Sensitivity Analysis for Averaged Asset Price Dynamics with Gamma Processes

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1 Sensiiviy Analysis for Averaged Asse Price Dynamics wih Gamma Processes REIICHIRO KAWAI AND ASUSHI AKEUCHI Absrac he main purpose of his paper is o derive unbiased Mone Carlo esimaors of various sensiiviy indices for an averaged asse price dynamics governed by he gamma Lévy process. he key idea is o apply a scaling propery of he gamma process wih respec o he Esscher densiy ransform parameer. Our framework covers no only he coninuous Asian opion bu also European discree Asian average srike Asian weighed average spread opions and geomeric average Asian opions. Numerical resuls are provided o illusrae he effeciveness of our formulas in Mone Carlo simulaions relaive o finie difference approximaion. Keywords: Asian opions Esscher ransform gamma process Greeks Lévy process Malliavin calculus. 2 Mahemaics Subec Classificaion: 6E7 6G5 9B7 6H7. Inroducion In mahemaical finance i has been widely known ha he logarihmic derivaives of he densiy funcion of sochasic differenial equaions correspond essenially o he so-called Greeks ha is sensiiviy indices wih respec o various model parameers of asse price dynamics. he pioneer work in his direcion is of Fournié e al. whose approach is based upon he inegraion-bypars formula developed in he Malliavin calculus on he Wiener space. hey provide a sysemaic approach o he derivaion of various Greeks formulas for a relaively general diffusion asse price Published in Saisics and Probabiliy Leers 2) 8) Eddress: reiichiro.kawai@gmail.com. Posal Address: Deparmen of Mahemaics Universiy of Leiceser Leiceser LE 7RH UK. Corresponding Auhor. Address: akeuchi@sci.osaka-cu.ac.p. Posal Address: Deparmen of Mahemaics Osaka Ciy Universiy Sugimoo Osaka Japan.

2 dynamics. Since hen here has long been a naural and nonrivial quesion wheher a similar Malliavin calculus approach is applicable in he case of ump processes. he invesigaion of he Malliavin calculus for ump processes is iniiaed in Bismu 4 based upon he Girsanov densiy ransform wih a view owards he sudy of he exisence and he smoohness of he densiy. Various ypes of Malliavin calculus and logarihmic derivaives have been sudied on he Poisson space or on he Wiener-Poisson space. In paricular he exisence of weighs for he logarihmic derivaives daes back o he work of Bicheler Gravereaux and Jacod 3 while he uniqueness and he closed forms of he weigh have no been esablished in he general framework. For pracical consideraions however i is mos imporan o have closed bu no necessarily unique weighs on hand in order o design an efficien Mone Carlo evaluaion. In his direcion Greeks formulas are obained in Davis and Johansson 9 and Cass and Friz 8 for ump diffusion processes. heir approach consiss of condiioning on he ump componen and hen performing he Malliavin calculus echniques on he diffusion componen. In oher words heir models are required o be a superposiion of independen ump and diffusion componens where he diffusion one mus no be degenerae. On he oher hand El-Khaib and Privaul applied he Malliavin calculus focused on he Poisson arrival imes due o Carlen and Pardoux 5 while Bally e al. ook a unified approach considering he derivaives wih respec o boh he Poisson arrival imes and he ampliude of he umps. Kawai and Kohasu-Higa 3 derives formulas for Lévy process models of ime-changed Brownian moion ype by employing he Gaussian Malliavin calculus condiionally on he ime-changing process. akeuchi 7 sudied he same problem for he sochasic differenial equaions wih umps via a maringale approach. Kawai and akeuchi 4 sudied he compuaion of he Greeks of European payoffs for an asse price dynamics defined wih gamma processes and wih Brownian moions possibly ime-changed by one of he gamma processes. In Kawai and akeuchi 4 various well known financial models such as he Black- Scholes model and he variance gamma model of Madan Carr and Chang 5 are wihin is framework. he key idea here is o apply a scaling propery of he gamma disribuion wih respec o he Esscher ransform. In his paper we ake an approach again based on he scaling propery no of he gamma disribuion as in Kawai and akeuchi 4 bu of he gamma Lévy process. his raher simple exension widens he applicabiliy o he averaged asse price dynamics defined wih he gamma process. A ypical example of ineres is he coninuous) Asian opion which looks a he ime average of underlying asse price dynamics unlike only he erminal value in he European opions. Meanwhile many ohers such as European discree Asian average srike Asian weighed average and spread opions can easily be accommodaed in our framework. In addiion Asian opions of geomeric average ype are also wihin our scope. his work can also be hough of as 2

3 a considerable generalizaion of Kawai and akeuchi 4 in he sense ha some of he European Greeks formulas here can be recovered from our resuls. Finally our approach aken in his paper can also be applied in various sochasic sysems involving gamma processes. he res of he paper is organized as follows. Secion 2 recalls generaliies on he gamma process and inroduces our asse price dynamics model wih assumpions o be imposed on is characerizing parameers. In Secion 3 we derive formulas of he Greeks for disconinuous payoff funcion of averaged asse price dynamics defined wih he gamma process. he derivaion of our formulas enails raher lenghy proofs of somewha rouine naure. o avoid overloading he paper we omi nonessenial deails in some insances. We close his sudy wih some numerical resuls o illusrae remarkable improvemens in Mone Carlo simulaions in erms of esimaor variance relaive o he finie difference esimaion. 2 Preliminaries Le us begin wih general noaions which will be used hroughou he ex. For each posiive ineger k k indicaes he parial derivaive wih respec o k-h argumen. We fix ΩF P) as our underlying he probabiliy space. We denoe by he expecaion under he probabiliy measure P. We denoe by δ x dy) he Dirac dela measure on R wih concenraion a x R while B ε := ε+ε) is he open ball around he origin wih radius ε >. Le {Y : } be a one-sided pure-ump Lévy process in + ) wih he Lévy measure νdz) = a e bz z dz z + ) where a > and b >. he sochasic process {Y : } is hen called a gamma process wih parameer ab) whose marginal has he gamma disribuion wih characerisic funcion e iyy = exp e iyz ) νdz) = iy ) a. b + ) Moreover he marginal densiy funcion a ime is given in he form where Γp) is he gamma funcion of order p >. f P y) = ba Γa) ya e by y + ) 2.) hroughou he paper we fix >. Define Λ := { λ R : e λy < + } = b). For λ Λ define a new probabiliy measure Q λ by dq λ dp := eλy = exp λy F e λy ϕ λ) 3

4 where ϕ λ) := a ln b/b λ) ) and where F ) is he naural filraion generaed by {Y : }. his measure change is he simples form of he Girsanov ransformaion ofen called he Esscher ransform in mahemaical finance and acuarial science. he following is he key ool for our discussions. Lemma 2.. he laws of he process Y under Q λ and of he process Y λ) := by /b λ) under P are idenical. Proof. I is well known ha for Lévy processes he ideniy of finie dimensional disribuions implies he ideniy of random processes in law. Fix = n = and ζ ζ n R arbirarily. Due o he independence of he incremens under he measure P we have ) ) n n E Qλ exp i ζ k Y k = exp i k Y k + λy k= k=ζ λ ) a b n = i n k= ζ ) a k + λ ) λ = b b ) n b = exp i ζ k b λ Y k which proves he claim. k= ) a ) Le us inroduce our asse price dynamics model and wih some examples of pracical ineres wihin our scope. Le Z := {Z : } be a nondecreasing sochasic process in R wih Z = P-a.s. independen of he process Y := {Y : }. Our asse price dynamics model {S : } is defined by S := S expθy + κz + cθκ) where S + ) θ + ) κ R and c : + ) R R wih k cθκ) being well defined for k = 2. Le us impose some assumpions on he parameers θ and b and on he process Z. Assumpion 2.2. We henceforh assume he following; i) 4θ < b ii) here exiss a consan p > 2 such ha expp κ Z ) < +. Remark 2.3. Le us propose some candidaes for he process Z which makes our asse price dynamics model of adequae pracical ineres. 4

5 he firs one is he variance gamma model of Madan Carr and Chang 5 by seing Z also o be a gamma process wih a negaive κ. he model has araced he aenion of finance praciioners and have hus appeared ofen in he compuaional finance lieraure for example Carr and Madan 7 and Fu 2. o be precise se θ = and κ = and he gamma processes Y and Z should be characerized respecively wih ab) = ν µ p ν) ) and ab) = ν µ n ν) ) for some ν > where µ p = 2 θ 2 + 2σ 2 /ν + θ/2 and µ n = 2 θ 2 + 2σ 2 /ν θ/2. he inverse Gaussian process is anoher ineresing candidae for Z. I is also a subordinaor induced by he inverse Gaussian disribuion whose marginal a ime is idenical in law o he firs ime ha a Brownian moion wih drif ν reaches he posiive level ha is e yz = exp 2y + ν 2 ν ). More deails on he inverse Gaussian process may be found in Barndorff- Nielsen 2 and Carr e al Sensiiviy formulas Le µ d) be a finie Borel measure over. For simpliciy in noaion we henceforh wrie S λ) H λ) := S exp := J λ) := L λ) := M λ) := θy λ) S λ) Y λ) + κz + cθκ) F λ) ) Z µd) I λ) := S λ) := S λ) Z µd) K λ) := by λ) a + ) by λ) λ) I a + by λ) a + θf λ) ) H λ) θf λ) + Iλ) F λ) 2 F λ) + Hλ) F λ) 2 F λ) ) 2 Iλ) F λ) ) 2 Hλ) F λ) S λ) λ)) Y µd) = 23) λ)) Y µd) = ) ) F λ) θf λ) + Fλ) F λ) 2 F λ) ) 2 for λ Λ and. For furher simpliciy when λ = we suppress he superscrip ) of Y ) S ) F ) H ) I ) J ) K ) L ) and M ). We begin wih our main resuls. heorem 3.. Le Φ : R + R be a measurable funcion such ha ΦF ) 2 is locally uniformly bounded in S θ and κ. hen we have i) Sensiiviy wih respec o S S ΦF ) = S ΦF )K. 5

6 ii) Second derivaive wih respec o S 2 S 2 ΦF ) = S 2 ΦF ) 3bY a + )K F θf ) 2bY a + ) 2 F + a θf ) 2 + F2 F 3 F 3 +3 F2 F2 2 F 4 3 F F 2 F 2 ) S S ΦF ). iii) Sensiiviy wih respec o θ ) θ E by a P ΦF ) = ΦF ) + cθκ)l. θ iv) Sensiiviy wih respec o κ κ ΦF ) = ΦF )M + 2 cθκ)l). Remark 3.2. In he conex of mahemaical finance he sensiiviy i) and ii) are called Dela and Gamma respecively. Boh of he formulas iii) and iv) correspond bes o he Vega ha is he sensiiviy wih respec o he Black-Scholes volailiy parameer. Our framework covers several payoffs of pracical ineres. Firs seing µd) = δ d) reduces o he European framework. Wih his seing our resuls indeed recover he European formulas derived in Kawai and akeuchi 4. As a maer of course µd) = d corresponds o he Asian opion also called average price opion) while in real financial rading he coninuous average is simply impossible o compue and is hus no very ineresing. Meanwhile we can deal wih arihmeic average discree Asian) ype which is raher he genuine Asian from a pracical poin of view by seing µd) = N N k= δ k d) o be numerically esed laer or a lile more generally a weighed arihmeic average by µd) = N k= w k) N k= w k δ k d) for reals {w k } k N. Opions of average srike Asian ype and of spread ype respecively wih µd) = d δ d) and µd) = δ /2 d) + δ d) ) are also wihin our framework. he proof of heorem 3. is divided ino several preliminary resuls. he firs one is concerned wih he momens of Y λ) and of S λ). Lemma 3.3. Fix and λ Λ. I holds ha for p a+ ) while for p p b λ)/θ) S λ)) p = Y λ)) p Γa + p) = b λ) p Γa) pθ b λ ) a S p exp pcθκ) ) ) exp pκ Z. 6

7 Proof. he firs asserion is sraighforward by he densiy 2.) while he second follows from he independence of Y and Z he exponenial inegrabiliy of Z in Assumpion 2.2 and again he densiy 2.). We fix ε b/2) in wha follows. Noe ha B ε Λ and p b λ)/θ > 2 under Assumpion 2.2. he following lemma laer plays an imporan role for he he inerchange of he derivaive wih respec o λ) and he expecaion. Lemma 3.4. he random variables S λ) and λ B ε. and Y λ) are in L p b λ)/θ Ω) uniformly in Proof. he asserion is he direc consequence of Assumpion 2.2 and Lemma 3.3. he following lemma is also an imporan ool. Lemma 3.5. For k = 2 and for λ B ε i holds ha λ S λ) λ)) k Y Z µ d) = S λ) λ)) ) k Y Z µ d) P-a.s. λ λ)) k Y Z is coninuous in λ so is /λ)m λ) Proof. Clearly for each M λ) := S λ) M λ) λ) θy + k ) /b λ). Since Y λ) is nondecreasing in we have for h b λ) ) ξ Mξ ) 2 ξ =λ+δh dδ µd) S 2 exp 2θY ξ ) + 2 κ Z + 2 cθκ) ) Y ξ ) ) 2k Z 2 θy ξ ) ) 2 + k µ ) b ξ By he aylor heorem i holds ha h M λ+h) µd) M λ) µd) ) which ends o as h. he proof is complee. = λ Mλ) µd) ξ Mξ ) ξ =λ+δh λ Mλ) Le us invesigae he inegrabiliy of he random variable F λ) of negaive order. Lemma 3.6. For p p a )/2) i holds ha F λ)) p sup < +. λ B ε 7 ξ =λ+h. = ) µd)

8 Proof. Choose conugae exponens qr) wih q a /p) and r a /a p)2 ) and pick ζ pq/a ). Observe ha ) F λ) p p µζ ) p µζ ) ζ S λ) Y λ) ) p µζ ) p Y λ) pq /q ζ ζ ) p µd) ) S λ) pr /r µd)) where we have used he nonnegaiviy of he inegrand he nonnegaiviy and he nondecreasingness of he process Y he Jensen inequaliy and he Hölder inequaliy. Finally he finieness of he above las line follows from he momen formulas of Lemma 3.3. We invesigae he inegrabiliy of he random variable F λ) H λ) I λ) order. and J λ) of posiive Lemma 3.7. For p p /2 b ε)/2θ) ) i holds ha ) 3 sup F λ) p ) = + H λ) p = + λ B ε I λ) ) p ) + J λ)) p < +. Proof. By choosing conugae exponens qr) and uv) wih q 2) and v 2) we have by he Jensen inequaliy 3 = F λ) ) p + = H λ) ) p + c p p µ ) p µ c p lp p µ ) p Y λ) I λ) ) p ) S λ) + J λ)) p Y λ) ) Z k l µd)) p ) pqu qu EP e prz /r which is well defined by Lemma 3.3 and Assumpion 2.2. S λ) ) pqv qv µd) Proof of heorem 3.. We sar wih he smoohness assumpion Φ Cb 2 R;R) which can be removed laer by sandard densiy argumens. For deails we refer o Kawai and Kohasu-Higa 3 and Kawai and akeuchi 4.) hroughou he proof we le λ B ε. i) By Lemma 2. i holds ha Φ F λ) ) F λ) S θf λ) = e λy e λy ΦF ) F S θf. 3.) Differeniaing boh sides of 3.) a λ = yields he resul provided ha he differeniaion wih respec o λ and he expecaion are commuaive on he boh hand sides which we will prove 8

9 in wha follows. Concerning he lef hand side of 3.) by he aylor heorem and Lemma 3.5 i holds ha sup λ ε E P Φ F λ) ) F λ) λ S θf λ) sup λ ε sup λ ε Φ F δλ) Φ F δλ) ) F δλ) + sup Φ F δλ) λ ε where F ΦF ) S θf ) ) F δλ) S b δλ) + Φ F δλ) S b δλ) Φ F ) F ) U δλ) S b δλ) ΦF ) U) S b ) Φ F ) F S b + ΦF ) U) S b ) U δλ) S b δλ) Φ F ) F S b ΦF ) U) dδ S b dδ S b dδ F λ) U λ) := Fλ) 2 λ)) 2 Fλ) F θf λ). he firs inegral on he righ hand side ends o zero as ε by he Cauchy-Schwarz inequaliy wih he boundedness of Φ and Lemma 3.4. For he second inegral again by he Cauchy- Schwarz inequaliy wih F λ) 2 Y λ) F λ) P-a.s. i remains o show ha for λ B ε and for λ) k = F Y λ) ) k /F λ) ) 2 < +. By choosing conugae exponens pqr) wih p p a )/4 ) and q p /4 b ε)/4θ) ) we have λ) F Y λ) F λ) ) k ) 2 F λ) ) 2p /p EP F λ) ) 2q /q EP Y λ) ) 2kr /r which is uniformly bounded over λ ε by Lemma 3.3 Lemma 3.6 and Lemma 3.7. he inerchange of he derivaive wih respec o λ and he expecaion on he righ hand side of 3.) can be shown by he Cauchy-Schwarz inequaliy wih ΦF ) 2 < + and wih in L 2 Ω) as λ. e λy ϕ λ) λ Y a b ii) Define Ψx) := Φ x)x. Observe ha 2 ΦF ) = Φ F ) F = Φ F ) S S S 2 = S 2 F S ) 2 Ψ F ) Φ F ) ) F = S 2 ΨF ) ΦF ))K 9

10 where he firs and he fourh equaliies holds by he resul i) he second by Lemma 3.5 and he hird by Φ x)x 2 = Ψ x)x Φ x)x. he desired resul can be derived by aking derivaive a λ = of he ideniy Φ F λ) ) bf λ) K λ) = θf λ) e λy ΦF e λy ) bf K θf where he inerchange of he differeniaion wih respec o λ and he expecaion and he exisence of he formulas can be usified in a similar manner o he proof of i). iii)-iv) he desired formulas can be derived by aking derivaive a λ = respecively of Φ F λ) ) ) + cθκ) Iλ) e λy F λ) = ΦF e λy ) + cθκ) I ) F Φ F λ) ) H λ) + 2 cθκ) J λ) e λy = ΦF e λy ) H + 2 cθκ) J. F F λ) We omi usifying he inerchange of he derivaive wih respec o λ and he expecaion and he proof of he exisence of he formulas. We nex presen Greeks formulas for geomerically averaged asse price dynamics. Fix N N and < 2 N. Define G λ) := N S λ) S λ) S λ) N 2 = S exp θỹ λ) + κ Z + cθκ). where Ỹ λ) := N k= Y λ) k /N Z := N k= Z k /N := N k= k/n. Wihou essenial loss of generaliy we have ruled ou he case = a priori. Noice ha he above geomeric averaging framework reduces o he European seing by puing N = and N =. As before we suppress he superscrip ) ha is Ỹ := Ỹ ) and G := G ). hen Greeks formulas are as follows. heorem 3.8. Le Φ : R + R be a measurable funcion such ha ΦG) 2 is locally uniformly bounded in S θ and κ. If a N > 2 i holds ha ΦG) = ΦG) by a + S S θỹ θ ΦG) = ΦG) by a + S cθκ) θ κ ΦG) = ΦG) by a + ) Z θỹ + S 2 cθκ) S ΦG) S ΦG).

11 If a N > 4 hen i holds ha 2 S 2 ΦG) = S 2 ΦG) by a + ) 2 + a + 2θ 2 Ỹ 2 ) θ 2Ỹ 2 S S ΦG). Proof. o avoid overloading his proof wih nonessenial deails we only give he proof of he Dela formula a a skechy level. Wih he addiional regulariy assumpion Φ Cb 2 R;R) we ge he formula by differeniaing he ideniy Φ G λ) ) e λy = E Ỹ λ) P ΦG) e λy Ỹ a λ =. he inerchange of he derivaive wih respec o λ and he expecaion and he removal of he regulariy of Φ can be usified as before wih he help of Y 2 /Ỹ 2 Y 2 /Y 2 N = EP Y Y N ) 2 Y 2 + 2EP Y Y N Y + < + where we have used he independence of he incremens. Le us close our sudy wih numerical examples o illusrae he performance of our Greeks formulas in Mone Carlo esimaions. We consider a digial payoff ΦF ) = F > C) for some C > wih he weighing measure µd) = N N k= δ k N d) so ha ΦF ) is an equidisan discree Asian opion. We divide ino N = 2 inervals. Le us emphasize ha as discussed in Remark 3.2 our approach is rigorously valid for his discree Asian opions no as a numerical approximaion of coninuously averaged Asians. Fix S = C = = a = b = 6 and θ =. In our experimens he process Z is an inverse Gaussian Lévy) process which is inroduced in Remark 2.3. We se κ = and ν = 3. Concerning he generaion of a inverse Gaussian process we follow an algorihm proposed in Rydberg 6. By seing cθκ) = aln θ/b)+ 2κ + ν 2 ν ) we can induce he model risk-neural. For simpliciy we omi he risk-free discouning. Figure presens numerical resuls of he sensiiviies /S 2 /S 2 /θ and /κ of he expecaion ΦF ). For clear comparisons he convergence of he finie difference mehod is also provided. he figures and he variance raios evidenly indicae a faser convergence of our Greeks formulas. Noe also ha in general he finie difference approximaion yields some bias. N N Acknowledgemens he auhors are graeful o Co-Edior-in-Chief Prof.Hira Koul for his careful reading and valuable suggesions. his work was suppored in par by JSPS Gran-in-Aid for Scienific Research

12 .6 3.e e MC FD MC FD 2.e S E P ΦF ) vraio= 44 and ε = e-3. 2 E S 2 P ΦF ) vraio= 33 and ε = e MC FD MC FD θ ΦF ) vraio= 9 and ε = 5e-3. κ ΦF ) vraio= 366 and ε = 2e-2. Figure : vraio indicaes he variance raio in he finie difference esimaion X+ε)θ) X ε)θ) 2εθ VarFinie Difference) VarMalliavin Calculus). he quaniy ε is he incremen or X+ε)θ) 2Xθ)+X ε)θ) εθ) and was largely carried ou while RK was based a Cener for he Sudy of Finance and Insurance Osaka Universiy Japan. References Bally V. Bavouze M. Messaoud M. 27) Inegraion by pars formula for locally smooh laws and applicaions o sensiiviy compuaions Ann. Appl. Probab Barndorff-Nielsen O.E. 998) Processes of normal inverse Gaussian ype. Finance Soch

13 3 Bicheler K. Gravereaux J. B. Jacod J. 987) Malliavin calculus for processes wih umps Gordon and Breach Science Publishers New York. 4 Bismu J.M. 983) Calcul des variaions sochasique e processus de saus. Z. Wahrsch. Verw. Gebiee Carlen E. Pardoux E. 99) Differenial calculus and inegraion by pars on Poisson space In: Albeverio S. Blanchard Ph. esard D. Eds.) Sochasics algebra and analysis in classical and quanum dynamics Kluwer Acad. Publ. pp Carr P. Geman H. Madan D. Yor M. 23) Sochasic Volailiy for Lévy processes. Mah. Finance Carr P. Madan D. 999) Opion pricing and he fas Fourier ransform. J. Compu. Finance Cass. R. Friz P. K. 27) he Bismu-Elworhy-Li formula for ump-diffusions and applicaions o Mone Carlo mehods in finance available a arxiv:mah/643v3. 9 Davis M.H.A. Johansson M.P. 26) Malliavin Mone Carlo Greeks for ump diffusions. Sochasic Process. Appl El-Khaib Y. Privaul N. 24) Compuaions of Greeks in a marke wih umps via he Malliavin calculus. Finance Soch Fournié E. Lasry J. Lebuchoux J. Lions P. ouzi N. 999) Applicaions of Malliavin calculus o Mone Carlo mehods in finance. Finance Soch Fu M.C. 27) Variance-Gamma and Mone Carlo In: Fu M. C. Jarrow R. A. Yen J.-Y. J. Ellio R. J. Eds.) Advances in Mahemaical Finance Birkhäuser pp Kawai R. Kohasu-Higa A. 29) Compuaion of Greeks and mulidimensional densiy esimaion for asse price models wih ime-changed Brownian moion o appear in Appl. Mah. Finance. 4 Kawai R. akeuchi A. 29) Greeks formulae for an asse price model wih gamma processes under revision. 5 Madan D. Carr P. Chang E. 998) he variance gamma process and opion pricing. European Finance Review

14 6 Rydberg.H. 997) he normal inverse Gaussian Lévy process: simulaion and approximaion. Comm. Sais. Sochasic Models akeuchi A. 29) he Bismu-Elworhy-Li ype formulae for sochasic differenial equaions wih umps under revision. 4