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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

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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

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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

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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

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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

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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

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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

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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

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Adequacy customer/product (which payo¤ ?)

Choice of l (examples)

J.-L. Prigent (THEMA)

Bogota Seminar, Course N 2 December 2012

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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)

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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

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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

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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

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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)

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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)

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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)

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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)

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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

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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)

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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)

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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)

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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 )],

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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)

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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)

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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

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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)

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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

pγ

follows: w (p ) =

= 0, 69 and

1 , with, for example, γ

(p γ +(1 p )γ ) γ

γ+ = 0, 61.

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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 +

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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 .

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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)

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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

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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

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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 .

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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)

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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 ) =

pγ

(p γ + (1

1

.

p )γ ) γ

We deduce:

wγ0 (p ) =

pγ

1

(γ

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 ).

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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)

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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)

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,

(17)

(18)

(19)

2 December 2012

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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

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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

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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)

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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.

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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

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Examples (Kahneman and Tversky)(3)

Optimal payo¤ (CPT case)

J.-L. Prigent (THEMA)

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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

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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)

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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)

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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)

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