SLIDE 1
Backward and Forward Preferences Undergraduate scholars: Gongyi - - PowerPoint PPT Presentation
Backward and Forward Preferences Undergraduate scholars: Gongyi - - PowerPoint PPT Presentation
Backward and Forward Preferences Undergraduate scholars: Gongyi Chen, Zihe Wang Graduate mentor: Bhanu Sehgal Faculty mentor: Alfred Chong University of Illinois at Urbana-Champaign Illinois Risk Lab Illinois Geometry Lab Final Presentation
SLIDE 2
SLIDE 3
Two-period Binomial Model
Let (Ω, F, P) be the probability space. Let F =
- Ft
- t=0, T
2 ,T be the filtration generated by
ξ∗ =
- ξ∗
t
- t=0, T
2 ,T.
The two random variable ξ∗
T 2 and ξ∗
T are given by
ξ∗
T 2 =
ξu∗
T 2 ,
pu
0, T
2 = P(ξ∗ T 2 = ξu∗ T 2 |F0)
ξd∗
T 2 ,
pd
0, T
2 = P(ξ∗ T 2 = ξd∗ T 2 |F0)
ξ∗
T =
ξu∗
T ,
pu
T 2 ,T = P(ξ∗
T = ξu∗ T |F T
2 )
ξd∗
T ,
pd
T 2 ,T = P(ξ∗
T = ξd∗ T |F T
2 )
SLIDE 4
Two-period Binomial Model
Suppose there are two types of assets, a risk-free asset that offers a return of r and a risky asset. Denote the price of the risky asset to be S∗ =
- S∗
t
- t=0, T
2 ,T.
The arbitrage-free conditions are 0 < ξd∗
T 2 < er T 2 < ξu∗ T 2 and
0 < ξd∗
T < er T
2 < ξu∗
T .
SLIDE 5
Two-period Binomial Model
The following relationships for the price hold in the two period binomial model. S∗
T 2 =
S∗
0ξu∗
T 2 ,
p0, T
2 = pu
0, T
2
S∗
0ξd∗
T 2 ,
p0, T
2 = pd
0, T
2
S∗
T =
S∗
T 2 ξu∗
T ,
p T
2 ,T = pu T 2 ,T
S∗
T 2 ξd∗
T ,
p T
2 ,T = pd T 2 ,T
where S∗
0 ∈ F0, S∗
T 2 ∈ F T 2 and S∗
T ∈ FT.
SLIDE 6
Two-period Binomial Model
S∗ S∗
0 ξu∗ T 2
S∗
0 ξd∗ T 2
S∗
0 ξu∗ T 2
ξd∗
T
S∗
0 ξd∗ T 2
ξ∗u
T
S∗
0 ξu∗ T 2
ξu∗
T
S0ξd∗
T 2
ξd∗
T
pu
0, T 2
pd
T 2 ,T
pd
0, T 2
pu
T 2 ,T
pu
T 2 ,T
pd
T 2 ,T
t = 0 t = T t = T
2
SLIDE 7
Two-period Binomial Model
Consider an investor who holds α shares of risky asset S, and risk-free asset B, with initial wealth x. Let
- X ∗α;x
t
- t=0, T
2 ,T be a wealth process of the investor
employing control α with initial wealth x at t = 0. Therefore, X ∗α;x = x, e−r T
2 X ∗α;x T 2
= x + α0(S∗
T 2 e−r T 2 − S∗
0),
e−rTX ∗α;x
T
= e−r T
2 X ∗α;x T 2
+ α T
2 (S∗
Te−rT − S∗
T 2 e−r T 2 ).
SLIDE 8
Two-period Binomial Model
For simplicity, we’ll use the discount price of the risky
- asset. Denote it by S =
- St
- t=0, T
2 ,T =
- e−rtS∗
t
- t=0, T
2 ,T.
Let the up and down factor for discounted price to be
- ξt
- t= T
2 ,T =
- ξ∗
t e−r T
2
t= T
2 ,T.
The arbitrage free conditions become 0 < ξd
T 2 < 1 < ξu T 2 and
0 < ξd
T < 1 < ξu T.
SLIDE 9
Two-period Binomial Model
Define a discounted wealth process to be
- X α;x
t
- t=0, T
2 ,T,
changing S∗ to S. X α;x = x, Xα;x
T 2
= x + α0(S T
2 − S0),
Xα;x
T
= X α;x
T 2
+ α T
2 (ST − S T 2 ).
SLIDE 10
Two-period Binomial Model
Denote another wealth process
- X α;x,t
s
- s=t...T be a wealth
process of the investor employing control α with the initial wealth x at time t. Therefore, X
α;x, T
2
T
= x + α T
2 (ST − S T 2 ),
X α;x
T
= X α;x
T 2
+ α T
2 (ST − S T 2 ) = X
α;X α;x
T 2
, T
2
T
, X
α0,α T
2
;x T
= X
α T
2
;X
α0;x T 2
, T
2
T
.
SLIDE 11
Two-period Binomial Model
The value function at time T is given by V(x, T) = UT(x). The objective function is E[UT(X α;x
T
)]. Therefore, his value function at time t = 0 is V(x, 0) = sup
α E[UT(X α;x T
)]. And the value function at t = T
2 is given by
V(x, T 2 ) = sup
α E[UT(X α;x, T
2
T
)|F T
2 ].
SLIDE 12
Two-period Binomial Model
By the definition and Dynamic Programming Principle, V(x, T) = UT(x), V(x, T 2 ) = sup
α T
2
E[V(X
α T
2
;x; T
2
T
, T)|F T
2 ],
V(x, 0) = sup
α0
E[V(X α0;x
T 2
, T 2 )]. We can solve the value functions recursively, and this is why V =
- V(ω, x, t)
- ω∈Ω,x∈R,t=0, T
2 ,T are coined as backward
preferences. V(x, 0)
α∗
⇐ V(x, T
2 ) α∗
T 2
⇐ V(x, T).
SLIDE 13
Two-period Binomial Model
Assume V(x, T) = UT (x) = −e−γx, γ > 0. V(x, T 2 ) = −e
−γx−EQ[h T
2
|F T
2
]
, where h T
2 =
hu
T 2
= qu
T ln
- qu
T
pu
T 2 ,T
- + qd
T ln
- qd
T
pd
T 2 ,T
- ,
pu
T 2 ,T = P(p T 2 ,T = pu T 2 ,T |ξ T 2 = ξu T 2
); hu
T 2
= qu
T ln
- qu
T
pu
T 2 ,T
- + qd
T ln
- qd
T
pd
T 2 ,T
- ,
pu
T 2 ,T = P(p T 2 ,T = pu T 2 ,T |ξ T 2 = ξd T 2
), and qu
T , qd T are the conditional probabilities of going up and down under measure Q in
[ T
2 , T].
The optimal strategy is α∗
T 2
=
ln
pu
T 2 ,T qu T
- −ln
pd
T 2 ,T qd T
- γS T
2
(ξu
T −ξd T )
.
SLIDE 14
Two-period Binomial Model
V(x, 0) = −e
−γx−EQ[h0+h T
2
],
where h0 = qu
T 2 ln
qu
T 2
pu
0, T 2
- + qd
T 2 ln
qd
T 2
pu
0, T 2
- and qu
T 2 , qd T 2 are the
conditional probabilities of going up and down under measure Q in [0, T
2 ].
The optimal strategy α∗
0 = ln
pu
0, T 2 qu T 2
- −ln
pu
0, T 2 qd T 2
- −hu
T 2
+hd
T 2
γS0(ξu
T 2
−ξd
T 2
)
.
SLIDE 15
Two-period Binomial Model
First, if the stock price goes up initially at t = T
2 (ξ T
2 = ξu T 2 ),
SLIDE 16
Two-period Binomial Model
And if the price of the risky asset goes up at t = T (ξT = ξu
T);
Or if the price of risky asset goes down at t = T, (ξT = ξd
T),
SLIDE 17
Two-period Binomial Model
Second, if the stock price goes down initially at t = T
2 (ξ T
2 = ξd T 2 ),
SLIDE 18
Two-period Binomial Model
And if the price of the risky asset goes up at t = T (ξT = ξu
T);
Or if the price of risky asset goes down at t = T (ξT = ξd
T),
SLIDE 19
Multi-period Model
Consider there are N periods, where t = 0, T
N , 2 T N ...T. For
simplicity, denote h = T
N 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 risky asset.
SLIDE 20
Multi-period Model
∀t = 0, h, ..., (N − 1)h, in the interval [t, t + h], the price could go up or down with respective probabilities, (ξt+h|Ft) =
- ξu
t+h,
pu
t,t+h = P(ξt+h = ξu t+h|Ft)
ξd
t+h,
pd
t,t+h = P(ξt+h = ξd t+h|Ft)
where ξu
t+h, ξd t+h and pu t,t+h, pd t,t+h ∈ Ft.
SLIDE 21
Multi-period Model
∀t = 0, h, ..., (N − 1)h, we used the money account as the numeraire. The arbitrage-free condition is 0 < ξd
t+h < 1 < ξu t+h.
The risk-neutral probability of realizing up in [t, t + h] is qu
t,t+h = Q(ξt+h = ξu t+h|Ft) = 1−ξd
t+h
ξu
t+h−ξd t+h .
We assume ξu
t+h, ξd t+h, pu t,t+h, pd t,t+h, qu t,t+h and qd t,t+h ∈ F0
to simplify the simulation process.
SLIDE 22
Multi-period Model
Denote
- αt
- t=0,h,...,(N−1)h be a stochastic control process. Let
X α;x =
- X α;x
t
- t=0,h,...,(N−1)h,T be the wealth process of the
investor using the control α with initial wealth x at time 0. Therefore, X α;x = x, X α;x
t+h = X α;x t
+ αt(St+h − St), ∀t = 0, h, ..., (N − 1)h.
SLIDE 23
Multi-period Model
∀s = 0, h, ...T, let X α;x,s =
- X α;x,s
t
- t=s,s+h,...,T, be the wealth
process of the investor using the control α with initial wealth x at time s. Therefore, X α;x,0 = X α;x, X α;x,s
s
= x, X α;x,s
t+h
= X α;x,s
t
+ αt(St+h − St), ∀t = s, s + h, ..., (N − 1)h. And more importantly, X α;x,t
T
= X
α;X α;x,t
t+h ,t+h
T
, ∀t = s, s + h, ..., (N − 1)h.
SLIDE 24
Multi-period Model
The value function at time T is given by V(x, T) = UT(x). The objective function is E[UT(X α;x
T
)]. Therefore, his value function at time t = 0 is V(x, 0) = sup
α E[UT(X α;x T
)]. ∀t = 0, h, ..., T, value function at time t is given by V(x, t) = sup
α E[UT(X α;x,t T
)|Ft].
SLIDE 25
Multi-period Model
V(x, t) = sup
α E[UT(X α;x,t T
)|Ft], = sup
αt,αt+h,...,α(N−1)h
E[UT(X
αt,αt+h,...,α(N−1)h;x,t T
)|Ft], = sup
αt,αt+h,...,α(N−1)h
E[E[UT(X
αt,αt+h,...,α(N−1)h;x,t T
)|Ft+h]|Ft], = sup
αt,αt+h,...,α(N−1)h
E[E[UT(X
αt+h,...,α(N−1)h;X αt ;x,t
t+h
,t+h T
)|Ft+h]|Ft], = sup
αt
E[ sup
αt+h,...,α(N−1)h
E[UT(X
αt+h,...,α(N−1)h;X αt ;x,t
t+h
,t+h T
)|Ft+h]|Ft], = sup
αt
E[V(X αt;x,t
t+h , t + h)|Ft].
SLIDE 26
Multi-period Model
By Dynamic Programming Principle, we can derive the value functions backwardly,
V(x, 0)
α∗
⇐ 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.
SLIDE 27
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 = qu
t,t+h ln
- qu
t,t+h
pu
t,t+h
- + qd
t,t+h ln
- qd
t,t+h
pd
t,t+h
- .
The optimal strategy is α∗
t =
ln pu
t,t+h
qu
t,t+h
- − ln
pd
t,t+h
qd
t,t+h
- −
- EQ[(N−1)h
i=t+h hi|Fu t+h] − EQ[(N−1)h i=t+h hi|Fd t+h])
γSt(ξu
t+h − ξd t+h
- .
SLIDE 28
Multi-period Model
SLIDE 29
Future Goals
Derive the mechanism of forward preference. Solve the preference and optimal investment strategy explicitly. Visualize forward preference and optimal investment strategy and compare them with backward case.
SLIDE 30
Reference
[1] Marek Musiela and Thaleia Zariphopoulou (2004): A valuation algorithm for indifference prices in incomplete
- markets. Finance and Stochastics 8, 399–414.