Lecture notes on Stochastic Di erential Equations and Financial Mathematics for the course in Modeling and Simulation

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1 Lecure noes on Sochasic Di erenial Equaions and Financial Mahemaics for he course in Modeling and Simulaion Thomas Önskog Ocober 9, 8 The aim of his par of he course is o give an inroducion o he heory of sochasic di erenial equaions (SDE) and show how his heory can be useful in nancial modelling. Since mos SDEs canno be solved analyically, we shall also sudy some numerical mehods for sochasic di erenial equaions. We wish o solve he following problem. Problem Le X () denoe he price per uni of a given asse a ime. Consider a xed nancial derivaive, for example he European call opion, which gives he owner he righ, bu no he obligaion, o buy a uni of he asse a a speci ed price K a a xed fuure ime T. The payo of he opion is (X (T ) K) + : X (T ) K; if X (T ) > K; ; if X (T ) K: () We would like o deermine he arbirage free price of his derivaive a ime given ha he price per uni of he asse is hen X () x. By arbirage free we mean ha neiher he seller nor he buyer of he opion should be able o gain money wih probabiliy one. In order o deermine he price we need:. A mahemaical model for he price process (he Black-Scholes equaion). Mahemaical echniques o handle he price dynamics (sochasic calculus) Sochasic processes and he Wiener process We model he sock price X () as a sochasic process. A sochasic process is a funcion (rajecory) X : [; )! R ha evolves randomly over ime (Here is a space of oucomes, meaning ha he randomness of X is deermined by he choice of! ). In paricular, we wan X o be a Markov process, which means ha given he presen value, he fuure behaviour of X is independen of he pas values of X. In nancial erms his means ha all pas informaion is incorporaed in he presen sock price. This is known as marke e ciency and is a reasonable assumpion. Now le X be any sochasic process and le d be an in niesimal ime sep. Then, we de ne he incremen of X over he ime inerval [; + d] as dx () X ( + d) X () : () For any xed, he incremen dx () is a random variable. If he incremens dx (s) and dx () over disjoin inervals [s + ds] and [ + d] are independen, we say ha X has independen incremens. If he disribuion of every incremen is normal, we say ha X is a Gaussian process. Equipped wih hese de niions we are ready o de ne he Wiener process, which is he sochasic process describing he behaviour of he well-known Brownian moion

2 De niion The Wiener process W : [; )! R is a Gaussian sochasic process wih independen incremens for which. W () wih probabiliy ;. E [W () W (s)] ; h 3. E (W () W (s)) i s: From he de niion i follows ha he densiy of W () W (s) is given by x p W () W (s) (x) p exp ; (3) ( s) ( s) for all > s. One can show ha he rajecories of he Wiener process are coninuous, bu a he same ime nowhere di ereniable. Sochasic inegrals Le (; x) and (; x) be wo given funcions, denoed drif and di usion coe ciens below. The sochasic process X is a sochasic di erenial if is local dynamics is given by dx () (; X ()) d + (; X ()) dw () ; (4) X () X : (5) This means ha during he ime inerval [; + d] he process X is driven by a locally deerminisic velociy (; X ()) and an independen Gaussian disurbance erm ampli ed by he facor (; X ()). How should we rea his expression mahemaically? If we divide boh sides formally by d, we obain he sochasic di erenial equaion dx () d (; X ()) + (; X ()) () ; (6) where dw () () ; (7) d is he formal ime derivaive of he Wiener process. Bu as we noed above he Wiener process is nowhere di ereniable and hence does no exis as an ordinary funcion. Insead we wrie (4) in inegraed form as X () x + Z (s; X (s)) ds + Z (s; X (s)) dw (s) : (8) The lefmos inegral is a Riemann inegral for each rajecory of X, bu he righmos inegral is no even a Lebesgue-Sieljes inegral, since he coninuous rajecories of he Wiener process can be shown o vary in niely on any bounded ime inerval. We need a new heory for his sochasic inegral (also known as an Iô inegral). The main resricion on he inegrands (; x) in he Iô inegral is ha hey are only allowed o depend on he pas and presen behaviour of he Wiener process. We shall sae his crierion more precisely below. De niion 3 Le W be he Wiener process. Then we le F W be a se conaining all informaion generaed by he Wiener process up o he ime. If a sochasic variable Z is compleely deermined by F W, we wrie Z F W.

3 De niion 4 Le W be he Wiener process. F W : if Then a sochasic process X is adaped o he lraion X () F W ; for all : (9) Inuiively speaking, an adaped process does no look ino he fuure, so for example X () Z W (s) ds and X () sup W (s) ; () s are adaped o F W :, whereas X 3 () sup W (s) ; () s+ is no. De niion 5 For < T < he space L T f : [; T ]! R such ha. f is F W -adaped,. h E (f ()) i d < ; of Iô inegrable funcions conains all measurable funcions The Iô inegral is de ned in wo seps - rs for simple funcions and hen for general funcions in L T. The subse ST of simple funcions in L T conains all funcions g L T, for which here exiss a deerminisic pariion a < < ::: < n b, such ha g is piecewise consan, ha is For g S T we de ne Z b a g () g k ; for k < k+ : () nx g () dw () g k (W ( k+ ) W ( k )) : (3) For general f L T, here exiss a sequence g (n), n g(n) ST, such ha as n!. For each g (n) he Iô inegral Z b a k E g (n) () f () d! ; (4) Z n Z b a g (n) () dw () ; (5) is well de ned, by means of (3). Since L T random variable. Hence, we can de ne is a complee space, he random variables Z n converge o a Z b a f () dw () lim n! Z b a g n () dw () : (6) The Iô inegral sais es he following relaions. Noe ha he rs wo relaions provides a mean o deermine he expecaion and variance of an Iô inegral. 3

4 Theorem 6 (Properies of he Iô inegral) For any f L T, < T <, he Iô inegral sais es h R i T. E f () dw () ; R T. E f () dw () R T h(f E ()) i d; 3. R T g () dw () F W T : 3 Sochasic calculus Consider a sochasic di erenial X given by dx () (; X ()) d + (; X ()) dw () : (7) Le he funcion U (; x) be given and de ne he sochasic process Y by Y () U (; X ()) : (8) Can Y be expressed as a sochasic di erenial? The answer o his quesion is a rmaive and is provided by he Iô formula, which we prove below. Consider an in niesimal Wiener incremen dw () W ( + d) W (). By de niion, we know ha E [dw ()] and V ar (dw ()) d. (9) and, furhermore, we can use he densiy of he Wiener process o derive h E (dw ()) i d and V ar (dw ()) 3 (d) : () This means ha o rs order in d, he sochasic par of (dw ()) (quani ed by he variance) is negligible compared o he deerminisic par (quani ed by he expecaion). Hence, in a rs order expansion we can consider (dw ()) o be a deerminisic variable equal o d and we deduce he following rules of muliplicaion: (d) d dw () dw () d and (dw ()) d: () Nex, we apply hese rules o a second order Taylor expansion of dy (). dy () du (; d dx () (d) d dx () U (dx d @U U ( d + dw ()) (3) (d) d ( d + dw ()) d + dw () U ( d + dw ()) (4) (d) (d) U d dw () (d) + d dw () + (dw ()) (6) d dw () ; (7) where he rules of muliplicaion were used in he las sep and he evaluaion poin (; X ()) of U and is derivaives were suppressed. This proves he Iô formula, saed below. 4

5 Theorem 7 (Iô formula) Le X be a sochasic di erenial given by dx () (; X ()) d + (; X ()) dw () ; (8) and le U : [; T ] R! R be a wo imes coninuously di ereniable funcion. Then Y () U (; X ()) is a sochasic di erenial dy () (; X ()) + (; X (; X ()) + (; X U (; X ()) d + (; X (; X ()) dw () : (9) Alernaively we can wrie @U (; X ()) d + (; X ()) dx () U (; X ()) (dx ()) ; (3) and use he rules of muliplicaion () o deermine (dx ()). Example 8 In his example we derive he soluion o he sochasic di erenial equaion dn () a () N () d + b () N () dw ; (3) known as geomeric Brownian moion. Noe ha his he sochasic counerpar o he exponenial growh model dn () a () N () ; (3) d which has he soluion! N (T ) N () exp a () d As a consequence, we expec he soluion o (3) o be a randomly disurbed exponenial funcion. Indeed, dividing boh sides of (3) by N () and inegraing, we obain dn () N () a () d + b () dw () ) : Z dn () T N () a () d + b () dw () : (33) Guided by he form of he inegral erm on he lef hand side of he second equaliy of (33), we apply he Iô formula o he funcion U (; x) log x; (34) and arrive a d (log @U (; N ()) d + (; N ()) dn () + + N () dn () + (dn ()) N () dn () N () (b () N ()) N () d dn () N U (; N ()) (dn ()) (b ()) d: (35) Hence Z dn () T N () (f ()) d (log N ()) + d log N () N () + (b ()) d: (36) 5

6 Combining (33) and (36), we ge or log N () N () a () d + b () dw () N (T ) N exp a () which in he special case of consan a and b reduces o N (T ) N () exp a One can show ha E [N (T )] E [N ()] exp Z T (b ()) d; (37) Z! T (b ()) d + b () dw () ; (38) a () d, so, as expeced, he expecaion of N (T ) coincides wih he soluion o he deerminisic exponenial growh model. 4 The Black-Scholes equaion b T + bw (T ) : (39)! The geomeric Brownian moion can be applied o model sock prices. Hence, le X () denoe he price per uni of a given sock a ime and suppose ha X is given by dx () () X () d + () X () dw () ; (4) where ; : [; T ]! R are some given funcions. We inerpre he drif coe cien as he ineres rae per uni ime and he di usion coe cien as he volailiy, ha is he srengh of he ucuaions of he marke. The price of a European call opion for he sock X can be found by he Black-Scholes equaion. Theorem 9 (Black-Scholes equaion) The price p (; x) of a European call opion for a sock saisfying (4) wih iniial condiion X () x, is given by he parial di erenial equaion wih ( () p (; x) + () x (; x) + (; x) () p (; x) ; for < < T; x p (T; x) (x K) + ; for x R: (4) Here K is he price, for which he owner of he opion may purchase one uni of he sock X a ime T. Proof. For simpliciy, we prove his heorem for consan and. The proof is similar for non-consan and. Le X be given by (4) and assume he exisence of a risk-free paper B saisfying Consider he porfolio replicaing he opion db () B () d; (4) I () p (; X ()) + () X () + () B () ; (43) for () ; () R. We assume ha he porfolio is self- nancing in he sense ha no money is brough in or aken ou of he porfolio. Mahemaically his can be wrien d ( () X () + () B ()) () dx () + () db () ; (44) 6

7 for he porfolio () X () + () B (). For noaional clariy we suppress he evaluaion poin (; X ()) of p and is derivaives in he calculaions below. Applying he Iô formula o I (), we obain di () dp + d ( () X () + () dx p (dx ()) + () dx () + () @ X p (X ()) + () X () + () B () + () X () dw () : (45) Now choose () so ha he porfolio becomes riskless (only he deerminisic par remains). This can be achieved by seing Then @p X p (X ()) + () X () + () B () p (X ()) + () B () d: (46) If we now suppose ha he opporuniy of arbirage is precluded, so ha he probabiliy of gaining money wihou risk is zero, hen we mus have di () I () d, so by combining (43) and (46), (X ()) + () B () d di () I () d ( p + () X () + () B ()) d p X () + () B () d; (47) which, afer rearranging he erms and seing x X (), reduces o he parial di + + p p; (48) for < T, where T is he exercise ime of he opion. A ime T he cos of exercising he opion is K so he erminal condiion corresponding o (48) mus be and he proof is complee. p (T; x) (x K) + ; (49) For non-consan and we canno be cerain ha here exiss a analyical soluion o he Black-Scholes equaion and o deermine he opion price we mus use numerical mehods for parial di erenial equaions, such as nie di erenial or nie elemen mehods. Anoher alernaive is o use he represenaion formula supplied by he Feynman-Kac formula derived in he nex secion. 5 Connecing sochasic and parial di erenial equaions Here we derive a represenaion in erms of sochasic di erenial equaions o he soluion o he following parial di erenial equaion wih prescribed iniial (; x) + AF (; x) ; for < < T; x F (T; x) (x) ; for x R; (5) 7

8 where he di erenial operaor A is given by AF (; x) (; (; x) + ( F x)) (; x) : (5) Le F be a soluion o (5), x he poin (; x) and de ne he process X as he sochasic di erenial We apply he Iô formula on F (s; X (s)) and obain F (T; X (T )) + df (s; X (s)) dx (s) (s; X (s)) ds + (s; X (s)) dw (s) ; (5) X () x: (53) F (; (s; X (s)) + (s; X (s; X (s)) + ( (s; F (s))) (s; X (s; X (s)) dw (s; X (s)) + AF (s; X F is he soluion o (5), so (54) reduces o (X (T )) F (; x) Taking expecaions of boh sides, we deduce (s; X (s)) ds + (s; X (s; X (s)) dw (s) : (54) (s; X (s; X (s)) dw (s) : (55) F (; x) E ;x [ (X (T ))] ; (56) where he indices ; x means ha he process X sais es X () x. Formula (56) represens he soluion o (5) as an expecaion of a funcional of he soluion o a sochasic di erenial equaion. Similarly we can show ha he soluion o he parial di erenial equaion (; x) + AF (; x) (x) F (; x) ; for < < T; x F (T; x) (x) ; for x R; (57) is given by F (; x) E ;x " (X (T )) exp (X (s)) ds!# ; (58) where X sais es (5). This resul is known as he Feynman-Kac formula. Comparing (57) wih A given by (5) o he Black-Scholes equaion (4), i is clear ha he wo parial di erenial equaions coincide if we se F (; x) p (; x) and (x) (x K) +. Hence, by he Feynman-Kac formula, he opion price can be represened p (; x) E ;x (X (T ) K) + exp ( i (T )) : (59) Since X is a geomeric Brownian moion, whose exac soluion is known and was derived in Example 8, we can wrie p (; x) as follows. 8

9 p (; x) E xe (r + )(T )+W T K exp ( (T )) Z R xe (r )(T )+y + K exp ( (T )) p (T exp ) y dy; (6) (T ) where we used he densiy of he Wiener process in he las sep. If (6) canno be solved analyically, he represenaion (59) neverheless makes i possible o approximae he opion price using numerical mehods for sochasic di erenial equaions. We devoe he nal secion o his ask. 6 Numerical mehods Firs, we de ne he simples numerical approximaion for sochasic di erenial equaions, he Euler approximaion. Consider he sochasic di erenial dx (; X ) d + (; X ) dw ; (6) on T, wih iniial value X. For a given ime discreizaion < < ::: < N T of he inerval [; T ], he Euler approximaion of X, is a sochasic process X de ned by X n+ X n + n ; X n (n+ n ) + n ; X n Wn+ W n ; (6) for n ; ; :::; N wih iniial value X X. We shall use he noaion n n+ n ; (63) for he nh ime incremen and max n; (64) n for he maximal ime incremen. We will only consider he case of equidisan pariions, so n TN; (65) for every n ; ; ; :::; N. The main di erence compared o he deerminisic Euler approximaion for ordinary di erenial equaions is ha an incremen W n W n+ W n of he Wiener process has o be generaed a every ime sep. We know ha hese incremens are Gaussian wih mean zero and variance n, so W n can be drawn from he disribuion N (; n ). In all applicaions considered in his course we will be ineresed in approximaing E [g (X T )], ha is expecaions of funcionals of he soluion X. As his does no require knowledge of he whereabous of he rajecories of X bu merely knowledge of he disribuion of X, his case is known as weak approximaion. Le X T;j, for j M, be M samples of X T, for example calculaed by means of he recursive relaion in (6). We use hese M samples o consruc he sample mean M MX g X T;j ; (66) j which we can use as an approximaion of E [g (X T )]. The error arising by his approach can be decomposed ino wo pars as E [g (X T )] M MX g X T;j E g (XT ) g X T + E MX g X T g X T;j : (67) {z } M j discreizaion error {z } saisical error j 9

10 The discreizaion error is deerminisic and deermined by he choice of ime discree approximaion X. The saisical error is almos he same for all choices of X and is variance can be shown o decrease as M. In addiion here is always he risk ha he resul is in uenced by roundo errors, especially in he case of non-linear equaions. However, in mos cases, an immense amoun of rajecories is required o obain a low saisical error so he roundo error can ofen be negleced in comparison o he oher wo ypes of error. To compare he performance of di eren ime discree approximaion, we shall use he concep of weak convergence order. De niion Le X be a ime discree approximaion of X wih maximum sep size. We say ha X converges weakly o X a ime T wih order if, for each g C, where C is some class of es funcions g : R! R, here exiss a consan C, independen of, and a consan > such ha E g (XT ) g X T C ; (68) for each (; ). I can be shown ha he weak convergence order of he Euler approximaion is. As i, for weak convergence, su ces ha he disribuion of X ends o he disribuion of X, i is no necessary o use Gaussian variables in he Euler approximaion and in fac i holds ha he simpli ed weak Euler approximaion given by X n+ X n + n ; X n n + n ; X n c Wn ; (69) where he c W n is he wo-poin disribued random variable wih P W c n p n ; (7) also converges weakly wih order. For a sochasic di erenial given as dx (X ) d + (X ) dw ; (7) one can show ha he following order weak Iô-Taylor scheme converges weakly wih order. X n+ X n + n + W n + + (n ) n W n + (W n ) n ; (7) where all funcions are evaluaed a X n. One drawback wih he order weak Iô-Taylor scheme is ha i requires derivaives of he drif and di usion coe ciens o be calculaed a every ime sep. Jus as for ordinary di erenial equaions here are Runge-Kua like numerical mehods which avoid he use of derivaives. The simples such approximaion is he explici order weak scheme X n+ X n + (X ) + X n n + (X + ) + (X ) + X n W f n + (X + ) (X ) 4 f W n 4 n p ; (73) n

11 wih supporing values and X X n + X n n + X n f Wn ; (74) X X n + X n n X n p n : (75) Here W f n may be chosen as a N (; n ) random variable or as he hree-poin disribued random variable given by P W f n p 3 n 6 ; P W f n 3 : (76) In his scheme we need o evaluae he drif coe cien a wo poins and he di usion coe cien a hree poins per ime sep. The simples weak implici Euler approximaion X n+ X n + n+ ; X n+ n + n ; X n c Wn ; (77) where W c n is chosen according o (7) is weakly convergen wih order. Weak implici approximaions are paricularly useful in he case of si sochasic di erenial equaions. The predicor-correcor mehods are wo-sep versions of implici schemes and are mainly used due o heir numerical sabiliy. In he rs sep of a predicor-correcor mehod we predic he value of X o be ex n+ by, for example, he Euler approximaion. In he second sep we correc his value by applying an implici approximaion made explici hrough he use of he prediced value X e n+ insead of X n+ on he righ hand side of he implici scheme. In he case of a sochasic di erenial given by (7) he simples order predicor-correcor mehod has he correcor X n + exn+ X n+ X n + n + X n Wn c ; (78) and predicor where c W n is chosen according o (7). ex n+ X n + X n n + X n c Wn ; (79) A di eren way o obain high orders of weak convergence is o use exrapolaion mehods. Le X denoe a ime discree approximaion wih equidisan ime sep. Suppose ha we have simulaed he funcional h i E g X T ; (8) wih an order weak approximaion, for example he weak Euler or simpli ed weak Euler approximaion. Suppose furher ha we repea he simulaion wih he ime sep o simulae he funcional h i E g : (8) X T Then he following combinaion of (8)-(8) V g; E h i g X T h E g X T i ; (8) in an order weak exrapolaion. Similarly saring from an order weak approximaion, he order 4 weak exrapolaion is given by Vg;4 h i 3E g X T h E g X T i h + E g X 4 T i : (83)

12 7 References. Tomas Björk, Sochasic Calculus, Lecure noes for he course Numerical Mehods for Sochasic Di erenial Equaions given in Sockholm and Lund, Sweden, spring. hp:// Jonahan Goodman, Kyoung-Sook Moon, Anders Szepessy, Raúl Tempone, and Georgios Zouraris. Sochasic and Parial Di erenial Equaions wih Adapive Numerics. Technical repor, Royal Insiue of Technology (KTH), Sockholm, Sweden, Peer E. Kloeden and Eckhard Plaen. Numerical Soluion of Sochasic Di erenial Equaions. Springer Bern Øksendal. Sochasic Di erenial Equaions, An Inroducion wih Applicaions. Springer, 5h ediion, 998.