The pricing problem with inside information:

Finite Ω ♦

The arguments of this section are inspired by King [18] (see Section 4.3).

This section considers the pricing problem of the seller of a contingent claim under a general ltration modeling full- or inside information.

Consider a nancial market based on a probability space (Ω,F, P) where the scenario spaceΩis nite. There areN risky assets in the market with price processesS_n(t),n= 1, . . . , N and one bond with price processS₀(t). The time t∈ {0,1, . . . , T}where T <∞. LetS denote the composed price process, and let(Ft)_t be the ltration generated by the price process. Hence, the composed price processS is adapted to the ltration (F_t)_t. The market can be modeled by a scenario tree as in Section 4.3.

Recall that a contingent claim is a nonnegative,FT-measurable random vari-able on the probability space(Ω,FT, P). Consider a contingent claimB in the market. Consider also a seller of this claim with a general information modeling ltration(Gt)^T_t=0 such thatG0={∅,Ω}and GT is the algebra corresponding to the partition{{ω1}, ...,{ωM}} (sometimes the ltration is denoted by (Gt)t to simplify notation). We assume that the price process S is adapted to (Gt)t, in order for the seller to know the price of each asset at any given time. Hence the ltration(Ft)tis nested in(Gt)t.

It will turn out that the setM^a(S,G)is important: Q∈ M^a(S,G)if

ˆ Q is a probability measure on (Ω,GT) (recall thatFT =GT by assump-tion).

ˆ Qis absolutely continuous with respect toP.

ˆ The price process S is aQ-martingale w.r.t. the ltration(Gt)_t.

6.1. THE PRICING PROBLEM WITH INSIDE INFORMATION: FINITEΩ ♦105 SoM^a(S,G)is the set of absolutely continuous martingale measures w.r.t. (Gt)_t. This is (to my knowledge) a new term.

The pricing problem of this seller is

min v

subject to

S0·H0 ≤ v

Bk ≤ Sk·H_a(k) for allk∈ N_T^G,

Sk·Hk = Sk·H_a(k) for allk∈ N_t^G, 1≤t≤T−1, (6.1)

where the minimization is done with respect to v ∈ R and Hk ∈ R^N+1 for k ∈ N_t^G, t= 0, ..., T −1. Here, N_t^G denotes the set of time t-vertices (nodes) in the scenario tree representing the ltration (Gt)t, andBk denotes the value of the claim B in the node k ∈ NT. Recall that a(k)denotes the ancestor of vertex k, see Section 4.3.

Hence, the seller's problem is: Minimize the price v of the claim B such that the seller is able to pay B at timeT from investments in a self-nancing, adapted portfolio that costs less than or equal to v at time 0. Note that the portfolio process H has been translated, so that H is adapted to (G_t)_t, not predictable (which is assumed in many papers in mathematical nance). This is just a simplication of notation.

Problem (6.1) is actually a linear programming (LP) problem, and one can nd the dual of this problem using standard LP-duality techniques. However, it turns out to be easier to nd the dual problem via Lagrange duality techniques (see Section 5.4). The LP-dual problem is a special case of the Lagrange dual problem, so LP-duality theory implies that there is no duality gap (this can also be shown via the Slater condition, since choosing v := 1 + sup_ω∈ΩB(ω) and putting everything in the bank account is a strictly feasible solution of problem (6.1)). Note also that since problem (6.1) is a linear programming problem, the simplex algorithm is an ecient method for computing optimal prices and optimal trading strategies for specic examples.

Problem (6.1) is equivalent to

min v

subject to

S0·H0−v ≤ 0

Bk−Sk·Ha(k) ≤ 0 for allk∈ N_T^G

Sk·(Hk−H_a(k)) ≤ 0 for allk∈ N_t^G, 1≤t≤T−1,

−Sk·(Hk−Ha(k)) ≤ 0 for allk∈ N_t^G, 1≤t≤T−1, (6.2) which is of a form suitable for the Lagrange duality method.

Let y₀ ≥ 0, z_k ≥ 0 for all k ∈ N_T^G and y_k¹, y²_k ≥ 0 for all k ∈ N_t^G, t ∈ {1, . . . , T −1} be the Lagrange multipliers. Then, the Lagrange dual problem is

sup_y0,z,y1,y²≥0infv,H {v+y0(S0·H0−v) +P vector of theyk's) is a free variable (i.e. the sign of the components ofy is not clear a priori).

Consider each of the minimization problems separately. In order to have a feasible dual solution, all of these minimization problems must have optimal value greater than−∞. By considering the second equation above for the bond, i.e. forS_k⁰, one sees that in order to have a feasible dual solution

y₀S₀⁰= X

m∈CG(0)

y_mS⁰_m

6.1. THE PRICING PROBLEM WITH INSIDE INFORMATION: FINITEΩ ♦107 must hold. Since the market is assumed to be normalized, soS_k⁰= 1for allk,

1 =y₀= X

m∈CG(0)

y_m.

Also, from the second equation above, the same type of argument implies that

From the last dual feasibility equation (considered for n = 0), P

k∈N_T^Gzk = P

k∈N_T−1^G y_k= 1andz_k ≥0for allk(sincez is a Lagrange multiplier). There-fore,{z_k}_k∈NG

T can be identied with a probability measureQ(on the terminal vertices of the scenario tree) such that theQ-probability of ending up in terminal vertex k iszk. Then, the condition ykSk =P

m∈CG(k)ymSm for all k∈ N_t^G, is a martingale condition of the formS(t−1) =E[St|Gt−1](from the denition of conditional expectation), which can be shown by induction to imply the general martingale condition in this discrete time case. This proves that any feasible dual solution is inM^a(S,G). The converse also holds: TakeQ∈ M^a(S,G), and dene z_m :=Q(ω_m)for m = 1, ..., M, y_k := P

m∈CG(k)z_m for k ∈ N_T^G₋₁ and y_k :=P

m∈CG(k)y_m fork ∈ N_t^G, 0≤t≤T−2. It can be checked (from these denitions) that this is a feasible dual solution.

Hence, the Lagrange dual problem can be rewritten sup

Q∈M^a(S,G)

E_Q[B]

where M^a(S,G) denotes the set of martingale measures with respect to (Gt)t

which are absolutely continuous with respect to the original measureP. As explained previously, LP-duality (or the Slater condition) implies that there is no duality gap. Hence, the optimal primal value, i.e. the seller's price of the contingent claim B, is equal to the optimal dual value, that is sup_Q∈Ma(S,G)EQ[B].

From general pricing theory (see Karatzas and Shreve [17], Theorem 6.2), a seller who has the ltration (Ft)t (the original ltration) will oerB at the price sup_Q∈Me(S,F)E_Q[B] (in a normalized market). Here, M^e(S,F) denotes the set of all equivalent martingale measures w.r.t. the ltration (Ft)_t. This appears to be a dierent price than the one derived above. However, it turns out that

The following lemma is a slight adaptation of Proposition 2.10 in Schacher-mayer [40].

Lemma 6.1 Assume there existsQ^∗∈ M^e(S,G). For anyg∈ L^∞(Ω,F, P), EQ[g]≤0 for allQ∈ M^a(S,G).

if and only if

E_Q[g]≤0 for allQ∈ M^e(S,G).

Proof:

By assumption, there exists at least one Q^∗ ∈ M^e(S,G) (and hence also a Q ∈ M^a(S,G), since M^e(S,G) ⊆ M^a(S,G)). For any Q ∈ M^a(S,G) and λ∈(0,1),λQ^∗+ (1−λ)Q∈ M^e(S,G). Hence,M^e(S,G)is dense inM^a(S,G),

and the lemma follows.

Lemma 6.2 ♦

Assume there exists aQ∈ M^e(S,G). Then sup

Q∈M^a(S,G)

EQ[B] = sup

Q∈M^e(S,G)

EQ[B].

Proof: ♦

ˆ To prove that sup_Q∈Ma(S,G)E_Q[B] ≤sup_Q∈Me(S,G)E_Q[B]: Dene x :=

sup_Q∈Ma(S,G)E_Q[B]. ThenE_Q[B]≤x for all Q∈ M^a(S,G). But from Lemma 6.11 (with g = B−x), this implies that EQ[B] ≤xfor all Q ∈ M^e(S,G), sosup_Q∈Me(S,G)EQ[B]≤x, and hence the inequality follows.

ˆ The opposite inequality follows from that M^e(S,G)⊆ M^a(S,G).

Hence, in particular

sup

Q∈M^a(S,F)

EQ[B] = sup

Q∈M^e(S,F)

EQ[B]. (6.3)

if we assume that there is no arbitrage with respect to (Ft)t, so there exists Q∈ M^e(S,F) =M^e(S).

Note that the arguments above would go through in the same way for the buyer's problem, so for a general ltration(Gt)t, the buyer's price of the claimB isinf_Q∈Ma(S,G)EQ[B]. Note also that by assuming that both seller and buyer in the market have the same full information ltration(Ft)t, and that the market is complete, we know that there is only one equivalent martingale measure (see Øksendal [27], chapter 12), and hence the buyer and seller will agree onE_Q[B]

(whereQis the single equivalent martingale measure) as the price ofB.

6.1. THE PRICING PROBLEM WITH INSIDE INFORMATION: FINITEΩ ♦109

In document Convex duality and mathematical finance (sider 112-117)