A method for Hedging in continuous time

A metho d for hedging in Con tin uous Time Y oa v F reund No v em b er 1, 2018 1 In tro duction This article gives an analysis o f the NormalHedge a lgorithm in c o n tin uous time. The Nor malHedge a lgorithm is des crib ed and a nalyzed in discrete time in [CFH]. The contin uous time analysis is mathematically cleaner, simpler a nd tighter tha n the discrete time a na lysis. T o mo tiv ate the contin uous time fra mework consider the problem of p ortfolio management. Supp ose we are managing N different financial instruments allowed to define a desired distribution o f our wealth among the instruments. W e ignor e the details of the buy and sell order s that hav e to be pla ced in order to r each the desired distribution, we also ignore issues that hav e to do with trans action co sts, buy-sell s pr eads and the like. W e a ssume that at each moment the buy a nd s ell prices for a unit of a par ticular instrumen t are the s ame a nd that there ar e no transa ction costs. Our goal is to find an alg o rithm for manag ing the p or tfolio distribution. In other words, we are lo oking for a mapping from past prices to a distribution over the instruments. As we are consider ing co ntin uous time, the pa st can b e arbitr a rily clos e to the present. F ormally sp eaking, we say that the p or tfolio distr ibution is “causal” or “una nt icipa ting ” to r emov e the p ossibly of defining a po rtfolio which is a function of the future gains as his would clear ly b e a cheat. W e ar e interested in co ns idering contin uous time, b ecause instrument prices can fluctuate very ra pidly . T o mo de l this very ra pid fluctuation we use a type of sto chastic pro cess ca lled an Itˆ o pro cess to mo del the log of the price as a function o f time. Intuitiv ely , an Itˆ o pro cess is a linear co mb inatio n o f a differentiable pro cess and white noise. A mor e fo r mal definition is g iven b elow. T o read more about Itˆ o pro cess es see [O sk03]. Our algorithm and its analysis do not make any additional assumption on the price mov ement o f the instruments. Of co urse, with no a dditio na l assumption we cannot have any guar antees regar ding o ur future wealth. F or example, if the pr ice of a ll of the instruments dec reases a t a particula r momen t by 10%, our wealth will necessa ry decrea se by 10%, r egardles s o f our wealth distr ibution. How ever, s urprisingly enough, we c an g ive a guarantee on the r e gr et asso c ia ted with our metho d without any additional assumptions . Regret quantifies the difference betw een our wealth at time t and the wealth we would have had if we inv ested all of our money in the b est one of the N ins tr ument s. Sp ecifically , denote the log price of instrument i at time t by X i t and assume that the initial unit price for all instruments is one, i.e. X i 0 = log (1) = 0 . Let G t be the log of our wealth at time t . W e define our regre t a t time t as R t = max i =1 ,...,N X i t − G t . Int uitively , the r egret is larg e if by inv esting all of our money in a particula r instr ument (whose identit y is known only in hind-sight) we w ould hav e made muc h more money than what we actually have. The main result of this pap er is a n algor ithm for which the r egret is b ounded by p 2 ct (ln N + 1) wher e c is the amount of r andom fluctuatio ns (white noise) in the instr ument price s . It sta nds to reaso n that the b o und dep ends on the num b er of instruments N , b eca use the la r ger the p o tential impact of price fluctuations. If the instr umen t prices are ar e all simple r andom walks (brownian motion in contin uous time) the e x pe c ted pr ice of the b est instrument is prop ortional to √ ct ln N . 1 W e can g et p otentially tig hter b ounds if w e consider the set of b est instruments. Suppo se w e sort X i t for a par ticular time t from the larg est to the s ma llest. W e say that the ǫ − q uanti l e of the pr ices is the v alue x such that ⌊ ǫN ⌋ of X i t are lar ger than x . W e prov e a b ound of p 2 ct (ln(1 /ǫ ) + 1) on the reg ret of our algorithm relative to the ǫ - quantile for a ny ǫ > 0 . This gener a lized b o und can b e used when we ar e hedging ov er an infinite, even uncountably infinite set of instruments. Case in p oint. Supp o se that the instruments that we are combining ar e themselves p ortfolio s. The set of fixed p or tfolio s ov er N > 1 ins truments co nsists of the N − 1 dimensio nal simplex, which is a n uncountably infinite set. Cov er’s universal p or folios a lgorithm [Cov91 , CO96] uses this set as the s et o f p or tfolios to b e combined. W e can apply our algorithm to this set and guarantee that our reg ret relative to the top ǫ - quantile of fixed re balanced po rtfolios is s mall. 2 Hedging in Con tin uous time The por tfolio mana gement problem is defined as follows. Let X i t for i = 1 , . . . , N define the log-prices of N instrumentss as a function of time. The initial unit price is one, thus X i 0 = 0. X i t is an Itˆ o pro ces s. More sp ecifically , le t dW t denote a n N dimensional Wiener pr o cess where e a ch co or dinate is an indep endent pro cess with unit v ariance . Then the differential of the total g ain co rresp onding to the i th instr ument is : dX i t = ˆ a i ( t ) dt + N X j =1 ˆ b i,j ( t ) dW j t (1) Where ˆ a i ( t ) , ˆ b i,j ( t ) are adapted (non-anticipatory) sto chastic pro ces ses that are Itˆ o-integrable with resp ect to W j t . In mathematical finance terms ˆ a i corres p o nd to pric e drift and ˆ b i,j corres p o nd to diffusion or pric e volatility . The volatility o f the i th instrument at time t is defined as ˆ V i ( t ) = N X j =1  ˆ b i,j ( t )  2 (2) and the maxima l volitilit y at time t is defined a s V M ( t ) = max i ˆ V i ( t ) (3) An “ a ggreg ating strategy” is a n p ortfolio management p olicy that defines how to distr ibute the wealth among the N instrumen ts as a function of their past p er formance. Mathematically sp eak ing, a p olicy is a sto chastic pr o cess that is a n a dapted (non-anticipativ e) function of the pas t prices X i t N i =1 . Given an instantiation of the sto chastic pro cesses X i t N i =1 the a ggreg ating strategy defines N stochastic pr o cesses P i t such that for all t P i t ≥ 0 a nd P i P i t = 1. The cumulativ e gain o f the master a lgorithm is defined to b e 0 a t t = 0 , for t ≥ 0 it is defined by the differential dG t = N X j =1 P i t dX j t The regr et of the ma ster algor ithm relative to the i th instr ument is defined to b e zer o at t = 0 and is otherwise defined by the differential dR i t = dX i t − dG t And we can co m bine the last t wo equatio ns to get: N X i =1 P i t dR i t = 0 (4) 2 As R i t is a linear combination of X i t it is also an Itˆ o pr o cess and ca n b e expre s sed as dR i t = a i ( t ) dt + N X j =1 b i,j ( t ) dW j t (5) where a i ( t ) = ˆ a i ( t ) − N X k =1 P k t ˆ a k ( t ) (6) and b i,j ( t ) = ˆ b i,j ( t ) − N X k =1 P k t ˆ b k,j ( t ) (7) Similarly to X i t we define the diffusion r a te of R i t to b e V i ( t ) = N X j =1  b i,j ( t )  2 (8) W e prove a n uppe r b ound on V i ( t ) Lemma 1 ∀ t, V i ( t ) ≤ 2 V M ( t ) Pro of: W e use b j and ˆ b j the N dimensional vectors  b 1 ,j , . . . , b N ,j  and D ˆ b 1 ,j , . . . , ˆ b N ,j E resp ectively .Using this notation we rew r ite Eq uations (7) and (8) as b i ( t ) = ˆ b i ( t ) − N X k =1 P k t ˆ b k ( t ); V i ( t ) = k b i ( t ) k 2 2 Equations (3) and (8) imply that k ˆ b i ( t ) k 2 2 ≤ V M ( t ) for all i . It follows tha t the no rm of the co nvex combination is als o b ounded:      N X k =1 P k t ˆ b k ( t )      2 2 = N X k =1 P k t    ˆ b k ( t )    2 2 ≤ V M ( t ) F rom which it follows that      ˆ b i ( t ) − N X k =1 P k t ˆ b k ( t )      2 2 ≤ 2 V M ( t ) 3 Normalhedge NormalHedge is a par ticular ag grega ting stra tegy which is defined as follows. W e define a p otential function tha t dep ends o n tw o v a r iables, x a nd c : φ ( x, c ) = ( exp  x 2 2 c  ( x > 0 ) 1 ( x ≤ 0 ) 3 W e will use the following pa rtial der iv atives o f φ ( x, c ): φ ′ ( x, c ) . = ∂ ∂ x φ ( x, c ) = ( x c exp  x 2 2 c  ( x > 0) 0 ( x ≤ 0) φ ′′ ( x, c ) . = ∂ 2 ∂ x 2 φ ( x, c ) = (  1 c + x 2 c 2  exp  x 2 2 c  ( x > 0) 0 ( x < 0) and φ c ( x, c ) . = ∂ ∂ c φ ( x, c ) = ( − x 2 c 2 exp  x 2 2 c  ( x > 0 ) 0 ( x ≤ 0 ) The NormalHedge s tr ategy is defined by the following conditio ns that should ho ld for ev ery t ≥ 0. If R i t ≤ 0 for all 1 ≤ i ≤ N then P i t = 1 / N . Other wise P i t and c ( t ) ar e defined by the following equations. 1 N N X i =1 φ ( R i t , c ( t )) = e (9) P i t = φ ′ ( R i t , c ( t ) P N j =1 φ ′ ( R j t , c ( t ) (10) 4 Analysis W e intro duce a new notion of reg ret. F or a given time t we order the cumulative gains X i t for i = 1 , . . . , N from highest to low est and define the r e gr et of the agr e gating s t r ate gy t o the top ǫ -quant ile to be the differe nc e betw een G ( t ) and the ⌊ ǫN ⌋ -th element in the sorted list. Lemma 2 At any time t , the r e gr et to the b est instru ment c an b e b oun de d as: max i R i,t ≤ p 2 c ( t )(ln N + 1) Mor e over, for any 0 ≤ ǫ ≤ 1 and any t , the re gr et to the top ǫ - quantile of instrument s is at most p 2 c ( t )(ln(1 /ǫ ) + 1) . Pro of: The first part of the lemma follows fro m the fact that, for any i ∈ E t , exp  ( R i,t ) 2 2 c ( t )  = exp  ([ R i,t ] + ) 2 2 c ( t )  ≤ N X i ′ =1 exp  ([ R i ′ ,t ] + ) 2 2 c ( t )  ≤ N e which implies R i,t ≤ p 2 c ( t )(ln N + 1). F or the second part of the le mma, let R i,t denote the re gret of our algor ithm to the instrument with the ǫN -th highest price at time t . Then, the total p otential of instruments with regr ets gre a ter than or equal to R i,t is at lea st: ǫN exp  ([ R i,t ] + ) 2 2 c ( t )  ≤ N e from which the se c ond par t of the lemma follows. W e quote Itˆ o’s for mu la , as stated in [Osk03] (Theorem 4.2.1) Theorem 3 (Itˆ o) L et dX ( t ) = udt + v dB ( t ) b e an n-dimensional It ˆ o pr o c ess. L et g ( t, x ) = ( g 1 ( t, x ) , . . . , g p ( t, x )) b e a C 2 map fr om [0 , ∞ ) × R n into R p . The t he pr o c ess Y ( t, ω ) = g ( t, X ( t )) 4 is again an Itˆ o pr o c ess, whose c omp onent numb er k , Y k , is given by d Y k = ∂ g k ∂ t ( t, X ) dt + X i ∂ g k ∂ x i ( t, X ) dX i + 1 2 X i,j ∂ 2 g k ∂ x i ∂ x j ( t, X ) dX i dX j wher e dB i dB j = δ i,j dt, dB i dt = dtdB i = 0 . W e now g ive the main theo rem, which characterizes the rate of increas e of c ( t ). Theorem 4 With pr ob ability one with r esp e ct to the Weiner pr o c ess ∀ t, dc ( t ) dt ≤ 6 V M ( t ) Pro of: W e deno te the p o tential cor resp onding to the i th instrument by p otential by Φ i t , i.e. Φ i t = φ ( R i t , c ( t )) Using Itˆ o’s for mula we can derive a n e quation fo r the differential d Φ i t : d Φ i t =   dc ( t ) dt φ c ( R i t , c ( t )) + a i ( t ) φ ′ ( R i t , c ( t )) + 1 2   N X j =1 ( b i,j ( t )) 2   φ ′′ ( R i t , c ( t ))   dt + N X j =1 b i,j ( t ) φ ′ ( R i t , c ( t )) dW j t =   dc ( t ) dt φ c ( R i t , c ( t )) + 1 2   N X j =1 ( b i,j ( t )) 2   φ ′′ ( R i t , c ( t ))   dt + dR i t φ ′ ( R i t , c ( t )) (11) W e s um Equatio n (11) ov er a ll instruments. As c ( t ) is chosen so that the a verage p otential is constant, the differ ent ia l of the av er age p o tential is zero. W e thus get: 0 = N X i =1 d Φ i t (12) = N X i =1   dc ( t ) dt φ c ( R i t , c ( t )) + 1 2   N X j =1 ( b i,j ( t )) 2   φ ′′ ( R i t , c ( t ))   dt + N X i =1 dR i t φ ′ ( R i t , c ( t )) (13) F rom Equa tion (4) w e know that the last ter m is equal to zero. Removing this ter m and reor ganizing the equation we ar rive a t an expr ession for the rate of change of c ( t ): dc ( t ) dt = − P N i =1  P N j =1 ( b i,j ( t )) 2  φ ′′ ( R i t , c ( t )) 2 P N i =1 φ c ( R i t , c ( t )) we plug in the definitions of V i ( t ), φ c and φ ′ to get: dc ( t ) dt = P i ; R i t > 0 V i ( t )  1 c ( t ) + ( R i t ) 2 c ( t ) 2  exp  ( R i t ) 2 2 c ( t )  2 P i ; R i t > 0 ( R i t ) 2 c ( t ) 2 exp  ( R i t ) 2 2 c ( t )  Multiplying the en umera to r a nd denominator by c ( t ), using the bound V i ( t ) ≤ V M and denoting x i . = R i t / p c ( t ) we get the inequa lit y dc ( t ) dt ≤ V M ( t ) P i ; x i > 0 (1 + x 2 i ) e x 2 i / 2 P i ; x i > 0 x 2 i e x 2 i / 2 (14) The maximum of the ratio o n the r ight hand side under the constra int (1 / N ) P i ; x i > 0 e x 2 i / 2 = e is achiev ed when x i = √ 2 for all i . Plugging this v a lue back into equation (refeqn:final) yields the s ta tement of the theorem. 5 5 references There are man y go o d sourc es for sto chastic differential eq uations a nd the Itˆ o calculus. One which I found particularly app ea ling is a set of lecture notes for a cours e on “Sto chastic Calculus, Filtering, and Sto chastic Control” b y Ramon v a n Ha ndel, av a ilable from the web her e: http:/ /www. princeton.edu/ ∼ rvan/acm217/ACM217.pdf References [CFH] Kamalik a Chaudhuri, Y oav F reund, and Daniel H su. A parameter-free hedging algorithm. Arxiv/09 03.285 1. [CO96] T. M. Cov er and E . O rdentlic h. Universal p ortfolios with side info r mation. IEEE T r ansactions on Information The ory , Mar ch 199 6 . [Cov91] Thomas M. Cover. Universal p ortfolios . Mathematic al Financ e , 1(1):1–2 9, January 19 9 1. [Osk03] Brent Oskendal. Sto chastic D iffer ential Equations . Spring e r, six th editio n edition, 20 03. 6