backward and forward preferences presentation.pdffinal presentation december 13th, 2018....

Backward and Forward Preferences

Undergraduate scholars: Gongyi Chen, Zihe WangGraduate mentor: Bhanu Sehgal

Faculty mentor: Alfred ChongUniversity of Illinois at Urbana-Champaign

Illinois Risk Lab Illinois Geometry LabFinal Presentation

December 13th, 2018

Introduction

Consider an investor who has a portfolio consisting of twoassets and he wants to optimize his wealth.Select a time period [0,T ], adopt the binomial model, andpre-specify the time T utility function of the investor at time0.After the exogenous triplet is pre-committed at time 0, wethen solve for optimal investment strategies and derive thevalue functions/preferences before time T .We will then use hypothetical value for the model and S&P500 data to simulate the value functions.

Two-period Binomial Model

Let (Ω,F ,P) be the probability space.Let F =

Ft

t=0, T2 ,T

be the filtration generated by

ξ∗ =ξ∗t

t=0, T2 ,T

.

The two random variable ξ∗T2

and ξ∗T are given by

ξ∗T2

=

ξu∗T2, pu

0, T2

= P(ξ∗T2

= ξu∗T2|F0)

ξd∗T2, pd

0, T2

= P(ξ∗T2

= ξd∗T2|F0)

ξ∗T =

ξu∗T , pu

T2 ,T

= P(ξ∗T = ξu∗T |F T

2)

ξd∗T , pd

T2 ,T

= P(ξ∗T = ξd∗T |F T

2)


Suppose there are two types of assets, a risk-free assetthat offers a return of r and a risky asset.Denote the price of the risky asset to be S∗ =

S∗t

t=0, T2 ,T

.

The arbitrage-free conditions are 0 < ξd∗T2< er T

2 < ξu∗T2

and

0 < ξd∗T < er T

2 < ξu∗T .


The following relationships for the price hold in the two periodbinomial model.

S∗T2

=

S∗0ξu∗T2, p0, T

2= pu

0, T2

S∗0ξd∗T2, p0, T

2= pd

0, T2

S∗T =

S∗T2ξu∗

T , p T2 ,T

= puT2 ,T

S∗T2ξd∗

T , p T2 ,T

= pdT2 ,T

where S∗0 ∈ F0, S∗T2∈ F T

2and S∗T ∈ FT .


S∗0

S∗0 ξ

u∗T2

S∗0 ξ

d∗T2

S∗0 ξ

u∗T2ξd∗

T

S∗0 ξ

d∗T2ξ∗u

T

S∗0 ξ

u∗T2ξu∗

T

S0ξd∗T2ξd∗

T

pu0, T

2

pdT2 ,T

pd0, T

2

puT2 ,T

puT2 ,T

pdT2 ,T

t = 0 t = Tt = T2


Consider an investor who holds α shares of risky asset S, andrisk-free asset B, with initial wealth x .Let

X ∗α;x

t

t=0, T2 ,T

be a wealth process of the investoremploying control α with initial wealth x at t = 0. Therefore,

X ∗α;x0 = x ,

e−r T2 X ∗α;x

T2

= x + α0(S∗T2e−r T

2 − S∗0),

e−rT X ∗α;xT = e−r T

2 X ∗α;xT2

+ α T2

(S∗T e−rT − S∗T2e−r T

2 ).


For simplicity, we’ll use the discount price of the riskyasset. Denote it by S =

St

t=0, T2 ,T

=

e−rtS∗t

t=0, T2 ,T

.

Let the up and down factor for discounted price to beξt

t= T2 ,T

=ξ∗t e−r T

2

t= T2 ,T

.

The arbitrage free conditions become 0 < ξdT2< 1 < ξu

T2

and

0 < ξdT < 1 < ξu

T .


Define a discounted wealth process to be

Xα;xt

t=0, T2 ,T

,changing S∗ to S.

Xα;x0 = x ,

Xα;xT2

= x + α0(S T2− S0),

Xα;xT = Xα;x

T2

+ α T2

(ST − S T2

).


Denote another wealth process

Xα;x ,ts

s=t ...T be a wealth

process of the investor employing control α with the initialwealth x at time t . Therefore,

Xα;x , T

2T = x + α T

2(ST − S T

2),

Xα;xT = Xα;x

T2

+ α T2

(ST − S T2

) = Xα;Xα;x

T2, T

2

T ,

Xα0,α T

2;x

T = Xα T

2;Xα0;xT2

, T2

T .


The value function at time T is given by

V (x ,T ) = UT (x).

The objective function is

E[UT (Xα;xT )].

Therefore, his value function at time t = 0 is

V (x ,0) = supα

E[UT (Xα;xT )].

And the value function at t = T2 is given by

V (x ,T2

) = supα

E[UT (Xα;x , T

2T )|F T

2].


By the definition and Dynamic Programming Principle,

V (x ,T ) = UT (x),

V (x ,T2

) = supα T

2

E[V (Xα T

2;x ; T

2

T ,T )|F T2

],

V (x ,0) = supα0

E[V (Xα0;xT2

,T2

)].

We can solve the value functions recursively, and this is whyV =

V (ω, x , t)

ω∈Ω,x∈R,t=0, T

2 ,Tare coined as backward

preferences.

V (x ,0)α∗0⇐ V (x , T

2 )α∗T

2⇐ V (x ,T ).


Assume V (x ,T ) = UT (x) = −e−γx , γ > 0.

V (x ,T2

) = −e−γx−EQ[h T

2|F T

2],

where

h T2

=

hu

T2

= quT ln

(qu

Tpu

T2 ,T

)+ qd

T ln(

qdT

pdT2 ,T

), pu

T2 ,T

= P(p T2 ,T

= puT2 ,T|ξ T

2= ξu

T2

);

huT2

= quT ln

(qu

Tpu

T2 ,T

)+ qd

T ln(

qdT

pdT2 ,T

), pu

T2 ,T

= P(p T2 ,T

= puT2 ,T|ξ T

2= ξd

T2

),

and quT , qd

T are the conditional probabilities of going up and down under measure Q in[ T

2 ,T ].

The optimal strategy is α∗T2

=ln

( puT2 ,T

quT

)−ln

( pdT2 ,T

qdT

)γS T

2(ξu

T−ξdT )

.


V (x ,0) = −e−γx−EQ[h0+h T

2],

where h0 = quT2

ln( qu

T2

pu0, T

2

)+ qd

T2

ln( qd

T2

pu0, T

2

)and qu

T2, qd

T2

are the

conditional probabilities of going up and down under measureQ in [0, T

2 ].

The optimal strategy α∗0 =

ln( pu

0, T2

quT2

)−ln( pu

0, T2

qdT2

)−hu

T2

+hdT2

γS0(ξuT2−ξd

T2

).


First, if the stock price goes up initially at t = T2 (ξ T

2= ξu

T2

),


And if the price of the risky asset goes up at t = T (ξT = ξuT );

Or if the price of risky asset goes down at t = T , (ξT = ξdT ),


Second, if the stock price goes down initially at t = T2 (ξ T

2= ξd

T2

),


And if the price of the risky asset goes up at t = T (ξT = ξuT );

Or if the price of risky asset goes down at t = T (ξT = ξdT ),

Multi-period Model

Consider there are N periods, where t = 0, TN ,2

TN ...T . For

simplicity, denote h = TN so that t = 0,h,2h...,T .

Let (Ω,F ,P,F) be the filtered probability space.Suppose there are only one risk-free asset and one riskyasset.

Multi-period Model

∀t = 0,h, ..., (N − 1)h, in the interval [t , t + h], the price could goup or down with respective probabilities,

(ξt+h|Ft ) =

ξu

t+h, put ,t+h = P(ξt+h = ξu

t+h|Ft )

ξdt+h, pd

t ,t+h = P(ξt+h = ξdt+h|Ft )

where ξut+h, ξ

dt+h and pu

t ,t+h,pdt ,t+h ∈ Ft .

Multi-period Model

∀t = 0,h, ..., (N − 1)h, we used the money account as thenumeraire.

The arbitrage-free condition is 0 < ξdt+h < 1 < ξu

t+h.The risk-neutral probability of realizing up in [t , t + h] is

qut ,t+h = Q(ξt+h = ξu

t+h|Ft ) =1−ξd

t+hξu

t+h−ξdt+h

.

We assume ξut+h, ξ

dt+h, pu

t ,t+h,pdt ,t+h,q

ut ,t+h and qd

t ,t+h ∈ F0to simplify the simulation process.

Multi-period Model

Denoteαt

t=0,h,...,(N−1)h be a stochastic control process. LetXα;x =

Xα;x

t

t=0,h,...,(N−1)h,T be the wealth process of theinvestor using the control α with initial wealth x at time 0.Therefore,

Xα;x0 = x ,

Xα;xt+h = Xα;x

t + αt (St+h − St ), ∀t = 0,h, ..., (N − 1)h.

Multi-period Model

∀s = 0,h, ...T , let Xα;x ,s =

Xα;x ,st

t=s,s+h,...,T , be the wealth

process of the investor using the control α with initial wealth xat time s. Therefore,

Xα;x ,0 = Xα;x ,

Xα;x ,ss = x ,

Xα;x ,st+h = Xα;x ,s

t + αt (St+h − St ), ∀t = s, s + h, ..., (N − 1)h.

And more importantly,

Xα;x ,tT = X

α;Xα;x,tt+h ,t+h

T , ∀t = s, s + h, ..., (N − 1)h.

Multi-period Model

The value function at time T is given by

V (x ,T ) = UT (x).

The objective function is

E[UT (Xα;xT )].

Therefore, his value function at time t = 0 is

V (x ,0) = supα

E[UT (Xα;xT )].

∀t = 0,h, ...,T , value function at time t is given by

V (x , t) = supα

E[UT (Xα;x ,tT )|Ft ].

Multi-period Model

By Dynamic Programming Principle, we can derive thevalue functions backwardly,

V (x ,0)α∗

0⇐ V (x ,h)α∗

h⇐ ...α∗

(N−1)h⇐ V (x , (N − 1)h)α∗

(N−1)h⇐ V (x ,T ).

This is the reason why V =

V (ω, x , t)ω∈Ω,x∈R,t=0,h,...,T

are coined as backward preferences.

Multi-period Model

∀t = 0,h, ..., (N−1)h, assume V (x ,T ) = UT (x) = −e−γx , γ > 0, then

V (x , t) = −e−γx−EQ[∑(N−1)h

i=h hi |Ft

],

where ht = qut,t+h ln

(qu

t,t+hpu

t,t+h

)+ qd

t,t+h ln

(qd

t,t+h

pdt,t+h

).

The optimal strategy is

α∗t =ln( pu

t,t+hqu

t,t+h

)− ln

( pdt,t+h

qdt,t+h

)−(EQ[∑(N−1)h

i=t+h hi |Fut+h]− EQ[

∑(N−1)hi=t+h hi |Fd

t+h])

γSt (ξut+h − ξd

t+h

) .

Multi-period Model

Future Goals

Derive the mechanism of forward preference.Solve the preference and optimal investment strategyexplicitly.Visualize forward preference and optimal investmentstrategy and compare them with backward case.

Reference

[1] Marek Musiela and Thaleia Zariphopoulou (2004): Avaluation algorithm for indifference prices in incompletemarkets. Finance and Stochastics 8, 399–414.

backward and forward preferences presentation.pdffinal presentation december 13th, 2018....

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