Course2(Bogota)(structured products) .pdf



Nom original: Course2(Bogota)(structured products).pdf

Ce document au format PDF 1.4 a été généré par LaTeX with beamer class version 3.06 / pdfTeX-1.11a, et a été envoyé sur fichier-pdf.fr le 29/06/2013 à 01:21, depuis l'adresse IP 197.27.x.x. La présente page de téléchargement du fichier a été vue 1143 fois.
Taille du document: 797 Ko (76 pages).
Confidentialité: fichier public


Aperçu du document


Do we need to include options in the …nancial portfolio ?
by Jean-Luc Prigent
(talk based on joint works with several co-authors)
THEMA, University of Cergy-Pontoise, France

Bogota Seminar, Course N 2
December 2012

J.-L. Prigent ()

Bogota Seminar, Course N 2 December 2012
/ 76
Do we need to include options in the …nancial portfolio ?

Presentation schedule

Overview of structured products
Complexity and fair pricing
Adequacy customer/product
Optimal positioning (standard case)
Behavorial …nance
Optimal positioning (ambiguity theory)
Optimal positioning (prospect theory)
Optimal positioning (regret theory)

J.-L. Prigent (THEMA)

Bogota Seminar, Course N 2 December 2012
Do we need to include options in the …nancial portfolio ?
/ 76

Overview of structured products (de…nition)
“Financial engineering” is most often based on the use of already
existing components that are combined to create new (complex)
…nancial instruments, in order to better …t customers’needs.
One of the most prominent groups of newly introduced …nancial
instruments is termed “structured products”, proposed to enhance
portfolio returns. They are issued by …nancial institutions and are
intended to private or institutional investors. They can be traded on
an organized exchange or sold directly by their issuing bank, who will
quote bid and ask prices.
The demand for structured products has quickly increased. For
example, the outstanding investment on this type of fund have been
estimated at a total value which borders the 500 billions of Euros (for
European investors only) in 2007. On December 31, 2010, in France,
the outstanding of funds with a formula (about 700 "fonds à formule"
in France) amounted to 61.8 billion euros, that is 4.62% of total
outstanding of French funds.
J.-L. Prigent (THEMA)

Bogota Seminar, Course N 2 December 2012
/ 76
Do we need to include options in the …nancial portfolio ?

Overview of structured products (simpli…ed typology)

We can attempt to de…ne them as products combining at least two
basic instruments one of which is an option (Das, 2000; Fabozzi,
1998).
A possible (and maybe too simple nowadays) typology: Stoimenov
and Wilkens (2005), for "equity-linked structured products".
Equity-linked structured products
Plain vanilla option components
Exotic option components
Classic, Corridor, Guarantee
Turbo, Barrier, Rainbow
We can also add Asian options.

J.-L. Prigent (THEMA)

Bogota Seminar, Course N 2 December 2012
Do we need to include options in the …nancial portfolio ?
/ 76

Portfolio Insurance (objectives)
Portfolio insurance has two main objectives:
- First, to allow the investor to recover at maturity at least a given
percentage p of his initial investment, usually 100% (this limits
downside risk in bearish …nancial market, dramatically relevant during
…nancial crisis);
- Second, to allow investors to bene…t from potential market rises
(this allows investors to bene…t from bullish markets).
The portfolio is usually indexed on stock or bond market indices (for
example, for life-insurance funds) but the risky benchmark asset can
also be a credit portfolio (see the principal protected structured credit
products) or a fund of hedge funds (see the protected securities linked
to hedge funds).

J.-L. Prigent (THEMA)

Bogota Seminar, Course N 2 December 2012
/ 76
Do we need to include options in the …nancial portfolio ?

Overview of structured products (basic example)
OBPI: Static hedging strategy and guarantee only at maturity T
(speci…ed for each contract)
The value VTOBPI of the portfolio is given by (a = number of shares):
VTOPPI = a ST + (K

ST ) + = a K + ( ST

K )+

aK = pV0

Then, at any time:
VtOPPI = pV0 e

r (T

t)

+ a C (t, St , K )

Thus the initial weight invested on the risky asset corresponds to
1 pe rT (function of both the interest rate level and the
horizon time).
Usually p = 100%. For example, for T = 7 years, 1
for r = 2% and ' 19% for r = 3%.
J.-L. Prigent (THEMA)

e

rT

' 13%

Bogota Seminar, Course N 2 December 2012
/ 76
Do we need to include options in the …nancial portfolio ?

Overview of structured products (basic example)
Capped OBPI strategy: (to reduce insurance costs)
VTCOPPI = Min a0 ST + (K 0

J.-L. Prigent (THEMA)

ST )+ , qV0

Bogota Seminar, Course N 2 December 2012
Do we need to include options in the …nancial portfolio ?
/ 76

Overview of structured products (problems)
However, two main problems arise:
- First, investment banks have to determine the costs of creating such
a given structured product. This is typically done by using standard
arbitrage-theoretical tools, assuming that …nancial markets are perfect
from active management of these …nancial institutions.
- Second, customer’s potential utility gains induced by buying a given
…nancial structured product would be evaluated. For this second
point, the risk aversion is indeed crucial to describe the investors
behavior.
As mentioned by Breuer and Perst (2007), “structured products are
combinations of derivatives and underlying …nancial assets which
exhibit structures with special risk/return pro…les that may not be
otherwise attainable on the capital market without signi…cant
transaction costs being incurred – at least for private investors (see,
e.g., Das, 2000).” Therefore, investors may agree to pay an (implicit)
additional cost to have access to such products.
J.-L. Prigent (THEMA)

Bogota Seminar, Course N 2 December 2012
/ 76
Do we need to include options in the …nancial portfolio ?

Complexity and fair pricing
Most empirical studies on structured products focus on European
markets, especially Switzerland, Germany and the Netherlands.
The adopted approach consists in comparing prices in the primary or
secondary market to theoretical fair values. The fair values of
embedded options are computed from the implied volatilities of
similar publicly traded options.
Several authors have analyzed the pricing of such products: Chen and
Kensinger (1990); Chen and Sears (1990); Wasserfallen and Schenk
(1996); Wilkens et al. (2003); Stoimenov and Wilkens (2005). They
document a signi…cant pricing bias in favor of the issuing institution
(average of 2% above their theoretical values).
In general, more complex products incorporate higher implicit
premiums (up to 6%) (see also Bertrand-Prigent (2012) for french
…nancial structured products).
J.-L. Prigent (THEMA)

Bogota Seminar, Course N 2 December 2012
Do we need to include options in the …nancial portfolio ?
/ 76

Adequacy customer/product (regulation)
The Markets in Financial Instruments Directive (MiFID): European
Union law that provides harmonised regulation for investment services
(across the 27 Member States of the European Union plus Iceland,
Norway and Liechtenstein) (November 2007).
The main objectives of the directive are to increase competition and
consumer protection in investment services.
Among key aspects: Client categorisation.
Firms are required to categorise clients as "eligible counterparties",
professional clients or retail clients (these have increasing levels of
protection).
Clear procedures to assess their suitability for each type of investment
product. The appropriateness of any investment advice or suggested
…nancial transaction must still be veri…ed before being given.
Among problems to implement the directive: how to take account of
attitude towards risk?
J.-L. Prigent (THEMA)

Bogota Seminar, Course N 2 December 2012
/ 76
Do we need to include options in the …nancial portfolio ?

Adequacy customer/product (static case)
Obviously, other options can be introduced to provide a percentage of
the potential market rise:
VT = pV0 + (c + l [(St )0

t T]

) such that c + l (.)

0.

The choice of a particular payo¤ l may depend on:
Market predictions: rise or drop of the …nancial market, volatility
levels, etc.
The type of risky assets: stock index, hedge fund, etc.
The insurance cost associated to each chosen derivative: lookback
options, corridor options, etc.
Note that, for standard portfolio insurance, l (.) is increasing w.r.t.
the risky asset (it corresponds to the second objective).

J.-L. Prigent (THEMA)

Bogota Seminar, Course N 2 December 2012
Do we need to include options in the …nancial portfolio ?
/ 76

Adequacy customer/product (which payo¤ ?)
Choice of l (examples)

J.-L. Prigent (THEMA)

Bogota Seminar, Course N 2 December 2012
/ 76
Do we need to include options in the …nancial portfolio ?

Optimal positioning (standard case)

Introduced by Leland (1980), and Brennan and Solanki (1981): Who
should buy portfolio insurance ?
The portfolio value is a function of a given benchmark (usually a
…nancial index).
Its payo¤ maximizes the investor’s expected utility.
It is characterized by the investor’s risk aversion.
It involves option-based strategies (Carr and Madan, 2001).

J.-L. Prigent (THEMA)

Bogota Seminar, Course N 2 December 2012
Do we need to include options in the …nancial portfolio ?
/ 76

Optimal positioning (main result)
The optimal payo¤ with insurance constraints on the terminal wealth
is solution of the following problem:
Maxh EQ [U (h(ST )]
V0 = e
h ( ST )

rT

(1)

EQ [h(ST )]

h0 (ST )

Result: (Bertrand-Lesne-Prigent, 2001) There exists an unconstrained
optimal payo¤ h associated to a Lagrange coe¢ cient λc such that
the optimal payo¤ h is given by:
h

= Max (h0 , h ).

(2)

The parameter λc can also be considered as a Lagrange multiplier
associated to a non insured optimal portfolio but with a modi…ed
initial wealth. Indeed, when h is greater than the insurance ‡oor h0 ,
then h = h . Otherwise, h = h0 .
See also El Karoui, Jeanblanc and Lacoste (2005) for the American
case.
Bogota Seminar, Course N 2 December 2012
J.-L. Prigent (THEMA)

Do we need to include options in the …nancial portfolio ?

/ 76

Optimal positioning (hedging)
Carr and Madan (1997): we can explicitly identify the investment
strategy that achieves a twice di¤erentiable given payo¤ h.
Suppose for example that the interest rate is non stochastic.
The portfolio h(S ) is duplicated by an unique initial position of
h(S0 ) h0 (S0 )S0 unit discount bonds, h0 (S0 ) shares and h(K )dK
out-of-the-money options of all strikes K :
h ( S ) = [ h ( S0 )

+

Z S0
0

h 0 ( S0 ) S0 ] + h 0 ( S0 ) S
h00 (K )(K

S )+ dK +

Z ∞
S0

(3)
h00 (K )(S

K )+ dK .

Generally, h0 is increasing and h also. Therefore, the optimal payo¤
is an increasing function of the benchmark. If h is not di¤erentiable,
it is approximated by a sequence of twice di¤erentiable payo¤
functions hn . Then, since the payo¤ hn are twice di¤erentiable, hn are
duplicated by initial positions of hn (S0 ) hn0 (S0 )S0 unit discount
bonds, hn0 (S0 ) shares and hn (K )dK out-of-the-money options of all
strikes K .
Bogota Seminar, Course N 2 December 2012
J.-L. Prigent (THEMA)

Do we need to include options in the …nancial portfolio ?

/ 76

Optimal positioning (fundamental example)
The interest rate r is assumed to be constant and the stock price S
follows a GBM:
1 2
St = S0 exp (µ
σ )t + σWt ,
2
where W standard Brownian motion.
Radon-Nikodym density g = dQ/dP function of ST .
La densité de Radon-Nikodym
4.5
4
3.5

Densité g

3
2.5
2
1.5
1
0.5
0
50

J.-L. Prigent (THEMA)

100
S

150

Bogota Seminar, Course N 2 December 2012
/ 76
Do we need to include options in the …nancial portfolio ?

Optimal positioning (optimal payo¤)
Assume that the utility function of the investor is a CRRA utility:
U (v ) =

v1 φ
,
1 φ

The optimal payo¤ is given by:
hEU (s ) = R

0

V0 e rT
g (s )

1 φ
φ

g (s )

1
φ

.

f (s )ds

Therefore, hEU (s ) satis…es:
hEU (s ) = d

s m with d = cχ

1
φ

, and m =

κ
> 0.
φ

Note that h is increasing. This property is satis…ed for all concave
utilities, as soon as the density g is decreasing, for instance within the
Black-Scholes asset pricing framework.
J.-L. Prigent (THEMA)

Bogota Seminar, Course N 2 December 2012
/ 76
Do we need to include options in the …nancial portfolio ?

Optimal positioning (EU concavity/convexity criterion)

Corollary
The concavity/convexity of the optimal payo¤ is determined by the
µ r
comparison between the relative risk-aversion γ and the ratio κ = σ2 ,
which is the Sharpe ratio divided by the volatility σ:
i) hEU is concave if κ < φ (low performance or/and high risk aversion);
ii) hEU is linear if κ = φ (very special case);
iii) hEU is convex if κ > φ (good performance or/and weak risk aversion).

J.-L. Prigent (THEMA)

Bogota Seminar, Course N 2 December 2012
/ 76
Do we need to include options in the …nancial portfolio ?

Optimal positioning (EU convex)

Optimal portfolio profile

Option positioning

2500

2400
2200

2000

2000

Portfolio value

portfolio value

1800

1500

1000

1600
1400
1200
1000

500

800
600

0

80

100
S

120

400
50

60

70

80

90

100

110

120

130

140

S

Portfolio value for φ = 1/2

Corresponding buy/sell options

According to the …nancial values and the relative risk aversion, we can get
a convex payo¤ ( where µ = 7% ). The (approximated) corresponding
position on option markets is for example
VT = 3400BT + 40ST + 32.7(K ST )+ with strike K = 105.
J.-L. Prigent (THEMA)

Bogota Seminar, Course N 2 December 2012
/ 76
Do we need to include options in the …nancial portfolio ?

Optimal positioning (EU concave)

Option Positioning
1400

1350

1350

1300

1300

1250

1250

Portfolio Value

Portfolio value

Optimal portfolio profile
1400

1200

1150

1200

1150

1100

1100

1050

1050

1000

1000

950
70

80

90

100

110

120

130

140

950
70

80

90

100

110

120

130

140

S

S

Portfolio value for φ = 1/2

Corresponding buy/sell options

For µ = 3% , the (approximated) corresponding position on option
markets is for example VT = 615 BT + 5.25 ST 0.75 (K ST )+ with
strike K = 100.

J.-L. Prigent (THEMA)

Bogota Seminar, Course N 2 December 2012
/ 76
Do we need to include options in the …nancial portfolio ?

Optimal positioning (insurance constraints)

h (ST ) = Max (h0 (ST ), h (ST )) with h (ST ) = d

ST m ,

Thus: for example for h0 constant,
h (ST ) = h0 + Max (dST m

h0 , 0).

Therefore, power and polynomial options are indeed interesting!
"On the pricing of power and other polynomials options", Journal of
Derivatives, summer 2006, (Quittard-Pinon-Macovschi).

J.-L. Prigent (THEMA)

Bogota Seminar, Course N 2 December 2012
/ 76
Do we need to include options in the …nancial portfolio ?

Behavorial …nance (EU or non EU ?)

Many empirical and theoretical studies have been developed against
the expected utility theory (see e.g. Allais, 1953; Kahneman and
Tversky, 1979; Epstein and Schneider, 2003).
EU is based on the Independence Axiom. But this axiom may fail
empirically and can induce some paradoxes (Allais, 1953).
Additionally, the expected utility theory implies that the utility
function must simultaneously represent the choice among possible
outcomes, and also model the attitude towards risk. Thus, for
example, an investor who has a decreasing marginal utility is
necessarily risk-averse (Cohen and Tallon, 2000).

J.-L. Prigent (THEMA)

Bogota Seminar, Course N 2 December 2012
/ 76
Do we need to include options in the …nancial portfolio ?

Behavorial …nance (alternative criteria)

During the 80’s, alternative theories have been introduced by slightly
modifying or relaxing the original axioms:
The Weighted Utility Theory of Chew and MacCrimmon (1979);
The Non-Linear Expected Utility Theory of Machina (1982);
The Utility with Ambiguity of Gilboa and Schmeidler (1989)
(introduced by Ellsberg, 1961);
The Regret Theory of Loomes and Sugden (1982) and Bell (1982)
The Anticipated Utility Theory of Quiggin (1982);
The Prospect Theory of Kahnemann and Tversky (1979, 1992);

J.-L. Prigent (THEMA)

Bogota Seminar, Course N 2 December 2012
/ 76
Do we need to include options in the …nancial portfolio ?

Optimal positioning (ambiguity theory)
Maccheroni et al. (2006) consider the following representation of
preferences:
For all random variables X and Y which represent results or
consequences and with values in [ M, M ], we have:

MinP2∆

Z

X
U (X )dP + C (P)

Y ,
MinP2∆

Z

U (Y )dP + C (P).

The function U corresponds to decision risk attitude. Index C
represents the individual attitude towards ambiguity. ∆ is the set of
multiple priors (probability scenarii).
This representation of preferences includes both the multiple priors
preferences of Gilboa and Schmeidler (1989) and the multiplier
preferences of Hansen and Sargent (2000, 2001). C (P) = ξR (P, Q)
the relative entropy with respect to given probability distribution Q.
J.-L. Prigent (THEMA)

Bogota Seminar, Course N 2 December 2012
/ 76
Do we need to include options in the …nancial portfolio ?

Optimal positioning (optimization with ambiguity)

The utility U of the investor is supposed to be increasing and
di¤erentiable.
For the optimal positioning, the portfolio value V is a function h of
the basic assets: V = h(ST ).
Therefore, the investor has to solve the following maximization
problem: search h such that
Maxh MinP2∆ (EP [U (h(ST ))] + C [P]) .
under
V0 = e

J.-L. Prigent (THEMA)

rT

EQ [h(ST )],

Bogota Seminar, Course N 2 December 2012
/ 76
Do we need to include options in the …nancial portfolio ?

Optimal positioning under ambiguity (assumptions)

We assume that h 2 H = L2 (R+ , PST (ds ), PST 2 ∆), which is the
set of the measurable functions with squares that are integrable on
R+ with respect to all the distributions PST (ds ) belonging to the set
∆. Under mild assumption about the payo¤s and the set ∆ of
multipriors, we can deduce a …rst general result.
Assumption 1: The utility function U is strictly concave and continuous;
Assumption 2: For any h 2 H, the functional EP [U (h(ST ))] + C [P] is
continuous and quasiconvex on ∆.
Assumption 3: We search the solution in the subset of continuous
functions belonging to H;
Assumption 4: The set ∆ of multipriors is compact.

J.-L. Prigent (THEMA)

Bogota Seminar, Course N 2 December 2012
/ 76
Do we need to include options in the …nancial portfolio ?

Optimal positioning under ambiguity (main result)
Proposition
(Ben Ameur-Prigent 2013) Under Assumptions (1,2,3,4), the optimal
payo¤ h exists and corresponds to the optimal solution for a given
PST 2 ∆.

Proof.
By Assumptions (1,2,3), the functional (EP [U (h(ST ))] + C [P]) is a real
function which is continuous with respect to P 2 ∆.
Therefore, if ∆ is compact (Assumption 4), for any given payo¤ h, the
solution of MinP2∆ (EP [U (h(ST ))] + C [P]) is reached at a probability
P(h)2∆.
Using MinimaxTheorem (Sion, 1958: Saddle-point), we deduce the
result.
J.-L. Prigent (THEMA)

Bogota Seminar, Course N 2 December 2012
Do we need to include options in the …nancial portfolio ?
/ 76

Optimal positioning under ambiguity (examples)
Case 1 (C = 0) It corresponds to the criterion maxmin expected utility of
Gilboa and Schmeidler (1989).
The optimal solution for the CRRA case is given by:
κ

h (ST ) = d.STφ ,
for the pair (µ, σ) that minimizes the absolute value of the Sharpe ratio
µ r
θr = σ .
Case 2. (C (P) = ξR (P, Q)). Let Pµ0 be the reference probability
µ µ
distribution. Denote θ µ0 = σ 0
The optimal solution minimizes
1 φ

V0 e rT
1 φ

J.-L. Prigent (THEMA)

exp

1 2 1 φ
θ T
2 r
φ

1
+ ξθ 2µ0 T .
2

Bogota Seminar, Course N 2 December 2012
/ 76
Do we need to include options in the …nancial portfolio ?

Optimal positioning (cumulative prospect theory, CPT)
Tversky and Kahneman (1992) have introduced both speci…c utility
functions for losses and gains and a transformation function of the
cumulative distributions (special case of Rank Dependent Expected
Utility).
There exist two functions, w and w + de…ned on [0, 1], and an
utility type function v such that the utility V on the lottery
L = f(x1 , p1 ), ..., (xn , pn )g with x1 < ... < xm < 0 < xm +1 < ... < xn
is de…ned as follows: de…ne Φ and Φ+ by: Φ1 = w (p1 ) and
Φn+ = w + (pn ),
Φi
Φi+

= w
∑ij =1 pj
= w + ∑nj=i pj

J.-L. Prigent (THEMA)

w
w+

∑ij =11 pj , 8i 2 f2, ..., m g ,
∑nj=i +1 pj , 8i 2 fm + 1, ..., ng .

(4)

Bogota Seminar, Course N 2 December 2012
/ 76
Do we need to include options in the …nancial portfolio ?

Optimal positioning (CPT with pdf)
V is given by: V (L) = V (L) + V + (L) with
m

V (L) =

∑ v (xi )Φi and V + (L) =

i =1

n



v (xi )Φi+ .

(5)

i =m +1

When the probability distribution F has a pdf f on [ M, M ], and the
0
0
functions w and w + have derivatives w and w + , then: (de
Palma, Picard and Prigent, 2010)
V (L) =

Z 0

M

v (x ) w

0

[F (x )]f (x )dx +

Z M

0

v (x ) w + [1

F (x )]f (x )dx.

0

(6)
As in Quiggin (1982), both functions w and w + can be chosen as

follows: w (p ) =
= 0, 69 and
1 , with, for example, γ
(p γ +(1 p )γ ) γ

γ+ = 0, 61.
J.-L. Prigent (THEMA)

Bogota Seminar, Course N 2 December 2012
/ 76
Do we need to include options in the …nancial portfolio ?

Optimal positioning (CPT: utility and for probability
transformation)

Utility of gain and loss

Fonction de pondération

0.6

1
0.9

0.4

0.8

Utility

0.2

0.7
0.6

0

0.5

-0.2
0.4

-0.4

0.3
0.2

-0.6

0.1

-0.8
-0.25 -0.2 -0.15 -0.1 -0.05

0 0.05
Loss/Gain

0.1

0.15

0.2

0.25

0

0

0.2

0.4

0.6

0.8

1

Utility function U (Loss/Gain) Weighting functions w and w +

J.-L. Prigent (THEMA)

Bogota Seminar, Course N 2 December 2012
/ 76
Do we need to include options in the …nancial portfolio ?

Optimal positioning (assumptions on probability weighting)
We asssume that there exists a functional Φ which modi…es the
historical probability, taking account of the investor’s behavior.
When basic assets are assumed to have pdf, another functional ϕ,
de…ned on the set of probability density functions (pdf), can be
associated to the functional Φ.
The function ϕ satis…es the following conditions:
Assumption 1: for any pdf f , ϕ(f ) is positive.
Assumption 2: for any pdf f ,

Z

ϕ(f ) (x ) dx = 1.

Rn

Assumption 3: for any probability distribution P with pdf f , the
distribution Φ [P] has a pdf equal to ϕ(f ) = f ψ(F ), where F denotes the
cdf associated to f .

J.-L. Prigent (THEMA)

Bogota Seminar, Course N 2 December 2012
Do we need to include options in the …nancial portfolio ?
/ 76

Optimal positioning (wealth optimization)
The utility U of the investor is supposed to be injective and piecewise
twice-di¤erentiable. Denote by J the inverse of the marginal utility
U 0 , de…ned on the range of U 0 .
e equal to Φ [P], where Φ
Introduce the “behavioral” probability P
denotes the functional which modi…es the historical probability.
Max EPe [VT ] under V0 = e

rT

EQ [VT ].

The expectation EPe [VT ] is equal to
Z

U [v ] fV (v )ψ(FV (v ))dv .

(7)

(8)

For the optimal positioning, the portfolio value V is a function of the basic
assets: V = h(ST ) and the investor solves:
Maxh EPe [U [h(ST )]] under V0 = e

J.-L. Prigent (THEMA)

rT

EQ [h(ST )].

(9)

Bogota Seminar, Course N 2 December 2012
/ 76
Do we need to include options in the …nancial portfolio ?

Optimal positioning (wealth as function of S)
b
Assume that ST has a pdf denoted by f . De…ne the transformation ψ
by:
b [F (s )] = ψ [FV (h(s )] .
ψ
(10)
Then, the expectation EPe [U (h(ST )] is equal to
Z

b [F (s )] ds.
U [h(s )] f (s )ψ

(11)

Suppose as usual that the function h ful…ls:
Z

R+

h2 (s )f (ds ) < ∞.

Denote also by g the density of MT with respect to P, de…ned on the
set of the values of ST . Then, the optimization problem is reduced to:
Maxh 2L2 (R+ , PS
with V0 =
J.-L. Prigent (THEMA)

Z

R+

T

)

Z

R+

h(s )g (s )f (s )ds,

b [F (s )] ds,
U [h(s )] f (s )ψ

Bogota Seminar, Course N 2 December 2012
/ 76
Do we need to include options in the …nancial portfolio ?

Optimal positioning (RDEU, main result)
If the optimal payo¤ h is injective, then it is given by: (Prigent, 2010)
b (F )),
h = J (λg /ψ

(12)

where λ is the scalar Lagrange multiplier such that
V0 =

Z

R+3

b (F )(x ))g (x )f (x )dx.
J (λg /ψ

The previous result can be extended if there exists an increasing
sequence (si )1 si n such that 8i, the payo¤ h is injective on
[si , si +1 [. Then, the optimal payo¤ h satis…es:
b (F )),
h = J (λg /ψ

(13)

and is characterized both by the determination of the appropriate
sequence of real numbers (si )1 si n and the corresponding Lagrange
parameter.
Thus, the ratio of the densities "risk-neutral density / behavorial
density " plays a key role.
J.-L. Prigent (THEMA)

Bogota Seminar, Course N 2 December 2012
/ 76
Do we need to include options in the …nancial portfolio ?

Properties of the optimal portfolio (RDEU case) (1)

The properties of the optimal payo¤ h as function of the benchmark
S can be analyzed.
Consider a given set [a, b ].
- If the utility function U is concave on this set, h is an increasing
b (F ) is a
function of the benchmark ST if and only if the ratio g /ψ
decreasing function of ST .
- If the utility function U is convex on this set, h is an increasing
b (F ) is an
function of the benchmark ST if and only if the ratio g /ψ
increasing function of ST .

J.-L. Prigent (THEMA)

Bogota Seminar, Course N 2 December 2012
/ 76
Do we need to include options in the …nancial portfolio ?

Properties of the optimal portfolio (RDEU case) (2)
b are di¤erentiable, then,
If we assume that f , g and ψ
h 0 (s ) =

U 0 (h(s ))
U 00 (h(s ))

!
b 0 (F (s ))
g 0 (s ) ψ
+
f (s ) .
b (F (s ))
g (s )
ψ

(14)

According to the sign of U 0 (h(s ))/U 00 (h(s )) (U convex/concave),
the optimal payo¤ h is increasing according to the comparison of
b 0 (F (s ))
g 0 (s )
ψ
with
f (s ).
b (F (s ))
g (s )
ψ

Di¤erentiating twice with respect to s, we get: (To risk tolerance)
h00 (s ) = [To0 (h(s )) +
J.-L. Prigent (THEMA)

Y 0 (s )
]
Y (s )2

[To (h(s ))Y 2 (s )].

(15)

Bogota Seminar, Course N 2 December 2012
/ 76
Do we need to include options in the …nancial portfolio ?

Examples (anticipated utility)(1)
Assume also that the utility function of the investor is a CRRA utility:
α
U (v ) = vα .
The functional ϕ(fV ), which characterizes the probability transformation,
is de…ned by ϕ(fV )(v ) = fV (v )ψ(FV )(v ) with:
ψ(FV )(v ) = wγ0 [FV (v )].
Consider the case:
wγ (p ) =



(p γ + (1

1

.

p )γ ) γ

We deduce:
wγ0 (p ) =



1



1)p + γ (1

(p γ + (1

Consider also the functions iwγ (p ) = wγ (1
J.-L. Prigent (THEMA)

p ) γ + p (1
p )γ )

1
γ +1

p )γ

1

.

p ) and iwγ0 (p ) = wγ0 (1

(16)
p ).

Bogota Seminar, Course N 2 December 2012
/ 76
Do we need to include options in the …nancial portfolio ?

Examples (anticipated utility)(3)
Under the assumptions on …nancial parameters and on the behavorial
g (s )
g (s )
parameter γ = 0.61, both functions w 0 (F )(s ) and w 0 (1 F )(s ) have
γ

S

γ

S

the following graphs:

Function

g (.)
0
w γ (F S )(.)

Function

(max for s = 99.5)
(max for s
Ratios of the weighting transformations
J.-L. Prigent (THEMA)

g (.)
0
w γ (1 F S )(.)

= 106.1)

Bogota Seminar, Course N 2 December 2012
Do we need to include options in the …nancial portfolio ?
/ 76

Examples (anticipated utility)(4)
b is such that:
The probability transformation Ψ
0

b (FS )(s ) = wγ [FS (s )], if h is increasing,
ψ
0

b (FS )(s ) = wγ [1
ψ

FS (s )], if h is decreasing.

Since the utility is concave (α = 0.5), the optimal solution h (.) is a
g (.)
g (.)
decreasing function of the ratios w 0 (F )(.) and w 0 (1 F )(.) .Thus, it is
S

γ

γ

S

necessarily decreasing on the interval ]0, s ] and increasing on
g (.)
]s , ∞[. On [s , s ], w 0 (gF(.))(.)
.Therefore:
0
w (1 F )(.)
γ

h (s ) =

γ

S

V0 e rT

R∞
0

with

S

g (s )
b (F S )(s )
ψ

1
α 1

0

g (s )f (s )ds

b (FS )(s ) = wγ [1
ψ
0

g (s )
b (FS )(s )
ψ

FS (s )], if s 2]0, s ],

1
α 1

b (FS )(s ) = wγ [FS (s )], if if s 2]Bogota
ψ
s , Seminar,
∞[. Course N

J.-L. Prigent (THEMA)

Do we need to include options in the …nancial portfolio ?

,

(17)

(18)
(19)

2 December 2012
/ 76

Optimal positioning (anticipated theory)
(U concave) For the numerical base case with φ = 0.5 and γ = 0.61,
we get: (straddle type)
Profil du portefeuille optimal

Stratégie d'options correspondante

2500

3000

2500

Valeurs du portefeuille

Valeurs du portefeuille

2000

1500

1000

2000

1500

1000

500
500

0

80

100
S

120

Portfolio value
for α = 1/2, γ = 0.61

0

80

100
S

120

Corresponding option strategy

The (approximated) corresponding position on option markets looks like a
straddle VT = 500 + qc (ST K )+ + qp (K ST )+ with qc ' 25 and
qp ' 65,with strike K = 106 (for the standard straddle,
qc =Course
qp ).
Bogota Seminar,
N 2 December 2012
J.-L. Prigent (THEMA)

Do we need to include options in the …nancial portfolio ?

/ 76

Optimal positioning (anticipated theory)(comments)
According to the …nancial values and the behavorial parameters, we
get an optimal payo¤ which is …rst decreasing and then increasing.
For small risky asset values (s 80%.S0 ), the cdf F (s ) is smaller than
0.44. For high risky asset values (s 120%.S0 ), the term (1 F (s ))
is also smaller than 0.44. Therefore, respectively for these two cases,
0
0
wγ [FS (s )] and wγ [1 FS (s )] are higher than FS (s ) and (1 FS (s )).
This overweighting of relatively “rare” events leads the investor to
adopt an option positioning based on realizations of relatively extreme
events (signi…cant drops or rises of the underlying asset). This
strategy usually corresponds to an anticipation of a high volatility.
Note that, for an investor who maximizes the standard expected
utility, such anticipation leads to a more convex payo¤ but it must be
always increasing.

J.-L. Prigent (THEMA)

Bogota Seminar, Course N 2 December 2012
/ 76
Do we need to include options in the …nancial portfolio ?

Optimal positioning (anticipated theory) (Insurance
constraint)
Using previous general results and the same assumptions on …nancial
and behavorial parameters, we get the following …gure for an
insurance function h0 (ST ) = 90%.V0 + 5%.ST .
Profil du portefeuille optimal
2400

2200

2200

2000

2000

Valeurs du portefeuille

Valeurs du portefeuille

Profil du portefeuille optimal assuré
2400

1800

1600

1400

1200

1600

1400

1200

1000

800

1800

1000

80

100
S

120

Insured portfolio
(expected utility )
J.-L. Prigent (THEMA)

800
70

80

90

100
S

110

120

130

Insured portfolio
(anticipated utility)

Bogota Seminar, Course N 2 December 2012
Do we need to include options in the …nancial portfolio ?
/ 76

Examples (Kahneman and Tversky)(1)
According to Tversky and Kahneman (1992) who introduce the
cumulative prospect theory (CPT), the investor has not the same kind
of utility U when she su¤ers from losses (U convex) and when she
bene…ts from gains (U concave). Assume that the utility function of
the investor is de…ned by:
U (v ) = a (v
U (v ) = a (v

v )+

(v

v ) α1

, for v v ,
α1
(v v ) α2
v )+
, for v > v .
α2

with 1 < α1 , 0 < α2 < 1 and a = v (α1 1 ) .
We deduce:
1

J (y ) = v

(a

J (y ) = v + (y
J.-L. Prigent (THEMA)

y ) α1
a)

1

1
α2 1

, for y

a,

, for y > a.

Bogota Seminar, Course N 2 December 2012
Do we need to include options in the …nancial portfolio ?
/ 76

Examples (Kahneman and Tversky)(2)
Consider the following behavorial parameters: α1 = 1.5, α2 = 0.5,
and wγ = 0.69, wγ+ = 0.61.

J.-L. Prigent (THEMA)

Bogota Seminar, Course N 2 December 2012
Do we need to include options in the …nancial portfolio ?
/ 76

Examples (Kahneman and Tversky)(3)

Optimal payo¤ (CPT case)

J.-L. Prigent (THEMA)

Bogota Seminar, Course N 2 December 2012
/ 76
Do we need to include options in the …nancial portfolio ?

Optimal positioning (CPT)
Jin and Zhou (2008): shape of the optimal payo¤.
2000
1500
1000
500
140
120
100
80

J.-L. Prigent (THEMA)

Bogota Seminar, Course N 2 December 2012
/ 76
Do we need to include options in the …nancial portfolio ?

Introduction (regret theory and applications)
Recent surge in the interest in regret theory, in particular in
economics and …nance:
- Braun and Muermann (2004) show that regret can explain the commonly
observed preference for low deductibles in personal insurance markets;
- Gollier and Salanié (2006): asset pricing and portfolio choice;
- Barberis, Huang, Thaler (2006) show how regret theory can explain why
people often invest too little in stocks;
- Muermann, Mitchell, Volkman (2006) examine the impact of regret
aversion on the willingness to pay for a pension guarantee;
- Muermann and Volkmann (2007) illustrate how regret aversion can
explain the disposition e¤ect: investors have a preference for selling
winning stocks too early and holding losing stocks too long (see Shefrin
and Statman, 1985).
- Michenaud and Solnik (2008) show that regret can explain empirically
observations that many investors do not adopt a full hedging policy with
respect to currency risk.
J.-L. Prigent (THEMA)

Bogota Seminar, Course N 2 December 2012
/ 76
Do we need to include options in the …nancial portfolio ?

Regret Theory (1)

Regret theory (RT) is based on the di¤erence between the utility of
the actual outcome and the utility that the individual would have
received if he had made another choice.
This axiomatic decision theory can potentially explain violations of
usual axioms of the standard EU theory.
RT supposes that individuals are « rational » but that their decisions
are not only based on potential bene…ts (as for EU criterion) but also
on expected regret with respect to other possible oucomes.
Therefore, RT includes two types of aversions: risk-aversion φ and
regret-aversion RA.
We prove that the ratio RA/φ plays a key role.

J.-L. Prigent (THEMA)

Bogota Seminar, Course N 2 December 2012
/ 76
Do we need to include options in the …nancial portfolio ?

Regret Theory (2)

It can explain "preference reversal" phenomenon, which is in
contradiction with transitivity (see studies of psychologists such as
Lichtenstein and Slovic, 1971).
Among all usual axioms about preferences, transitivity is considered
as one of the most rational. However, some theorists have suggested
that this axiom can be released (see Georgescu-Roegen, 1936, 1958).
The most signi…cant alternative theory to standard EU which can
take account of regret has been developed by Loomes and Sugden
(1982, 1985), Bell (1982) and Fishburn (1982).
Loomes and Sugden propose a regret/rejoice function for pairs of
lotteries (the selected and the reference lotteries).

J.-L. Prigent (THEMA)

Bogota Seminar, Course N 2 December 2012
/ 76
Do we need to include options in the …nancial portfolio ?


Course2(Bogota)(structured products).pdf - page 1/76
 
Course2(Bogota)(structured products).pdf - page 2/76
Course2(Bogota)(structured products).pdf - page 3/76
Course2(Bogota)(structured products).pdf - page 4/76
Course2(Bogota)(structured products).pdf - page 5/76
Course2(Bogota)(structured products).pdf - page 6/76
 




Télécharger le fichier (PDF)


Course2(Bogota)(structured products).pdf (PDF, 797 Ko)

Télécharger
Formats alternatifs: ZIP



Documents similaires


course2 bogota structured products
fairpricing
4 performancelossaversion
les agents de croyance ou de desir
3 universalperformance
8 portfoliooptihedge