Saisics 441 (Fall 214) November 19, 21, 214 Prof Michael Kozdron Lecure #31, 32: The Ornsein-Uhlenbeck Process as a Model of Volailiy The Ornsein-Uhlenbeck process is a di usion process ha was inroduced as a model of he velociy of a paricle undergoing Brownian moion We know from Newonian physics ha he velociy of a (classical) paricle in moion is given by he ime derivaive of is posiion However, if he posiion of a paricle is described by Brownian moion, hen he ime derivaive does no exis The Ornsein-Uhlenbeck process is an aemp o overcome his di culy by modelling he velociy direcly Furhermore, jus as Brownian moion is he scaling limi of simple random walk, he Ornsein-Uhlenbeck process is he scaling limi of he Ehrenfes urn model which describes he di usion of paricles hrough a permeable membrane In recen years, however, he Ornsein-Uhlenbeck process has appeared in finance as a model of he volailiy of he underlying asse price process Suppose ha he price of a sock {S, } is modelled by geomeric Brownian moion wih volailiy and drif µ so ha S saisfies he SDE ds = S db + µs d However, marke daa indicaes ha implied volailiies for di eren srike prices and expiry daes of opions are no consan Insead, hey appear o be smile shaped (or frown shaped) Perhaps he mos naural approach is o allow for he volailiy funcion of ime so ha S saisfies he SDE ds = ()S db + µs d () o be a deerminisic This was already suggesed by Meron in 1973 Alhough i does explain he di eren implied volailiy levels for di eren expiry daes, i does no explain he smile shape for di eren srike prices Insead, Hull and Whie in 1987 proposed o use a sochasic volailiy model where he underlying sock price {S, } saisfies he SDE and he variance process {v, ds = p v S db + µs d } is given by geomeric Brownian moion dv = c 1 v db + c 2 v d wih c 1 and c 2 known consans The problem wih his model is ha geomeric Brownian moion ends o increase exponenially which is an undesirable propery for volailiy Marke daa also indicaes ha volailiy exhibis mean-revering behaviour This lead Sein and Sein in 1991 o inroduce he mean-revering Ornsein-Uhlenbeck process saisfying dv = db + a(b v )d where a, b, and are known consans This process, however, allows negaive values of v 2 1
In 1993 Heson overcame his di culy by considering a more complex sochasic volailiy model Before invesigaing he Heson model, however, we will consider he Ornsein- Uhlenbeck process separaely and prove ha negaive volailiies are allowed hereby verifying ha he Sein and Sein sock price model is flawed We say ha he process {X, } is an Ornsein-Uhlenbeck process if X saisfies he Ornsein-Uhlenbeck sochasic di erenial equaion given by dx = db + ax d (21) where and a are consans and {B, } is a sandard Brownian moion Remark Someimes (21) is called he Langevin equaion, especially in physics conexs Remark The Ornsein-Uhlenbeck SDE is very similar o he SDE for geomeric Brownian moion; he only di erence is he absence of X in he db erm of (21) However, his sligh change makes (21) more challenging o solve The rick for solving (21) is o muliply boh sides by he inegraing facor e a and o compare wih d(e a X ) The chain rule ells us ha d(e a X )=e a dx + X d(e a )=e a dx ae a X d (22) and muliplying (21) by e a gives so ha subsiuing (23) ino (22) gives e a dx = e a db + ae a X d (23) d(e a X )= e a db + ae a X d ae a X d = e a db Since d(e a X )= e a db,wecannowinegraeoconcludeha and so e a X X = X = e a X + e as db s e a( s) db s (24) Observe ha he inegral in (24) is a Wiener inegral Definiion 81 ells us ha e a( s) db s N, e ( s) ds = N, e 1 In paricular, choosing X = x o be consan implies ha X = e a x + e a( s) db s N xe a, 2 (e 1) 2 2
Acually, we can generalize his slighly If we choose X N(x, 2 )independenlyof {B, }, henexercise312ellsusha X = e a X + e a( s) db s N xe a, 2 e 2 (e 1) + = N xe a, 2 2 + e 2 Exercise 21 Suppose ha {X, } is an Ornsein-Uhlenbeck process given by (24) wih X = Ifs<,compueCov(X s,x ) We say ha he process {X, saisfies he SDE } is a mean-revering Ornsein-Uhlenbeck process if X dx = db +(b X )d (25) where and b are consans and {B, } is a sandard Brownian moion The rick for solving he mean-revering Ornsein-Uhlenbeck process is similar Tha is, we muliply by e and compare wih d(e (b X )) The chain rule ells us ha and muliplying (25) by e gives so ha subsiuing (27) ino (26) gives d(e (b X )) = e dx + e (b X )d (26) e dx = e db + e (b X )d (27) d(e (b X )) = e db e (b X )d + e (b X )d = e db Since d(e (b X )) = e db,wecannowinegraeoconcludeha and so e (b X ) (b X )= X =(1 e )b + e X + e s db s Exercise 22 Suppose ha X N(x, 2 )isindependenof{b, disribuion of X given by (28) e s db s (28) } Deerminehe Exercise 23 Use an appropriae inegraing facor o solve he mean-revering Ornsein- Uhlenbeck SDE considered by Sein and Sein, namely dx = db + a(b X )d Assuming ha X = x is consan, deermine he disribuion of X and conclude ha P{X < } > forevery> Hin: X has a normal disribuion This hen explains our earlier claim ha he Sein and Sein model is flawed 2 3
As previous noed, Heson inroduced a sochasic volailiy model in 1993 ha overcame his di culy Assume ha he asse price process {S, } saisfies he SDE ds = p v S db (1) + µs d where he variance process {v, } saisfies dv = p v db (2) + a(b v )d (29) and he wo driving Brownian moions {B (1), } and {B (2), } are correlaed wih rae, ie, dhb (1),B (2) i = d The p v erm in (29) is needed o guaranee posiive volailiy when he process ouches zero he sochasic par becomes zero and he non-sochasic par will push i up The parameer a measures he speed of he mean-reversion, b is he average level of volailiy, and is he volailiy of volailiy Marke daa suggess ha he correlaion rae is ypically negaive The negaive dependence beween reurns and volailiy is someimes called he leverage e ec Heson s model involves a sysem of sochasic di erenial equaions The key ool for analyzing such a sysem is he mulidimensional version of Iô s formula Theorem 24 (Version V) Suppose ha {X, } and {Y, } are di usions defined by he sochasic di erenial equaions and dx = a 1 (, X,Y )db (1) dy = a 2 (, X,Y )db (2) + b 1 (, X,Y )d + b 2 (, X,Y )d, respecively, where {B (1), } and {B (2), } are each sandard one-dimensional Brownian moions If f 2 C 1 ([, 1)) C 2 (R 2 ), hen df(, X,Y )= f(, X,Y )d + f 1 (, X,Y )dx + 1 2 f 11(, X,Y )dhxi where he parial derivaives are defined as + f 2 (, X,Y )dy + 1 2 f 22(, X,Y )dhy i + f 12 (, X,Y )dhx, Y i f(, x, y) = @ f(, x, y), @ f 1(, x, y) = @ @x f(, x, y), f 11(, x, y) = @2 f(, x, y) @x2 f 2 (, x, y) = @ @y f(, x, y), f 22(, x, y) = @2 @y f(, x, y), f 12(, x, y) = @2 f(, x, y), 2 @x@y and dhx, Y i is compued according o he rule dhx, Y i =(dx )(dy )=a 1 (, X,Y )a 2 (, X,Y )dhb (1),B (2) i 2 4
Remark In a ypical problem involving he mulidimensional version of Iô s formula, he quadraic covariaion process hb (1),B (2) i will be specified However, wo paricular examples are worh menioning If B (1) = B (2),hendhB (1),B (2) i =d, whereasifb (1) and B (2) are independen, hen dhb (1),B (2) i = Exercise 25 Suppose ha f(, x, y) = xy Using Version V of Iô s formula (Theorem 24), verify ha he produc rule for di usions is given by d(x Y )=X dy + Y dx +dhx, Y i Thus, our goal in he nex few lecures is o price a European call opion assuming ha he underlying sock price follows Heson s model of geomeric Brownian moion wih a sochasic volailiy, namely 8 >< ds = p v S db (1) + µs d, dv = p v db (2) + a(b v )d, >: dhb (1),B (2) i = d 2 5