Metodología del diario de observación participante y no participante

The transaction costs are supposed to be proportional to the amount of the transaction. In that case, it can be shown that the superreplication price of a European call option is the price of the underlying.

5.3.6 Variance hedging

Let us assume that r = 0. A self-financing portfolio is such that its value V satisfies dVt= rVtdt + θt(dSt− rStdt) = θtdSt (5.5) or Vt= V0+ Z _t 0 θsdSs A strategy hedges H if VT = H.

Under the hypothesis that P is a risk neutral probability, F¨ollmer and Sondermann succeed to minimize the variance

E((VT − H))2)

The proof is based on the orthogonal decomposition ; if H is any contingent claim, it can be written as H = h + Z _T 0 θsdSs+ M > T where M>

T is the terminal value of a martingale orthogonal to S (i.e. such that E(Mt>St) = 0)

This method was extended by Rheinlander and Schweizer and by Laurent, Pham and Gourieroux to the general case. It can be proved that this lead to the choice of a particular e.m.m., called the minimal martingale measure.

5.3.7 Remaining risk

Let us assume that r = 0. Let (α, θ) be any strategy and Vt= αt+θtStits value. We do not assume

here that the strategy is self-financing. The gain process associated to this strategy is defined as Gt=

Z _t

0 θsdSs the cost process up to time t is

Ct= Vt−

Z _t

0 θsdSs A strategy hedges H if VT = H.

If the market is complete, there exists a strategy for which H = VT = V0+ GT

so that the corresponding cost process satisfies

CT = V0 = EQ(H) .

Following Schweizer, we define the remaining risk as

Rt= E[(CT − Ct)2|Ft)]

A strategy is risk-minimizing if it minimizes the remaining risk at any time.

5.3.8 Reservation price

A different approach is initiated by Hodges and Neuberger [5], and studied in El Karoui and Rouge [11], Hugonnier [6] and Bouchard-Denize [1].

Let x be the initial endowment of an agent and U a utility function. The reservation price of the contingent claim H is defined as the infimum of h such that

sup E[U (X_Tx+h− H)] > sup E[U (X_Tx)]

The agent selling the option starts with an initial endowment x + h, he gets an optimal portfolio with terminal value X_Tx+h and he has to deliver the contingent claim H.

5.3.9 Davis approach

Another way, studied by Davis [2] (1997), is to value options for an agent endowed with a particular utility function. Related results have been obtained by a number of authors in various contexts. Davis defines the price of the contingent claim ζ using a marginal rate : suppose that ζ is traded at price p. An investor invests an amount of δ in ζ and kep this position till maturity. Her final wealth is X_Tx−δ,π+ δ

p(ζ)ζ. Her investment program is W (δ, x, p) = sup

pi

E(U (Xx−δ,π+ δ p(ζ)ζ)) Definition 5.3.1 Assume that the equation

∂W

∂δ (0, p, x) = 0 has a unique solution p∗_{. The fair price of ζ is defined as p}∗_.

Theorem 5.3.2 Let V (x) = sup_πE(U (X_Tπ,x)) = E(U (X_Tπ∗,x)). Assume that V is differentiable and that V0(x) > 0. Then p∗ satisfies p∗ = E(U

0_(Xπ∗_,x

T )ζ)

V0_(x)

It can be proved that this lead to the choice of a particular e.m.m., called the Davis martingale measure.

5.3.10 Minimal entropy

Another way to price options is to choose a particular equivalent martingale measure, e.g. the minimal entropy measure as in Frittelli [6] (1996).

Let S be the dynamics of the prices and M(P ) the set of e.m.m., i.e., the set of probability equivalent to P such that S is a Q martingale. We assume that this set is not empty. For any Q ∈ M(P ), the entropy of Q with respect to P is defined as

H(Q|P ) = E_P(dQ dP ln

dQ dP)

Theorem 5.3.3 Let ML ln L = {Q ∈ M : H(Q|P ) < ∞. There exists a unique probability

Qe_{∈ M}

L ln L such that

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