On the paper Is Itô calculus oversold? by A. Izmailov and B. Shay

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1 On he paper Is Iô calculus oversold? by A. Izmailov and B. Shay M. Rukowski and W. Szazschneider March, 1999 The main message of he paper Is Iô calculus oversold? by A. Izmailov and B. Shay is, we quoe: However, when applied o he nonlinear ineres rae models such as Cox- Ingersoll-Ross, Iô calculus only gives approximae resuls...) This surprising claim is firs suppored in he inroducion by few words on he diffusion approximaion, in which he auhors seem o sugges ha he Sraonovich inegral gives a more exac approximaion han he Iô inegral. We are, of course, aware abou he difference in boh kinds of inegrals, however, i is unclear o us o wha exen his is relaed o he main opic of he paper. Also, since he auhors only menion vaguely he diffusion approximaion, wihou clarifying his imporan issue, a serious discussion wih heir saemens in his regard is no possible. Le us only menion ha he Cox-Ingersoll-Ross CIR) diffusion-ype model is well known o be a good diffusion approximaion for several branching processes. Since many oher auhors saemens are vague and frequenly confusing, we shall commen on he purely mahemaical aspecs of he paper only. Many references from physical lieraure are given in he lis of references of he original paper; heir relevance if any) o he main opic of he paper is oally obscure, however. I is known ha he Wiener process can model he moion of a paricle e.g., he Rayleigh gas) only in very paricular cases. Therefore, he use of he Wiener process is a raher crude approximaion of majoriy of real siuaions. For a deailed discussion of his poin and furher references, he ineresed reader is referred o Szazschneider 1993). A. Secion The CIR model and procedure In his secion, he auhors consider he shor-erm rae process r), which is governed by he following sochasic differenial equaion r) =a) θ) r) ) d + r)σ) dw. 1) where a, θ and σ are coninuous funcions i is convenien o assume ha σ) > for every ). Typically, he sochasic inegral is here undersood in he Iô sense. I seems o us ha he auhors follow his convenion oherwise, hey wouldn be able o derive heir equaion 3) hrough an applicaion of Iô s rule. In principle, i is possible o inerpre he sochasic inegral in 1) in he sense of Sraonovich; his would give rise o a differen model of he shor-erm rae. 1

2 By definiion, he T -mauriy bond price is se o saisfy B, T )=E P e T ) ru) du F, ) where F is he filraion generaed by he underlying Wiener process sandard Brownian moion) W. We shall hus wrie F = F W. I should be observed ha in he CIR model he filraion generaed by W coincides wih he filraion generaed by he shor-erm rae process r; formally F W = F r for every R +. Acually his propery holds in mos shor-erm rae models pu forward in exising lieraure. Therefore B, T )=E P e T ) ru) du F r. 3) I should be noiced ha he probabiliy measure P is auomaically he maringale measure for he erm srucure model inroduced above; indeed, i is obvious ha for any mauriy T he discouned bond price B, T )/B, where B =exp r u du) follows a maringale under P. A.1 The CIR Procedure The so-called CIR procedure is presened incorrecly in he paper. Since he auhors sugges ha here are some approximaions in his procedure, we shall describe i carefully. Sep 1. One does no assume ha R, T ) is he funcion of he curren value r) of he shor erm rae. In fac, one shows ha he bond price B, T ) is a funcion of ime argumen and he shor-erm rae process r). This is an almos immediae consequence of formula ), combined wih he Markov propery of he process r. The auhors discussion abou he missed correlaion see Page 53) is hus erroneous. Sep. We apply Iô s rule o derive he dynamics ha is, he semimaringale represenaion) of he process v, r)). In fac, we search here for he drif erm raher han for he diffusion erm, since he knowledge of he exac form of he diffusion erm is no necessary for he derivaion of he parial differenial equaion PDE) which governs he value funcion v, r). We obain dv, r)) = v, r)) + a) θ) r) ) ) v r, r)) + 1 r)σ )v rr, r)) d +v r, r)) r)σ) dw. Sep 3. In fac, his sep is inexisen in he correc procedure. Sep 4. We simply observe ha under P we have db, T )=r)b, T ) d + ζ dw 4) for some predicable process ζ. Formula 4) is a consequence of he so-called predicable represenaion propery of he Brownian filraion, applied o he maringale B, T )/B

3 and combined wih he Iô formula. Combining formula 4) wih he represenaion obained in Sep, we ge for every s R + s v, r)) + a) θ) r) ) ) v r, r)) + 1 r)σ )v rr, r)) rv, r)) d = s ζ v r, r)) r)σ) ) dw. Since he lef-hand side in he las formula defines a process of finie variaion, while he righ-hand side manifesly follows a coninuous local) maringale, i is clear ha boh processes vanish. In paricular, his yields for every s R + s v, r)) + a) θ) r) ) ) v r, r)) + 1 r)σ )v rr, r)) rv, r)) d =. This leads o he parial differenial equaion for v, r) v, r)+a) θ) r ) v r, r)+ 1 rσ )v rr, r) rv, r) = 5) wih he erminal condiion vr, T )=1forr R +. I should be noiced ha no assumpions oher han dynamics 1) and formula ) were used o derive he PDE 5). Therefore, here is no reason o suspec ha he soluion o his PDE would give an incorrec resul for he bond price. Le us menion ha he rigorous approach o he exended CIR model, as defined by 1), can be found in Rogers 1995). B. Secion Analysis of he CIR Procedure To jusify heir conjecure, he auhors consider he special case when he shor-erm rae equals r) =σ W, [,T], 6) where σ is a sricly posiive consan. I is worhwhile o noice ha formula 9) in he paper makes no sense; he differenial d r) should have he form d r) =σd W. For obvious reasons, formula 6) above provides a much beer represenaion of he process r inroduced in his secion. The auhors insis ha he direc calculaions of he expeced value I := E P e T ru) du) = E P e T σ W du) u gives a differen resul han he PDE approach. Le us menion here ha he PDE approach should be correcly referred o as he Feynman-Kac procedure raher, han he CIR procedure. Cox, Ingersoll and Ross 1985) simply used in he conex of bond valuaion he well-known procedure, which is presened in mos exbooks on sochasic calculus. Any person familiar wih he opic easily recognizes ha he discrepancy suggesed by he auhors is simply no possible provided, of course, ha all calculaions are done properly). 3

4 I is noeworhy o menion ha I is simply he Laplace ransform of he linear funcional of he squared Wiener process. No sochasic inegraion whasoever wheher of he Iô ype or of any oher kind) is direcly involved in he definiion of I. B.1 Direc Approach. The auhors make here a simple error; namely, hey have miscalculaed he expeced value E P The righ answer should hus read σ W u du ) 3 = σt)6. I =1 1 σt) σt) σt) and no I =1 1 σt) σt) σt) as is claimed in he paper see formula 1) herein). In fac, sandard calculaions show ha for any u s s Wu) = su +u, and Ws W u) = s)ep W s Wu) + EP W 4 s Wu ) = su +u +s u +1su for any u s. Consequenly, ) s E P σ Wv dv =!σ 4 s W ) u duds = 7 1 σt)4 and E P ) 3 s σ Wv dv =3!σ 6 Ws Wu) dudsd = σt)6. B. PDE Approach. To derive he PDE for he bond price, one needs o find he sochasic differenial equaion which governs he shor-erm rae process. To his end, we apply Iô s formula o 7) and we ge sgn x) =1ifx, and equals 1 oherwise) dr) =σ dw =σ W sgn W ) dw + σ d, or equivalenly dr) =σ dw =σ r) d W + σ d, 4

5 where he process W, which is given by he formula W = sgn W u ) dw u, is anoher Brownian moion under P. Consequenly, he PDE for he bond price has he form cf. formula 5) above) v, r)+σ v r, r)+rσ 1 v rr, r) rv, r) =. 7) As was already explained, no approximaion of any kind is involved in he derivaion of he PDE 7). Therefore, an exac soluion of his PDE, subjec o he erminal condiion vr, T )=1, does necessarily saisfy he equaliy v,t)=i. The explici soluion provided in he paper see formula 14) herein) only suppor his self-eviden saemen. Finally, le us menion ha his formula is known since he fories; i is usually referred o as he Cameron-Marin formula. The ineresed reader may consul he original paper by Cameron and Marin 1945), or he monograph by Revuz and Yor 1994) see Page 45 herein) in his regard. B.3 Maringale Approach. I should be noiced he Cameron-Marin formula can be easily derived wih he use of he so-called exponenial maringales. Once again, he reader is referred o Revuz and Yor 1994) see Pages herein). Le us reproduce here heir argumens, adaped o he simple case analysed here. For σ =1, we wish o evaluae he condiional expecaion: I := E P e T ) W u du F. Le f :[,T] R be an auxiliary coninuous funcion, coninuously differeniable in,t). I is well known ha for any T we have { J f) :=E P exp fu)w u dw u 1 ) } f u)wu du F =1. Since dw =W dw +, we deduce ha { 1 T J f) =E P exp fu) dwu 1 fu) du 1 ) } f u)w u du F. 8) The Iô inegraion by pars formula yields: fu) dwu = ft )W T f)w f u)wu du. 5

6 Plugging he las expression ino 8) and rearranging, we obain J f) =e 1 { fu) du ft )W E P exp T f)w 1 f u)+f u) ) W u du ) F }. In he nex sep, we choose he funcion f in such a way ha Then we ge J f) =e 1 f u)+f u) =, ft )=. 9) { fu) du E P exp 1 ) } f)w Wu du F =1, and finally 1 T I =exp fu) du + 1 ) f)w. I remains o deermine explicily he funcion f ha solves equaion 9). To his end, we se fu) =h u)/hu) for some wice coninuously differeniable funcion h, where by convenion we se h) =1. In view of 9), i is easily seen ha h saisfies he ordinary differenial equaion h u) =hu), u, T ), wih suiable boundary condiions. Solving he las equaion, we obain hu) =Ae u + Be u, and hus fu) = h u) hu) = Ae u Be u ) Ae u + Be u where he values of consans A and B are found from he equaliies recall ha h) =1 and ft )=) Ae + Be =1, Ae T Be T =. Observe now ha fu) du = h u)h 1 u) du =, d ln hu) =lnht ) ln h) =lnht ) and f) =h ). We hus finally obain ) 1 I =exp ln ht )+ 1 h )W = ht )exp 1 h )W ). Elemenary calculaions now lead o he well-known explici formula for I cf. equaliy 14) in he paper). I is worhwhile o noice ha he same approach can be applied o he exended CIR model, as defined by he dynamics 1) see Szazschneider and Flores-López 1997)). The only difference is ha he explici soluion for he corresponding Surm- Liouville ordinary differenial equaion is usually no available one needs hus o employ approximae soluion which can be obained hrough an appropriae numerical procedure). 6

7 References [1] Cameron, W.T., Marin, W.T. 1945) Evaluaions of various Wiener inegrals by use of cerain Surm-Liouville differenial equaion. Bull. Amer. Mah. Soc. 51, [] Cox, J.C., Ingersoll, J.E., Ross, S.A. 1985) A heory of he erm srucure of ineres raes. Economerica 53, [3] Izmailov, A., Shay, B. 1999) Is Iô calculus oversold? Risk 1), [4] Revuz. D., Yor, M. 1994) Coninuous Maringales and Brownian Moion. Springer, Berlin Heidelberg New York. [5] Rogers, L.C.G. 1995) Which model for erm-srucure of ineres raes should one use? In: IMA Vol.65: Mahemaical Finance, M.H.A.Davis e al., eds. Springer, Berlin Heidelberg New York, pp [6] Szazschneider W., Flores-López, M.A. 1997) Pricing ineres rae derivaives in exended CIR model. Maringale approach. Working paper, Anahuac Universiy. [7] Szazschneider W. 1993) The moion of a agged paricle and nonhomogeneous media in R 1. Journal of Saisical Physics 7, Marek Rukowski Warsaw Universiy of Technology, -661 Warszawa, Poland markru@alpha.mini.pw.edu.pl Wojciech Szazschneider Universidad Anáhuac, Huixquilucan, Mexico, C.P wojciech@anahuac.mx 7