Comparison of insider’s optimal strategies, three different types of side information RIMS symposium, the 7th workshop on Stochastic Numerics, June 27-29, 2005 Caroline HILLAIRET Monique PONTIER CMAPX U.M.R. CNRS C 5583, L.S.P. Ecole Polytechnique Universit´e Paul Sabatier 91 128 Palaiseau cedex 31 062 TOULOUSE cedex 04 FRANCE FRANCE
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Caroline’s paper [16] deals with the following problem on a financial market. This one is built with a riskless asset and d risky assets, which are driven by a m−dimensional Brownian motion and a n−dimensional Poisson process. The traders have different information on the market, we say that they are in an asymmetric information situation: (i) initial strong information : at the early beginning, on time t = 0, the insider knows L ∈ FT +ε , a value which will be revealed only after the end of the period T. (ii) progressive strong information : the insider knows L + εt at time t, he knows something more than a non insider trader, and this information is more and more precise as time evolves; for instance, the variance of random variable εt goes to 0 when t goes to T. (iii) weak information : he knows (or bets) the real law of L. In each case, L could be the price of a risky asset after time T. The purpose is to obtain optimal strategies in each case and to compare them, analytically if possible, numerically with simulations if not.
1
THE MODEL
On a filtered probability space (Ω, A, FT := (Ft ), t ∈ [0, T ], P), we consider W, a mdimensional Brownian motion, and N, a n-dimensional Poisson process, with intensity κ. 1
d = m + n. On such a space, there exists: B(0) = 1; riskless asset (bond) : dB(t) = B(t)r(t)dt, d X i i i d risky assets : dP = P [b dt + σji (t)d(W ∗ , N ∗ )∗j (t)]. − t t t j=1
.
The filtration FT is generated by the process (W, N ), completed and c`ad. We suppose that r, b, σ are FT -predictable and such that the SDE admits a unique strong solution. Moreover, σ is invertible, strictly non degenerate, Ft is the public information at time t, and we denote Gt the insider’s information at time t.
On such a market, the agents can invest and their aim is to optimize the utility of their terminal wealth in the set of admissible portfolios π, i.e. π is predictable with respect to the agents’ own filtration, namely GT , Z T ||σt∗ πt ||2 dt < ∞, 0
and discounted associated wealth (self-financing hypothesis)
dBt−1 Xtπ = Bt−1 hπt , bt − rt 1d idt + Bt−1 πti (σji dWtj + σki dNtk ),
(1)
X has to be bounded below, initial wealth X0 ∈ L+ 1 (G0 ), XT ≥ 0. Henceforth, we suppose there is no Poisson component in the drivers, for sake of simplicity, but the results are quite analogous in case of a Poisson component. In the non informed case, with logarithmic utility of the terminal wealth, we get as optimal wealth and optimal portfolio: ˆ t = X0 (Y 0 )−1 ; π ˆ t (σ ∗ )−1 (σt )−1 (bt − rt 1d ) = X ˆ t (σ ∗ )−1 l0 (t). Bt−1 X ˆt = X t t t 0 dP /Ft ] where P0 is risk neutral probability measure, Here Yt0 = EP [ dP dYt0 = −Yt0 σt−1 (bt − rt 1d )dWt , and finally dWt0 = dWt + σt−1 (bt − rt 1d )dt is a (F , P0 ) − Brownian motion. Remark that since σ is invertible, P0 is the unique risk neutral probability measure, so the market is both complete and viable. In case of side information, π is no more F −predictable and thus we need first to make ˆ = Y P in sense to wealth equation (1), then to exhibit risk neutral probability measure Q each case. ˆ risk neutral probability measure a The sketch of the proof is as following: for Q necessary and sufficient condition for admissibility is: EQˆ [BT−1 XTπ /G0 ] ≤ X0 . Thus, to optimize π 7→ EP [ln XTπ /G0 ], we introduce the Lagrangian function L(XTπ , λ) = EP [ln XTπ − λYT (BT−1 XTπ − X0 )/G0 ], and this yields the result. 2
2
THREE TYPES of SIDE INFORMATION
2.1
Initial strong information
In this case the trader’s information is L = PT1+ε , for instance, from the beginning, we say that he has a strong information. The tool is the initial enlargement of filtration, the filtration F is enlarged with the knowledge of L, so we get the “strong” filtration as following GtS := ∩s>t (Fs ∨ σ(L)), 0 ≤ t ≤ T. To make the wealth equation meaningful, we suppose: (H S ) P{L ∈ ./Ft } ∼ P{L ∈ .}, ∀t ∈ [0, T ], a.s. So let be p the density of probability: P{L ∈ dx/Ft } = p(t, x)P{L ∈ dx}. Using Jacod’s results [19], we get the predictable representation of this martingale as following: dp(t, x) = p(t, x)β(t, x)dWt , we define ρSt := β(t, L), thus dWtS = dWt − ρSt dt is a (G S , P) Brownian motion. Let the Dol´eans exponential S −1 S S d(Z S )−1 t = −(Zt ) ρt dWt .
Thus we get a risk neutral probability measure for the insider’s point of view: ˆ S := Y 0 (Z S )−1 P = Y S P. Q As announced, a necessary and sufficient condition for admissibility is EQˆ S [BT−1 XTπ /G0S ] ≤ X0 . So we get a Lagrangian function to modelize the optimization problem under constraint: L(XTπ , λ) = EP [ln XTπ − λYTS (BT−1 XTπ − X0 )/G0S ].
2.2
Progressive strong information
Some agents have an additional information more and more precise as time evolves. They know at time t the random variable Lt = f (L, εt ), with V ar(εt ) decreasing as t goes to T, ε is a noise. The tool is the progressive enlargement of filtration as Corcuera et al. [6] do it, not as in [23]. So we get the “progressive enlarged” filtration as following GtP := ∩s>t (Fs ∨ σ(Lu , u ≤ s)), 0 ≤ t ≤ T. To make the wealth equation meaningful, we suppose: (H P ) f is measurable, L is FT −measurable, (εt , t ≤ T ) and FT are independent, 3
P{L ∈ ./Ft }
