Exponential utility maximization under partial information

Transcription

1 Exponential utility maximization under partial information Marina Santacroce Politecnico di Torino Joint work with M. Mania AMaMeF 5-1 May, 28 Pitesti, May 1th, 28

2 Outline Expected utility maximization under partial information: semimartingale setting Semimartingale model Expected utility and partial information Equivalent problem and solution Value process and BSDE Assumptions Equivalent problem Value process and BSDE F S G Two examples with explicit solution. F S G : Diffusion model 1: Change point problem. F S G : Diffusion model 2.

3 The model Let S = (S t, t [, T ]) be a continuous semimartingale which represents the price process of the traded asset. (Ω, A, A = (A t, t [, T ]), P), where A = A T and T < is a fixed time horizon. Assume the interest rate equal to zero. The price process S admits the decomposition S t = S + N t + λ ud N u, λ N T < a.s., where N is a continuous A -local martingale and λ is a A -predictable process (Structure condition).

4 Expected utility maximization and partial information Denote by G = (G t, t [, T ]) a filtration smaller than A G t A t, for every t [, T ]. G represents the information available to the hedger. We consider the exponential utility maximization problem with random payoff H at time T when G is the available information, to maximize E[ e α(x+ T π uds u H) ] over all π Π(G ). x represents the initial endowment (without loss of generality we take x = ) Π(G ) is a certain class of strategies (G - predictable and S- integrable processes). ( πudsu, t [, T ]) represents the wealth process related to the self-financing strategy π. This problem is equivalent to to minimize E[e α( T π uds u H) ] over all π Π(G ).

5 In most papers, under various setups, (see, e.g., Lakner (1998), Pham and Quenez (21), Zohar (21)) expected utility maximization problems have been considered for market models where only stock prices are observed, while the drift can not be directly observed. = under the hypothesis F S G. We consider the case when G does not necessarily contain all information on the prices of the traded asset i.e. S is not a G -semimartingale in general. = In this case, we solve the problem in 2 steps: Step 1: Prove that the expected exponential maximization problem is equivalent to another maximization problem of the filtered terminal net wealth (reduced problem) Step 2: Apply the dynamic programming method to the reduced problem. (In Mania et al. (28) a similar approach is used in the context of mean variance hedging).

6 Filtration F and decomposition of S w.r.t. F Let F = (F t, t [, T ]) be the augmented filtration generated by F S and G. S is a F -semimartingale: S t = S + λ (F) u d M u + M t, (Decomposition of S with respect to F ) M t = N t + (F) [λ u λ u ]d N u is F -local martingale where we denote by λ (F) the F predictable projection of λ. Note that M = N are F S -predictable.

7 Assumptions In the sequel we will make the following assumptions: A) M is G -predictable and λ (F) = λ (G ) λ where λ (G ) denotes the G predictable projection of λ, B) any G -martingale is a F -local martingale, C) the filtration G is continuous, D) for any G -local martingale m G, M, m G is G -predictable, E) H is an A T -measurable bounded random variable, such that P- a.s. E[e αh F T ] = E[e αh G T ], F) there exists a martingale measure for S (on F ) that satisfies the Reverse Hölder condition R L ln L (P). Note: When F S G A) B) D) and [the second part of] E) are automatically satisfied!

8 G projection processes Let Ŝt = E(St Gt) be the G -optional projection of St. Note that since λ (F) = λ (G ) = λ, we have: Ŝ t = E(S t G t) = S + where M t is the G -local martingale E(M t G t). Moreover, it is possible to show that LEMMA Under conditions A)-D), λ ud M u + M t. Ê t(m) = E(E t(m) G t) = E t( M).

9 We recall that our aim is Equivalent problem to minimize E[e α( T π uds u H) ] over all π Π(G ). (1) where the class of strategies is defined as Π(G ) = {π : G predictable, Ee α T π uds u <, π M BMO(G )} PROPOSITION Let conditions A)-E) be satisfied. Then the optimization problem (1) is equivalent to to minimize E[e α( T π udŝu H)+ α2 2 T π 2 u (1 κ2 u )d M u ], over all π Π(G ) (2) Moreover, for any π Π(G ) E[e α( T t H = 1 α ln E[eαH G T ], κ 2 t = d M t d M t. π uds u H) G t] = E[e α( T t π udŝu H)+ α2 2 T t πu 2 (1 κ2 u )d M u G t].

10 Sketch of the proof Let t =. Taking the conditional expectation with respect to F T and using condition E[e αh F T ] = E[e αh G T ], we have that ( E[e α( T π uds u H) ] = E e α ) T π uds u E[e αh F T ] Resorting to the F -decomposition of S, the previous lemma, the decomposition of Ŝ, we have that (3) is equal to E[E T ( απ M)e α2 2 = E[E(E T ( απ M) G T )e α2 2 = E[E T ( απ α2 M)e 2 T π 2 u d M u α T π u λud M u+α H] ( ) = E e α( T π uds u H). T π 2 u d M u α T π u λud M u+α H] T π 2 u d M u α T π u λud M u+α H] (3) = E[e α T π ud M u α2 2 T π 2 u d M u+ α2 2 T π 2 u d M u α T π u λud M u+α H] = E[e α( T π udŝu H)+ α2 2 T π 2 u (1 κ2 u )d M u ].

11 Remarks Let V t = ess inf π Π(G ) E[e α( T t π udŝu H)+ α2 2 be the value process related to the equivalent problem. Note that: c V t C for two positive constants: T t πu 2 (1 κ2 u )d M u G t], H is bounded V t C. R L ln L (P) condition V t c (by duality arguments). If H is G T -measurable, then H = H and problem (1) is equivalent to minimize E[e α( T π α2 T udŝu H)+ π 2 2 u (1 κ2 u )d M u ].

12 Main result THEOREM Under assumptions A)-F) the value process V related to the equivalent problem (2) is the unique bounded strictly positive solution of the following BSDE Y t = Y Y T = E[e αh G T ] (ψ uκ 2 u + λ uy u) 2 Y u d M u + ψ ud M u + L t (4) Moreover the optimal strategy exists in the class Π(G ) and is equal to π t = 1 α ( λ t + ψtκ2 t Y t ). (5)

13 F S G. What happens if F S t G t? G t = F t F S t G t The F decomposition of S is the G decomposition of S : S t = S + λ ud M u + M t, M M loc (G ) Note Mt = M t and κ 2 t = 1 for all t A), B), D) and E[e αh F T ] = E[e αh G T ] are satisfied = If G is continuous, the initial problem (1) is equivalent to to minimize E(e α( T π uds u H) ) over all π Π(G ).

14 COROLLARY Let F S G A. Assume G continuous; the Reverse Hölder condition R L ln L (P); H to be a bounded A T -measurable random variable. Then, the value process V is the unique bounded positive solution of the BSDE Y t = Y (ψ u + λ uy u) 2 Y u d M u + Moreover, the optimal strategy is equal to π t = 1 α ( λ t + ψt Y t ). ψ udm u +L t, Y T = E(e αh G T ). (6) REMARK: In the case of full information G t = A t, we also have and Equation (6) takes on the form Y t = Y M t = M t = N t, λt = λ t, Y T = e αh (ψ u + λ uy u) 2 Y u d N u + ψ udn u + L t, Y T = e αh.

15 Diffusion model 1: Change point problem. We consider a market with one risky traded asset with returns dynamics: ds t = µi (t τ) dt + σdw t. The drift of the return process S changes value from to µ at some random time τ: the change-point τ admits a known prior distribution (exponential) and is independent of W τ cannot be observed only S is observed: = G t = Ft S and G t A t. H is a positive bounded function of S T and of another variable independent of FT S (the terminal value of a non traded asset) the strategy π t is interpreted as a dollar amount invested in the stock. = we give an explicit solution of the problem in terms of the a posteriori probability process p t = P(τ t Ft S ) which satisfies the SDE p t = p + µ σ (See Shiryayev (1978).) p u(1 p u)d W u + γ (1 p u)du.

16 Comments: The optimal wealth process is the sum of two components: a hedging fund (which is zero if H = ) an investment fund (which depends on the a posteriori probability process). REMARK: This fact is a consequence of wealth independent risk aversion of the exponential utility function. µ If H = and τ is deterministic (τ = t ) the optimal strategy is I ασ 2 (t t ). Moreover, if t =, we obtain Merton s optimal strategy π = µ. ασ 2

17 Diffusion model 2. We consider a diffusion market model consisting of two risky assets with the following dynamics ds t =µ(t, η)dt + σ(t, η)dw 1 t, dη t =b(t, η)dt + a(t, η)dw t, (7) W 1 and W are standard Brownian motions with constant correlation ρ; η represents the price of a nontraded asset (e.g. an index); S denotes the process of returns of the tradable asset. Problem: An agent is hedging a contingent claim H trading with the liquid asset S and using only the information on η. Under suitable conditions on µ, σ, b, a and H i.e. 1) T µ 2 (t,η) dt is bounded, σ 2 (t,η) 2) σ 2 >, a 2 >, 3) equation (7) admits a unique strong solution, 4) H bounded and measurable with respect to the σ-algebra F η T σ(ξ), where ξ is a random variable independent of S 1) 4) imply that our assumptions A) F) are satisfied = we give an explicit expression of the optimal amount of money that should be invested in the liquid asset.

18 In particular, the value process related to the optimization problem is equal to V t = ( E Q[e (1 ρ2 )(α H 1 2 T ) t θu 2 1 du) F η 1 ρ t ] 2. (8) The optimal strategy π is identified by ( ) πt 1 ρh t = θ t + ασ(t, η) (1 ρ 2 )(c + hud W, (9) u) h t is the integrand of the integral representation e (1 ρ2 )(α H 1 T T θ 2 2 t dt) = c + h td W t. Q, defined by d Q = E dp T ( ρ θ W ) is a new measure and W t = W t + ρ θudu is a Q-Brownian motion.

19 Comparison What happens in the full information case? Problem: An agent is trading with a portfolio of stocks in order to hedge a contingent claim written on the non traded asset, using all market information. This problem was earlier studied in, e.g., Musiela and Zariphopoulou (24), Henderson and Hobson (25), Davis (26), for a model (7) with constant coefficients µ and σ. Assuming the Markov structure of b and a, Musiela and Zariphopoulou gave an explicit expression for the related value function. Let S t and η denote the price processes of the traded and non traded assets d S t = S t(µ(t, η)dt + σ(t, η)dw 1 t ), dη t =b(t, η)dt + a(t, η)dw t, We consider to minimize ( E e α( T π t d S t H) ) over all π Π(F S,η ) (1) where π t represents the number of stocks held at time t and is adapted to the filtration F S,η t.

20 Applying our results to the case of full information, under suitable conditions on the coefficients of the model [i.e. 1) 3)] and assuming H = f (η) and bounded we recover the result of Musiela and Zariphopoulou in a non Markovian setting we find that the the optimal amount of money depends only on the observation coming from the non traded asset. Indeed, the two optimization problems (partial/full information) are equivalent: the corresponding value processes coincide and the optimal strategies of these problems are related by the equality π partial t = π full t S t. = This means that, if H = f (η), the optimal dollar amount invested in the assets is the same in both problems and is based only on the information coming from the non traded asset η.

21 Appendix References M.H.A. Davis, Optimal hedging with basis risk. From stochastic calculus to mathematical finance, Springer, Berlin, 26, F. Delbaen, P. Grandits, T. Rheinländer, D. Samperi, M. Schweizer and C. Stricker, Exponential Hedging and Entropic Penalties. Mathematical Finance 12, N.2, 22, V. Henderson, Valuation of claims on nontraded assets using utility maximization, Math. Finance 12, N.4, 22, V. Henderson and D.G. Hobson, Real options with constantrelative risk aversion, Journal of Economic Dynamics and Control 29, N.7, 25, P. Lakner, Optimal trading strategy for an investor: the case of partial information. Stochastic Proccess. Appl. 76, 1998, M. Mania, R. Tevzadze and T. Toronjadze, Mean-variance Hedging Under Partial Information (to appear in SIAM J. Control Optim.) M. Musiela and T. Zariphopoulou, An example of indifference prices under exponential preferences, Finance Stochast. 8, 24, H. Pham and M. C. Quenez, Optimal portfolio in partially observed stochastic volatility models. Ann. Appl. Probab. 11, N.1, 21, A.N. Shiryayev, Optimal stopping rules. Springer, New York-Heidelberg, R. Tevzadze, Solvability of backward stochastic differential equations with quadratic growth. Stochastic Proccess. Appl. 118, 28,