Leech: el principio de cortesía y la escala de coste-beneficio

a complete market.

3.2.1 A continuous time two assets model

As in Black and Scholes’ model, there are two assets, a bond with price S0

t = ert and a risky asset

with price

dSt= St(µdt + σdWt) , S0 = s .

Then, the market is complete. If an investor holds portfolio (αt, θt) and consumes at rate ctat time

t, his wealth V , with Vt= αtSt0+ θtSt evolves according to the stochastic differential equation

dVt= αtSt0rdt + θtdSt− ctdt = rVtdt + θt(dSt− rStdt) − ctdt ,

(3.2)

due to the self financing condition dVt= αtdSt0+ θtdSt− ctdt.

Assume that his objective is to maximize EP U (VT) +

Z _T

0 u(cs)ds

!

,

where u and U are strictly increasing, strictly concave and sufficiently differentiable functions. We shall solve this problem by two methods. The first one, called the martingale method is very useful to compute the optimal consumption and terminal bequest in a complete market, while the second called the dynamic programming method gives the hedging portfolio.

3.2.2 Historical probability

Let Lt= exp[−κWt−1₂κ2t]. Using Itˆo’s formula, the process

e−rtVtLt

can be shown to be a martingale. Therefore, its expectation is constant. In particular, Ee−rTLTVT



= x (3.3)

This is the budget constraint.

Reciprocally, if a positive random variable (the terminal wealth that the agent would like to obtain, or the payoff he would receive) VT is given such that (3.3) holds, thanks to the martingale

property the current wealth is given by via VtLte−rt= E



e−rTLTVT|Ft



and the portfolio which hedges this terminal wealth is given via a representation theorem. I other words, the market being complete, it is possible to hedge he contingent claim H = VT.

Solving the problem

The Lagrangian of the constrained problem is EP h U (VT) − λ  VTe−rTLT − x i .

The first order condition is (the derivative with respect to the terminal wealth equals 0) U0(V_T∗) = λe−rTLT

where λ is such that the budget constraint holds, i.e.,

Ehe−rTL_T(U0)−1(λe−rTL_T)i= x. Hence V∗

T may more or less easily be computed from the above formula. The hedging portfolio

is much harder to be computed. It is obtained from a representation theorem.

Examples

If U (x) = ln(x), then I(y) = y−1_{. The optimal terminal wealth is V}∗

T = (λe−rTLT)−1, where the

parameter λ is adjusted so that the budget constraint holds E(e−rTL_T(λe−rTL_T)−1) = x or

λ = 1/x The optimal wealth is obtained via

VtLte−rt= E  e−rTLT(λe−rTLT)−1|Ft  = 1/λ = x i.e. Vt= xertL−1t

Hedging portfolio associated with the optimal utility for the log case. The optimal wealth is Vt= xertL−1t = x exp[rt + κWt+1₂κ2t] = x exp[rt + κ2t] exp[κWt−1₂κ2t]

so that dVt = [r + κ2]Vtdt + VtκdWt = rVtdt + Vtκ[dWt+ κdt] = rVtdt + Vtκ_σ[σdWt+ (µ − r)dt] = rVtdt + Vt κ σSt St[σdWt+ (µ − r)dt] = rVtdt + Vt κ σSt [dSt− rStdt]

so that the hedging portfolio is θ = Vt_σSκ

t.It is interesting to remark that we obtain that e rt_L−1

t is

3.2.3 The Dynamic programming method

Let us now present the Dynamic programming method. The optimal portfolio is obtained in terms of the value function that we shall next define. Assume that the investor’s wealth equals x at time t and that he invests the proportion π of his wealth in the risky asset (this number is related with the portfolio by πX = θS), then his wealth fulfills the stochastic differential equation

dX_st,x,π = rX_st,x,πds + πsXst,x,π[σdWs+ (µ − r)ds] , s ≥ t

(3.4)

with initial condition X_tt,x,π = x. Let v the value function be defined by v(t, x) = sup

π E

n

U (X_Tt,x,π)o

where the supremum runs over the portfolio and where X_Tt,x,πis the terminal wealth of the investor. The value function satisfies the dynamic programming equation

v(t, x) = sup

π E

n

v(τ, X_τt,x,π)o

for any time τ . When it is smooth enough, it fulfills the Hamilton Jacobi Bellman equation

     ∂v ∂t + sup_π ( (rx + (µ − r)π) ∂v_{∂x +} 1 2σ 2_π2 ∂2v ∂x2 ) = 0 v(T, x) = U (x)

Therefore, the optimal portfolio π∗ is the value of π for which the supremum is attained and is the solution of a quadratic problem. Hence π∗ _{is the function of wealth defined by the relation}

π∗(t, x) = −µ − r σ2 ∂v ∂x(t, x) ∂2_v ∂x2(t, x) !₋₁ , .

In the particular case where U (x) = ln x, it may be proven that v(t, x) = p(t) ln x where p(t) fulfills a differential equation, hence π∗ _{is constant. The optimal portfolio is therefore to invest a}

fixed multiple of wealth in the risky asset at all dates.