The Structure of General Mean-Variance Hedging Strategies

Transcription

1 The Srucure of General Mean-Variance Hedging Sraegies Jan Kallsen TU München (join work wih Aleš Černý, London) Pisburgh, February 27,

2 Quadraic hedging S H (discouned) asse price process (discouned) coningen claim How o hedge he risk from selling he claim? Hedging error: v + ϕ S T H v ϕ (discouned) iniial endowmen dynamic rading sraegy Quadraic hedging: min v,ϕ E ( (v + ϕ S T H) 2) v ϕ variance-opimal iniial endowmen variance-opimal hedging sraegy 2

3 Quadraic hedging S H (discouned) asse price process (discouned) coningen claim How o hedge he risk from selling he claim? Hedging error: v + ϕ S T H v ϕ (discouned) iniial endowmen dynamic rading sraegy Quadraic hedging: min v,ϕ E ( (v + ϕ S T H) 2) v ϕ variance-opimal iniial endowmen variance-opimal hedging sraegy 2

4 Quadraic hedging viewed differenly funcional analyic poin of view: L 2 -projecion of H on {v + ϕ S T : v R, ϕ admissible} Is {v + ϕ S T : v R, ϕ admissible} closed? (cf. Mona & Sricker 1995, Delbaen e al. 1997, Choulli e al. 1998, Delbaen & Schachermayer 1996) or compare o linear regression (= one-period model): min v,ϕ E ( (v + ϕs H) 2) for random variables H, S soluion: ϕ = Cov(H, S) Var(S) 3

5 Quadraic hedging viewed differenly funcional analyic poin of view: L 2 -projecion of H on {v + ϕ S T : v R, ϕ admissible} Is {v + ϕ S T : v R, ϕ admissible} closed? (cf. Mona & Sricker 1995, Delbaen e al. 1997, Choulli e al. 1998, Delbaen & Schachermayer 1996) or compare o linear regression (= one-period model): min v,ϕ E ( (v + ϕs H) 2) for random variables H, S soluion: ϕ = Cov(H, S) Var(S) 3

6 Variance-opimal hedging in general Case 1: S maringale (Föllmer & Sondermann 1986) use Galchouk-Kunia-Waanabe decomposiion Case 2: deerminisic mean-variance radeoff process of S (Schweizer 1994) use Föllmer-Schweizer decomposiion Case 3: arbirary S (e.g. Schweizer 1996, Rheinländer & Schweizer 1997, Gourieroux e al. 1998,..., Arai 2005) 4

7 Case 1: S maringale Galchouk-Kunia-Waanabe decomposiion: H = V 0 + ξ S T + R T, where R maringale, orhogonal o S (i.e. RS maringale) Mean value process of he opion: V := E(H F ) Variance-opimal hedge: v = V 0, ϕ = ξ = d V, S d S, S Hedging error: ( (v E + ϕ S T H ) 2 ) = E ( V ϕ S, V ϕ S T ) 5

8 Case 2: deerminisic mean-variance radeoff process of S Mean-variance radeoff process: ˆK = ˆλ A S, where ˆλ = and S = S 0 + M S + A S Doob-Meyer decomposiion of S da S d M S, M S Föllmer-Schweizer decomposiion: H = V 0 + ξ S T + R T, where R maringale, orhogonal o he maringale par M S of S Mean value process of he opion: V := E Q (H F ), where Q minimal (signed) maringale measure wih densiy process E ( ˆλ M S ) Variance-opimal hedge: v = V 0, ξ = d V, S d S, S, ϕ = ξ + λ(v v ϕ S ), where λ = das d S, S Hedging error: ( (v E + ϕ S T H ) 2 ) = E ( E ( ˆK) V ξ S, V ξ S T ) 1 E ( ˆK) T 6

9 Case 3: arbirary S (Schweizer 1996) Variance-opimal hedge: v := E Q (H), ϕ = ϱ ã(v + ϕ S ), where Q variance-opimal (signed) maringale measure (VOMM) wih densiy dq dp = E ( ã S) T E(E ( ã S) T ) backward sochasic differenial equaions for adjusmen process ã and ϱ (Rheinländer & Schweizer 1997) for coninuous S Mean value process of he opion: V := E Q (H F ) where Q variance-opimal maringale measure (VOMM) Variance-opimal hedge: v = V 0, ξ = ϕ = ξ + ã(v v ϕ S ) d V, S Q, d S, S Q How o obain he adjusmen process ã? 7

10 Case 3: arbirary S (Schweizer 1996) Variance-opimal hedge: v := E Q (H), ϕ = ϱ ã(v + ϕ S ), where Q variance-opimal (signed) maringale measure (VOMM) wih densiy dq dp = E ( ã S) T E(E ( ã S) T ) backward sochasic differenial equaions for adjusmen process ã and ϱ (Rheinländer & Schweizer 1997) for coninuous S Mean value process of he opion: V := E Q (H F ), where Q variance-opimal maringale measure (VOMM) Variance-opimal hedge: v = V 0, ξ = ϕ = ξ + ã(v v ϕ S ) d V, S Q, d S, S Q How o obain he adjusmen process ã? 7

11 Case 3: arbirary S (Černý & K 2005) Key idea: change of measure P P (deermined by characerisic equaion) Föllmer-Schweizer decomposiion relaive o P : H = V 0 + ξ S T + R T, where R P -maringale, orhogonal o he P -maringale par of S Mean value process of he opion: V := E Q (H F ), where Q variance-opimal (signed) maringale measure (VOMM) = minimal maringale measure relaive o P Variance-opimal hedge: v = V 0, ξ = d V, S P d S, S P das ϕ = ξ + ã(v v ϕ S ), where ã = d S, S P adjusmen process and S = S 0 + M S + A S Doob-Meyer decomposiion of S relaive o P Hedging error: (v E( + ϕ S T H ) 2 ) = E (L V ξ S, V ξ S P, T ) 8

12 Case 3: arbirary S (Černý & K 2005) Key idea: change of measure P P (deermined by characerisic equaion) Föllmer-Schweizer decomposiion relaive o P : H = V 0 + ξ S T + R T, where R P -maringale, orhogonal o he P -maringale par of S Mean value process of he opion: V := E Q (H F ), where Q variance-opimal (signed) maringale measure (VOMM) = minimal maringale measure relaive o P Variance-opimal hedge: v = V 0, ξ = d V, S P d S, S P das ϕ = ξ + ã(v v ϕ S ), where ã = d S, S P adjusmen process and S = S 0 + M S + A S Doob-Meyer decomposiion of S relaive o P Hedging error: (v E( + ϕ S T H ) 2 ) = E (L V ξ S, V ξ S P, T ) 8

13 Case 3: arbirary S (Černý & K 2005) Key idea: change of measure P P (deermined by characerisic equaion) Föllmer-Schweizer decomposiion relaive o P : H = V 0 + ξ S T + R T, where R P -maringale, orhogonal o he P -maringale par of S Mean value process of he opion: V := E Q (H F ), where Q variance-opimal (signed) maringale measure (VOMM) = minimal maringale measure relaive o P Variance-opimal hedge: v = V 0, ξ = d V, S P d S, S P das ϕ = ξ + ã(v v ϕ S ), where ã = d S, S P adjusmen process and S = S 0 + M S + A S Doob-Meyer decomposiion of S relaive o P Hedging error: (v E( + ϕ S T H ) 2 ) = E (L V ξ S, V ξ S P, T ) 8

14 The equaions for he opporuniy-neural measure P is called opporuniy process. L = inf E ( (1 (1 ]],T ]] ϑ) S T ) 2 ϑ ) F I is he unique semimaringale such ha 1. L, L are (0, 1]-valued, 2. L T = 1, 3. he join characerisics (b S,L, c S,L, F S,L, A) of (S, L) solve where and b := b S + csl c := c S + b L = L b 2 c, 1 L + x 2 ( some (unpleasan) inegrabiliy condiions hold. x y L F S,L y ) S,L L F (d(x, y)) (d(x, y)), 9

15 The equaions for he opporuniy-neural measure P is called opporuniy process. L = inf E ( (1 (1 ]],T ]] ϑ) S T ) 2 ϑ ) F I is he unique semimaringale such ha 1. L, L are (0, 1]-valued, 2. L T = 1, 3. he join characerisics (b S,L, c S,L, F S,L, A) of (S, L) solve where and b := b S + csl c := c S + b L = L b 2 c, 1 L + x 2 ( some (unpleasan) inegrabiliy condiions hold. x y L F S,L y ) S,L L F (d(x, y)) (d(x, y)), 9

16 In his case we define 1. he adjusmen process ã := b c, 2. he densiy process of P relaive o P : Z P := L E(L 0 )E ( b 2 c A ), 3. and he densiy process of Q relaive o P : Z Q := LE ( ã S). E(L 0 ) 10

17 Opporuniy process L in specific siuaions discree ime: backward recursion L T = 1, L 1 = E(L F 1 ) ( E( S L F 1 ) ) 2 E(( S ) 2 L F 1 ), adjusmen process ã = E( S L F 1 ) E(( S ) 2 L F 1 ) affine sochasic volailiy models (S, v) Try L = exp(α() + v β()) wih α, β deerminisic ordinary differenial equaions for α, β 11

18 Opporuniy process L in specific siuaions discree ime: backward recursion L T = 1, L 1 = E(L F 1 ) ( E( S L F 1 ) ) 2 E(( S ) 2 L F 1 ), adjusmen process ã = E( S L F 1 ) E(( S ) 2 L F 1 ) affine sochasic volailiy models (S, v) Try L = exp(α() + v β()) wih α, β deerminisic ordinary differenial equaions for α, β 11