COMPUTING DELTAS WITHOUT DERIVATIVES. This Version : May 23, 2015

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1 COMPUING DELAS WIHOU DEIVAIVES D.. BAÑOS, S. DUEDAHL,. MEYE-BANDIS, AND F. POSKE hi Verion : May 23, 215 Abtract. A well-known application of Malliavin calculu in Mathematical Finance i the probabilitic repreentation of option price enitivitie, the o-called Greek, a expectation functional that do not involve the derivative of the pay-off function. hi allow for numerically tractable computation of the Greek even for dicontinuou pay-off function. However, while the pay-off function i allowed to be irregular, the coefficient of the underlying diffuion are required to be mooth in the exiting literature, which for example exclude already imple regime witching diffuion model. he aim of thi article i to generalie thi application of Malliavin calculu to Itô diffuion with irregular drift coefficient, whereat we here focu on the computation of the Delta, which i the option price enitivity with repect to the initial value of the underlying. o thi purpoe we firt how exitence, Malliavin differentiability, and (Sobolev) differentiability in the initial condition of trong olution of Itô diffuion with drift coefficient that can be decompoed into the um of a bounded but merely meaurable and a Lipchitz part. Furthermore, we give explicit expreion for the correponding Malliavin and Sobolev derivative in term of the local time of the diffuion, repectively. We then turn to the main objective of thi article and analye the exitence and probabilitic repreentation of the correponding Delta for lookback and Aian type option. We conclude with a imulation tudy of everal regime-witching example. Key word and phrae: Greek, Delta, option enitivitie, Malliavin calculu, Bimut- Elworthy-Li formula, irregular diffuion coefficient, trong olution of tochatic differential equation, relative L 2 -compactne MSC21: 6H1, 6H7, 6H3, 91G6. 1. Introduction hroughout thi paper, let > be a given time horizon and (Ω, F, P ) a complete probability pace equipped with a one-dimenional Brownian motion {B t } t, ] and the filtration {F t } t, ] generated by {B t } t, ] augmented by the P -null et. Further, we will only deal with random variable that are Brownian functional, i.e. we aume F := F. One of the mot prominent application of Malliavin calculu in financial mathematic concern the derivation of numerically tractable expreion for the o-called Greek, which are important enitivitie of option price with repect to involved parameter. he firt paper to addre thi application wa 15], which ha conecutively triggered an active reearch interet in thi topic, ee e.g. 14], 4], 1]. See alo 7], 11] and reference therein for a related approach baed on functional Itô calculu. Suppoe the rik-neutral dynamic of the underlying aet of a European option i driven by a tochatic differential equation (for hort SDE) of the form dx x t = b(x x t )dt + σ(x x t )db t, X x = x, where b : and σ : are ome given drift and volatility coefficient, repectively. Let Φ : denote the pay-off function and the expectation EΦ(X x )] the rik-neutral price at time zero of the option with maturity >. For notational implicity we aume the dicounting rate to be zero. In thi paper we will focu on the Delta x EΦ(Xx )], (1.1) which i a meaure for the enitivity of the option price with repect to change of the initial value of the underlying aet. A i well known, the Delta ha a particular role among the Greek a it 1

2 2 D.. BAÑOS, S. DUEDAHL,. MEYE-BANDIS, AND F. POSKE determine the hedge portfolio in many complete market model. If the drift b( ), the volatility σ( ), and the pay-off Φ( ) are ufficiently regular to allow for differentiation under the expectation, the Delta can be computed in a traight-forward manner a ] E x Φ(Xx ) = EΦ (X x ) Z ], (1.2) where the firt variation proce Z t := x Xx t i given by { Z t = exp b (X x ) 1 2 (σ (X x )) 2] } d + σ (X x ) db, (1.3) and where Φ, b, σ denote the derivative of Φ, b, σ, repectively. For example, requiring that Φ, b, σ are continuouly differentiable with bounded derivative would allow (1.2) to hold (we refer to 18] for condition on b and σ that guarantee the exitence of the firt variation proce), and the expectation in (1.2) could be approximated e.g. by Monte Carlo method. In mot realitic ituation, though, traight-forward computation a in (1.2) are not poible. In that cae, one could combine numerical method to approximate the derivative and the expectation in (1.1), repectively, to compute the Delta. However, in particular for dicontinuou pay-off Φ a i the cae for a digital option thi procedure might be numerically inefficient, ee for example 15]. At that point, the following reult for lookback option obtained with the help of Malliavin calculu appear to be ueful, where the option pay-off i allowed to depend on the path of the underlying at finitely many time point. heorem 1.1 (Propoition 3.2 in 15]). Let b( ) and σ( ) be continuouly differentiable with bounded Lipchitz derivative, σ( ) > ɛ >, and Φ : m be uch that the pay-off Φ(X x 1,..., X x m ), 1,..., m (, ], of the correponding lookback option i quare integrable. hen the Delta exit and i given by ] x EΦ(Xx 1,..., X x m )] = E Φ(X x 1,..., X x m ) a(t) σ 1 (Xt x ) Z t db t, (1.4) where Z t i the firt variation proce given in (1.3) and a(t) i any quare integrable determinitic function uch that, for every i = 1,..., m, i a()d = 1. While for notational implicity we preent the above reult for one-dimenional X x we remark that in 15] the extenion to multi-dimenional underlying aet and Brownian motion i conidered. If the option i of European type, i.e. the pay-off Φ(X x ) depend only on the underlying at, then (1.4) i the probabilitic repreentation of the pace derivative of a olution to a Kolmogorov equation which i alo referred to a Bimuth-Elworthy-Li type formula in the literature due to 13], 6]. he trength of (1.4) i that the Delta i expreed again a an expectation of the pay-off multiplied by the o-called Malliavin weight a(t) σ 1 (Xt x ) Z t db t. Computing the Delta by Monte-Carlo via thi reformulation then guarantee a convergence rate that i independent of the regularity of the pay-off function Φ and the dimenionality. Note that the Malliavin weight i independent of the option pay-off, and thu the ame weight can be employed in the computation of the Delta of different option. Alo, in 14] and 3] the quetion of how to optimally chooe the function a(t) with repect to computational efficiency i conidered. While the repreentation (1.4) ucceed to handle irregular pay-off by getting rid of the derivative of Φ, the regularity aumption on the coefficient b and σ driving the dynamic of the underlying diffuion are rather trong. Conider for example an extended Black and Schole model where the tock pay a dividend yield that witche to a higher level when the tock value pae a certain threhold. hen, again with the rik-free rate equal to zero for implicity, the logarithm of the tock price i modelled by the following dynamic under the rik-neutral meaure: dx x t = b(x x t )dt + σdb t, X x = x,

3 COMPUING DELAS WIHOU DEIVAIVES 3 where σ > i contant and the drift coefficient b : i given by b(x) := λ 1 1 (,) (x) λ 2 1, ) (x), for dividend yield λ 1, λ 2 + and a given threhold. In 9], a (more complex) irregular drift b i interpreted a tate-dependent fee deducted by the inurer in the evolution of variable annuitie intead of dividend yield. Already, thi imple regime-witching model i not covered by the reult in heorem 1.1 ince the drift coefficient i not continuouly differentiable. Or allow for tate-dependent regime-witching of the mean reverion rate in an extended Orntein-Uhlenbeck proce: dx x t = b(x x t )dt + σdb t, X x = x, where σ > i contant and the drift coefficient b : i given by b(x) := λ 1 x1 (,) (x) λ 2 x1, ) (x) for mean reverion rate λ 1, λ 2 + and a given threhold (here the mean reverion level i et equal to zero). hi type of model capture well, for intance, the evolution of electricity pot price, which witche between o-called pike regime on high price level with very fat mean reverion and bae regime on normal price level with moderate peed of mean reverion, ee e.g. 5], 17], 26] and reference therein. Alternatively, an extended Orntein-Uhlenbeck proce with tate-dependent regime-witching of the mean reverion level (low and high interet rate environment) i an intereting modification of the Vašíček hort rate model. Note that in that cae the Delta i rather a generalied ho, i.e. a enitivity meaure with repect to the hort end of the yield curve. We oberve that alo thee two extended Orntein-Uhlenbeck procee are not covered by the reult in heorem 1.1. Motivated by thee example, thi paper aim at deriving an analogou reult to heorem 1.1 when the underlying i driven by an SDE with irregular drift coefficient. More preciely, we will conider SDE dx x t = b(t, X x t )dt + db t, t, X x = x, (1.5) where we allow for time-inhomogeneou drift coefficient b :, ] in the form b(t, x) = b(t, x) + ˆb(t, x), (t, x), ], (1.6) for b merely bounded and meaurable, and ˆb Lipchitz continuou and at mot of linear growth in x uniformly in t, i.e. there exit a contant C > uch that ˆb(t, x) ˆb(t, y) C x y (1.7) ˆb(t, x) C(1 + x ) (1.8) for x, y and t, ]. Adding the Lipchitz component ˆb(t, x) in (1.6) i motivated by the fact that many drift coefficient intereting for financial application are of linear growth. At preent we are not able to how our reult for general meaurable drift coefficient of linear growth, but only for thoe where the irregular behavior remain in a bounded pectrum. However, from an application point of view thi cla i very rhich already, and in particular it contain the regime witching example from above. In (1.5) we conider a contant volatility coefficient σ(t, x) := 1, but we will ee at the end of Section 3 (heorem 3.8) that our reult apply to many SDE with more general volatility coefficient which can be reduced to SDE of type (1.5) (which for example i poible for volatility coefficient a in heorem 1.1). In order to be able to apply Malliavin calculu to the underlying diffuion, the firt thing we need to enure i that the olution of SDE (1.5) i a Brownian functional, i.e. we are intereted in the exitence of trong olution of (1.5). Definition 1.2. A trong olution of SDE (1.5) i a continuou, {F t } t, ] -adapted proce {X x t } t, ] that olve equation (1.5).

4 4 D.. BAÑOS, S. DUEDAHL,. MEYE-BANDIS, AND F. POSKE emark 1.3. Note that the uual definition of a trong olution require the exitence of a Brownian-adapted olution of (1.5) on any given tochatic bai. However, an {F t } t, ] - adapted olution {Xt x } t, ] on the given tochatic bai (Ω, F, P, B) can be written in the form Xt x = F t (B ) for ome family of functional F t, t, ], (ee e.g. 24] for an explicit form of F t ). hen for any other tochatic bai (ˆΩ, ˆF, ˆP, ˆB) one get that Xt x := F t ( ˆB ), t, ], i a ˆB-adapted olution to SDE (1.5). So once there i a Brownian-adapted olution of (1.5) on one given tochatic bai, it follow that there indeed exit a trong olution in the uual ene. hi jutifie our definition of a trong olution above. o purue our objective we proceed a follow in the remaining part of the paper. In Section 2 we recall ome fundamental concept from Malliavin calculu and local time calculu which compoe central mathematical tool in the following analyi. We then analye in Section 3 the exitence and Malliavin differentiability of a unique trong olution of SDE with irregular drift coefficient a in (1.5) (heorem 3.1). It i well known that the SDE i Malliavin differentiable a oon a the coefficient are Lipchitz continuou (ee e.g. 28]); for merely bounded and meaurable drift coefficient Malliavin differentiability wa hown only recently in 25], (ee alo 23]). Here, we extend idea introduced for bounded coefficient in 25] to drift coefficient of type (1.6). Unlike in mot of the exiting literature on trong olution of SDE with irregular coefficient our approach doe not rely on a pathwie uniquene argument (Yamada-Watanabe heorem). Intead, we employ a compactne criterium baed on Malliavin calculu together with local time calculu to directly contruct a trong olution which in addition i Malliavin differentiable. Alo, we are able to give an explicit expreion for the Malliavin derivative of the trong olution of (1.5) in term of the integral of b (and not the derivative of b) with repect to local time of the trong olution (Propoition 3.2). We mention that while exitence and Malliavin differentiability of trong olution could be extended to analogue multi-dimenional SDE a in 23], the explicit expreion of the Malliavin derivative i in general only poible for one-dimenional SDE a conidered in thi paper. Moreover, in thi paper we replace argument that are baed on White Noie analyi in 25] and 23] by alternative proof which might make the text more acceible for reader who are unfamiliar with concept from White Noie analyi. Next, we need to analye the regularity of the dependence of the trong olution in it initial condition and to introduce the analogue of the firt variation proce (1.3) in cae of irregular drift coefficient. Uing the cloe connection between the Malliavin derivative and the firt variation proce, we find that the trong olution i Sobolev differentiable in it initial condition (heorem 3.4). Again, we give an explicit expreion for the correponding (Sobolev) firt variation proce which doe not include the derivative of b (Propoition 3.5). In Section 4 we develop our main reult (heorem 4.2) which extend heorem 1.1 to SDE with irregular drift coefficient. o thi end, one ha to how in the firt place that the Delta exit, i.e. that EΦ(X x 1,..., X x m )] i continuouly differentiable in x. At thi point the explicit expreion for the Malliavin derivative and the firt variation proce are eential. In the final repreentation of the Delta we then have gotten rid of both the derivative of the pay-off Φ and the derivative of the drift coefficient b in the firt variation proce, whence the title Computing Delta without Derivative of the paper. In addition to Delta of lookback option ( a in heorem 1.1, we further ) 2 conider Delta of Aian option with pay-off of the type Φ 1 Xu x du for 1, 2, ] and ome function Φ :. In cae the tarting point of the averaging period of the Aian pay-off lie in the future, i.e. 1 >, we are able to give analogue reult to the one of lookback option. If the averaging period tart today, i.e. 1 =, the Malliavin weight in the expreion for the Delta would include a general Skorohod integral which i neither numerically nor mathematically tractable in our analyi (except for linear coefficient a in the Black and Schole model where the Skorohod integral turn out to be an Itô integral). However, we are till able to tate two approximation reult for the Delta in thi cae. In Section 5 we conider ome example and compute the Delta in the concrete regime-witching model mentioned above. We do a mall imulation tudy and compare the performance to a finite difference approximation of the Delta in the ame pirit a in 15].

5 COMPUING DELAS WIHOU DEIVAIVES 5 We conclude the paper by an appendix with ome technical proof from Section 3 which have been deferred to the end of the paper for better readability. Notation: We ummarie ome of the mot frequently ued notation: C 1 () denote the pace of continuouly differentiable function f :. C (, ] ), repectively C (), denote the pace of infinitely many time differentiable function on, ], repectively, with compact upport. For a meaurable pace (S, G) equipped with a meaure µ, we denote by L p (S, G) or L p (S) the Banach pace of (equivalence clae of) function on S integrable to ome power p, p 1. L p loc () denote the pace of locally Lebegue integrable function to ome power p, p 1, i.e. U f(x) p dx < for every open bounded ubet U. W 1,p loc () denote the ubpace of Lp loc () of weakly (Sobolev) differentiable function uch that the weak derivative f belong to L p loc (), p 1. For a progreive proce Y we denote the Doléan-Dade exponential of the correponding Brownian integral (if well defined) by ( ) ( E b(u, Y u )db u := exp b(u, Y u )db u 1 2 ) b 2 (u, Y u )du, t, ]. (1.9) For Z L 2 (Ω, F ) we denote the Wiener-tranform of Z in f L 2 (, ]) by ( )] W(Z)(f) := E ZE f()db. We will ue the ymbol to denote le or equal than up to a poitive real contant C > not depending on the parameter of interet, i.e. if we have two mathematical expreion E 1 (θ), E 2 (θ) depending on ome parameter of interet θ then E 1 (θ) E 2 (θ) if, and only if, there i a poitive real number C > independent of θ uch that E 1 (θ) CE 2 (θ). 2. Framework Our main reult centrally rely on tool from Malliavin calculu a well a integration with repect to local time both in time and pace. We here provide a concie introduction to the main concept in thee two area that will be employed in the following ection. For deeper information on Malliavin calculu the reader i referred to i.e. 28, 21, 22, 1]. A for theory on local time integration for Brownian motion we refer to i.e. 12, 29] Malliavin calculu. Denote by S the et of imple random variable F L 2 (Ω) in the form ( ) F = f h 1 ()db,..., h n ()db, h 1,..., h n L 2 (, ]), f C ( n ). he Malliavin derivative operator D acting on uch imple random variable i the proce DF = {D t F, t, ]} in L 2 (Ω, ]) defined by ( n ) D t F = i f h 1 ()db,..., h n ()db h i (t). i=1 Define the following norm on S: 1/2 F 1,2 := F L 2 (Ω) + DF L 2 (Ω;L 2 (, ])) = E F 2 ] 1/2 + E D t F dt] 2. (2.1) We denote by D 1,2 the cloure of the family of imple random variable S with repect to the norm given in (2.1), and we will refer to thi pace a the pace of Malliavin differentiable random variable in L 2 (Ω) with Malliavin derivative belonging to L 2 (Ω).

6 6 D.. BAÑOS, S. DUEDAHL,. MEYE-BANDIS, AND F. POSKE In the derivation of the probabilitic repreentation for the Delta, the following chain rule for the Malliavin derivative will be eential: Lemma 2.1. Let ϕ : m be continuouly differentiable with bounded partial derivative. Further, uppoe that F = (F 1,..., F m ) i a random vector whoe component are in D 1,2. hen ϕ(f ) D 1,2 and m D t ϕ(f ) = i ϕ(f )D t F i, P a.., t, ]. i=1 he Malliavin derivative operator D : D 1,2 L 2 (Ω, ]) admit an adjoint operator δ = D : Dom(δ) L 2 (Ω) where the domain Dom(δ) i characteried by all u L 2 (Ω, ]) uch that for all F D 1,2 we have ] E D t F u t dt C F 1,2, where C i ome contant depending on u. For a tochatic proce u Dom(δ) (not necearily adapted to {F t } t, ] ) we denote by δ(u) := u t δb t (2.2) the action of δ on u. he above expreion (2.2) i known a the Skorokhod integral of u and it i an anticipative tochatic integral. It turn out that all {F t } t, ] -adapted procee in L 2 (Ω, ]) are in the domain of δ and for uch procee u t we have δ(u) = u t db t, i.e.the Skorokhod and Itô integral coincide. In thi ene, the Skorokhod integral can be conidered to be an extenion of the Itô integral to non-adapted integrand. he dual relation between the Malliavin derivative and the Skorokhod integral implie the following important formula: heorem 2.2 (Duality formula). Let F D 1,2 and u Dom(δ). hen ] ] E F u t δb t = E u t D t F dt. (2.3) he next reult, which i due to 8] and central in proving exitence of trong olution in the following, provide a compactne criterion for ubet of L 2 (Ω) baed on Malliavin calculu. Propoition 2.3. Let F n D 1,2, n = 1, 2..., be a given equence of Malliavin differentiable random variable. Aume that there exit contant α > and C > uch that for t t, and up E F n 2 ] C, n up E D t F n D t F n 2] C t t α n up n up t E D t F n 2] C. hen the equence F n, n = 1, 2..., i relatively compact in L 2 (Ω). We conclude thi review on Malliavin calculu by tating a relation between the Malliavin derivative and the firt variation proce of the olution of an SDE with mooth coefficient that i eential in the derivation of heorem 1.1. We give the reult for the cae when the volatility coefficient i equal to 1, but the analogue reult i valid for more general mooth volatility coefficient. Aume the drift coefficient b(t, x) in the SDE (1.5) fulfil the Lipchitz and linear growth condition (1.7)-(1.8). hen it i well known that there exit a unique trong olution

7 COMPUING DELAS WIHOU DEIVAIVES 7 Xt x, t, ], to equation (1.5) that i Malliavin differentiable, and that for all t the Malliavin derivative D Xt x fulfil, ee e.g. 28, heorem 2.2.1] D X x t = 1 + b (u, X x u)d X x udu, (2.4) where b denote the (weak) derivative of b with repect to x. Further, under thee aumption the trong olution i alo differentiable in it initial condition, and the firt variation proce x Xx t, t, ], fulfil (ee e.g. 18] for differentiable coefficient and 2] for an extenion to Lipchitz coefficient) x Xx t = 1 + Solving equation (2.4) and (2.5) thu yield the following propoition. b (u, X x u) x Xx udu. (2.5) Propoition 2.4. Let Xt x, t, ], be the unique trong olution to equation (1.5) when b(t, x) fulfil the Lipchitz and linear growth condition (1.7)-(1.8). hen Xt x i Malliavin differentiable and differentiable in it initial condition for all t, ], and for all t we have { } D Xt x = exp b (u, Xu)du x (2.6) and A a conequence, where all equalitie hold P -a.. x Xx t { } = exp b (u, Xu)du x. (2.7) x Xx t = D X x t x Xx, (2.8) 2.2. Integration with repect to local-time. Let now X x be a given (trong) olution to SDE (1.5). In the equel we need the concept of tochatic integration over the plane with repect to the local time L Xx (t, y) of X x. For Brownian motion, the local time integration theory in time and pace ha been introduced in 12]. We extend thi local time integration theory to more general diffuion of type (1.5) by reorting to the Brownian etting under an equivalent meaure where X x i a Brownian motion. o thi end, we notice the following fact that i extenively ued throughout the paper. emark 2.5. he adon-nikodym denity ( dq dp = E b(, X x )db ) define a probability meaure Q equivalent to P under which X x i Brownian motion tarting in x. Indeed, becaue b i of at mot linear growth we obtain by Grönwall inequality a in the proof of Lemma A.1 a contant C t,x > uch that Xt x C t,x (1 + B t ). One can thu find a equiditant partition = t < t 1... < t m = uch that E exp {i+1 t i }] b 2 (, X x )d E exp {i+1 t i ( C1 + C 2 B + C 3 B 2) } ] d < for all i =,..., m 1, where C 1, C 2 and C 3 are ome poitive contant. hen it i well-known, ee e.g. 16, Corollary 5.16], that Q i an equivalent probability meaure under which X x i Brownian motion by Giranov theorem.

8 8 D.. BAÑOS, S. DUEDAHL,. MEYE-BANDIS, AND F. POSKE We now define the feaible integrand for the local time-pace integral with repect to L Xx (t, y) by the Banach pace (H x, ) of function f :, ] with norm ( f x = 2 + f 2 1 y x 2 (, y) exp ( 2π 2 1 y x f(, y) ( 2π exp ) ) 1/2 dyd ) y x 2 dyd. 2 We remark that thi pace of integrand i the ame a the one introduced in 12] for Brownian motion (i.e. the pecial cae when the X x i a Brownian motion), except that we have in a traight forward manner generalied the pace in 12] to the ituation when the Brownian motion ha arbitrary initial value x. We denote by f :, ] a imple function in the form f (, y) = f ij 1 (yi,y i+1](y)1 (j, j+1](), 1 i n 1,1 j m 1 where ( j ) 1 j m i a partition of, ] and (y i ) 1 i n and (f ij ) 1 i n,1 j m are finite equence of real number. It i readily checked that the pace of imple function i dene in (H x, ). he local time-pace integral of an imple function f with repect to L Xx (dt, dy) i then defined by f (, y)l Xx (d, dy) := := f ij (L Xx ( j+1, y i+1 ) L Xx ( j, y i+1 ) L Xx ( j+1, y i ) + L Xx ( j, y i )). 1 i n 1 1 j m 1 Lemma 2.6. For f H x let f n, n 1, be a equence of imple function converging to f in H x. hen f n(, y)l Xx (d, dy), n 1, converge in probability. Further, for any other approximating equence of imple function the limit remain the ame. Proof. Define Fn Xx := f n(, x)l Xx (d, dx). Now conider the equivalent meaure Q from emark 2.5 under which X x i Brownian motion. Define F Xx := f(, (d, dx) to be x)lxx the time-pace integral of f with repect to the local time of Brownian motion X x under Q, which exit a an L 1 (Q)-limit of Fn Xx, n 1 by the Brownian local time integration theory introduced in 12] (ince f n, n 1 converge to f in H x ). We how that Fn Xx, n 1 converge in probability to F Xx under P. Indeed, ( ( ) )] E1 F Xx Fn Xx ] = E 1 F Bx Fn Bx E b(, B x )db E E ( ) 1+ b(, B x )db ) C E (1 F Bx Fn Bx 1/(1+) ] n 1+ ( E 1 F Bx Fn Bx ) 1+ ] 1+, (2.9) where, in analogy to the notation F Xx and Fn Xx above, the notation F Bx and Fn Bx refer to the correponding integral with repect to local time of Brownian motion B x under P, and where in the firt equality we have ued that (F Bx, Fn Bx ) ha the ame law under P a (F Xx, Fn Xx ) under Q. he inequalitie follow by Lemma A.1 for ome > uitably mall. Further, by 12] we know that Fn Bx, n 1 converge to F Bx in L 1 (P ), which implie the convergence in (2.9). Hence Fn Xx, n 1 converge to F Xx in the Ky-Fan metric d(x, Y ) = E1 X Y ], X, Y L (Ω), which characterie convergence in probability. Finally, again by 12], F Xx i independent of the approximating equence f n, n 1.

9 COMPUING DELAS WIHOU DEIVAIVES 9 Definition 2.7. For f H x the limit in Lemma 2.6 i called the time-pace integral of f with repect to L Xx (dt, dx) and i denoted by f(, y)lxx (d, dy). Further, for any t, ] we define f(, y)lxx (d, dy) := f(, y)i,t]()l Xx (d, dy). emark 2.8. We notice that the drift coefficient b(t, x) in (1.6), which i of linear growth in x uniformly in t, i in H x, and thu the local time integral of b(t, x) with repect to L Xx (dt, dy) exit for any x. If X x i a Brownian motion B we have the following decompoition due to 12] that we employ in the contruction of trong olution, and that alo contitute the foundation in the contruction of the local time integral in 12]. heorem 2.9. Let f H. hen f(, y)l Bx (d, dy) = = (2.1) f(, B x )db + f(, B x )dw f(, B B x ) t t d, where B t = B t, t i time-revered Brownian motion, and W, defined by B t = B + W t i a Brownian motion with repect to the filtration of B. B d, We conclude thi ubection by tating three further identitie for the local time integral of a general diffuion X x which will be ueful later on. Lemma 2.1. Let f H x be Lipchitz continuou in x. hen for all t, ] f(, y)l Xx (d, dy) = f (, X x )d. (2.11) where f denote the (weak) derivative of f(t, y) with repect to y. If f H x i time homogeneou (i.e. f(t, y) = f(y) only depend on the pace variable) and locally quare integrable, then for any t, ] f(, x)l Xx (d, dx) = f(, X x ), X x ] t. (2.12) and f(, y)l Xx (d, dy) = 2F (X x t ) 2F (x) 2 f(x x )dx x (2.13) where F i a primitive function of f and b(, X x ), X x ] t i the generalied covariation proce m ) ) f(, X x ), X x ] t := P lim (f(t m k, Xt x ) mk f(tm k 1, Xt (X x ) x mk 1 tk X x tk 1 m k=1 where for every m {t m k }m k=1 i a partition of, t] uch that lim m (2.13) can be conidered a a generalied Itô formula. up t m k t m k 1 =. Note that k=1,...,m Proof. If X x = x + B, then identitie (2.11)-(2.13) are given in 12]. For general X x, we conider the identitie under the equivalent meaure Q from emark 2.5. hen, by the contruction of the local time integral outlined in Lemma 2.6, the integral in the identitie are the one with repect to Brownian motion X x, for which we know the identitie hold by 12] (where uch identitie are given in the cae x = but one can eaily extend them to the cae of the Brownian motion tarting at an arbitrary x ).

10 1 D.. BAÑOS, S. DUEDAHL,. MEYE-BANDIS, AND F. POSKE 3. Exitence, Malliavin, and Sobolev differentiability of trong olution In thi ection we prepare the neceary theoretical ground to develop the probabilitic repreentation of Delta. Being notationally and technically rather heavy, the proof of thi ection are deferred to Appendix A for an improved flow and readability of the paper. We firt tudy the exitence and Malliavin differentiability of a unique trong olution of SDE (1.5) before we turn to the differentiability of the trong olution in it initial condition and the correponding firt variation proce. We tate the firt main reult of thi ection: heorem 3.1. Suppoe that the drift coefficient b :, ] i in the form (1.6). hen there exit a unique trong olution {X x t } t, ] to SDE (1.5). In addition, X x t i Malliavin differentiable for every t, ]. he proof of heorem 3.1 employ everal auxiliary reult preented in Appendix A. he main tep are: (1) Firt, we contruct a weak olution X x to (1.5) by mean of Giranov theorem, that i we introduce a probability pace (Ω, F, P ) that carrie ome Brownian motion B and a continuou proce X x uch that (1.5) i fulfilled. However, a priori X x i not adapted to the filtration {F t } t, ] generated by Brownian motion B. (2) Next, we approximate the drift coefficient b = b + ˆb by a equence of function (which alway exit by tandard approximation reult) b n := b n + ˆb, n 1, (3.1) uch that { b n } n 1 C (, ] ) with up n 1 b n C < and b n b in (t, x), ] a.e. with repect to the Lebegue meaure. By tandard reult on SDE, we know that for each mooth coefficient b n, n 1, there exit a unique trong olution X n,x to the SDE dx n,x t = b n (t, X n,x t )dt + db t, t, X n,x = x. (3.2) We then how that for each t, ] the equence X n,x t converge weakly to the conditional expectation EXt x F t ] in the pace L 2 (Ω; F t ) of quare integrable, F t -meaurable random variable. (3) By Propoition 2.4 we know that for each t, ] the trong olution X n,x t, n 1, are Malliavin differentiable with D X n,x t { = exp } b n(u, Xu n,x )du, t, n 1, (3.3) where b n denote the derivative of b n with repect to x. We will ue repreentation (3.3) to employ a compactne criterion baed on Malliavin calculu to how that for every t, ] the et of random variable {X n,x t } n 1 i relatively compact in L 2 (Ω; F t ), which then allow to conclude that X n,x t converge trongly in L 2 (Ω; F t ) to EXt x F t ]. Further we obtain that EXt x F t ] i Malliavin differentiable a a conequence of the compactne criterion. (4) In the lat tep we how that EXt x F t ] = Xt x, which implie that Xt x i F t -meaurable and thu a trong olution. Moreover, we how that thi olution i unique. Notation: In the following we ometime include the drift coefficient b into the equence {b n } n by putting b := b + ˆb := b + ˆb = b. he next important reult i an explicit repreentation of the Malliavin derivative of the trong olution X x t, t, ]. For mooth coefficient b we can explicitly expre the Malliavin derivative in term of the derivative of b a tated in (3.3). For general, not necearily differentiable coefficient b, we are till able to give an explicit formula which now only involve the coefficient b in a local time integral:

11 COMPUING DELAS WIHOU DEIVAIVES 11 Propoition 3.2. For t, the Malliavin derivative D Xt x of the unique trong olution Xt x to equation (1.5) ha the following explicit repreentation: { } D Xt x = exp b(u, y)l Xx (du, dy) P-a.., (3.4) where L Xx (du, dy) denote integration in pace and time with repect to the local time of X x, ee Section 2.2 for definition. Next, we turn our attention to the tudy of the trong olution Xt x a a function in it initial condition x for SDE with poible irregular drift coefficient. he firt reult etablihe Hölder continuity jointly in time and pace. Propoition 3.3. Let X x t, t, ] be the unique trong olution to the SDE (1.5). hen for all, t, ] and x, y K for any arbitrary compact ubet K there exit a contant C = C(K, b, ˆb ) > uch that E X x t X y 2] C( t + x y 2 ). In particular, there exit a continuou verion of the random field (t, x) Xt x continuou trajectorie of order α < 1/2 in t, ] and α < 1 in x. with Hölder If the drift coefficient b i regular, then we know by Propoition 2.4 that Xt x i even differentiable a a function in x. he firt variation proce x Xx i then given by (2.7) in term of the derivative of the drift coefficient and i cloely related to the Malliavin derivative by (2.8). In the following we will derive analogou reult for irregular drift coefficient, where in general the firt variation proce will now exit in the Sobolev derivative ene. Let U be an open and bounded ubet. he Sobolev pace W 1,2 (U) i defined a the et of function u :, u L 2 (U) uch that it weak derivative belong to L 2 (U). We endow thi pace with the norm u 1,2 = u 2 + u 2 where u tand for the weak derivative of u W 1,2 (U). We ay that the olution X x t, t, ], i Sobolev differentiable in U if for all t, ], X t belong to W 1,2 (U), P -a.. Oberve that in general X t i not in W 1,2 (), e.g. take b. heorem 3.4. Let b :, ] be a in (1.6). Let Xt x, t, ] be the unique trong olution to the SDE (1.5) and U an open, bounded et. hen for every t, ] we have (x X x t ) L 2 (Ω, W 1,2 (U)). We remark that uing analogue technique a in 27] one could even etablih that the trong olution give rie to a flow of Sobolev diffeomorphim. hi, however, i beyond the cope of thi paper. Similarly a for the Malliavin derivative, we are able to give an explicit repreentation for the firt variation proce in the Sobolev ene that doe not involve the derivative of the drift coefficient by employing local time integration. Propoition 3.5. Let b :, ] be a in (1.6). hen the firt variation proce (in the Sobolev ene) of the trong olution Xt x, t, ] to SDE (1.5) ha the following explicit repreentation x Xx t { = exp } b(u, y)l Xx (du, dy) dt P a.. (3.5) A a direct conequence of Propoition 3.5 together with Propoition 3.2 we obtain the following relation between the Malliavin derivative and the firt variation proce, which i an extenion of Propoition 2.4 to irregular drift coefficient and which i a key reult in deriving the deired expreion for the Delta.

12 12 D.. BAÑOS, S. DUEDAHL,. MEYE-BANDIS, AND F. POSKE Corollary 3.6. Let Xt x, t, ], be the unique trong olution to (1.5). hen the following relationhip between the patial derivative and the Malliavin derivative of Xt x hold: for any, t, ], t. x Xx t = D X x t x Xx P a.. (3.6) emark 3.7. Note that by Lemma 2.1 the Malliavin derivative in (3.4) and the firt variation proce in (3.5) can be expreed in variou alternative way. Firtly, we oberve that by formula (2.11) the local time integral of the regular part ˆb in b can be eparated and rewritten in the form b(u, y)l Xx (du, dy) = b(u, y)l X x (du, dy) + ˆb (u, Xu)du x a.. (3.7) If in addition b(t, ) i locally quare integrable and continuou in t a a map from, ] to L 2 loc () or even time-homogeneou, then by Lemma 2.1 alo the local time integral aociated to the irregular part b can be reformulated in term of the generalied covariation proce a in (2.12) or in term of the generalied Itô formula a in (2.13), repectively. In particular, thee reformulation appear to be ueful for imulation purpoe. We conclude thi ection by giving an extenion of all the reult een o far to a cla of SDE with more general diffuion coefficient. heorem 3.8. Conider the time-homogeneou SDE dx x t = b(x x t )dt + σ(x x t )db t, X x = x, t, (3.8) where the coefficient b : and σ : are Borel meaurable. equire that there exit a twice continuouly differentiable bijection Λ : with derivative Λ and Λ uch that a well a Suppoe that the function b : given by Λ (y)σ(y) = 1 for a.e. y, Λ 1 i Lipchitz continuou. b (x) := Λ ( Λ 1 (x) ) b(λ 1 (x)) Λ ( Λ 1 (x) ) σ(λ 1 (x)) 2 atifie the condition of heorem 3.1. hen there exit a Malliavin differentiable trong olution X x to (3.8) which i (locally) Sobolev differentiable in it initial condition. Proof. he proof i obtained directly from Itô formula. See 25]. 4. Exitence and derivative-free repreentation of the Delta We now turn the attention to the tudy of option price enitivitie with repect to the initial value of an underlying aet with irregular drift coefficient. Notably, we will conider lookback option with path-dependent (dicounted) pay-off in the form Φ(X x 1,..., X x m ) (4.1) for time point 1,..., m (, ], ome function Φ : m, and where the evolution of the underlying price proce under the rik-neutral pricing meaure i modelled by the trong olution X x of SDE (1.5) with poibly irregular drift b a in (1.6). In particular, for m = 1 the pay-off (4.1) i aociated to a European option with maturity 1. hen the current option price i given by E Φ(X x 1,..., X x m ) ] and the main objective of thi ection i to etablih exitence and a derivative-free, probabilitic repreentation of the Delta x E Φ(X x 1,..., X x m ) ].

13 COMPUING DELAS WIHOU DEIVAIVES 13 After having analyed lookback option, we will alo addre the problem of computing Delta of Aian option with (dicounted) path-dependent pay-off in the form ( ) 2 Φ Xu x du (4.2) 1 for 1, 2, ] and ome function Φ :. We tart with a preliminary reult which how that in cae of a mooth pay-off function with compact upport the Delta exit for a large cla of path dependent option that include both lookback a well a Aian option. Lemma 4.1. Let Xt x, t, ], be the trong olution to SDE (1.5) and {X n,x t } n 1 the correponding approximating trong olution of SDE (3.2). Let Φ C ( m ) and conider the function and u n (x) := E u(x) := E Φ ( Φ ( Xu n,x µ 1 (du), Xu n,x µ 2 (du),..., Xu n,x µ m (du) )] Xuµ x 1 (du), Xuµ x 2 (du),..., Xuµ x m (du) where µ 1,..., µ m are finite meaure on, ] independent of x. Conider alo the function m ( ) ] ū(x) := E i Φ Xuµ x 1 (du), Xuµ x 2 (du),..., Xuµ x m (du) x Xx uµ i (du) i=1 where x Xx i the firt variation proce of X x introduced in (3.5). hen and u n (x) n u(x) for all x, u n(x) n ū(x) uniformly on compact ubet K, where u n denote the derivative. A a reult, we obtain that u C 1 () with u = ū. In particular, we obtain the reult for lookback option by taking µ i = δ ti the Dirac meaure concentrated on t i, i = 1,..., m, and for Aian option by taking m = 1 and µ 1 = du. Proof. Firt of all, oberve that the expreion in (4.5) i well-defined. hi can be een by uing Cauchy-Schwarz inequality, the fact that Φ C ( m ), and Corollary A.9. It i readily checked that u n (x) u(x) for all x ince Φ i mooth by uing the mean-value theorem and the fact that X n,x t Xt x in L 2 (Ω) a n for every t, ] (ee heorem A.6). We introduce the following hort-hand notation for the m-dimenional random vector aociated to a proce Y : ( ) h(y, ) := Y u µ 1 (du), Y u µ 2 (du),..., Y u µ m (du). For the mooth coefficient b n we have u n C 1 (), n 1, and ince i Φ are bounded for all i = 1,..., m and by dominated convergence we have m ( ) ] u n(x) = E i Φ h(x n,x, ) u µ i (du). i=1 x Xn,x Moreover, we can take integration with repect to µ i (du), i = 1,...m, outide the expectation. hu m ( ) ] u n(x) = E i Φ h(x n,x, ) x Xn,x u µ i (du). i=1 )] (4.3) (4.4) (4.5)

14 14 D.. BAÑOS, S. DUEDAHL,. MEYE-BANDIS, AND F. POSKE Hence u n(x) ū(x) = =: m i=1 m i=1 ( ) E i Φ h(x n,x, ) x Xn,x u i Φ ( h(x x, ) ) ] x Xx u µ i (du) F n,i (u, x)µ i (du) where F n,i (u, x) denote the expectation in the integral. We will how that for any i = 1,..., m and compact ubet K, up F n,i (u, x) n. (u,x), ] K Indeed, by plugging in expreion (3.5) for the firt variation proce and Giranov theorem we get F n,i (u, x) E i Φ ( h(b x, ) ) { exp i Φ ( h(b x, ) ) exp { u u E i Φ ( h(b x, ) ) ( ) E b(u, Bu)dB x u ( { u } exp b n (v, y)l Bx (dv, dy) } ( b n (v, y)l Bx (dv, dy) E b n (u, B x u)db u ) } ( ) )] b(v, y)l Bx (dv, dy) E b(u, Bu)dB x u { exp u + E i Φ ( h(b x, ) ) { u } exp b n (v, y)l Bx (dv, dy) ( ( ) ( )) ] E b n (u, Bu)dB x u E b(u, Bu)dB x u := I n + II n b(v, y)l Bx (dv, dy)}) ] Here, we will how etimate for II n, for I n the argument i analogou. Similarly to how we obtain the etimate II 1 n + II 2 n in the proof of Lemma A.5, uing inequality e x 1 x (e x + 1) we get II n E i Φ ( h(b x, ) ) { U n exp u + E i Φ ( h(b x, ) ) { U n exp =: II 1 n + II 2 n, u } ( b n (v, y)l Bx (dv, dy) E } b n (v, y)l Bx (dv, dy) E b n (u, B x u)db u ) ] ( ) ] b(u, Bu)dB x u where U n := ( b n (u, B x u) b(u, B x u))db u 1 2 (b 2 n(u, B x u) b 2 (u, B x u))du. We will now how that IIn 1 a n uniformly in x on a compact ubet K. he convergence of IIn 2 then follow immediately, too. Denote p = 1+ with > from Lemma A.1 and ue Hölder inequality with exponent 1 + on the Doléan-Dade exponential, then employ

15 COMPUING DELAS WIHOU DEIVAIVES 15 formula (2.11) on ˆb in b n = b n + ˆb and ue Cauchy-Schwarz inequality ucceively. A a reult, ( ) 1+ 1/(1+) IIn 1 E E b n (u, Bu)dB x u E i Φ ( h(b x, )) ] 1/(2p) E Un 8p ] 1/(8p) E exp { 4p u ] } 1/(4p) { bn (v, y)l Bx (dv, dy) E exp 8p u ˆb (v, B x v )dv}] 1/(8p). he firt and fourth factor are bounded uniformly in n and x K by emark A.2 and Lemma A.3, repectively. he econd and and fifth factor can be controlled ince i Φ, i = 1,..., m and ˆb are uniformly bounded. It remain to how that up E U n 8p ] n x K for any compact ubet K. Uing Minkowki inequality, Burkholder-Davi-Gundy inequality and Hölder inequality we can write E U n 8p ] E b n (u, Bu) x b(u, Bu) x 8p ]du + E b 2 n(u, Bu) x b 2 (u, Bu) x 8p ]du. (4.6) hen write the integrand of the firt term in (4.6) a E b n (u, Bu) x b(u, Bu) x 8p ] = 1 b n (u, y) b(u, y) 8p e (y x)2 2u dy. 2πu Uing Cauchy-Schwarz inequality on b n (u, y) b(u, y) 8p e y2 4u we obtain E b n (u, Bu) b(u, x Bu) x 8p ] 1 ( ) 1/2 ( e x2 2u b n (u, y) b(u, y) 16p e y2 2u dy 2πu ) 1/2 e y2 xy 2u +2 u dy. hen for each u, ], by taking the upremum over x K and by dominated convergence, we get up E b n (u, Bu) x b(u, Bu) x 8p ] n, x K and hence the reult follow. Similarly, one can argue for the econd term in (4.6). In um, up F n,i (u, x) n (u,x),t] K for all i = 1,..., m and hence u n(x) n ū(x) uniformly on compact et K, and thu u C 1 () with u = ū. We come to the main reult of thi paper, which extend heorem 1.1 to lookback option written on underlying with irregular drift coefficient. In particular, when plugging in expreion (3.5) for the firt variation proce, we ee that the formula for the Delta in (4.8) below involve neither the derivative of the pay-off function Φ nor the derivative of the drift coefficient b. We obtain thi reult for pay-off function Φ L q w( m ), where { } L q w( m ) := f : m meaurable : f(x) q w(x)dx < (4.7) m for the weight function w defined by w(x) := exp( 1 2 x 2 ), x m, and where the exponent q depend on the drift b. Note that all pay-off function of practical relevance are contained in thee pace. heorem 4.2. Let X x be the trong olution to SDE (1.5) and Φ : m a function in L 4p w ( m ), where p > 1 i the conjugate of 1 + for > in Lemma A.1. hen, for any < 1 m, the price u(x) := E Φ(X x 1,..., X x m ) ]

16 16 D.. BAÑOS, S. DUEDAHL,. MEYE-BANDIS, AND F. POSKE of the aociated lookback option i continuouly differentiable in x, and it derivative, i.e. the Delta, take the form ] u (x) = E Φ(X x 1,..., X x m ) a() x Xx db (4.8) for any bounded meaurable function a : uch that, for every i = 1,..., m, i=1 i a()d = 1. (4.9) Proof. Aume firt Φ C ( m ). hen by Lemma 4.1 with µ i = δ ti, i = 1,..., m, we know that u(x) = E Φ(X x 1,..., X x m ) ] i continuouly differentiable with derivative m u (x) := E i Φ(X x 1,..., X x m ) ] x Xx i. Now, by Corollary 3.6, we have for any i = 1,..., m x Xx i = D X x i x Xx for all i. (4.1) Alo recall that D X x i = for i. So, for any function a : atifying (4.9) we have A a reult, u (x) = x Xx i = m E i Φ(X x 1,..., X x m ) i=1 = E a()d X x i x Xx d. a()d Φ(X x 1,..., X x m ) x Xx d ] a()d X x i x Xx d ] where in the lat tep we could ue the chain rule for the Malliavin derivative backward, ee Lemma 2.1, ince Φ(X x 1,..., X x m ) i Malliavin differentiable due to heorem 3.1. hen a() x Xx i an F -adapted Skorokhod integrable proce by Corollary A.9 with p = 2, o the duality formula for the Malliavin derivative (ee heorem 2.2) yield ] u (x) = E Φ(X x 1,..., X x m ) a() x Xx db. Finally, we extend the reult to a pay-off function Φ L 4p w ( m ). By tandard argument we can approximate Φ by a equence of function Φ n C ( m ), n, uch that Φ n Φ in L 4p w ( m ) a n. Now define u n (x) := EΦ n (X x 1,..., X x m )] and ū(x) := EΦ(X x 1,..., X x m ) a() x Xx db ]. hen u (Φn n(x) ū(x) = E (X x 1,..., X x m ) Φ(X x 1,..., X x m ) ) ] a() x Xx db Φn E (X x,..., 1 Xx ) m Φ(Xx,..., 1 Xx ) 2] 1/2 m E a() x Xx 2 d CE Φ n (B x 1,..., B x m ) Φ(B x 1,..., B x m ) ( )] 1/2 2 E b(u, Bu)dB x u where we have ued Cauchy-Schwarz inequality, Itô iometry, Corollary A.9 and Giranov theorem in thi order. hen we apply Hölder inequality with 1 + for a mall enough > and, ] 1/2,

17 COMPUING DELAS WIHOU DEIVAIVES 17 ue Lemma A.1 to get u n(x) ū(x) CE CE Φn ] (B x,..., 1 Bx ) m Φ(Bx,..., 1 Bx ) (1+) m Φn ] (B x,..., 1 Bx ) m Φ(Bx,..., 1 Bx ) (1+) m. E E ( ) 1+ b(u, Bu)dB x u 1 2(1+) For the lat quantity, denote by P t (y) := 1 2πt e y2 /(2t), y the denity of B t, and et := and y := x. ecall that < 1 m. Uing the independent increment of the Brownian motion we rewrite Φn E (B x 1,..., B x m ) Φ(B x 1,..., B x m ) ] 2 1+ = Φ n (y 1,..., y m ) Φ(y 1,..., y m ) 2 m Furthermore, with t := min i=1,...,m 1 (t i+1 t i ) Φn E (B x 1,..., B x m ) Φ(B x 1,..., B x m ) ] m P i i 1 (y i y i 1 )dy 1 dy m. (2πt ) m/2 Φ n (y 1,..., y m ) Φ(y 1,..., y m ) 2 1+ m m e y 2 i 4( i i 1 ) + y i y i 1 i i 1 y2 i 1 i=1 2( i i 1) dy 1 dy m. By applying Cauchy-Schwarz inequality we obtain E Φ n (B x 1,..., B x m ) Φ(B x 1,..., B x m ) ] 2 1+ i=1 e y 2 i 4( i i 1 ) ( ) 1/2 (2πt ) m/2 Φ n (y 1,..., y m ) Φ(y 1,..., y m ) 4 1+ e y 2 2 dy1 dy m m ( m ) e y 2 1/2 i 2( i i 1 ) + 2y i y i 1 i y2 i 1 i 1 ( i i 1) dy 1 dy m m i=1 =: I n II. For the econd factor we have ( II e x2 e y xy 1 m m i=2 ) 1/2 e (y i y i 1 ) 2 2 dy 1 dy m and hence up II <. x K hu, ince factor I n converge to by aumption, we can approximate ū uniformly in x on compact et by mooth pay-off function. So u C 1 () and u = ū. Next, we conider Aian option with pay-off given by (4.2). If 1 > we are able to give the analogou reult to heorem 4.2 by approximating the Aian pay-off with lookback pay-off: Corollary 4.3. Let X x be the trong olution to SDE (1.5) and Φ : a function in L 4p w () where w i defined in (4.15) further below and where p > 1 i the conjugate of 1 + for > in

18 18 D.. BAÑOS, S. DUEDAHL,. MEYE-BANDIS, AND F. POSKE Lemma A.1. hen for any 1, 2 (, ] with 1 < 2, the price ( )] 2 u(x) = E Φ Xudu x of the aociated Aian option i continuouly differentiable in x, and it derivative, i.e. the Delta, take the form ( ) 2 ] 1 u (x) = E Φ X x d a() x Xx db (4.11) for any bounded meaurable function a : uch that a()d = 1. (4.12) Proof. Aume firt that Φ C 1 (), and conider a erie of partition of 1, 2 ] with vanihing meh, i.e. let { 1 = t m < t m 1 <... < t m m = 2 } m=1 with lim m up i=1,...,m (t m i t m i 1 ) =. hen we may write the integral uing iemann um a follow 2 X x d = lim Xt x m(tm i i t m i 1). hen 1 ( ) 2 Φ X x d = lim 1 By heorem 4.2 we have m Φ u (x) = lim m E i=1,...,m m i=1,...,m Xt x m(tm i i t m i 1) =: lim ˆΦ m (Xt x,..., m Xx 1 t ) m m ] a m () x Xx db ˆΦ m (X x t m 1,..., Xx t m m ). where a m i a bounded meaurable function uch that m i a m ()d = 1 for each i = 1,..., m. hen ] u (x) = lim E ˆΦ m (Xt x,..., m m Xx 1 t ) a m m () x Xx db ( ) 2 1 = E Φ X x d 1 a() x Xx db where a i a function uch that 1 a()d = 1. For a general pay-off Φ, we approximate Φ in L 4p w () by a equence of function {Φ n } n C() 1 and define u(x) := EΦ( ( 2 ) 1 X x 2 d)] and ū(x) := E Φ 1 X x 1 d a() x Xx db ]. Conider u n (x) = EΦ n ( 2 1 X x d)]. Finally, imilarly a in heorem 4.2 one ha u n (x) u(x) a n for all x and ( ) ( 2 ) ] 1/p 2 u n(x) ū(x) E Φ n B x d Φ B x 2p d, 1 which goe to zero uniformly in x K on compact et K a n by uing the fact that 2 1 B x d ha a Gauian ditribution with mean x( 2 1 ) and variance ( 2 1 )1 2 which explain the weight w. emark 4.4. From the proof of Corollary 4.3 it follow that the Delta (4.11) of an Aian option can be approximated by the Delta ( m ) ] 2 E Φ Xt x i (t i t i 1 ) a() x Xx db (4.13) i=1 1 ],

19 COMPUING DELAS WIHOU DEIVAIVES 19 of a lookback option for a fine enough partition 1 = t < t 1 < < t m = 2, where i a()d = 1 for each i = 1,..., m.. From a numerical point of view, thi might make a difference ince the function a in (4.13) can be choen to have upport on the full egment, 2 ], while in (4.11) the function a can only have upport on, 1 ]. If the averaging period of the Aian option tart today, i.e. 1 =, then formula (4.11) doe not hold anymore. Intead, one can derive alternative cloed-form expreion for the Aian delta for mooth diffuion coefficient, ee e.g. 15] and 3], which potentially can be generalied to irregular drift coefficient. However, except for linear coefficient (Black & Schole model), thee expreion involve tochatic integral in the Skorokhod ene which are, in general, hard to imulate. Intead, we here propoe to enlarge the tate pace by one dimenion and to conider a perturbed Aian pay-off. In that cae we are able to derive a probabilitic repreentation for the correponding Delta that only include Itô integral. More preciely, we conider the (trong) olution to the perturbed two-dimenional SDE dx x t = b(t, X x t )dt + db t, X x = x, dy ɛ,x,y t = X x t dt + ɛdw t, Y ɛ,x,y = y, t, (4.14) for ɛ >, where W i a one-dimenional Brownian motion independent of B. he idea i now to conider the perturbed Aian pay-off with averaging period, 2 ], 2 (, ] a a European pay-off on Y ɛ,x,y 2 : ( ) ( 2 ) 2 Φ X x d Φ(Y ɛ,x, 2 ) = Φ X x d + ɛw 2. We then get the following reult, where we now conider the lightly differently weighted pay-off function pace } L {f q w () := : meaurable: f(x) q w(x)dx < for the weight function w defined by ( x 2 ) w(x) = exp 2 2 (2 2, /3 + 1) x. (4.15) heorem 4.5. Let Y ɛ,x,y be the econd component of the trong olution to (4.14) and Φ L 4p w (), where p > 1 i the conjugate of 1+ for > in Lemma A.1. For a given maturity time 2 (, ] and < ɛ 1, the price u ɛ (x) := EΦ(Y ɛ,x, 2 )] of the aociated perturbed Aian option i continuouly differentiable in x, and it derivative, i.e. the Delta, take the form u ɛ(x) = E Φ(Y ɛ,x, 2 ) ( a() x Xx db + ɛ 1 a() where a :, ] i a bounded meaurable function uch that )] x Xx udu dw, (4.16) a()d = 1. Proof. he proof i a traight forward generalization of the proof of heorem 4.2 to the (particularly imple) two-dimenional extenion (4.14) of the underlying SDE. herefore, we here only give the main tep. Firt oberve that the trong olution (Xt x, Y ɛ,x,y t ) i clearly differentiable in y, and by heorem 3.4 alo (weakly) differentiable in x, and we get ( ) ( ) X x D t x,y Y ɛ,x,y = x Xx t, t x Xx udu 1 for all t, ], where D x,y denote the derivative.