IMA Preprint Series # 2056

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1 MULTISCALE ASYMPTOTICS OF PARTIAL HEDGING By Gerard Awanou IMA Preprin Series # 056 ( July 005 ) INSTITUTE FOR MATHEMATICS AND ITS APPLICATIONS UNIVERSITY OF MINNESOTA 400 Lind Hall 07 Church Sree S.E. Minneapolis, Minnesoa Phone: 61/ Fax: 61/ URL: hp://

2 MULTISCALE ASYMPTOTICS OF PARTIAL HEDGING GERARD AWANOU Absrac. We consider he problem of parial hedging of an European derivaive under he assumpion ha he volailiy is sochasic, driven by wo diffusion processes, one fas mean revering and he oher varying slowly. For opions wih long mauriies ypically beyond 90 days, he singular perurbaion analysis in [Parial Hedging in a Sochasic Volailiy Environmen, M. Jonsson and K.R. Sircar, Mahemaical Finance, 1, pp , 00] ignores he slow facor. In his paper, we invesigae he full wo facors model and show how an addiional erm can be included in he approximae value funcions and sraegies. 1. Inroducion We consider he problem of shorfall risk minimizaion in sochasic volailiy models under he assumpion ha he volailiy is driven by wo diffusions, one slowly varying and one fas mean revering. Following he mehodology of [3], he shorfall risk minimizaion problem is ransformed ino a sae dependen uiliy maximizaion problem. The opimal sraegies depend on he soluion of a high dimensional HJB equaion saisfied by he value funcion of he uiliy problem. The PDE saisfied by he Legendre ransform of he value funcion is derived along wih he one saisfied by he smalles opimizer in he Legendre ransform. Tha opimizer can be hough as he inverse of marginal uiliy and is shown o be he price of a modified claim. We perform he asympoical analysis on he inverse of he marginal uiliy and derive approximae sraegies and value funcions.. Muliscale sochasic volailiy models Le (Ω, F, P ) denoe a probabiliy space which describes a financial marke wih ime horizon T and a risky asse whose price X X() saisfies dx = µx d + σ X dw (0), where W (0) is a sandard one-dimensional Brownian moion, µ and σ are he mean and volailiy of he sock respecively. We consider in his paper a wo-facor sochasic volailiy model σ = f(y, Z ) for a smooh, bounded and posiive funcion f which is also bounded away from zero. The process Y is a fas mean revering diffusion process. To fix ideas, we shall assume ha i is a Gaussian Ornsein-Uhlenbeck process wih rae of mean reversion α and invarian disribuion a Gaussian wih mean m and variance ν = β /α; dy = α(m Y ) d + β dw (1), 1

3 GERARD AWANOU where W (1) is a sandard one-dimensional Brownian moion wih d W (0), W (1) = ρ 1 d and ρ 1 is consan, ρ 1 < 1. Pu α = 1 so ha β = ν/, > 0 being he ime scale of he process. Fas mean reversion occurs when is small and he variance ν is moderae. The process Z is a slowly varying diffusion process dz = δc(z ) d + δh(z ) dw (), where δ > 0 is a small parameer, c and g are smooh and a mos linearly growing a infiniy. We assume ha d W (0) d W (1), W () = ρ d, W () = ρ 1 d, wih consans ρ and ρ 1. More explicily, he correlaion beween he hree Brownian moions can be described as dw (0) = db 1 dw (1) = ρ 1 db ρ 1 db dw () = ρ 1 db 1 + ρ 1 db + 1 ρ ρ 1 db 3 where ρ 1 is a consan which saisfy ρ + ρ 1 < 1 and (B 1, B, B 3 ) is a sandard hree dimensional Brownian moion. Noice ha we have ρ 1 = ρ 1 ρ + ρ 1 1 ρ 1. In summary we have (1) dx = µx d + σ X dw (0) dy = α(m Y ) d + β dw (1) dz = δc(z ) d + δh(z ) dw () 3. Shorfall risk minimizaion Le F, 0 T be he σ-field generaed by {X u, Y u, Z u, 0 u }. For simpliciy, we shall assume ha F 0 is rivial and F T = F. Le also g be a coninuous funcion from R ino R +. We consider an invesor who has an iniial capial v 0 and is willing o inves π dollars in he sock. His goal is o hedge a coningen claim H = g(x T ), and we shall assume ha he is using self-financing sraegies so ha he value V of his porfolio is given by V = v + 0 π u X u dx u, 0 T. We require π, 0 T predicable wih π /X inegrable. If he iniial capial is no high enough, here is a possibiliy of shorfall, V T < H. For a given iniial value x of he risky asse, we are ineresed in sraegies which minimize he shorfall risk E{l(H V T ) + }

4 MULTISCALE ASYMPTOTICS OF PARTIAL HEDGING 3 under he consrain V 0 for all, where l is a posiive convex funcion wih l(0) = 0 and x + =max(x, 0). To fix ideas we shall assume ha l(u) = 1 u. Le us define he sae dependen uiliy funcion U(x, v) = 1 [h(x) ((h(x) v) + ) ]. I is easy o see ha he problem is equivalen o maximize E{U(X T, V T )} under he consrain V 0 for all. HJB equaion. Recall (1), he dynamics of (X, Y, Z ) and noice ha he porfolio value process saisfies We nex define dv = π µ d + π f(y, Z ) dw (0). H(, x, y, z, v) = sup π E,x,y,z,v {U(X T, V T )} where E,x,y,z,v is he expecaion condiional o X 0 = x, Y 0 = y, Z 0 = z, V 0 = v. We also inroduce he infiniesimal generaor L x,y,z H = µxh x + α(m y)h y + δc(z)h z + 1 x f(y, z) H xx () + βρ 1 xf(y, z)h xy + δh(z)ρ xf(y, z)h xz + 1 β H yy + δh(z)βρ 1 H zy + 1 δh(z) H zz. By he Bellman principle, H saisfies he following HJB equaion H + L x,y,z H + max π {πµh v + πxf(y, z) H xv + βρ 1 πf(y, z)h vy I is no difficul o find ha he maximum occurs a (3) so + δh(z)ρ πf(y, z)h zv + 1 π f(y, z) H vv } = 0. π = µh v + xf(y, z) H xv + βρ 1 f(y, z)h vy + δh(z)ρ f(y, z)h zv f(y, z) H vv H + L x,y,z H [µh v + f(y, z) xh xv + ρ 1 βf(y, z)h vy + δh(z)ρ f(y, z)h zv ] f(y, z) H vv = 0 in he domain x > 0, < y <, < z <, v > 0 and < T and erminal condiion H(T, x, y, z, v) = U(x, v). Legendre ransform. We now use he Legendre ransform o ransform his nonlinear PDE o a PDE wih several linear erms. Le us assume ha H is convex in v and define he convex dual Ĥ of H wih respec o v by (4) We also define (5) Ĥ(, x, y, z, r) = sup v>0 {H(, x, y, z, v) rv}. g(, x, y, z, r) = inf{v > 0 H(, x, y, z, v) rv + Ĥ(, x, y, z, r)}

5 4 GERARD AWANOU which is in some sense he smalles opimizer in (4). If H is sufficienly smooh, g = Hv 1, so ha g is he inverse of he marginal uiliy. We have he following equaions which relae H o Ĥ. Ĥ(, x, y, z, r) = rg(, x, y, z, r) + H(, x, y, z, g(, x, y, z, r)) H(, x, y, z, v) = vĝ(, x, y, z, v) + Ĥ(, x, y, z, ĝ(, x, y, z, v)) H v (, x, y, z, g(, x, y, z, r)) = r H v (, x, y, z, v) = ĝ(, x, y, z, v) Ĥ r (, x, y, z, ĝ(, x, y, z, v)) = v Ĥ r (, x, y, z, r) = g(, x, y, z, r) H v = r H = Ĥ H x = Ĥx H y = Ĥy H z = Ĥz H vv = 1 H xv = Ĥxr H yv = Ĥyr H zv = Ĥzr H xx = Ĥxx Ĥ xr H xy = Ĥxy ĤxrĤyr H yy = Ĥyy Ĥ yr H xz = Ĥxz ĤxrĤzr H zz = Ĥzz Ĥ zr H yz = Ĥyz ĤyrĤzr From he PDE of H and he above relaionships beween H and following equaions for Ĥ, Ĥ, we ge he (6) Ĥ + L x,y,z Ĥ µrxĥxr µrρ 1β µr δh(z)ρ f(y, z)ĥyr f(y, z) Ĥ yr Ĥ zr + 1 µ r f(y, z) Ĥrr 1 β (1 ρ 1) 1 δh(z) (1 ρ )Ĥ zr 1 ρ 1ρ 1 β δh(z)ĥyrĥzr = 0, wih erminal condiion Ĥ(T, x, y, z, r) = sup v>0 {H(T, x, y, z, v) rv} = sup v>0 {U(x, v) rv} := Û(x, v). The PDE for g is obained from he one of Ĥ, by differeniaion and using g(, x, y, z, r) = Ĥr(, x, y, z, r).

6 MULTISCALE ASYMPTOTICS OF PARTIAL HEDGING 5 We have g +L x,y,z g µxg x ρ 1βµ f(y, z) g y µ δh(z)ρ f(y, z) µxrg xr ρ 1βµ f(y, z) rg yr β (1 ρ 1) g y g yr g r 1 ρ 1ρ 1 β δh(z) g z g yr µr δh(z)ρ (7) g r f(y, z) 1 ρ 1ρ 1 β δh(z) g y µ g zr + g r + 1 δh(z) (1 ρ ) g z g gr rr + wih erminal condiion Hedging sraegies. g z + µ f(y, z) r g r g zr δh(z) (1 ρ ) g z g r g zr f(y, z) r g rr + 1 (1 ρ 1)β g y 1 ρ 1ρ 1 β δh(z) g yg z g r g rr = 0, g(t, x, y, z, r) = inf v>0 {U(x, v) rv + Û(x, r)} := G(x, r). g gr rr Using he expression of he opimal sraegy (3), relaions beween he derivaives of Ĥ and H, one can wrie π in erms of g and hen ge he opimal sraegies. Explicily (8) π = µ f(y, z) rg r + xg x + βρ 1 f(y, z) g y + δh(z)ρ f(y, z) g z. Pricing a modified claim. I can be shown ha (6) is he HJB equaion for he conrol problem Ĥ(, x, y, z, r) = inf γ,η E,x,y,z,r {Û(X T, Y T, Z T, R γ,η T )}, where (X, Y, Z ) saisfy (1) and R γ,η dr γ,η is defined by µ = f(y, Z ) Rγ,η db 1 + γ db + η db3. Moreover i is possible o choose γ and η so ha he opimal sraegy is he perfec hedge of he claim G(X T, R γ,η T ). To see his, le us assume ha a ime, he price of he claim is given by g(, X, Y, Z, R γ,η ) for a smooh funcion g o be deermined wih g(t, x, y, z, r) = G(x, r). We hen consider he porfolio which consiss of he claim and unis of he he sock a ime Π = g(, X, Y, Z, R γ,η ) X. Using Iô s Lemma, we ge an expression for, γ and η for he porfolio o be insaneneously riskless in erms of g and a PDE for g which urns ou o be he same as he one of g. Since hey have he same erminal condiion, hey are equal.

7 6 GERARD AWANOU Explicily, we have βρ 1 = g x + X f(y, Z ) g δh(z )ρ y + X f(y, Z ) g z γ = β 1 ρ g y 1 + g z δh(z )ρ 1 g r g r η = δh(z ) 1 ρ ρ g z 1. g r 4. Muliscale asympoics Recall ha α = 1 and β = ν/. We firs define he following linear and nonlinear operaors (9) L 0 = ν y + (m y) y L 1 = ρ 1 ν (f(y, z)x x y µ f(y, z) r y r µ ) f(y, z) y L = + 1 ( f(y, z) x x + µ 1 f(y, z) r r + r ) r µxr x r (10) (11) M 1 = µρ h(z) f(y, z) M = c(z) z + h(z) M 3 = νρ 1 h(z) z y z µρ h(z) f(y, z) r z r + h(z)ρ xf(y, z) x z z ( ) gyr g y N L yr (g) = g y g g r gr rr = ( g ) y r g r ( ) gzr g z N L zr (g) = g z g g r gr rr = ( ) g z r g r ( gyr g z + g zr g y N L yzr (g) = g ) yg z g g r gr rr = ( ) gy g z r g r Using () and (7), g is seen o saisfy he PDE ( 1 L 0+ 1 ) ( δm1 L 1 + L g + + δm + (1) + ν ) δ M 3 g (1 ρ 1)N L yr (g) + δh(z) (1 ρ )N L zr (g) δ + (1 ρ 1)ρ 1 ν h(z)n L yzr(g) = 0.

8 MULTISCALE ASYMPTOTICS OF PARTIAL HEDGING 7 Le us pu γ 1 = ν (1 ρ 1), γ = h(z) (1 ρ ), and γ 3 = (1 ρ 1)ρ 1 νh(z). We nex perform he asympoics in he regime where and δ are small independen parameers. The erm g j,k will be associaed wih he erm of order j δ k. Slow scale expansion. We firs make an expansion of g g,δ in powers of δ (13) g,δ = g 0 + δg 1 + δg +... We will use several imes he expansion N L yr (g,δ ) = N L yr (g0) δ [ ( y g0 r r g0 y g1 )] yg0 r g0 r g = N L yr (g0) δ ] [ y g0 Γ(g 0, g1) +..., r ( ) where Γ(g0, g1) = 1 rg 0 y g1 yg 0 rg 0 r g1. The order δ and δ 3 erms are sums of erms which have a facor y g0 and/or y g1 and will no be needed in he erms we are ineresed in. Subsiuing (13) ino (1), and expanding in powers of δ we ge from he firs wo erms, ( 1 L 0+ 1 ) L 1 + L g 0 + γ 1 (14) N L yr(g0) = 0, ( 1 L 0+ 1 ) L 1 + L g 1 + M 1 g0 + 1 M 3 g γ 3 N L yzr (g 0) (15) ) ( y g 0Γ(g0, g1) = 0. r We append o hese equaions, he boundary condiions Fas scale expansion. g 0(T, x, y, z, r) = g(t, x, y, z, r) = G(x, r) g 1(T, x, y, z, r) = 0 We nex expand g 0 and g 1 in powers of. Expansion of g 0. (16) As above g 0 = g 0 + g 1,0 + g,0 + 3 g3, N L yr (g0) = N L yr (g 0 ) ] [ y g 0 Γ(g 0, g 1,0 ) +... r

9 8 GERARD AWANOU Subsiuing ino (14), we ge ( ) 1 L 0 g 0 + γ 1 N L yr (g 0 ) + 1 ( [ L 0 g 1,0 + L 1 g 0 γ 1 y g 0 Γ(g 0, g 1,0 ) ] ) r ) + (L 0 g,0 + L 1 g 1,0 + L g 0 + ) (L 0 g 3,0 + L 1 g,0 + L g 1, = 0 From he lowes order erm, we ge L 0 g 0 + γ 1 N L yr (g 0 ) = 0. Noice ha 1 L 0 is he infiniesimal generaor of he OU process Y. Arguing as in [3] p. 17, we conclude ha g 0 does no depend on y. This implies ha a he nex order L 0 g 1,0 + L 1 g 0 = 0. Since g 0 does no depend on y and L 1 akes derivaives in y, L 1 g 0 = 0. This gives L 0 g 1,0 = 0 from which we conclude ha g 1,0 also does no depend on y. To see his, one can use he expression of κ in [1] p. 93, where κ is he soluion of L 0 κ = g. Wih L 1 g 1,0 = 0, we ge from he zeroh order erm L 0 g,0 + L g 0 = 0. We know from [1] p. 91, ha a necessary condiion for his Poisson equaion o be solvable in g,0 is ha L g 0 = 0, where, denoes he average wih respec o he invarian disribuion of Y. The zeroh order approximaion is hen aken o be soluion of (17) where L g 0 = 0 g 0 (T, x, y, z, r) = G(x, r), L = + 1 ( σ(z) x x + µ 1 σ (z) r r + r ) µxr r x r as g 0 does no depend on y and we have defined (18) σ(z) := f 1 (., z) σ (z) := 1 f (., z). Finally from he erm of order, we have L 0 g 3,0 + L 1 g,0 + L g 1,0 = 0. This is again a Poisson equaion in g 3,0 and we have he cenering condiion L 1 g,0 + L g 1,0 = 0. Using a sandard muliscale argumen, e.g. [] p. 8 for differen operaors, g 1,0 is seen as he soluion of he problem L g 1,0 = L 1 L 1 0 (L L ) g 0 g 1,0 (T, x, y, z, r) = 0. As in [3] p. 1-, an explici expression of g 1,0 can be given in erms of g 0 and cerain marke group parameers. In fac he problem is he same up o renaming

10 he variables. MULTISCALE ASYMPTOTICS OF PARTIAL HEDGING 9 Expansion of g 1. (19) g 1 = g 0,1 + g 1,1 + g,1 + 3 g3, Since g 0 does no depend on y, (15) becomes ( 1 L 0+ 1 ) (0) L 1 + L g 1 + M 1 g0 + 1 M 3 g γ 3 N L yzr (g 0) = 0. Again since g 0 and g 1,0 do no depend on y, we have N L yzr (g 0) = O( ) Therefore subsiuing (19) and (16) ino (0) we obain 1 L 0g 0,1 + 1 ) (L 0 g 1,1 + L 1 g 0,1 + M 3 g 0 ) + (L 0 g,1 + L 1 g 1,1 + L g 0,1 + M 1 g 0 + M 3 g 1, = 0 The firs order erm gives L 0 g 0,1 = 0 which implies as above ha g 0,1 does no depend on y. Since L 1 and M 3 akes derivaives in y, we have a he nex order L 0 g 1,1 = 0 from which we conclude ha g 1,1 also does no depend on y. The same argumens lead us o conclude ha from he zeroh order erm, he following equaion holds L 0 g,1 + L g 0,1 + M 1 g 0 = 0. The solvabiliy condiion for his Poisson equaion in g,1 leads us o conclude ha g 0,1 solves (1) L g 0,1 = M 1 g 0 wih zero erminal condiion. This equaion does no admi an exac soluion and 1 mus be solved numerically. Noice ha he erm M 1 involves f(y, z) and f(y,z) and so would require model specificaion. Having obained he firs hree erms in he asympoics, we can use (8) o ge he approximae hedging sraegies. 5. Conclusion The difference of his aricle wih [3] is ha we inroduce an approximaion erm in he asympoics o ake ino accoun a slow facor which invariably one would observe if one is ineresed in opions wih long mauriies. To implemen he mehod one should proceed as in [3].

11 10 GERARD AWANOU References [1] J.P. Fouque; G. Papanicolaou and R. Sircar, Derivaives in Financial Markes wih Sochasic Volailiy, Cambridge Universiy Press, 000. [] J.P. Fouque; G. Papanicolaou; R. Sircar and K. Solna, Muliscale sochasic volailiy asympoics, Muliscale Model. Simul., no. 1, pp. -4, 003. [3] M. Jonsson and K.R. Sircar, Parial Hedging in a Sochasic Volailiy Environmen, Mahemaical Finance, Vol. 1, pp , 00. Insiue for Mahemaics and is Applicaions, Universiy of Minnesoa, Minneapolis, MN, address: awanou@ima.umn.edu URL: hp://