: the asset on whch the opton s bult wrter: the seller of the opton holder: the buyer of the opton premum: the cost of the opton T: the maturty date of the opton K: the strke (or exercse) prce payoff: value of opton at strke tme t call opton: rght to buy underlyng for agreed prce K by date T put opton: rght to sell underlyng for agreed prce K by date T ccomucsdedu/~mholst/teachng/ 2

3 The contract between the wrter and the holder s generally as follows, although some of the detals wll vary between dfferent types of optons, such as Amercan, European, and Asan optons Requred Wrter Actons (at any tme 0 t T ): calls: Must delver underlyng for strke prce K (at any t T ) puts: Must buy underlyng for strke prce K (at any t T ) Possble Holder Actons (at any tme 0 t): Retan opton (do nothng) Sell opton at current market prce on market exchange (at any t < T ) Exercse the opton (at any t T ) Let the opton expre as worthless (at any t T ) The basc dea s that the wrter sells the opton, and the premum they receves compensates the rsk they take The holder buys the opton, and the premum they pay s for the chance to exercse the opton at the rght tme to earn back more than the premum In these notes we wll follow the approach n [4, 2] and buld mathematcal models optons prmarly from the perspectve of the holder (the models can also be rewrtten so as to be from the perspectve of the wrter) The dstnctons between Amercan, European, and Asan (and other exotc optons) are manly concernng these three features: 1) One underlyng asset, or a basket of underlyng assets 2) Permtted strke (exercse) tmes: any t [0, T ], or only at t = T 3) Instantaneous payoff, or a path-dependent payoff The common types of optons are then: Standard (Vanlla) Opton: one underlyng wth nstantaneous payoff Exotc Opton: an opton whch s not a standard opton Amercan Opton: A standard opton that can be exercsed at any t, 0 t T European Opton: A standard opton that can be exercsed only at t = T Asan Opton: An exotc opton, wth one or a basket of underlyngs, where the payoff depends on the average value of the asset durng opton lfetme Barrer Opton: An exotc opton, wth one or a basket of underlyngs, where the payoff depends on whether stock prce hts a prescrbed barrer durng opton lfetme We are gong to focus manly on standard optons n the sense lsted above, although all of the technques we wll develop can (and have been) extended to exotc optons, and more complex optons models 12 Basc Ideas and Varables for Modelng Standard Optons For the moment, let us restrct the dscusson to standard optons, whch from our dscusson above means: 1) One underlyng asset 2) Instantaneous payoff At ths pont we need to ntroduce some addtonal notaton n order to make thngs a bt more precse Frst, there are four key parameters that play a role n buldng a dfferental equaton model of an opton Each parameter s real-valued, meanng that t s a sngle real number for a partcular nstance of the model Ths wll be mportant from a modelng perspectve, snce t wll allow us to consder these parameters as all takng values n the set of real numbers R ccomucsdedu/~mholst/teachng/ 3

4 Parameter K T r σ Meanng The strke (or exercse) prce The maturty date of the opton (n years) Contnuously compounded rsk-free nterest rate (per year) Annual volatlty of the underlyng asset (per year) The parameters r and σ are market parameters that affect the prce of the opton V (t) The parameter r s the rsk-free nterest rate (per year) For example, f r = 007, then the nterest rate s 7% The parameter σ s the volatlty of the prce S(t), whch can be defned as the stndard devaton of fluctons n S(t), whch s typcally dvded by T for scalng purposes For example, f σ = 04, then the volatlty of 40% We wll assume that both r and σ are constant throughout the lfetme of the opton, so that r, σ R are fxed real numbers for all t [0, T ] Smlarly, for a partcular opton beng modeled, both K, T R are fxed real numbers for all t [0, T ] There are three varables that play a role n a dfferental equaton model of an opton The frst s the ndependent real varable t, takng values of tme (measured n years) n the range 0 t T, whch we wrte as t [0, T ] Next s the value of the underlyng asset, denoted as S(t), whch s a tme-dependent functon that represents nput to our model of the opton It s a real-value functon of ts sngle real ndependent varable t, and we equvalently wll wrte S(t), or S t, or smply as S The thrd varable s the man dependent varable we wsh to develop a model for, namely the value of the opton V Ths s a real-value functon of ts two varables S and t; we denote ths as V (S, t) to emphasze ts dependence on both t and S(t) However, t s also a functon of the four parameters K, T, r, and σ above, so that n prncpal we should thnk about V as a functon of all sx (the two varables and four parameters), or V = V (S, t; K, T, r, σ) Varable Meanng t The current tme t [0, T ] (n years) S(t) Spot (current, nstantaneous) prce of the underlyng asset S at tme t [0, T ] V (S, t) Instantaneous value V of the opton for asset S at tme t [0, T ] For a ratonal nvestor (e, an nvestor tryng to maxmze proft and mnmze costs), the value of a standard call opton, denoted V c, at ts exercse tme of t [0, T ] s clearly: { } 0, f St K (opton s worthless) V c (S t, t) = (11) S t K, f S t > K (opton s exercsed) Smlarly, for a ratonal nvestor, the value of a standard put opton, denoted V p, at ts exercse tme of t [0, T ] s clearly: { } 0, f St K (opton s worthless) V p (S t, t) = (12) K S t, f S t < K (opton s exercsed) These are called the payoff functons We can wrte the payoff functons more compactly as: V c (S t, t) = max{s t K, 0}, (call payoff functon) (13) V p (S t, t) = max{k S t, 0} (put payoff functon) (14) However, t s useful to defne the followng functon: g + = max{g, 0}, g R (15) ccomucsdedu/~mholst/teachng/ 4

5 Ths allows an even more compact representaton: V c (S t, t) = (S t K) +, (16) V p (S t, t) = (K S t ) + (17) Note that payoff functons of ths type are pontwse, meanng that ther value depends only on the current stock prce S t = S(t) at the exercse tme t [0, T ] Path-dependent payoff functons are used for exotc optons, and they cannot be wrtten n ths form snce ther values depend on the value of the stock S(t) at other tme ponts t [0, T ] The payoff functons (16) (17) are for standard optons (one underlyng asset wth nstantaeneous payoff) Amercan optons allow for exercsng at any t [0, T ], so that the payoff s then smply the value of these functons evaluated at tme t European optons requre exercsng only at t = T, so that ther payoff functons are always (16) (17) evaluated at t = T, or: V c (S T, T ) = (S T K) +, (18) V p (S T, T ) = (K S T ) + (19) When there s a need to draw a dstncton between European and Amercan optons (and payoffs), we wll denote then as V E for European optons, and V A for Amercan optons A ratonal nvestor wll only exercse the opton when t s proftable, leadng to the form of the payoff functons (16) (17) Snce European optons can only be exercsed at t = T, there s less opportunty to maxmze proft/mnmze cost when compared to Amercan optons Ths s a type of a pror bound or nequalty that relates the two opton values, and the opton values wth the call and stock values The followng a pror bounds can be derved for European and Amercan optons (V wth no superscrpt apples to both types of optons): 0 V c (S T, T ) S t, (110) 0 V p (S T, T ) K, (111) V E (S t, t) V A (S t, t), (112) S t K Vc A (S t, t), (113) K S t Vp A (S t, t), (114) Vp E (S t, t) Ke r(t t) (115) 13 The Black-Merton-Scholes Model of Standard Optons The Black-Merton-Scholes model of the market nvolves the followng basc assumptons: Assumpton 11 (Black-Merton-Scholes Market Model) The opton s a standard European opton, and: There s no arbtrage It s a frctonless market The underlyng asset prce obeys a geometrc Brownan moton The nterest rate and volatlty r and σ are constant for all t [0, T ] The no arbtrage assumpton or prncple s that arbtrage opportuntes are not avalable (or not allowed) A fctonless market means that there are no transacton fees, that the cost of lendng and borrowng money s equal, that all nformaton s avalable mmedately wth no delay, and that all assets and credts are avalable at any tme and n any amount The assumpton of constant nterest rate and volatlty s an dealzaton, but necessary of ths model ccomucsdedu/~mholst/teachng/ 5

6 The assumpton that the underlyng asset obeys a geometrc Brownan moton means that t follows ds t = µs t dt + σs t dw t, (116) wth constant µ and σ Ths s a specal (lnear) case of an Itǒ moton: dx t = a(x t, t)dt + b(x t, t)dw t, (117) where W t s a Wener process (tself a very specal case of an Itǒ moton, dx t = dw t, e a = 0 and b = 1) A dvdend wth contnuous yeld δ 0 can be ncluded nto the Brownan moton: ds t = (µ δ)s t dt + σs t dw t, (118) whch produces decrease n S at each nterval dt by amount δsdt 14 Dervaton of the Black-Scholes Equaton Usng Ito s Lemma Imagne that we have a portfolo at tme t wth α t shares of an asset wth value S t, and β t shares of bond B t Assume the bound s rskless so that At tme t, the wealth process of the portfolo s: db t = rb t dt (119) P t = α t S t + β t B t (120) We wsh the portfolo to hedge a European opton wth value V t and payoff V T at maturty t = T Therefore, we am to construct α t and β t so that the portfolo replcates the paoff, e P T = V T = payoff (121) Snce a European opton cannot be traded before maturty, our portfolo wll be assumed to be closed, wth no nvestment and no payout before t = T Ths means our portfolo has the self-fnance property: dp T = α t ds t + beta t db t (122) Thus, the change n value P t s due only to changes n the prces of S and B Note that together wth the no arbtrage assumpton, (121) (122) mply P t = V t, t [0, T ], (123) snce both have the same payout at t = T Therefore, the replcatng and self-fnancng portfolo s equvalent to the European opton, and duplcates the rsk of the opton Assume now that the value functon P t = V (S, t) s suffcently smooth (n fact, we need V C 2,1, meanng twce contnuously dfferentable n S, and ones contnuously dfferentable n t) Ths gves us access to Itǒ s Lemma, whch s bascally a stochastc verson of the followng (determnstc) chan rule: d g g(x(t), t) = dt x dx dt + g t The standard statement of ths Lemma s the followng ccomucsdedu/~mholst/teachng/ 6

7 Lemma 12 (Itǒ s Lemma) Suppose X t s an Itǒ process, and let g C 2,1 Then Y t = g(x t, t) s also an Itǒ process wth the same Wener process W t, and ( g dy t = x a + g t ) g 2 x 2 b2 + g x bdw t, where the dervatves and coffcents depend on X t and t If we apply Itǒ s Lemma to our geometrc Brownan moton (118) we obtan ( dp = µs V S + V t + 1 ) 2 σ2 S 2 2 V S 2 dt + σs V dw (124) S However, f we combne (118), (119), and (122) we obtan an alternate expresson: dp = (αµs + βrb) dt + ασsdw (125) The coeffcents n these two expressons for dp must match; matchng the dw coffcents gves an equaton for α: α t = V (S t, t) (126) S Matchng the dt coeffcents gves a relaton for β, where the Wenter term drops out The term βb can be elmnated usng (120) and (123) va: S V S Ths then gves fnally the Black-Scholes equaton + βb = V (127) V t σ2 S 2 2 V V + (r δ)s S2 S rv = 0, 0 < S <, 0 t < T, (128) V (T ) = P T (129) 15 Reformulaton of Black-Scholes for Numercal Soluton Now that we have the Black-Scholes equaton, how do we solve t n order to determne a good prcng for the opton, ether as a wrte or a holder? Unfortunately, except for very specal smplfed models, we cannot solve the Black-Scholes equaton (128) n closed form, meanng that we generally cannot wrte down a functn V (S, t) that solves (128) (129) Instead, what we wll now do s develop numercal methods, payng attenton to how they are desgned so that they are both accurate and relable The frst thng we need to do s reformulate (128) (129) nto a mathematcally equvalent form so that t s n a more standard form for use wth numercal methods To ths end, we wll use the followng change of varables: S = Ke x, t = T 2τ σ 2, q = 2r σ 2, q 2(r δ) δ = σ 2 (130) Under ths transformaton, the value functon V (S, t) can be wrtten n terms of the new varables x and τ as: ( V (S, t) = V Ke x, T 2τ ) σ 2 = v(x, τ), (131) where v(x, τ) = Ke Q y(x, τ), Q = 1 2 (q δ 1)x ( 1 4 (q δ 1) 2 + q)τ (132) ccomucsdedu/~mholst/teachng/ 7

8 Ths then gves us fnally the followng ntal value problem for the unknown functon y(x, τ): Some observatons: y τ = 2 y x 2, < x <, 0 τ < 1 2 σ2 T (133) 1) We frst solve (133) for y(x, τ), then form v(x, τ) va (132), and then fnally recover V (S, t) va (131) 2) The backward equaton for V (S, t) s now a forward equaton for y(x, τ) 3) The fnal tme for τ s now τ = 1 2 σ2 T 4) The ntal condtons for a standard call are: wth Usng the dentty V (S, T ) = (S K) + = K max{e x 1, 0} = Ke x 2 (q δ 1) y(x, 0), y(x, 0) = e x 2 (q δ 1) max{e x 1, 0} = we have the ntal condtons for y(x, τ): { e x 2 (q δ 1) (e x 1), x > 0, 0 x 0 e x 2 (q δ 1) (e x 1), = e x 2 (q δ+1) e x 2 (q δ 1), call: y(x, 0) = max{e x 2 (q δ+1) e x 2 (q δ 1), 0} (134) put: y(x, 0) = max{e x 2 (q δ 1) e x 2 (q δ+1), 0} (135) 5) Truncatng the nfnte doman < x < + s necessary for use of (most) numercal methods; we would lke to restrct the doman to a fnte nterval x mn x x max Ths can be based (for European optons) on Put-Call Party: S + V E p whch can be shown to hold Ths gves: V E c = Ke r(t t), V c (S, t) = S Ke r(t t), S, (136) V p (S, t) = Ke r(t t) S, S 0 (137) For European optons, one uses as boundary condtons the followng: where for a call opton, one has: and for a put opton, one has: y(x, τ) = r 1 (x, τ), x, (138) y(x, τ) = r 2 (x, τ), x +, (139) r 1 (x, τ) = 0, (140) r 2 (x, τ) = e 1 2 (q δ+1)x+ 1 4 (q δ+1) 2τ, (141) r 1 (x, τ) = e 1 2 (q δ 1)x+ 1 4 (q δ 1) 2τ, (142) r 2 (x, τ) = 0 (143) These are known as Drchlet boundary condtons, and are the last pece of the puzzle we need for developng numercal methods for the Black-Scholes equaton ccomucsdedu/~mholst/teachng/ 8

9 2 Fnte Dfference Methods for the Black-Scholes Equatons (Lectures 9-10) Snce the Black-Scholes equaton cannot be solved n closed form except for the smplest stuatons, n ths secton we wll develop basc some basc algorthms for solvng the Black-Scholes equatons numercally usng a computer Ths wll requre that we understand a lttle bt more about the Black-Scholes equatons, and about how to develop and use numercal methods for equatons of ths type Our focus n ths secton wll be on fnte dfference methods, ncludng ther basc formulaton and ther propertes Some key concerns are stablty, consstency, and convergence, and ths wll lead us to look at two dfferent types of methods, namely explct and mplct methods One of our key concerns wll be how to treat boundary condtons; whle the Black- Scholes model of European optons ft ncely nto the standard formulaton of fnte dfference methods, one of the dffcult features of usng the Black-Scholes model for Amercan optons s the appearance of a free boundary We wll examne some technques for handlng free boundares numercally, and look at some of the avalable software tools such as MATLAB 21 Fnte Dfference Methods: Basc Formulaton and Propertes To develop a basc understandng of fnte dfference methods, let us frst examne a smplfed verson of (133) that s tme-ndependent: d2 u = f, dx2 x (a, b) R, (21) u = u a, at x = a, (22) u = u b, at x = b (23) When there s only one ndependent varable (x n ths case), t s sometmes convenent to use the smplfed notaton u = du dx and u = d2 u, n whch case (21) would read: dx 2 u = f, x (a, b) R (24) As we noted earler n the notes, equaton (21) (or (24)), together wth the boundary condtons (22) (23) s referred to as a Boundary Value Problem (BVP) n Ordnary Dfferental Equatons (ODE) Snce the boundary condtons nvolve the two endponts of the nterval [a, b], ths problem s also sometmes called a Two-Pont Boundary Value Problem (n ODE) We would lke to fnd the unknown functon u(x), where u: [a, b] R, where we are gven the nterval [a, b] R wth a < b, we are gven the model functon f(x), where f : [a, b] R, and where we are also gven the known boundary values of the functon we are after, namely u(a) = u a and u(b) = u b If the functon f(x) were partcularly smple, such as f t were a polynomal or a trgonometrc functon, then technques exst that allow for determnng u(x) s closed form, meanng that we can smply wrte t down as an explct functon of f and the other nput data However, for the functon f whch would arse n mathematcal models of varous physcal or other phenomena, we would have no choce but to develop some approxmaton to the soluton u, such as s provded by numercal methods One of the smplest and easest numercal methods to understand and work wth are fnte dfference methods Let us now develop a basc fnte dfference method for (21) (23) Let us frst recall how we defned the dervatve when we frst encountered t n the frst course on the calculus: du dx = lm h 0 u(x + h) u(x) (25) h ccomucsdedu/~mholst/teachng/ 9

10 If we stop short n the lmt process and just make sure that h s small enough so that we get close to the dervatve, then we have a fnte dfference approxmaton to the dervatve: du(x) dx u(x + h) u(x) (26) h How good (e, accurate) s ths approxmaton? Snce we have revewed a bt about Taylor expanson earler n the notes, we can actually answer ths queston precsely To do so, we smply Taylor expand the term u(x + h) appearng above: u(x + h) u(x) h = [u(x) + u (x)h + u (x(ξ)) h2 2! ] u(x) h = u (x)h + u (x(ξ)) h2 2 h (27) (28) = u (x) + u (x(ξ)) h 2, (29) where we have used the shorthand notaton = u (x) + O(h), (210) u = du dx, u = d2 u dx 2, and where x(ξ) s the pont n the nterval (u, u+h) that allows the Taylor remander to appear as the sngle term nvolvng u (see our earler dscusson about Taylor expanson and remanders) Therefore, ths fnte dfference s a frst-order approxmaton to u Notce that we have also confrmed that n fact t does actually approxmate u, snce we showed that the dfference formula can be wrtten as u plus an error that goes to zero as h goes to zero Here s another dfference formula known as the centered dfference approxmaton to u : du(x) dx u(x + h) u(x h) (211) 2h We can agan check the accuracy, ths tme expandng both terms; let us just examne the numerator frst: u(x + h) u(x h) = [u(x) + u (x)h + u (x) h2 2! + u (ξ 1 ) h3 3! ] (212) [u(x) u (x)h + u (x) h2 2! u (ξ 2 ) h3 3! ] (213) = 2u (x)h + [u (ξ 1 ) + u (ξ 2 )]h 3 (214) = 2u (x)h + 2u (ξ)h 3 (215) = 2u (x)h + O(h 3 ), (216) where we have used the observaton that f u (x) s contnuous on [a, b], then there exsts ξ such that u(ξ) = [u (ξ 1 ) + u (ξ 2 )]/2, allowng us to combne the two Taylor remander terms nto a sngle term If we now look at the actual dfference formula, we fnd: u(x + h) u(x h) 2h = 2u (x)h + 2u (ξ)h 3 2h = u (x) + u (ξ)h 2 = u (x) + O(h 2 ) (217) Therefore, the centered dfference (211) also approxmates u, but t s second order: the error term goes to zero as h 2 ccomucsdedu/~mholst/teachng/ 10

11 By wrtng down more complex dfference formulas, possbly nvolvng more than two ponts, we can derve hgher and hgher order approxmatons to u However, our dfferental equaton (24) nvolves the second dervatve u ; s ther an analogous fnte dfference formula for a second dervatve? Yes, n fact there s a multtude of dfference formulas that can be derved for approxmatons of varous dervatves; the one we wll fnd most useful for the second dervatve s also a centered dfference formula, but now for approxmatng u : u(x + h) 2u(x) + u(x h) h 2 = u (x) + O(h 2 ) (218) To confrm that t has ths order of accuracy, t s agan a smple Taylor expanson of the two terms u(x + h) and u(x h) and then a cancellaton and combnng of terms, as we dd above (Feel free to check ths!) To use ths approxmaton to the dervatve throughout the nterval [a, b], our frst step s to create a fnte dfference grd or mesh of ponts out of the set of ndependent varable x: a = x 0 < x 1 < x 2 < < x n < x n+1 = b, x = a + h, h = b a n + 1 (219) Ths s a unform mesh of h ponts, snce the separaton between each of the mesh ponts s exactly h = (b a)/n + 1 If we now use the notaton u u(x ), then (218) reads u (x ) = u +1 2u + u 1 h 2 + O(h 2 ) (220) If we use ths expresson to appoxmate our dfferental equaton (24) at the sngle pont x, then we have: u +1 2u + u 1 h 2 = f + O(h 2 ), (221) where we have used the notaton f = f(x ) If we drop the O(h 2 ) error term n (221), and nstead fnd u satsfyng u +1 2u + u 1 h 2 = f, (222) then we no longer fndng u(x ) exactly; nstead we are buldng an approxmaton: u u(x ) As we have seen above, through Taylor expanson we can characterze the accuracy of ths approxmaton If we multply (222) through by h 2, and consder fndng such a u for each pont = 1,, n, then we have a lnear system of equatons for the u : u u u 1 = h 2 f, = 1,, n (223) Note that ths s a set of n equatons n n unknowns u, for = 1,, n Note that the left-hand sde of each equaton n (223) for = 2,, n 1 only nvolves unknowns u, where s between 1 and n However, the equatons for = 1 and = n are specal: u 2 + 2u 1 u 0 = h 2 f 1, (224) u n+1 + 2u n u n 1 = h 2 f n (225) ccomucsdedu/~mholst/teachng/ 11

12 In partcular, the pont u 0 n (224) and the pont u n+1 n (225) are actually known, and not part of the set of unknowns; they are gven by the boundary condtons (22) (23) If we use these condtons and move that known nformaton to the rght-hand sdes, then these two specal equatons become: u 2 + 2u 1 = h 2 f 1 + u a, (226) 2u n u n 1 = h 2 f n + u b (227) All together then we have a system of n equatons n n unknowns, whch approxmates both the dfferental equaton (21) and the boundary condtons (22) (23): u 2 + 2u 1 = h 2 f 1 + u a, (228) u u u 1 = h 2 f, = 2,, n 1, (229) We can wrte ths as the followng matrx system: 2u n u n 1 = h 2 f n + u b (230) AU = F, (231) where A R n n and U, F R n are defned as: A =, U = u 1 u 2 u n 1 u n, F = f 1 + u a f 2 f n 1 f n + u b (232) One can show that the matrx A has some partcularly nce propertes: 1) It s tr-dagonal; the only non-zero entres are on the man dagonal and the frst superand sub-dagonal 2) It s symmetrc: A j = A j 3) It s postve defnte: x T Ax > 0, x 0 We would need to solve ths matrx system (usng MATLAB s bultn solver, or usng another computer algorthm) to produce our approxmaton n the vector U R n, gvng fnally u u(x ), = 1,, n + 1, (233) wth of course equalty at the two end ponts = 0 and = n + 1, snce the values of u 0 and u n+1 are the known boundary values As we dscussed earler n the notes, tme-ndependent problems of ths type nvolvng more than one ndependent varable are called ellptc partal dfferental equatons Ellptc equatons represent models of statonary (tme-ndependent) phenomena, such as electrostatcs, elastostatcs, and so forth The BVP n ODE we dscussed above s the smplest case, and can be vewed as a one-dmensonal ellptc equaton The smplest and most common ellptc equaton s known as the Posson Equaton: u = f, x Ω R d, (234) u = 0, x on Ω (235) ccomucsdedu/~mholst/teachng/ 12

13 As we noted earler n the notes, equaton (234) together wth the boundary condtons (235) s referred to as a Boundary Value Problem (BVP) n Partal Dfferental Equatons (PDE) Here, Ω R d, known as the spatal doman, s the set over whch the ndependent varable x R d s allowed to range (For example, Ω mght be the sphere of radus one centered at the orgn, known as the unt sphere, or t mght be a cube) The set Ω s the boundary of Ω (eg the surface of the unt sphere, or the boundary of a cube) The unknown functon u s a real-value functon of the d varables x = [x 1,, x d ] T, so can be vewed as a functon of the form u: Ω R d R The functon f appearng on the rght n (234) s a forcng functon, and s a functon of the same form as u, so that f : Ω R d R The forcng functon represents the partcular mathematcal model, and drves the behavor of the soluton u The frst equaton (234) s the PDE, and the second equaton (235) s the boundary condton The fnal symbol appearng n (234) that we need to defne s the Laplacean operator ; t s just the mult-dmensonal analogue of the second dervatve appearng n our BVP n ODE (24) above: u = d =1 2 u x 2 (236) If there s only one ndependent varable, so that Ω = (a, b) R, then the Laplacean s reduced to smply the second dervatve n that one varable: u = d2 u dx 2 = u, (237) and the Posson equaton (234) (235) s exactly our BVP n ODE above (21) (23) A fnte dfference approxmaton of the mult-dmensonal case s developed exactly as for the one-dmensonal case; for example, f there are now two ndependent varables x and y, and the doman Ω = [a, b] [c, d] R 2, then we would begn by puttng down a dscrete mesh of ponts n both ndependent varables: a = x 0 < x 1 < x 2 < < x n < x n+1 = b, x = a + h x, h x = b a n x + 1, (238) c = y 0 < y 1 < y 2 < < y n < y n+1 = d, y = a + h y, h y = d c n y + 1, (239) where n x s the number of ponts n the x varable, and n y s the the number of ponts n the y varable, gvng the mesh spacngs h x and h y above A unform mesh would take h x = h y = h Dfference approxmatons are developed n the same way; for example, a centered dfference approxmaton to u n the case of two ndependent varables s: u = 2 u x u y 2 = u +1,j 2u,j + u 1,j h 2 where we have used the natural extenson to our earler notaton: + u,j+1 2u,j + u,j 1 h 2 + O(h 2 ), (240) u,j = u(x, y j ) (241) The accuracy of the approxmaton (and verfcaton that t does actually approxmate u) proceeds exactly as above; one performs Taylor expanson n each varable ndependently: u(x + h, y j ) = u(x, y j ) + u(x, y j ) h + x (242) u(x, y j + h) = u(x, y j ) + u(x, y j ) h + y (243) ccomucsdedu/~mholst/teachng/ 13

14 and then smplfes as before If we agan drop the O(h 2 ) error term n (240) and use t as a bass for buldng a system of equatons for an approxmaton u,j u(x, y j ), (244) then we agan produce a lnear system of n equatons n n unknowns: u +1,j 2u,j + u 1,j h 2 or more smply: u,j+1 2u,j + u,j 1 h 2 = f,j,, j = 1,, n, (245) 4u,j u +1,j u 1,j u,j+1 u,j 1 = h 2 f,j,, j = 1,, n (246) Ths s agan a lnear system of the form (231) Unlke the one-dmensonal case, there are now varous optons for orderng the unknowns n the vector U; what s known as the natural orderng vares from 1 to n frst, then ncrement j, then agan vary from 1 to n, untl all unknowns u,j are wrtten as a sngle array U, whch also determnes the orderng of the rght hand sde vector F : U = u 1,1 u 2,1 u n,1 u 1,2 u 2,2 u n,2 u 1,n u 2,n u n,n, F = f 1,1 f 2,1 f n,1 f 1,2 f 2,2 f n,2 f 1,n f 2,n f n,n (247) Ths then fxes the order of the equatons appearng n the matrx A The boundary condtons are ncorporated nto F n exactly the same way as n the one-dmensonal case Whle the matrx A produced n ths was does not have the smple tr-dagonal structure as n the one-dmensonal case, t does have a block tr-dagonal structure One solves the lnear system (231) as before, usng eg the bultn MATLAB solver, or some other algorthm 22 Explct and Implct Methods for Parabolc Equatons So far, we have developed fnte dfference methods for the BVP n ODE (21) (23) and also the BVP n PDE (234) (235) However the Black-Scholes problem took the form of a Parabolc partal dfferental equaton, wth one ndependent varable representng space (or the underlyng asset prce S), and a second ndependent varable representng tme (the lfetme of the opton, taken to be n the range [0, T ]) The fnal form of the Black-Scholes problem we derved earler, ccomucsdedu/~mholst/teachng/ 14

15 together wth the boundary condtons on a fnte doman and an ntal condton n the case of European optons, was: where agan we use the shorthand notaton u t = u xx, x (a, b), t (0, T ] (248) u(0, x) = u 0 (x), x (a, b), t = 0, (249) u(t, x) = u D (t, x), x = a or x = b, t (0, T ], (250) u = u(t, x), u t = u(t, x), u xx = 2 u(t, x) t x 2 (251) The frst equaton (248) s the PDE, whch s just the Black-Scholes equaton after the change of varables outlned earler The second equaton (249) s the ntal condton that we obtaned for European optons (by reversng the tme varable and takng the fnal condton as an ntal condton), and the thrd equaton (250) s the boundary condton we obtaned for the Black-Scholes equaton by approxmatng the unbounded nterval < x < by a bounded nterval a x b Our goal s then to buld a fnte dfference method for solvng ths Intal Boundary Value Problem (IBVP) n PDE If we put down a mesh of ponts n the x and t drectons, we can use the smple notaton: x = a + h, = 0,, n + 1, h = b a n + 1, (252) t j = 0 + jk, j = 0,, m + 1, k = T 0 m + 1, (253) u j = u(t j, x ), = 0,, n + 1, j = 0,, m + 1, (254) where we use the superscrpt n the specal case of the tme varable (Note that ths notaton easly allows for more than one spatal varable, whch we would handle wth addtonal subscrpts) Our approach wll be to use the frst-order dfference approxmaton we developed earler (26) to approxmate u t, and then to use the second order dfference approxmaton we saw earler (218) to approxmate u xx Puttng these two approxmatons together for the left- and rght-hand sdes of (248) result n: u j+1 u j k The value at the new tme pont = uj +1 2uj + uj 1 h 2, = 1,, n, j = 1,, m (255) u j+1 u(t j+1, x ) (256) s then determned by placng all of the other quanttes on the rght-hand-sde, gvng what s known as the Forward Euler Method: u j+1 = u j + k h 2 [uj +1 2uj + uj 1 ], = 1,, n, j = 1,, m (257) Notce however that we could have chosen to evaluate the terms on the rght at the new tme pont (256), whch gves: u j+1 u j k = uj uj+1 + u j+1 1 h 2, = 1,, n, j = 1,, m (258) ccomucsdedu/~mholst/teachng/ 15

16 Determnng the u j+1 now nvolves solvng a system of equatons, whch after rearrangng looks lke: u j+1 k h 2 [uj +1 2uj + uj 1 ] = uj, = 1,, n, j = 1,, m (259) Ths s known as the Backward Euler Method If we are wllng to solve equatons, then n fact we could consder some type of average of the rght-hand sde at the two tme ponts; ths gves a whole collecton of methods known as theta methods: u j+1 u j = θ k [ u j uj+1 + u j+1 1 h 2 ] + (1 θ) [ u j +1 2uj + ] uj 1 h 2, (260) where θ [0, 1] If θ = 0 then we get the Forward Euler Method, whereas f θ = 1 then we get the Backward Euler Method A very mportant case s θ = 1 2, whch gves the Crank-Ncholson Method: u j+1 u j k = uj uj+1 + u j+1 1 2h 2 + uj +1 2uj + uj 1 2h 2 (261) Placng the unknowns on the left and the known quanttes on the rght, we obtan: u j+1 k 2h 2 [uj uj+1 + u j+1 1 ] = uj k 2h 2 [uj +1 2uj + uj 1 ] (262) Note that the matrx A that arose earler has agan appeared In partcular, f we defne A and U k as follows: 2 1 u k u k 2, U k =, (263) A = then the Forward Euler, Backward Euler, and Crank-Ncholson methods can be wrtten n a smple matrx form as follows: U k+1 = [I + kh ] 2 A U k, (Forward Euler) (264) [I kh ] 2 A U k+1 = U k, (Backward Euler) (265) [ I k ] [ 2h 2 A U k+1 = I + k ] 2h 2 A U k, (Crank-Ncholson) (266) For Forward Euler, we just do a smple matrx-vector multplcaton and a vector addton to produce the new tme approxmaton U k+1 from the prevous tme approxmaton U k In the case of both Backward Euler and Crank-Ncholson, we must solve a lnear system of equatons nvolvng the matrx B = [ I u k n 1 u k n k ] βh 2 A, (267) wth β = 1 for Backward Euler, or β = 2 for Crank-Ncholson, and wth the two dfferent rght-hand sdes as ndcated above ccomucsdedu/~mholst/teachng/ 16

17 I wll lecture on the followng three addtonal topcs Tme permttng, I may also add some wrtten notes on these three topcs, and then repost my notes 23 Stablty, Consstency, Convergence 24 Amercan Optons as a Free Boundary Problem 25 Software Packages for usng Fnte Dfference Methods 3 Advanced Dfferental Equaton Technques (Lecture 10) The followng are four advanced topcs that I may lecture on durng the fnal lecture I wll probably not make wrtten notes coverng these four advanced topcs 31 Fnte Element Methods for the Black-Scholes Equatons 32 Exotc Optons and the Black-Scholes Framework 33 Nonlnear Optons Models 34 Numercal Methods for Exotc Optons References [1] W E Boyce and R C DPrma Elementary Dfferental Equatons and Boundary Value Problems John Wley & Sons, Inc, New York, NY, fourth edton, 1977 [2] R Seydel Tools for Computatonal Fnance Sprnger-Verlag, New York, NY, ffth edton, 2012 [3] I Stakgold and M Holst Green s Functons and Boundary Value Problems John Wley & Sons, Inc, New York, NY, thrd edton, 2011 [4] P Wlmott, S Howson, and J Dewynne The Mathematcs of Fnancal Dervatves Cambrdge Unversty Press, New York, NY, 2009 ccomucsdedu/~mholst/teachng/ 17