Math 3589 Spring 2016
Introduction to Financial Mathematics
Homework Assignment #8 Solutions
Exercise 16. Consider the two period binomial model, with the stock price
at time t = 0, S0 = 4, the “up factor” u = 2, “down factor” d = 1/2, and
risk free interest rate r = 1/4 so that p̃ = 1/2. Assume in each period the
probability P[H] = 3/4. Solve the two-period investors problem for ∆i , i = 0, 1,
if U (x) = −2e−x and X0 = 10.
Solution. Let
9 3 3 1
−2e−x4 + −2e−xx + −2e−x2 + −2e−x1
f (x1 , x2 , x3 , x4 ) =
16 16 16 16
and
16 1 1 1 1
g(x1 , x2 , x3 , x4 ) = x4 + x3 + x2 + x1 − 10 = 0.
25 4 4 4 4
For x̄ = (x1 , x2 , x3 , x4 ), define the Lagrangian function
L(x̄, λ) = f (x̄) − λg(x̄)
−18 −x4 6 −x3 6 −x2 2 −x1 16
= e − e − e − e −λ (x4 + x3 + x2 + x1 ) − 10 .
16 16 16 16 100
And we find the 5 equations
∂ 2 −x1 16
∂x1
L(x̄, λ) = 16 e − 100 λ=0
∂ 6 −x2 16
∂x2 L(x̄, λ) = 16 e − 100 λ = 0
∂ 6 −x3 16
∂x3
L(x̄, λ) = 16 e − 100 λ=0 (0.1)
∂
L(x̄, λ) = 18 e−x4 − 100
16
λ=0
∂x4 16
∂ L(x̄, λ) = 16 (x + x + x + x ) − 10 = 0.
∂λ 100 4 3 2 1
From here we see that x2 = x3 , and setting the first and second equal, we see
2 −x1 6
e = e−x2 =⇒ e−x1 = 3e−x2 =⇒ −x1 = ln(3) − x2 =⇒ x1 = x2 − ln(3).
16 16
Similarly,
18 −x4 6
e = e−x2 =⇒ 3e−x4 = e−x2 =⇒ x4 = x2 + ln(3).
16 16
Plugging these into the last equation, we get
16
(4x2 ) = 10 =⇒ x2 = 125/8.
100
, Solving for x4 , we find x4 ≈ 16.7236 and x1 ≈ 14.526.
Now we find
ln(3) ln(3)
∆1 [H] = ≈ 0.091, ∆1 [T ] = ≈ 0.366
12 3
and
X1 [H] − X1 [T ]
∆0 =
S1 [H] − S1 [T ]
4 1 125
+ ln(3) + 21 · 125
− 45
1 125 1 125
5 2 8 8 2
· 8
+ 2 8
− ln(3)
=
8−2
≈ 0.1464
Exercise 17. In the previous problem, what was the optimal percentage of
wealth invested in the stock each period?
Solution. The optimal percentage of the wealth invested in the stock in each
period is
∆0 S0 ∆1 (H)S1 (H) ∆1 (T )S1 (T )
(t = 0) : (t = 1, ω1 = H) : (t = 1, ω1 = T ) : .
X0 X1 (H) X1 (T )
We have
X1 [H] ≈ 12.939, X1 [T ] ≈ 12.0605, S1 [H] = 8, S1 [T ] = 2.
Therefore, the optimal percentages are
∆0 S0 0.1464 · 4
(t = 0) : ≈ ≈ 0.05856
X0 10
∆1 (H)S1 (H)
(t = 1, ω1 = H) : ≈ 0.05626
X1 (H)
∆1 (T )S1 (T )
(t = 1, ω1 = T ) : ≈ 0.06069.
X1 (T )
We see that, unlike the CRRA utility functions, the optimal investment
choices are dependent upon our wealth.
Exercise 19. Consider the N -period optimal investment problem with U (x) =
ln(x). Let ζ(ω) be the state price density and ζn , n = 0, 1, . . . , N be the state
price density process.
a. Show that for each sequence of coin tosses ω = ωm the optimal wealth will be
X0
XN (ωm ) = ζ(ωm)
.
b. Show that the optimal portfolio process is Xn = Xζn0 , n = 0, 1, . . . , N .
Solution a. We will use the formula
2N
X Z(ωm ) λZ(ωm )
N
I N
P(ωm ) − X0 = 0
m=1
(1 + r) (1 + r)
to solve for λ. First notice, if U = ln(x), U 0 (x) = x1 and therefore, I(x) = 1
x
,
thus, we have
2N
X Z(ωm ) 1
N
P(ωm ) = X0
m=1
(1 + r) λZ(ωm )
(1+r)N
2
Introduction to Financial Mathematics
Homework Assignment #8 Solutions
Exercise 16. Consider the two period binomial model, with the stock price
at time t = 0, S0 = 4, the “up factor” u = 2, “down factor” d = 1/2, and
risk free interest rate r = 1/4 so that p̃ = 1/2. Assume in each period the
probability P[H] = 3/4. Solve the two-period investors problem for ∆i , i = 0, 1,
if U (x) = −2e−x and X0 = 10.
Solution. Let
9 3 3 1
−2e−x4 + −2e−xx + −2e−x2 + −2e−x1
f (x1 , x2 , x3 , x4 ) =
16 16 16 16
and
16 1 1 1 1
g(x1 , x2 , x3 , x4 ) = x4 + x3 + x2 + x1 − 10 = 0.
25 4 4 4 4
For x̄ = (x1 , x2 , x3 , x4 ), define the Lagrangian function
L(x̄, λ) = f (x̄) − λg(x̄)
−18 −x4 6 −x3 6 −x2 2 −x1 16
= e − e − e − e −λ (x4 + x3 + x2 + x1 ) − 10 .
16 16 16 16 100
And we find the 5 equations
∂ 2 −x1 16
∂x1
L(x̄, λ) = 16 e − 100 λ=0
∂ 6 −x2 16
∂x2 L(x̄, λ) = 16 e − 100 λ = 0
∂ 6 −x3 16
∂x3
L(x̄, λ) = 16 e − 100 λ=0 (0.1)
∂
L(x̄, λ) = 18 e−x4 − 100
16
λ=0
∂x4 16
∂ L(x̄, λ) = 16 (x + x + x + x ) − 10 = 0.
∂λ 100 4 3 2 1
From here we see that x2 = x3 , and setting the first and second equal, we see
2 −x1 6
e = e−x2 =⇒ e−x1 = 3e−x2 =⇒ −x1 = ln(3) − x2 =⇒ x1 = x2 − ln(3).
16 16
Similarly,
18 −x4 6
e = e−x2 =⇒ 3e−x4 = e−x2 =⇒ x4 = x2 + ln(3).
16 16
Plugging these into the last equation, we get
16
(4x2 ) = 10 =⇒ x2 = 125/8.
100
, Solving for x4 , we find x4 ≈ 16.7236 and x1 ≈ 14.526.
Now we find
ln(3) ln(3)
∆1 [H] = ≈ 0.091, ∆1 [T ] = ≈ 0.366
12 3
and
X1 [H] − X1 [T ]
∆0 =
S1 [H] − S1 [T ]
4 1 125
+ ln(3) + 21 · 125
− 45
1 125 1 125
5 2 8 8 2
· 8
+ 2 8
− ln(3)
=
8−2
≈ 0.1464
Exercise 17. In the previous problem, what was the optimal percentage of
wealth invested in the stock each period?
Solution. The optimal percentage of the wealth invested in the stock in each
period is
∆0 S0 ∆1 (H)S1 (H) ∆1 (T )S1 (T )
(t = 0) : (t = 1, ω1 = H) : (t = 1, ω1 = T ) : .
X0 X1 (H) X1 (T )
We have
X1 [H] ≈ 12.939, X1 [T ] ≈ 12.0605, S1 [H] = 8, S1 [T ] = 2.
Therefore, the optimal percentages are
∆0 S0 0.1464 · 4
(t = 0) : ≈ ≈ 0.05856
X0 10
∆1 (H)S1 (H)
(t = 1, ω1 = H) : ≈ 0.05626
X1 (H)
∆1 (T )S1 (T )
(t = 1, ω1 = T ) : ≈ 0.06069.
X1 (T )
We see that, unlike the CRRA utility functions, the optimal investment
choices are dependent upon our wealth.
Exercise 19. Consider the N -period optimal investment problem with U (x) =
ln(x). Let ζ(ω) be the state price density and ζn , n = 0, 1, . . . , N be the state
price density process.
a. Show that for each sequence of coin tosses ω = ωm the optimal wealth will be
X0
XN (ωm ) = ζ(ωm)
.
b. Show that the optimal portfolio process is Xn = Xζn0 , n = 0, 1, . . . , N .
Solution a. We will use the formula
2N
X Z(ωm ) λZ(ωm )
N
I N
P(ωm ) − X0 = 0
m=1
(1 + r) (1 + r)
to solve for λ. First notice, if U = ln(x), U 0 (x) = x1 and therefore, I(x) = 1
x
,
thus, we have
2N
X Z(ωm ) 1
N
P(ωm ) = X0
m=1
(1 + r) λZ(ωm )
(1+r)N
2
