5030
Quantitative MAFS Modeling of Derivatives Securities
Solution to Homework Three
Course Instructor: Prof. Y.K. Kwok
1. F is generated by the partition P = {{3, 2}, {1, 1}, {2, 3}}.
(i) Since {2, 3} P and X (2) = 4 = X (3) = 9, X is not F -measurable.
(ii) Since {2, 3} P and X (2) = 2 = X (3) = 3, X is not F -measurable.
Dene the random variable X ( ) = max(, 3). Now, X ( ) = 3 for all , hence
X is F -measurable.
2. (a) Suppose F is generated by a partition P . It suces to show that this property
is valid for every B P . Consider
E [IB E [X |F ]] =
E [ X | B ] P ( )
B
= E [ X | B ] P (B )
=
X ( )(P ( )/P (B ))P (B )
B
=
X ( )P ( )
B
= E [XIB ].
J
E [X |Bj ]1Bj , and consider
(b) Recall E [X |F ] =
j =1
E [max(X1 , , Xn )|F ]
J
E [max(X1 , , Xn )|Bj ]1Bj
=
j =1
J
Kj
max(X1 (k,j ), , Xn (k,j ))P (k,j )1Bj
=
j =1 k=1
while
max(E [X1 |F ], , E [Xn |F ])
J
= max
Kj
J
Kj
X1 (k,j )P (k,j )1Bj , ,
j =1 k=1
j =1 k=1
1
Xn (k,j )P (k,j )1Bj .
It is obviously that the maximum value among the various sums of X (k,j ),
= 1, , n, cannot be greater than the value obtained by taking the maximum
value among X1 (k,j ), , Xn (k,j ) and performing the summation afterward.
Hence, we obtain the desired The result.
3. property E [Xt+1 Xt |Ft] = 0, t = 0, 1, . . . , T 1 does imply that X is a
martingale because
T 1
E [XT |Ft] =
E [Xs+1 Xs |Ft ] + E [Xt |Ft]
s=t
T 1
=
E [E [Xs+1 Xs |Fs ]|Ft] + Xt
s=t
= Xt ,
t = 0, 1, . . . , T 1.
4. Note that Nk+1 Nk is independent of Fk since the successive binomial trials are
independent and p = probability of success in the (k + 1)th trial, we have
E [Nk+1 (k + 1)p (Nk kp)|Fk ] = E [Nk+1 Nk p|Fk ]
= E [Nk+1 Nk p]
= 0.
Then from Problem 3, we deduce that Yk is a martingale.
5. Consider a portfolio consisting of 4 units of money market account with interest
rate r = 0.3 and shorting one unit of asset, then we have
V (0) = 4 S (0) = 0, V (2; i ) = 4(1 + r)2 S (2; i ) 0.76, i = 1, 2, 3, 4.
Hence this is an arbitrage opportunity.
6. We dene k to be the largest non-negative integer such that uk dnk S X , that is,
ln X
k lnSdn . It is seen that
u
d
X uj dnj S when j k
0
when j > k .
max(X uj dnj S, 0) =
The price formula for a European put option with terminal payo max(X S, 0)
for the n-period binomial model is given by
p=
n
j =0
n
Cj pj (1 p)nj max(X uj dnj S, 0)
Rn
k
n
k
n
Cj pj (1
= XR
p)
nj
j =0
where p =
n
Cj pj (1 p)nj
S
j =0
up
.
R
2
uj dnj
,
Rn