Mean-Variance Hedging on uncertain time horizon in a market with a jump

Mean-Variance Hedging on uncertain time horizon in a market with a jump Thomas LIM 1 ENSIIE and Laboratoire Analyse et Probabilités d Evry Young Researchers Meeting on BSDEs, Numerics and Finance, Oxford 2012 Joint work with Idris Kharroubi and Armand Ngoupeyou 1 Supported by the Chaire Risque de Crédit, Fédération Bancaire Française. Thomas Lim Mean-Variance Hedging 1 / 36

Introduction Progressive enlargement of filtrations and BSDEs with jumps (with I. Kharroubi), forthcoming in Journal of Theoritical Probability. A decomposition approach for the discrete-time approximation of FBSDEs with a jump I: the Lipschitz case (with I. Kharroubi). A decomposition approach for the discrete-time approximation of FBSDEs with a jump II: the quadratic case (with I. Kharroubi). Mean-Variance Hedging on uncertain time horizon in a market with a jump (with I. Kharroubi and A. Ngoupeyou). Thomas Lim Mean-Variance Hedging 2 / 36

Mean-variance hedging in literature [( T ) 2 ] inf E x + π s ds s ξ. π 0 There exist two approaches to solve mean-variance hedging problem with a deterministic finite horizon: martingale theory and projection arguments: Delbaen-Schachermayer, Gouriéroux-Laurent-Pham, Schweizer,... for the continuous case, and Arai for the semimartingale case, quadratic stochastic control and BSDE: Lim-Zhou, Lim,... for the continuous case and the discontinuous case (driven by a Brownian motion and a Poisson process). Jeanblanc-Mania-Santacroce-Schweizer combine tools from both approaches which allows them to work in a general semimartingale model. Thomas Lim Mean-Variance Hedging 3 / 36

Mean-variance hedging with random horizon For some financial products (e.g. insurance, credit-risk) the horizon of the problem is not deterministic [( T τ ) 2 ] inf E x + π s ds s ξ. π 0 We use a BSDE approach as in Lim and provide a solution to the mean-variance hedging problem with random horizon, dependent jump and continuous parts. Theoretical issue: no result for our BSDEs in this framework. Thomas Lim Mean-Variance Hedging 4 / 36

Outline 1 Preliminaries and market model The probability space Financial model Mean-variance hedging 2 Solution of the mean-variance problem by BSDEs Martingale optimality principle Related BSDEs A verification Theorem 3 How to solve the BSDEs Thomas Lim Mean-Variance Hedging 5 / 36

Outline Preliminaries and market model 1 Preliminaries and market model The probability space Financial model Mean-variance hedging 2 Solution of the mean-variance problem by BSDEs Martingale optimality principle Related BSDEs A verification Theorem 3 How to solve the BSDEs Thomas Lim Mean-Variance Hedging 6 / 36

Preliminaries and market model The probability space Settings Let (Ω, G, P) be a complete probability space equipped with W a standard Brownian motion with its natural filtration F := (F t ) t 0, τ a random time (we define the process H by H t := 1 τ t ). τ not always an F-stopping time. G smallest right continuous extension of F that turns τ into a G-stopping time: G := (G t ) t 0 where G t := ε>0 G t+ε, for all t 0, with G s := F s σ ( 1 τ u, u [0, s] ), for all s 0. Thomas Lim Mean-Variance Hedging 7 / 36

Preliminaries and market model Assumption on W and τ The probability space (H) The process W remains a G-Brownian motion. (Hτ) The process H admits an F-compensator of the form. τ 0 λ s ds, i.e. H. τ 0 λ s ds is a G-martingale, where λ is a bounded P(F)-measurable process. We then denote by M the G-martingale defined by t τ t M t := H t λ s ds = H t λ G s ds, 0 0 for all t 0, with λ G t := (1 H t )λ t. Thomas Lim Mean-Variance Hedging 8 / 36

Preliminaries and market model Financial model Financial market Financial market is composed by a riskless bond B with zero interest rate: B t = 1, a risky asset S modeled by the stochastic differential equation t S t = S 0 + S u (µ u du + σ u dw u + β u dm u ), t 0, 0 where µ, σ and β are P(G)-measurable processes satisfying (HS) (i) µ, σ and β are bounded, (ii) there exists a constant c > 0 s.t. (iii) 1 β t, t [0, T ], P a.s. σ t c, t [0, T ], P a.s. Thomas Lim Mean-Variance Hedging 9 / 36

Admissible strategies Preliminaries and market model Mean-variance hedging We consider the set A of investment strategies which are P(G)-measurable processes π such that [ T τ ] E π t 2 dt 0 <. We then define for an initial amount x R and a strategy π, the wealth V x,π associated with (x, π) by the process t Vt x,π = x + 0 π r S r ds r, t [0, T τ]. Thomas Lim Mean-Variance Hedging 10 / 36

Preliminaries and market model Mean-variance hedging Problem For x R, the problem of mean-variance hedging consists in computing the quantity [ V inf E x,π 2] π A T τ ξ, (1) where ξ is a bounded G T τ -measurable random variable of the form ξ = ξ b 1 T <τ + ξ a τ 1 τ T, where ξ b is a bounded F T -measurable random variable and ξ a is a continuous F-adapted process satisfying ess sup ξt a < +. t [0,T ] Thomas Lim Mean-Variance Hedging 11 / 36

Outline Solution of the mean-variance problem by BSDEs 1 Preliminaries and market model The probability space Financial model Mean-variance hedging 2 Solution of the mean-variance problem by BSDEs Martingale optimality principle Related BSDEs A verification Theorem 3 How to solve the BSDEs Thomas Lim Mean-Variance Hedging 12 / 36

Solution of the mean-variance problem by BSDEs Martingale optimality principle Sufficient conditions for optimality We look for a family of processes {(J π t ) t [0,T ] (i) J π T τ = V x,π T τ ξ 2, for all π A. (ii) J π 1 0 = J π 2 0, for all π 1, π 2 A. (iii) J π is a G-submartingale for all π A. : π A} satisfying (iv) There exists some π A such that J π is a G-martingale. Under these conditions, we have for any π A E(J π T τ ) = J π 0 = J π 0 E(J π T τ ). Thomas Lim Mean-Variance Hedging 13 / 36

Solution of the mean-variance problem by BSDEs Martingale optimality principle Sufficient conditions for optimality We look for a family of processes {(J π t ) t [0,T ] (i) J π T τ = V x,π T τ ξ 2, for all π A. (ii) J π 1 0 = J π 2 0, for all π 1, π 2 A. (iii) J π is a G-submartingale for all π A. : π A} satisfying (iv) There exists some π A such that J π is a G-martingale. Under these conditions, we have for any π A Therefore, we get E(J π T τ ) = J π 0 = J π 0 E(J π T τ ). J π 0 = E [ V x,π T τ ξ 2 ] = inf E[ V x,π π A T τ ξ 2 ]. Thomas Lim Mean-Variance Hedging 13 / 36

Solution of the mean-variance problem by BSDEs Related BSDEs SG is the subset of R-valued càd-làg G-adapted processes (Y t) t [0,T ] essentially bounded Y S := sup Y t <. t [0,T ] S,+ G is the subset of SG of processes (Y t) t [0,T ] valued in (0, ), such that 1 Y S <. L 2 G is the subset of R-valued P(G)-measurable processes (Z t) t [0,T ] such that ( [ T ]) 1 Z L 2 := E Z t 2 2 dt <. 0 L 2 (λ) is the subset of R-valued P(G)-measurable processes (U t ) t [0,T ] such that U L2 (λ) := ( [ T τ ]) 1 E λ s U s 2 2 ds 0 <. Thomas Lim Mean-Variance Hedging 14 / 36

Solution of the mean-variance problem by BSDEs Construction of J π using BSDEs Related BSDEs To construct such a family {(J π t ) t [0,T ], π A}, we set J π t := Y t V x,π t τ Y t 2 + Υ t, t 0, where (Y, Z, U), (Y, Z, U) and (Υ, Ξ, Θ) are solution in SG L2 G L2 (λ) to Y t = 1 + Y t = ξ + Υ t = for all t [0, T ]. T τ t τ T τ t τ T τ t τ f(s, Y s, Z s, U s )ds g(s, Y s, Z s, U s )ds h(s, Υ s, Ξ s, Θ s )ds T τ t τ T τ t τ T τ t τ Z s dw s Z s dw s Ξ s dw s T τ t τ T τ t τ T τ t τ U s dm s, (2) U s dm s, (3) Θ s dm s, (4) Thomas Lim Mean-Variance Hedging 15 / 36

Solution of the mean-variance problem by BSDEs Related BSDEs We are bounded to choose three functions f, g and h for which J π is a submartingale for all π A, there exists π A such that J π is a martingale. For that we would like to write J π as the sum of a martingale M π and a nondecreasing process K π that is constant for some π A. Thomas Lim Mean-Variance Hedging 16 / 36

Solution of the mean-variance problem by BSDEs Related BSDEs We are bounded to choose three functions f, g and h for which J π is a submartingale for all π A, there exists π A such that J π is a martingale. For that we would like to write J π as the sum of a martingale M π and a nondecreasing process K π that is constant for some π A. From Itô s formula, we get dj π t = dm π t + dk π t, where M π is a local martingale and K π is given by with dk π t := K t (π t )dt = ( A t π t 2 + B t π t + C t ) dt, A t := σ t 2 Y t + λ G t β t 2 (U t + Y t ), B t := 2(Vt τ π Y t )(µ t Y t + σ t Z t + λ G t β t U t ) 2σ t Y t Z t 2λ G t β t U t (Y t + U t ), C t := f(t) Vt τ π Y t 2 + 2Xt π (Y t g(t) Z t Z t λ G t U t U t ) + Y t Z t 2 +λ G t U t 2 (U t + Y t ) h(t). Thomas Lim Mean-Variance Hedging 16 / 36

Solution of the mean-variance problem by BSDEs Related BSDEs In order to obtain a nondecreasing process K π for any π A and that is constant for some π A it is obvious that K t has to satisfy min π R K t (π) = 0: K t := min π R K t(π) = C t B t 2 4A t. We then obtain from the expressions of A, B and C that with K t = A t V π t τ Y t 2 + B t (V π t τ Y t ) + C t, A t := f(t) µ ty t + σ t Z t + λ G t β t U t 2 σ t 2 Y t + λ G t β t 2 (U t + Y t ), { (µt Y t + σ t Z t + λ G t β t U t )(λ G t β t U t (Y t + U t ) + σ t Y t Z t ) B t := 2 σ t 2 Y t + λ G + g(t)y t β t 2 t (U t + Y t ) } Z t Z t λ G t U t U t, C t := h(t) + Z t 2 Y t + λ G t (U t + Y t ) U t 2 σ ty t Z t + λ G t β t U t (U t + Y t ) 2 σ t 2 Y t + λ G t β t 2 (U t + Y t ). Thomas Lim Mean-Variance Hedging 17 / 36

Solution of the mean-variance problem by BSDEs Expressions of the generators Related BSDEs For that the family (J π ) π A satisfies the conditions (iii) and (iv) we choose f, g and h such that for all t [0, T ]. A t = 0, B t = 0 and C t = 0, (µty +σtz+λtβtu)2 σ t 2 Y +λ t β t 2 (U+Y ), f(t, Y, Z, U) = [ g(t, Y, Z, U) = 1 Y t Z t Z + λ t U t U h(t, Υ, Ξ, Θ) = Z t 2 Y t + λ t (U t + Y t ) U t 2 Nonstandard Decoupled BSDEs Theorem ] (µty t+σtzt+λtβtut)(σty tz+λtβt(ut+y t)u) σ t 2 Y t+λ tβt 2(Ut+Y t) σty tzt+λtβtu t(ut+y t) 2 σ t 2 Y t+λ t β t 2 (U t+y t). The BSDEs (2)-(3)-(4) admit solutions (Y, Z, U), (Y, Z, U) and (Υ, Ξ, Θ) in SG L2 G L2 (λ). Moreover Y S,+ G. Thomas Lim Mean-Variance Hedging 18 / 36,

Solution of the mean-variance problem by BSDEs A verification Theorem Optimal strategy-sde of the optimal value portfolio A candidate to be an optimal strategy is which gives the implicit equation in π with D t := π t = arg min π R K t(π), (5) π t = ( Y t V x,π t ) Dt + E t, µty t +σtzt+λg t βtut and σ t 2 Y t +λ G t βt 2 (U t+y t ) E t := σty t Zt+λG t βtut(y t +Ut). σ t 2 Y t +λ G t βt 2 (U t+y t ) Integrating each side of this equality w.r.t. ds t S t leads to the following SDE t Vt ( = x + Yr Vr ) ds t r ds r Dr + E r 0 S r 0 S r, t [0, T ]. (6) Nonstandard SDE since D and E are not bounded. Thomas Lim Mean-Variance Hedging 19 / 36

Solution of the mean-variance problem by BSDEs A verification Theorem Optimal strategy-sde of the optimal value portfolio Proposition The SDE (6) admits a solution V which satisfies [ E Vt 2] <. sup t [0,T τ] Thomas Lim Mean-Variance Hedging 20 / 36

Solution of the mean-variance problem by BSDEs A verification Theorem From Itô s formula, we get dj π t = dm π t + dk π t, where M π is a local martingale and K π is given by with dk π t := K t (π t )dt = ( A t π t 2 + B t π t + C t ) dt, A t := σ t 2 Y t + λ G t β t 2 (U t + Y t ), B t := 2(Vt τ π Y t )(µ t Y t + σ t Z t + λ G t β t U t ) 2σ t Y t Z t 2λ G t β t U t (Y t + U t ), C t := f(t) Vt τ π Y t 2 + 2Xt π (Y t g(t) Z t Z t λ G t U t U t ) + Y t Z t 2 +λ G t U t 2 (U t + Y t ) h(t). Thomas Lim Mean-Variance Hedging 21 / 36

Solution of the mean-variance problem by BSDEs A verification Theorem Verification theorem Theorem The strategy π given by (5) belongs to the set A and is optimal for the mean-variance problem (1) [ V x,π E 2] T τ ξ [ V = min E x,π 2] π A T τ ξ. Thomas Lim Mean-Variance Hedging 22 / 36

Outline How to solve the BSDEs 1 Preliminaries and market model The probability space Financial model Mean-variance hedging 2 Solution of the mean-variance problem by BSDEs Martingale optimality principle Related BSDEs A verification Theorem 3 How to solve the BSDEs Thomas Lim Mean-Variance Hedging 23 / 36

How to solve the BSDEs A decomposition Approach: Data We consider a BSDE of the form Y t = ξ + T τ t τ terminal condition F (s, Y s, Z s, U s ) T τ t τ ξ = ξ b 1 T <τ + ξ a τ 1 τ T, where ξ b is an F T -measurable bounded r.v. and ξ a S F, T τ Z s dw s U s dh s, (7) t τ generator: F is a P(G) B(R) B(R) B(R)-measurable map and F (t, y, z, u)1 t τ = F b (t, y, z, u)1 t τ, t 0, where F b is a P(F) B(R) B(R) B(R)-measurable map. We then introduce the following BSDE Y b t = ξ b + T t F b (s, Y b s, Z b s, ξ a s Y b s )ds T t Z b s dw s. (8) Thomas Lim Mean-Variance Hedging 24 / 36

How to solve the BSDEs A decomposition Approach: Theorem Theorem Assume that BSDE (8) admits a solution (Y b, Z b ) SF BSDE L2 F. Then Y t = ξ + T τ F (s, Y s, Z s, U s ) T τ Z s dw s T τ t τ t τ t τ U s dh s, t [0, T ], admits a solution (Y, Z, U) S G L2 G L2 (λ) given by Y t = Yt b 1 t<τ + ξτ a 1 t τ, Z t = Zt b 1 t τ, U t = ( ξt a Yt b ) 1t τ, for all t [0, T ]. Thomas Lim Mean-Variance Hedging 25 / 36

How to solve the BSDEs Expressions of the generators (µty +σtz+λtβtu)2 σ t 2 Y +λ t β t 2 (U+Y ), f(t, Y, Z, U) = [ g(t, Y, Z, U) = 1 Y t Z t Z + λ t U t U h(t, Υ, Ξ, Θ) = Z t 2 Y t + λ t (U t + Y t ) U t 2 ] (µty t+σtzt+λtβtut)(σty tz+λtβt(ut+y t)u) σ t 2 Y t+λ tβt 2(Ut+Y t) σty tzt+λtβtu t(ut+y t) 2 σ t 2 Y t+λ t β t 2 (U t+y t)., Thomas Lim Mean-Variance Hedging 26 / 36

How to solve the BSDEs Solution to BSDE (f, 1) According to the general existence Theorem, we consider for coefficients (f, 1) the BSDE in F: find (Y b, Z b ) SF L2 F such that { dyt b (µt λ tβ t)yt b + σ tzt b + λ tβ t 2 } = t dt +Zt b dw t, YT b = 1. σ t 2 Yt b + λ t β t 2 λ t + λ ty b The generator of this BSDE can be written under the form { µt λ tβ t 2 Y b σ t 2 t λt βt 2 σ t 4 µt λtβt 2 λ t + λ tyt b + + 2(µt λtβt) σ t 2 (σ tz b t + λ tβ t) σtzt b λt βt 2 + λ tβ t + (λ tβ t µ t) σ t 2 Y b t + λ t β t 2 2 } σ t 2. Thomas Lim Mean-Variance Hedging 27 / 36

How to solve the BSDEs Introduction of a modified BSDE Let (Y ε, Z ε ) be the solution in SF L2 F to the BSDE { dy ε µt λ tβ t 2 t = Y σ t 2 t ε λt βt 2 σ t 4 µt λtβt 2 2(µt λtβt) + (σ tz σ t 2 t ε + λ tβ t) σ tz ε λt βt 2 λ t + λ tyt ε t + λ tβ t + (λ tβ t µ t) 2 } σ + t 2 dt + Z ε σ t 2 (Yt ε ε) + λ t β t 2 t dw t, YT ε = 1, where ε is a positive constant such that ( exp Question: Y ɛ ɛ? T 0 (λ t + µt ) ) λtβt 2 dt σ t 2 ε, P a.s. Thomas Lim Mean-Variance Hedging 28 / 36

Change of probability How to solve the BSDEs Define the process L ε by ( L ε (µt λtβt) t := 2 + 2 σt λtβ t + σ t λt βt 2 σ t (λ tβ 2 t µ ) t) + σ t 2 (Yt ε ε) + λ t β t 2 Since L ε BMO(P), we can apply Girsanov theorem: t W t := W t + L ε sds, 0 is a Brownian motion under the probability Q defined by dq := E( dp FT T 0 L ɛ tdw t ). σ t 2 Z ε t σ t 2 (Y ε t ε) + λ t β t 2. Thomas Lim Mean-Variance Hedging 29 / 36

Comparison under Q How to solve the BSDEs dy ε t = { λt β t 2 +λ t λ t Yt ε YT ε = 1. σ t µ 4 t λ t β t 2 µt λtβt 2 σ t Y ε (µ 2 t 2λ t β t λ tβ t) t σ t 2 λtβ t+(λ tβ t µ t) λ t βt 2 2 } σt 2 σ t 2 (Y dt Z ε t ε ε)+λt βt 2 t d W t, We remark that generator λ t y µ t λ t β t 2 σ t 2 y. Therefore, we get from a comparison theorem that [ ( T Yt ε ( E Q exp λs + µ s λ s β s 2 ) ) ] Ft ds t σ s 2 Moreover Z ɛ BMO(P). ɛ. Thomas Lim Mean-Variance Hedging 30 / 36

How to solve the BSDEs Solution to BSDE (g, ξ) We consider the associated decomposed BSDE in F: find (Y b, Z b ) SF L2 F such that { dy t b ((µt λ t β t )Yt b + σ t Zt b + λ t β t )(σ t Yt b Zt b + λ t β t ξt a λ t β t Yt b ) = Yt b ( σ t 2 Yt b + λ t β t 2 ) Z b t Z b Yt b t λ t Yt b ξt a + λ } t Yt b Yt b dt + Zt b dw t, YT b = ξ b. Thomas Lim Mean-Variance Hedging 32 / 36

Change of probability How to solve the BSDEs ( Define the process ρ by ρ t := Z t b σt (µ t λ t β t )Y b Yt b ρ BMO(P), we can apply Girsanov theorem W t := W t ) t +σt Z t b +λt βt. σ t 2 Yt b +λt βt 2 t 0 ρ sds is a Q-Brownian motion, where d Q dp F T := E( T 0 ρ tdw t ). Hence, BSDE can be written { dy b t = a t(yt b ξt a )dt + Zt b d W t, with a t := λt σt 2 Yt b Y b T τ = ξ b, b λt βt ((µt λt βt )Yt +σt Z t b ). Yt b( σt 2 Yt b+λt βt 2 ) We can prove that Y b defined by Y b t := E Q [ ( exp is solution of this BSDE. T t ) a udu ξ b + T t ( exp s t Since ) ] a udu a sξs a ds F t Thomas Lim Mean-Variance Hedging 33 / 36

How to solve the BSDEs Solution to BSDE (h, 0) We consider the associated decomposed BSDE in F: find (Υ b, Θ b ) SF L2 F such that Υ b t = T ( Z b t 2 Y b t t T τ t τ Ξ b s dw s. We can prove that Υ b defined by where R t := Z b t 2 Y b t is solution of this BSDE. + λ t ξt a Yt b 2 σty t b Zt b + λ tβ t(ξt a Yt b ) 2 ) λ sυ σ t 2 Yt b + λ t β t 2 s ds [ Υ b T ( t := E exp t s t ) ] λ udu R sds F t, + λ t ξt a Yt b 2 σty t b Zt b + λ tβ t(ξt a Yt b ) 2, σ t 2 Yt b + λ t β t 2 Thomas Lim Mean-Variance Hedging 35 / 36

How to solve the BSDEs Thanks! Thomas Lim Mean-Variance Hedging 36 / 36