Cross hedging, utility maximization and systems of FBSDE U. Horst, Y. Hu, P . Imkeller, A. R´ eveillac, J. Zhang HU Berlin, U Rennes http://wws.mathematik.hu-berlin.de/ ∼ imkeller Tamerza, October 27, 2010 Partially supported by the DFG research center M ATHEON in Berlin
C ROSS HEDGING , UTILITY MAXIMIZATION AND SYSTEMS OF FBSDE 1 1 Cross hedging, optimal investment, exponential utility for convex constraints: (N. El Karoui, R. Rouge ’00; J. Sekine ’02; J. Cvitanic, J. Karatzas ’92, Kramkov, Schachermayer ’99, Mania, Schweizer ’05, Pham ’07, Zariphopoulou ’01,...) maximal expected exponential utility from terminal wealth � T EU ( x + X π V ( x ) = sup T + H ) = sup E ( − exp( − α ( x + π s [ dW s + θ s ds ] + H ))) π ∈A π ∈A 0 wealth on [0 , T ] by investment strategy π : � T � T � π u , dS u π u [ dW u + θ u du ] = X π � = T , S u 0 0 H liability or derivative, correlated to financial market S π ∈ A subject to π taking values in C closed aim: use BSDE to represent optimal strategy π ∗ Partially supported by the DFG research center M ATHEON in Berlin
C ROSS HEDGING , UTILITY MAXIMIZATION AND SYSTEMS OF FBSDE 2 2 Martingale optimality Idea: Construct family of processes Q ( π ) such that Q ( π ) = constant , 0 Q ( π ) − exp( − α ( x + X π = T + H )) , (form 1) T Q ( π ) supermartingale , π ∈ A , π ∗ ∈ A . Q ( π ∗ ) martingale, for (exactly) one Then E ( Q ( π ) E ( − exp( − α [ x + X π T + H ])) = T ) E ( Q π ≤ 0 ) E ( Q ( π ∗ ) = ) 0 E ( − exp( − α [ x + X ( π ∗ ) = + H ])) . T Hence π ∗ optimal strategy. Partially supported by the DFG research center M ATHEON in Berlin
C ROSS HEDGING , UTILITY MAXIMIZATION AND SYSTEMS OF FBSDE 3 3 Solution method based on BSDE Introduction of BSDE into problem Find generator f of BSDE � T � T Y t = H − Z s dW s − f ( s, Z s ) ds, Y T = F, t t such that with Q ( π ) = − exp( − α [ x + X π t + Y t ]) , t ∈ [0 , T ] , t we have Q ( π ) = − exp( − α ( x + Y 0 )) = constant , (fulfilled) 0 Q ( π ) − exp( − α ( x + X π = T + H )) (fulfilled) (form 2) T Q ( π ) supermartingale , π ∈ A , Q ( π ∗ ) π ∗ ∈ A . martingale, for (exactly) one This gives solution of valuation problem. Partially supported by the DFG research center M ATHEON in Berlin
C ROSS HEDGING , UTILITY MAXIMIZATION AND SYSTEMS OF FBSDE 4 4 Construction of generator of BSDE How to determine f : Suppose f generator of BSDE. Then by Ito’s formula Q ( π ) − exp( − α [ x + X π = t + Y t ]) t � t s [ − π s θ s − f ( s, Z s ) + α Q ( π ) + M ( π ) αQ ( π ) 2( π s − Z s ) 2 ] ds, = + 0 t 0 with a local martingale M ( π ) . Q ( π ) satisfies (form 2) iff for q ( · , π, z ) = − f ( · , z ) − πθ + α 2( π − z ) 2 , π ∈ A , z ∈ R , we have q ( · , π, z ) ≥ 0 , π ∈ A (supermartingale) (form 3) π ∗ ∈ A q ( · , π ∗ , z ) = 0 , for (exactly) one (martingale) . Partially supported by the DFG research center M ATHEON in Berlin
C ROSS HEDGING , UTILITY MAXIMIZATION AND SYSTEMS OF FBSDE 5 4 Construction of generator of BSDE Now − f ( · , z ) − πθ + α 2( π − z ) 2 q ( · , π, z ) = 2( π − z ) 2 − ( π − z ) · θ + 1 − zθ − 1 − f ( · , z )+ α 2 αθ 2 2 αθ 2 = 2[ π − ( z + 1 − zθ − 1 − f ( · , z )+ α αθ )] 2 2 αθ 2 . = Under non-convex constraint p ∈ C : [ π − ( z + 1 αθ )] 2 ≥ dist 2 ( C, z + 1 αθ ) . with equality for at least one possible choice of π ∗ due to closedness of C . Hence (form 3) is solved by the choice (predictable selection) 2 dist 2 ( C, z + 1 α θ ) − z · θ − 1 2 α θ 2 α f ( · , z ) = (supermartingale) (form 4) dist ( C, z + 1 α θ ) = dist ( π ∗ , z + 1 π ∗ : α θ ) (martingale) . Partially supported by the DFG research center M ATHEON in Berlin
C ROSS HEDGING , UTILITY MAXIMIZATION AND SYSTEMS OF FBSDE 6 5 Summary of results, exponential utility Solve utility optimization problem EU ( x + X π sup T + H ) π ∈A by considering FBSDE dX π X π = π t [ dW t + θ t dt ] , 0 = x, t = Z t dW t + f ( t, Z t ) dt, Y T = H dY t with generator as described before; determine π ∗ by previsible selection; coupling through requirement of martingale optimality EU ( x + X π ∗ EU ( x + X π sup T + H ) = T + H ) , π ∈A U ′ ( x + X π ∗ + Y t ) martingale . t for general U : forward part depends on π ∗ , get fully coupled FBSDE Partially supported by the DFG research center M ATHEON in Berlin
C ROSS HEDGING , UTILITY MAXIMIZATION AND SYSTEMS OF FBSDE 7 6 Cross hedging, optimal investment, utility on R Lit: Mania, Tevzadze (2003) U : R → R strictly increasing and concave; maximal expected utility from terminal wealth EU ( x + X π (1) V ( x ) = sup T + H ) π ∈A wealth on [0 , T ] by investment strategy π : � T � T � π u , dS u π u [ dW u + θ u du ] = X π � = T , S u 0 0 H liability or derivative, correlated to financial market S , W d − dimensional Wiener process, W 1 first d 1 components of W π ∈ A subject to convex constraint π = ( π 1 , 0) , π 1 d 1 − dimensional, hence incomplete market aim: use FBSDE system to describe optimal strategy π ∗ Partially supported by the DFG research center M ATHEON in Berlin
C ROSS HEDGING , UTILITY MAXIMIZATION AND SYSTEMS OF FBSDE 8 7 Verification theorems Thm 1 Assume U is three times differentiable, U ′ regular enough. If there exists π ∗ solving (1), and Y is the predictable process for which U ′ ( X π ∗ + Y ) is square integrable martingale, then with Z = d dt � Y, W � ( π ∗ ) 1 = − θ 1 U ′ U ′′ ( X π ∗ + Y ) − Z 1 . Pf: E ( U ′ ( X π ∗ T + H ) |F · ) , Y = ( U ′ ) − 1 ( α ) − X π ∗ . α = Use Itˆ o’s formula and martingale property. Find � T � T f ( s, X π ∗ Y = H − Z s dW s − s , Y s , Z s ) ds, · · with U (3) s , Y s , Z s ) = − 1 U ′′ ( X π ∗ + Y ) | π ∗ f ( s, X π ∗ s + Z s | 2 − π ∗ s θ s . 2 Use variational maximum principle to derive formula for π ∗ . Partially supported by the DFG research center M ATHEON in Berlin
C ROSS HEDGING , UTILITY MAXIMIZATION AND SYSTEMS OF FBSDE 9 7 Verification theorems From preceding theorem derive the FBSDE system Thm 2 Assumptions of Thm 1; then optimal wealth process X π ∗ given as component X of solution ( X, Y, Z ) of fully coupled FBSDE system � · � · U ′ U ′ ( θ 1 U ′′ ( X s + Y s ) + Z 1 s ) dW 1 ( θ 1 U ′′ ( X s + Y s ) + Z 1 s ) θ 1 = x − s − X s ds, s s 0 0 � T Y = H − Z s dW s · � T U (3) U ′ 2 ( U ′′ ) 3 + U ′ s | 2 (( − 1 [ | θ 1 U ′′ )( X s + Y s ) + Z 1 s · θ 1 − s ) 2 · s | 2 U (3) − 1 2 | Z 2 U ′′ ( X s + Y s )] ds. (2) Pf: Use expression for f and formula for π ∗ from Thm 1. Partially supported by the DFG research center M ATHEON in Berlin
C ROSS HEDGING , UTILITY MAXIMIZATION AND SYSTEMS OF FBSDE 10 8 Representation of optimal strategy Invert conclusion of Thm 2 to give representation of optimal strategy Thm 3 Let ( X, Y, Z ) be solution of (2), U ( X T + H ) integrable, U ′ ( X T + H ) square integrable. Then ( π ∗ ) 1 = − θ 1 U ′ U ′′ ( X + Y ) + Z 1 is optimal solution of (1). Pf: By concavity for any admissible π U ( X π + Y ) − U ( X + Y ) ≤ U ′ ( X + Y )( X π − X ) . Now prove that U ′ ( X + Y )( X π − X ) = U ′ ( X π ∗ + Y )( X π − X π ∗ ) is a martingale! Partially supported by the DFG research center M ATHEON in Berlin
C ROSS HEDGING , UTILITY MAXIMIZATION AND SYSTEMS OF FBSDE 11 9 The complete case Formula representing π ∗ − → martingale representation U ′ ( X π ∗ + Y ) = U ′ ( x + Y 0 ) E ( − θ · W ) . Aim: show existence for fully coupled system of Thm 2. Crucial observation: P = X + Y solves forward SDE � · � · U (3) U ′ 2 U ′ 1 P = x + Y 0 − U ′′ ( P s ) dW s − ( U ′′ ) 3 ( P s ) ds. θ s 2 0 0 Idea: forward SDE � · � · U (3) U ′ 2 U ′ 1 P m = x + m − U ′′ ( P m ( U ′′ ) 3 ( P m s ) dW s − s ) ds θ s 2 0 0 has solution; now decouple again, by considering BSDE � T � T U (3) U ′ 2 ( U ′′ ) 3 + U ′ ( | θ s | 2 [ − 1 Y m = H − Z m U ′′ ]( P m s ) + Z m s dW s − s θ s ) ds. 2 · · Partially supported by the DFG research center M ATHEON in Berlin
C ROSS HEDGING , UTILITY MAXIMIZATION AND SYSTEMS OF FBSDE 12 9 The complete case Solve for ( Y m , Z m ) , use continuity of m �→ Y m to find m such that Y m = m . 0 0 This gives Thm 4 Assume U (3) U ′ 2 ( U ′′ ) 3 and U ′ U ′′ are Lipschitz and bounded. Then the system of FBSDE � · � · U ′ U ′ U ′′ ( X s + Y s ) + Z s ) dW 1 = x − ( θ s s − ( θ s U ′′ ( X s + Y s ) + Z s ) θ s ds, X 0 0 � T � T U (3) U ′ 2 ( U ′′ ) 3 + U ′ [ | θ s | 2 (( − 1 = H − Z s dW s − U ′′ )( X s + Y s ) + Z s · θ s )] ds (3) Y 2 · · has solution ( X, Y, Z ) such that U ( X T + H ) is integrable, U ′ ( X T + H ) square integrable. Partially supported by the DFG research center M ATHEON in Berlin
Recommend
More recommend
