Proposition: Suppose that U (c, y) = u(c − ηy) and u is DARA.

1. The allocation in the risky asset is decreasing in k.

2. The allocation in the risky asset is decreasing in (increasing in, independent of ) η if u is standard and relative risk aversion is uniformly larger than (smaller than, equal to) unity.

 

Proof: Implicitly differentiating (12) with respect to k, η, and α for ec = z0 + αex yields

 

y∗k = −(1 − η)u0((1 − η)y∗) k(1 − η)2u00((1 − η)y∗) + η2Eu00(z0 + αex − ηy∗),

y∗η = ku0((1 − η)y∗) + ηk(1 − η)y∗u00((1 − η)y∗) − η2y∗Eu00(z0 + αex − ηy∗)

ηk(1 − η)2u00((1 − η)y∗) + η3Eu00(z0 + αex − ηy∗) ,y∗α =ηEexu00(z0 + αex − ηy∗)

k(1 − η)2u00((1 − η)y∗) + η2Eu00(z0 + αex − ηy∗) (18)

 

Note that y∗k > 0, y∗η < 0, and y∗α < 0 at α = α∗ as u standard implies ηk(1 − η)y∗u00((1 − η)y∗) − η2y∗Eu00(z0 + αex − ηy∗) ≥ 0 and u DARA impliesE [˜xu00 (z0 + α∗˜x − ηy∗)] > 0.

Implicitly differentiating the first-order condition for α∗,

 

Illustrative portfolio example

 

To illustrate the effect that anticipatory feeling and ex-post disappointment have on the optimal decision, we consider the case U (c, y) = ln(c − ηy). Suppose that the return of the risky asset ˜x under the objective probability distribution takes a value x+ or x− with equal likelihood and x+ > 0 > x−. Solving the two first-order conditions (16) and (13) for y and α we derive, after some manipulation,

y∗ = k (k + 1) η · z0 and α∗ = − x+ + x− 2 (k + 1) x+x− · z0.

As in the EU model with CRRA, the optimal allocation in the risky asset is proportional to the initial wealth and it is strictly positive if the equity premium is strictly positive, i.e. x+ + x− > 0. It is decreasing in the intensity of anticipatory feeling (∂α∗/∂k < 0) and independent of the intensity of ex-post disappointment (∂α∗/∂η = 0). The optimal anticipated payoff is also proportional to the initial wealth and independent of the actual values of returns, x+ and x−. It is increasing in the intensity of anticipatory feeling (∂y∗/∂k > 0) and decreasing in the intensity of ex-post disappointment (∂y∗/∂η < 0).

 

Optimal insurance purchase

 

In this section, we apply our decision criterion to an insurance purchase decision. The agent is endowed with initial wealth z0 and is facing a loss of random size ˜l. He can buy coinsurance at a rate β for a premium (1 + λ) βE˜l where λ denotes the proportional loading factor. The agent chooses the coinsurance rate β to maximize his intertermporal welfare w (α) = max ymin≤y≤ymax kv (y) + EU ³z0 − (1 − α)˜l − (1 + λ) αE˜l, y´. This problem is equivalent to the portfolio allocation problem where full insurance, β = 1, is equivalent to investing all wealth into the risk-free asset, α = 0. We therefore obtain the following result which mirrors Proposition 7. Proposition 9 If insurance is actuarially fair (λ = 0) then full coverage is optimal. If insurance is actuarially unfair (λ > 0) then partial coverage is optimal.

This is a direct consequence of Machina (1982) who has shown that most classical results in insurance are obtained in his Generalized Expected Utility model as long as the ”local utility function” M is concave in outcomes. In our special case, concavity of M is implied by the concavity of U (c, y) in c, see (10).

We obtain the following comparative statics of the optimal insurance amount with respect changes in k and η.

Proposition: Suppose that U (c, y) = u(c − ηy) and u is DARA.

1. The amount of insurance coverage is increasing in k.

2. The amount of insurance coverage is increasing in (decreasing in, independent of ) η if u is standard and relative risk aversion is uniformly larger than (smaller than, equal to) unity.

 

If relative risk aversion is uniformly larger than one and u is standard, then individuals with anticipatory feelings and ex post disappointment buy more insurance compared to the EU model. This result may help explain individuals’ preferences for low deductibles - see e.g. Pashigian et al. (1966), Cohen and Einav (2005), Sydnor (2006).

 

We extend our previous example to the insurance purchase decision. Assume U (c, y) = ln(c −ηy) and suppose that there is a loss of size l with probability q and no loss with probability 1 − q where q < 1/ (1 + λ). Solving the first order conditions for y and β yields

 

y∗ = k (z0 − (1 + λ) ql) (k + 1) η and β∗ = (1 + λ) (1 − q) l − λz0 (1 + λ) (1 − (1 + λ) q) l + λk (z0 − (1 + λ) ql) (k + 1) (1 − (1 + λ) q) (1+λ) l where the first term in the sum is the optimal coinsurance rate predicted by the traditional EU model. In our model, the optimal insurance amount is increasing in the intensity of anticipatory feeling (∂β∗/∂k > 0) and independent of the intensity of ex-post disappointment (∂β∗/∂η = 0). Individuals with anticipatory feelings therefore buy more insurance than predicted by the EU model. The optimal anticipated payoff is increasing in the intensity of anticipatory feeling (∂y∗/∂k > 0) and decreasing in the intensity of ex-post disappointment (∂y∗/∂η < 0).

It is interesting that the smaller the “stakes” are, i.e. the larger z0 − (1 + λ) ql, the higher the anticipated payoff is. This implies that the individual’s degree of risk aversion is higher with smaller stakes, which implies a higher optimal amount of insurance coverage relative to the one predicted by the EU model. This is consistent with Rabin’s calibration theorem (2000) who shows that the EU model implies that a measurable degree of risk aversion over small stake gambles should exhibit an unreasonably high degree of risk aversion over large stake gambles. In our model, the degree of risk aversion is decreasing in the amount of the stakes as the individual lowers his anticipated payoff.

Pseudo code pour évaluer un call européen lookback avec un prix d’exercice fixe, des fixings discrets et une volatilité stochastique dans le monde de Black&Scholes, selon la simulation de Monte Carlo en utilisant la réduction de variance par des variables de contrôle Delta, Gamma et Véga et une variable antithétique.

 

 

‘Pseudo code pour évaluer un call européen lookback avec un prix d’exercice fixe,

‘des fixings discrets et une volatilité stochastique dans le monde de Black&Scholes selon ‘la simulation de Monte Carlo en utilisant la réduction de variance par des variables de ‘contrôle Delta, Gamma et Véga et une variable antithétique.

‘ Ces variables de contrôle réduisent l’erreur standard « se » d’un facteur de 12.

‘Pour atteindre cette réduction au moyen d’une simulation Monte Carlo simple,

‘il faudrait 144,000 simulations, soit en gros 3 heures 30 !!!

‘ici 1,000 simulations = 153’’ avec un pentium III - 100 simulations = 15’’

‘initialiser_paramètres {X, T, S, sigma, r, div, alpha, Vbar,xi, N, M}

 

'print call_value(100, 1, 100, 0.2, 0.06, 0.03, 5,0.05,0.02, 52, 100)

'17,2907686686696

 

 

'Calculer les constantes

dt=T/N

sig2=sigma^2

alphadt=alpha*dt

xisdt=xi*sqr(dt)

erddt=exp((r-div)*dt)

egam1=exp(2*(r-div)*dt)

egam2=-2*erddt+1

eveg1=exp(-alpha*dt)

eveg2=Vbar-Vbar*eveg1

sdt = Sqr(dt)

 

sum_ct = 0

sum_ct2 = 0

 

beta1=-0.88

beta2=-.42

beta3=-.0003

 

 

For j = 1 To M do 'for each simulation

 

 

st1 = S

st2 = S

vt=sig2

Maxst1=S

Maxst2=S

cv1 = 0

cv2 = 0

cv3=0

ct =0

 

For i = 1 To N do 'for each time step

 

'calculer les variables de contrôle

t1 = (i - 1) * dt

delta1 = lookback_delta(st1, X, r, div, T – t1, sqr(vt),maxst1)

delta2 = lookback_delta(st2, X, r, div, T - t1, sqr(vt),maxst2)

gamma1 = lookback_gamma(st1, X, r, div, T - t1, sqr(vt), maxst1)

gamma2 = lookback_gamma(st2, X, r, div, T - t1, sqr(vt), maxst2)

vega1 = lookback_vegaV(st1, X, r, div, T - t1, sqr(vt), maxst1)

vega2 = lookback_vegaV(st2, X, r, div, T - t1, sqr(vt), maxst2)

 

'simuler la variance du prix de l'action

epsilon = standard_normal_sample

vtn = vt +alphadt*(Vbar-vt) + xisdt*sqr(vt)* epsilon

 

'simuler le prix de l'action

epsilon = standard_normal_sample

stn1 = st1 * Exp((r-div-0.5*vt)*dt + sqr(vt)*sdt*epsilon)

stn2 = st2 * Exp((r-div-0.5*vt)*dt + sqr(vt)*sdt*(-epsilon))

 

'accumuler les variables de contrôle

cv1 = cv1 + delta1 * (stn1 - st1 * erddt) + _

    delta2 * (stn2 - st2 * erddt)

cv2 = cv2 + gamma1 * ((stn1 - st1) ^ 2 - st1 ^ 2 * (egam1*exp(vt*dt)+egam2))+ _

gamma2*((stn2-st2)^2-st2^2*(egam1*exp(vt*dt)+egam2))

cv3= cv3+vega1*((vtn-vt)-(vt*eveg1+eveg2-vt))+ _

vega2*((vtn-vt)-(vt*eveg1+eveg2-vt))

 

vt=vtn

st1=stn1

st2=stn2

 

 

 

if (st1>maxst1)

maxst1 =st1

if (st2>maxst2)

maxst2 = st2

 

 

next i

 

 

ct = 0.5 * (max(0, maxst1 - X) + max(0, maxst2 - X) + beta1 * cv1 + beta2 * cv2 + _

beta3 * cv3 )

 

sum_ct = sum_ct + ct

sum_ct2 = sum_ct2 + ct * ct

 

Next j

 

call_value = sum_ct / M * Exp(-r * T)

sd = Sqr((sum_ct2 - sum_ct * sum_ct / M) * Exp(-2 * r * T) / (M - 1))

se = sd / Sqr(M)

 

 

‘Pseudo code pour évaluer un call lookback avec un prix d’exercice fixe,

‘un prix maximum basé sur un fixing continu et une volatilité constante

‘ce code vous permet de calculer delta, gamma et vega à chaque fixing

 

‘initialiser_paramètres {X, T, S, sigma, r, div, Maxst}

 

'exemple à tester print call_value(100, 1, 100, 0.20, 0.06, 0.03, 105)

'20,1562223161423

 

If S <= 0 Then

GoTo Endfunction ‘exemple d’utilisation GoTo p.208/209 du livre

 

if X>=Maxst then

E=X, G=0

Else E=Maxst, G=exp(-r*(T-t1))*(Maxst-X)

 

call_fixed_strike_lookback= G + S*exp(-div*(T-t1))*N(y+sigma*sqr(T-t1))- _

X*exp(-r*(T-t1))*N(y) – S/b*(exp(-r*(T-t1))*(E/S)^B * N(y+(1-b)*sigma*sqr(T-t1)) _

-exp(-div*(T-t1))*N(y+sigma*sqr(T-t1)))

b= 2*(r-div)/(sigma*sigma)

y=(log(S/E)+((r-div)-0.5*sigma*sigma)*(T-t1))/(sigma*(T-t1))

‘N( ) = probabilité cumulative de la Normale(0,1)

 

‘formules pour calculer les deltas, gammas, vegas à chaque fixing

‘lookback_delta = C(S+dS)-C(S-dS)/(2*dS)

‘lookback_gamma=(C(S+dS)-2*C(S)+C(S-dS))/(dS*dS)

‘lookback_vega=C(sigma+dsigma)-C(sigma-dsigma)/(2*dsigma)

 

 

 

‘utiliser le timer de Visual Basic

Dim Debut, Fin, TempsTotal

    Debut = Timer    ' Définit l'heure de début.

 

‘ entre Debut et Fin vous taper votre code ici

 

Fin = Timer

    TempsTotal = Fin - Debut    ' Calcule la durée totale de la simulation.

    MsgBox "Longueur de la simulation " & TempsTotal & " seconde(s)"