Options Pricing and Arbitrage

This assignment delves into the world of options trading by exploring concepts like arbitrage opportunities, European and American option valuation, and put-call parity. It involves problem-solving related to calculating option values using binomial trees and risk-neutral valuation. Students are also tasked with analyzing the discrepancies between different valuation methods and understanding the implications for early exercise in American options.

c0=value of a European call at time0
p0=value of European put at time 0
s0 =value of stock at time 0
x=excercise price
r =risk free interest rate
T =duration of the option
D=dividend
Problem 1
a)
ct ≥ max ( 0 , st−D−X e−rt )
r =8 % , t=4 months , x=$ 70 , st =$ 75 , D=$ 1.50
Now
st −D− X e−rt=75−1.5−(70∗e
−0.08∗4
12 )
thisgives $ 3.6864
therefore
ct =max (0 , 3.6864)
which is $3.6864
b) Call selling for 43
Lower bound $ 3.6864
To gain arbitrage profit buy the call option at c=$3. Sell the stock short at $75.
Afterwards invest the proceeds 975−3 ¿=$ 72 at the rate of r =8 %.
At the expiration the arbitrage trader must close the short stock position if st > 70,the
trader should buy the stock through her call option.
The payoff to the arbitrage position should be given by
72 ( 1.08 ) + ( st −70 )=7.76
Problem 2
a) p=European put option price , maturity dateis 6 months
p=c+ x e−rt + D−S
¿ 5+50 e−0.05 +1−50
5+ 47.5614+1−50
¿ $ 3.5614
b) p+ s=3.5614+52=55.5614
c + x e−rt=5+ 47.56145=52.5614
an arbitrage opportunity exists with a risk-free profit of $3

Problem 3
a) ST =30 , T =6 months , p=1.08 , down=0.9 , r =5 % , X=32
34.992
D
B 32.4 E 29.16
A
30
27
C
F 24.3
The up with probability P and the down by probability (1− p)
The f_uu=0
F_ud=2.84
F_dd=7.7
Now
p= ert−d
u−d = e0.05∗0.25 −0.9
0.18 =0.6254
f =e−2∗rt [ p2 Fuu + p ( 1− p ) Fud + ( 1−p )2 Fdd ]
¿ e−0.025 [ ( 0.3911∗0 ) + ( 0.4685∗2.84 ) + ( 0.1403∗7.7 ) ]
¿ e−0.025∗2.41085=2.3513
b) Since the portfolios are the same, the terminal payoffs i.e. Fuu , Fud , Fdd are the same to
that of European put option.
D
B
E
A
C
F
Fuu=0 , Fud=2.84 , Fdd =7.7
Now at node B we compute F_u with p=0.6254
Fu=e−rt [ p Fuu + ( 1− p ) Fud ]
¿ e−0.05∗0.25 [ ( 0.6254∗0 )+ ( 0.3746∗2.84 ) ]
¿ 1.0506
In contrast to the European option we have the right to exercise the option here St > X
meaning its not favourable to exercise the option hence Fu=f u

At node C
Fd=e−rt [ p Fud + ( 1− p ) Fdd ]
¿ e−0.05∗0.25 ¿]
¿ 4.6606∗e−0.05∗0.25=4.6027
At node C, we can sell the option at $ 32 which gives a payoff of ( 32−27 ) =$ 5. This is higher
than the $ 4.6027 which is the price of the option at C. hence the value of the American
option is $5
Now at node A
f =e−rt [ p Fu + ( 1− p ) Fd ]
¿ e−0.0125 [ ( 0.6254∗1,0506 )+ ( 0.3746∗5 ) ]=$ 2.4986
c) S=30
up=8 % , 1.08
down=10 % , 0.9
r =5 D
x=32 34.992
B p
32.4 1-p E
p 29.16
A
30 p F24.84
1-p 27
C G
1-p 24.3
f =e−rt [ pfu+ (1− p ) fd ]
p= ert−d
u−d
calculating thevalue of the option at node B from one stop binomial tree
D 34.992 f ¿u=2.992
p
B
1-pE 28.84 f_d= 0
p=( e−0.05∗0.25
1.08−0.9 )
0.08758
0.18 =0.4866

f =e−0.05∗0.25 [ 0.4866∗2.992+ 0.5776∗0 ] =1.4378
Also, at node C
F 24.84 f_u = 0
C
G 24.3 f_d = 0
Since both the F and G are 0 valued f at C is zero.
Finally, at node A
B 32.4 f_u = 1.4378
A
C 27 f_d=0
f =e0.05∗0.25 [ 0.4866∗1.4378 ] =0.6910
Therefore, the value of the option today is $ 0.6910
d) Put-call parity check
C0+k e−rt= p0 +s0
c0=max ( 30−32,0 ) =0
p0=max ( 32−30,0 ) =2
Now 0+32 e−0.05∗0.5 =2+30
Here we get that 31.2099<32
The put call parity does not hold hence an arbitrage opportunity exist
e) Calculation of deltas
European put
A=5 %, B=8 % ,C=10 %
European call
A=5 %, B=8 % ,C=10 %
Problem 4
a) No arbitrage C0=X e−rt =p+s
c=40−35∗e
−0.05∗2
12 =$ 5.29
b) Using neutral valuation
p=max ( x−s)

max (32,400=40)
32∗0.92+40∗1.1
2 =36.72
c) The two approaches don’t give the same outcome as the risk neutral valuation value is
higher
d) The derivative can be exercised early in case of American price options as the
investor have the right to sell the stock at the market favourable price should an
arbitrage opportunity arise
Problem 5
a) In excel
b) In excel
c) In excel (rate used is 5% as the Libor rate)
d) In excel, (the put-call parity does not hold)
e) In excel
f) The prices given in Bloomberg-Excel