Basic Financial Math

all questions are in the financal math PDF. We use the book material is ” Stochastic Calculus For finance 1, the Binomial Asset pricing model” . The book I have gave in the second PDF
MTH 558 Fall 2018 Take-Home Midterm
Please read the following instructions carefully before you start:
• You are free to use your notes, homework, textbook, or any related books. However, you
can NOT discuss the problems with other people, and you have to do it INDEPENDENTLY.
Any similar solution will result a ZERO score.
• Due time is: Oct/29/2013, 5pm. Any late one will lose 50% of the total score.
• If without further specification, all problems are considered under some risk-neutral probability
{p, ˜ q˜}, and some actual probability {p, q}.
• GOOD LUCK!
Problem 1 (30 pts)
A chooser option is an option, which gives the option buyer a fixed period to decide whether the
derivative will be a European call or put option.
In particular, consider a binomial tree with S0 = 100, u = 120%, d = 80% and r = 10%.
Suppose a chooser option with strike price K = 105 has a specified decision time n = 2, when the
buyer has to decide whether this derivative will be a European call or put option. At the expiration
time N = 3 the option expires. If the buyer has chosen that it is a call option, then the payoff at
time 3 is (S3 − K)
+; otherwise, if the buyer has chosen that it is a put option, then the payoff at
time 3 is (K − S3)
+;
1. Determine the buyer’s decision at time n = 2. (Hint: the decision depends on ω1ω2.)
2. Price this chooser option at time 0, i.e., calculate V0.
3. Consider a European call option with the same strike price K = 105, and expiration time 3.
Price this call option at time 0, i.e., calculate V
C
0
.
4. Consider a European put option with the strike price K
1+r
, and expiration time 2. Price this
put option at time 0, i.e., calculate V
P
0
.
5. Compare V0 with V
C
0 + V
P
0
.
Problem 2. (30 pts)
Given a 3-step binomial tree with S0 = 20, u = 125%, d = 75% and r = 5%.
1. Calculate V
EC
0
, the price a European call option with strike price K = 22; and V
AC
0
, the
price of an American call option with the same strike price K = 22. Compare the two prices.
1
2. Calculate V
EP
0
, the price a European put option with strike price K = 22; and V
AP
0
, the
price of an American put option with the same strike price K = 22. Compare the two prices.
Problem 3. (20 pts)
Consider a 4-step binomial tree in a financial market with S0 = 100, u = 120%, d = 80% and
r = 10%. (The same binomial tree as in problem 1.) Suppose the actual probability is given as
p = P(H) = 1/2, q = 1/2. Define Y =
S0 + S1 + S2 + S3
4
. Verify the equation (3.2.5) in Lemma
3.2.6 in the textbook, for n = 2, by direct calculation.
Problem 4. (20 pts)
A process {Yn}
N
n=0 is called predictable if Yn only depends on (ω1, · · · , ωn−1) for all n. Particularly,
Y0 and Y1 should be constants. Suppose {Xn}
N
n=0 is a martingale under probability {p, q}, and
{Yn}
N
n=0 is a predictable process. Show that (Xn
k=1
(Xk − Xk−1)Yk
)N
n=1
is also a martingale under
{p, q}.
2