Time-delayed backward stochastic differential equations: the theory and applications

Transcription

1 Time-delayed backward stochastic differential equations: the theory and applications Warsaw School of Economics Division of Probabilistic Methods

2 BSDEs The backward stochastic differential equation: Y(t) = ξ + f(s,y(s),z(s))ds Z(s)dW(s), 0 t T. t t A solution is a pair (Y,Z). The terminal condition ξ is an exogenous random variable, The generator f at time s [0,T] depends on the current value of the solution and an additional source of randomness.

3 BSDEs with time-delayed generators The backward stochastic differential equation with a time-delayed generator: 12 Y(t) = ξ + f(s,y s,z s )ds Z(s)dW(s), 0 t T. t t The generator f at time s [0,T] depends on the past values of the solution Y s := (Y(s +u)) T u 0, Z s := (Z(s +u)) T u 0. 1 L. Delong, P. Imkeller, Backward stochastic differential equations with time delayed generators - results and counterexamples. The Annals of Applied Probability. 2 L. Delong, P. Imkeller, On Malliavin s differentiability of BSDE with time delayed generators driven by Brownian motions and Poisson random measures. Stochastic Processes and their Applications.

4 The time-delayed backward stochastic differential equation: Y(t) = ξ(y T,Z T )+ f(s,y s,z s )ds Z(s)dW(s), 0 t T. t t The terminal condition ξ and the generator f depend on the past values of the solution Y s := (Y(s +u)) T u 0, Z s := (Z(s +u)) T u 0.

5 Existence and uniqueness of a solution, Linear time-delayed BSDEs, Properties of the solutions. Applications to portfolio management, Applications to pricing, hedging of insurance and financial contracts, Applications to pricing principles and recursive utilities.

6 The existence and uniqueness of a solution

7 Existence and uniqueness (A1) The generator and the terminal condition are product measurable, F-adapted and Lipschitz continuous, in the sense that for a probability measure α on [ T,0] B([ T,0]) and with constants K 1,K 2 we have: E [ f(ω,t,y t,z t ) f(ω,t,ỹt, Z t ) 2] and [ K 1 E sup 0 u t 0 Y(u) Ỹ(u) 2 + Z(t +u) Z(t ] +u) 2 α(du), T E[ ξ(ω,y T,Z T ) ξ(ω,ỹ T, Z T ) 2 ] [ K 2 E Y(u) Ỹ(u) 2 + sup 0 u T for any square integrable processes (Y,Z),(Ỹ, Z), 0 Z(u) Z(u) ] 2 du,

8 Existence and uniqueness (A2) E [ 0 f(ω,t,0,0,0) 2 dt ] <, (A3) E[ ξ(ω,0,0,0) 2 ] <, (A4) f(ω,t,.,.) = 0 for ω Ω and t < 0.

9 The result on existence and uniqueness Theorem Assume that (A1)-(A4) hold. For a sufficiently small time horizon T or for a sufficiently small Lipschitz constant K 1, and for a sufficiently small Lipschitz constant K 2, the time-delayed backward stochastic differential equation has a unique solution.

10 Linear time-delayed BSDEs

11 Linear time-delayed BSDEs s Y(t) = ξ + KY(u)duds Z(s)dW(s), 0 t T t 0 t 1. K < 0 there exists a unique solution: Y(t) = E[ξ] e Kt +e Kt e KT +e KT Z(t) = t ξ = E[ξ]+ + 1 Z(s) ( e K(t s) +e K(t s) ) dw(s), 2 0 2M(t) e K(T t) +e, K(T t) 0 M(t)dW(t).

12 Linear time-delayed BSDEs s Y(t) = ξ + KY(u)duds Z(s)dW(s), 0 t T t 0 t 2. K > 0 and T K < π 2 there exists a unique solution: Y(t) = E[ξ] cos(t K) t cos(t K) + Z(s)cos((t s) K)dW(s), 0 M(t) Z(t) = cos((t t) K), ξ = E[ξ]+ 0 M(t)dW(t).

13 Linear time-delayed BSDEs s Y(t) = ξ + KY(u)duds Z(s)dW(s), 0 t T t 0 t 4. K > 0, T K = π 2, E[ξ] 0 there exists no solution.

14 Linear time-delayed BSDEs s Y(t) = ξ + KY(u)duds Z(s)dW(s), 0 t T t 0 t Z(t) = M(t) cos((t t) K) 1{t > 0}, ξ = E[ξ]+ M(t)dW(t) K > 0, T K = π 2, E[ξ] = 0 and E[ 0 Z(s) 2 ds] = + there exists no solution. 6. K > 0, T K = π 2, E[ξ] = 0 and E[ 0 Z(s) 2 ds] < there exist multiple solutions: Y(t) = Y(0)cos(t K)+ t 0 Z(s)cos((t s) K)dW(s).

15 Linear time-delayed BSDEs Y(t) = ξ t 1 s βy(u)duds Z(s)dW(s), 0 t T s 0 t There exists a unique solution. The case of β > 0: Y(t) = E[ξ] I 0(2 βt) I 0 (2 βt) + Z(t) = t 0 Z(s)ψ(s, t)dw(s), M(t) 2I 0 (2 βt) βtk 1 (2 βt)+2k 0 (2 βt) βti 1 (2 βt), ψ(s,t) = I 0 (2 βt)k 1 (2 βs)+k 0 (2 βt)i 1 (2 βs), ξ = E[ξ]+ 0 M(s)dW(s).

16 Linear time-delayed BSDEs s Y(t) = ξ + KZ(u)duds Z(s)dW(s), 0 t T t 0 t There exists a unique solution: Y(t) = E Q[ ξ F t ]+(T t)k dq dp FT ξ = E Q [ξ]+ = exp ( K 0 0 t 0 Z(s)dW Q (s), Z(s)ds, (T s)dw(s) K (T s) 2 ds ).

17 Linear time-delayed BSDEs Y(t) = ξ t 1 s βz(u)duds Z(s)dW(s), 0 t T s 0 t There exists a unique solution: Y(t) = E Q [ξ F t ] βln( T t t ) Z(s)ds, dq dp FT ξ = E Q [ξ]+ = exp ( Z(s)dW Q (s), βln( T s )dw(s) ( βln( T s )) 2 ds ).

18 Properties of the solutions

19 Properties of the solutions A comparison principle may not hold, a comparison holds up to a stopping time, If the terminal condition is bounded/non-negative, a solution may be unbounded/negative, A solution exists, but a measure solution may not exists, a measure solution exists up to a stopping time, A solution may not be conditionally invariant with respect to the terminal condition.

20 Malliavin s differentiability A special case of a BSDE with a time-delayed generator: Y(t) = ξ + t t ( 0 0 ) f ω,s, Y(s +v)α(dv), Z(s +v)α(dv) ds T T Z(s)dW(s), 0 t T.

21 Malliavin s differentiability (A5) E [ [0,T] D sξ 2 ds ] <, (A6) the mapping (y,z,u) f(ω,t,y,z,u) is continuously differentiable in (y, z, u), with uniformly bounded and continuous partial derivatives f y,f z,f u, (A7) for (t,y,z,u) [0,T] R R R we have that f(,t,y,z,u) D 1,2 (R) E [ Ds f(,t,0,0,0) 2 dtds ] <, [0,T] 0 Ds f(ω,t,ŷ,ẑ,) Ds f(ω,t,ỹ, z ) ( ) K ŷ ỹ + ẑ z.

22 Malliavin s differentiability Theorem Assume that (A1)-(A7) hold. For a sufficiently small time horizon T or for a sufficiently Lipschitz constant K there exist unique solutions (Y,Z) and (Y s,z s ) to the time-delayed BSDEs Y(t) = ξ + t t ( 0 0 ) f ω,s, Y(s +v)α(dv), Z(s +v)α(dv) ds T T Z(s)dW(s), 0 t T,

23 Malliavin s differentiability Y s (t) = D s ξ + t f s (r,yr s,zr s )dr Z s (r)dw(r), 0 s t T, t 0 ( f s (r,yr s,zr s ) = D s f ω,r, Y(r +v)α(dv), T f y (ω,r, Y(r +v)α(dv), 0 + f z (ω,r, T T 0 Y s (r +v)α(dv) T T 0 T ) Z(r +v)α(dv) 0 ) Y(r +v)α(dv), Z(r +v)α(dv) T 0 Z s (r +v)α(dv). T ) Z(r +v)α(dv)

24 Malliavin s differentiability The solution (Y s (t),z s (t)) is a version of (D s Y(t),D s Z(t)). The process (D t Y(t)) P is a version of Z(t).

25 Applications to portfolio management, pricing and hedging

26 Financial market The bank account: db(t) B(t) = r(t)dt, 0 t T, dr(t) = a(t)dt +b(t)dw(t), 0 t T, r(0) = r 0. The bond: dd(t) D(t) = µ(t)dt +σ(t)dw(t), 0 t T.

27 The wealth process The portfolio process: dx(t) = π(t) ( µ(t)dt+σ(t)dw(t))+ ( X(t) π(t) ) r(t)dt, 0 t T. The terminal liability: X(T) = ξ. The discounted portfolio process: dy(t) = Z(t)dW Q (t), 0 t T. The discounted terminal liability: Y(T) = ξ.

28 Hedging problems Common financial instruments: European and Asian options, swaps, swaptions, puts on bonds, caps/floors, ξ = (L(T 0,T) K) +, ξ = (D(T 0,T) K) +, Y(t) = ξ(w T ) t Z(s)dW Q (s), 0 t T. The terminal pay-off depends on the investment portfolio process or the applied investment strategy, Y(t) = ξ(w T,Y T,Z T ) t Z(s)dW Q (s), 0 t T.

29 Option Based Portfolio Insurance Invest x euros, Protect the initial investment and earn an additional return, Buy a bond paying x at maturity and a call option on λ units of a benchmark process S xd(0)+c(xλs(t) x) = x, But λ units of a benchmark process S and a put option on λ units of a benchmark process S xs(0)+p(x xλs(t)) = x.

30 Option Based Portfolio Insurance ξ = X(T) = X(0)+(λS(T) X(0)) +, Y(t) = e r(s)ds( 0 Y(0)+(Y(0)λS(T) Y(0)) +) Z(s)dW Q (s), 0 t T, t We expect to find multiple solutions of the time-delayed BSDE.

31 The applications of time-delayed BSDEs Portfolio management problems, Participating contracts,...the benefits are based on the return on a specified pool of assets held by the insurer......the best estimate should be based on the current assets held by the undertaking. Future changes of the asset allocation should be taken into account... Variable annuities, Asset-liability management problems in insurance. Solvency II Directive

32 Ratchet option The liability: ξ = γmax{x(0)e gt,x(t 1 )e g(t t 1),...,X(t n 1 )e g(t t n 1),X(T)}. The time-delayed BSDE: Y(t) = γmax { Y(0)e 0 r(s)ds+gt,y(t 1 )e t r(s)ds+g(t t 1 ) 1,..., Y(t n 1 )e t r(s)ds+g(t t n 1 ) } n 1,Y(T) Z(s)dW Q (s), 0 t T. t

33 Ratchet option Proposition Let B denote the following set { B = ω Ω : γmax { e gt D(0),e g(t t1) D(t 1 ),...,e g(t tn 1) D(t n 1 ),1 } } > If P(B) = 0,γe gt D(0) = 1 then there exist multiple solutions (Y,Z), which differ in Y(0), of the form Y(t) = γy(0)e gt t 0 r(s)ds D(t), 0 t T, Y(0)e gt 0 r(s)ds = Y(0)+ 0 Z(s)dW Q (s).

34 Ratchet option Definition 2. If P(B) = 0,γe gt D(0) < 1,γ = 1 then there exist multiple solutions (Y,Z), which differ in (Y(0), ( η(t m+1 ) ) m=0,1,..n 1 ), of the form Y(t 0 ) = Y(0), { Y(t m ) = γ max Y(tk )e tm r(u)du t k e g(t t k ) } D(t m ) k=0,1,...,m +E Q [ η(t m+1 ) F tm ], { Y(t m+1 ) = γ max Y(tk )e t m+1 t r(u)du k e g(t t k) } D(t m+1 ) k=0,1,...,m + η(t m+1 ) = Y(t m )+ tm+1 t m Z(s)dW Q (s), Y(t) = E Q [Y(t m+1 ) F t ], t m t t m+1,m = 0,1,...,n 1

35 Ratchet option Multi-period Option Based Insurance Portfolio, Semi-static hedging, The solution is derived solely from the time-delayed BSDE.

36 Ratchet option The liability: ξ = γ sup {X(s)e g(t s) }. s [0,T] The time-delayed BSDE: Y(t) = γ sup 0 t T t {Y(t)e t r(s)ds+g(t t) } Z(s)dW Q (s), 0 t T.

37 Ratchet option Proposition Let C denote the following set { { C = ω Ω : γ sup e g(t t) D(t) } > 1}. 0 t T 1. If P(C) = 0,γe gt D(0) = 1 then there exist multiple solutions (Y,Z), which differ in Y(0), of the form Y(t) = γy(0)e gt t 0 r(s)ds D(t), 0 t T, Y(0)e gt 0 r(s)ds = Y(0)+ 0 Z(s)dW Q (s).

38 Ratchet option Proposition 2. If P(C) = 0,γe gt D(0) < 1,γ = 1 then there exist multiple solutions (X,π), which differ in (X(0),U), of the form t X(t) = X(0)+ π(s) dd(s) t 0 D(s) + ( )db(s) X(s) π(s) 0 B(s), π(t) = γ sup {X(s)e g(t s) }D(t) 0 s t + U(t) ( ) X(t) γ sup {X(s)e g(t s) }D(t) 1{S(t) > 0} S(t) 0 s t with the process S defined as ds(t) = U(t) dd(t) D(t) +(S(t) U(t))dB(t), S(0) = s > 0. B(t)

39 Return smoothing The liability: ξ = X(0)βS +γ 1 T 0 e s r(u)du X(s)ds. The time-delayed BSDE: Y(t) = βy(0) S +γ 1 T 0 Y(s)ds Z(s)dW Q (s), 0 t T. t

40 Return smoothing Proposition If βe Q [ S]+γ = 1 then there exist multiple solutions (Y,Z), which differ in Y(0), of the form t Y(t) = Y(0)+ Z(s)dW Q (s), 0 t T, 0 with the F-predictable control Z(t) = 1 1 γ +γ t M(t), 0 t T, T and the process M derived from the martingale representation βy(0) S = βy(0)e Q [ S]+ M(t)dW Q (t). 0

41 Return smoothing A real-life product from the UK market: ( { ( X(1) ξ = X(0) 1+max g,β ( { 1+max g, β 2 ( { 1+max g, β T )}) X(0) 1 ( X(1) X(0) + X(2) )}) X(1) 2... ( X(1) X(0) + X(2) X(T) )}) X(1) X(T 1) T.

42 Minimum withdrawal rates Minimum guaranteed withdrawal benefit with an annuity conversion option: dx(t) = π(t)(µ(t)dt +σ(t)dw(t))+(x(t) π(t))r(t)dt X(T) = La(T), γ( sup X(s))dt, 0 t T, s [0,t] a(t) = E Q[ T e s T r(u)du ds F T ].

43 Minimum withdrawal rates The time-delayed BSDE: Y(t) = Lã(T)+ t γ ( { sup Y(u)e s r(v)dv}) u ds u [0,s] Z(s)dW Q (s), 0 t T. t For a sufficiently small γ or T there exists a unique non-zero solution.

44 Numerical schemes Picard iterations, Random walk approximations, Forward-backward structure of a solution, Convergence?

45 Minimum withdrawal rates Minimum guaranteed withdrawal benefit with an annuity conversion option: dx(t) = π(t)(µ(t)dt +σ(t)dw(t))+(x(t) π(t))r(t)dt γ sup {X(s)}dt, s [0,t] X(T) = γ sup {X(s)}a(T). 0 s T a(t) = E Q[ T e s T r(u)du ds F T ].

46 Applications to pricing principles and recursive utilities

47 Prices and utilities We can model a price or a utility as a solution to a BSDE: dy(t) = g(t,y(t),z(t))dt +Z(t)dW(t). The generator defines a local (subjective) valuation rule: E[dY(t) F t ] = g(t,y(t),z(t))dt, Var[dY(t) F t ] = Z 2 (t)dt.

48 A habit process t dy(t) = u(c(t))dt β Y(s)dsdt +Z(t)dW(t) 0 β > 0 an anticipation effect, β < 0 a disappointment effect.

49 Pricing principles Y(t) = E[e β(t t) ξ F t ] The local valuation rule: E[dY(t) F t ] = βy(t)dt, The price changes proportionally to the last price (a local expected value principle).

50 Pricing principles The price changes proportionally to the average of the past prices, The local valuation rule: E[dY(t) F t ] = 1 t t 0 βy(s)dsdt, A time-delayed BSDE with a generator of a moving average type.

51 Pricing principles The global valuation rule: t Y(t) = φ(t,t,t)v(t) V(s)φ (s,t,t)ds, 0 V(t) = E[ξ F t ].

52 Recursive utilities A habit process: dy(t) = u(c(t))dt β 1 t t 0 Y(s)dsdt +Z(t)dW(t).

53 Thank you very much Institute of Econometrics, Division of Probabilistic Methods Warsaw School of Economics lukasz.delong@sgh.waw.pl homepage : akson.sgh.waw.pl/delong/