BEA602 Finance Assignment 1: Options Strategies, Pricing, and Models

This assignment solution for BEA602 covers various aspects of financial options. It begins with analyzing straddle and strangle strategies, including payoff calculations and future value considerations. The solution then delves into option pricing models, comparing the binomial model to the Black-Scholes model, highlighting their applications and advantages. The binomial model is applied to price American call and put options, demonstrating the construction of stock trees and option valuation at different time periods. Further, the Black-Scholes model is employed to price an American call option, taking into account dividends and time to expiration. Finally, the assignment addresses portfolio construction, specifically creating a delta and gamma-neutral portfolio using call options, with detailed calculations to achieve neutrality. The assignment provides a comprehensive overview of options trading and valuation techniques.

Running head: 2019 BEA602 ASSIGNMENT 1 1
2019 BEA602 ASSIGNMENT 1
Name
Institution
Date

2019 BEA602 ASSIGNMENT 1 2
2019 BEA602 Assignment 1
Question 1
a) The aim of a long bottom straddle strategy is to hold or write a call and put on the same
underlying asset for the same maturity and strike price.
I n this case Strike Price=$ 22.50
Call Price=$ 1.75
Put Price=$ 1.75
Total payment for one Long Straddle=1.75+1.75=$ 3.5
Future value of Payment at expiration(8 months=0.66667 years)
¿ 3.5∗( e0.03∗0.66667 ) =3.5∗1.020201=$ 3.57
Price at expiration=S
Payoff on Long Put :
If S >¿=$ 22.50 , Payoff =0
If S < $ 22.50 , Payoff =( 22.50−S)
Payoff on Long Call:
If S <¿=$ 22.50 , Payoff =0
If S > $ 22.50 , Payoff =( S−22.50)
S A B C=A+B-3.57
Price at
Expiration
Payoff Long
Put
Payoff Long
Call
Net Profit from Long
Straddle
$15 $7.50 $0 $3.93

2019 BEA602 ASSIGNMENT 1 3
$16 $6.50 $0 $2.93
$17 $5.50 $0 $1.93
$18 $4.50 $0 $0.93
$19 $3.50 $0 ($0.07)
$20 $2.50 $0 ($1.07)
$21 $1.50 $0 ($2.07)
$22 $0.50 $0 ($3.07)
$22.5 $0 $0 ($3.57)
$23 $0 $0.50 ($3.07)
$24 $0 $1.50 ($2.07)
$25 $0 $2.50 ($1.07)
$26 $0 $3.50 ($0.07)
$27 $0 $4.50 $0.93
$28 $0 $5.50 $1.93
$29 $0 $6.50 $2.93
$30 $0 $7.50 $3.93
b) The aim of a short strangles strategy to hold or write call and put option on the same
underlying assets for the same maturity but at different strike price
¿ this case , Strike Price CALL Option=$ 22.50
Premium received=$ 1.75
Strike Price PUT Option=$ 20

2019 BEA602 ASSIGNMENT 1 4
Premium received=$ 0.75
Total Premium received $ 1.75+$ 0.75=$ 2.50
Future value of Payment at expiration(8 months=0.66667 years)
¿ 2.5∗( e0.03∗0.66667 ) =2.5∗1.020201=$ 2.55
Price at expiration=S
Payoff on Short Put ( Strike Price=$ 20)
If S >¿=$ 20 , Payoff =0
If S < $ 20 , Payoff =(S−20)
Payoff on Short Call (Strike Price=$ 22.50)
If S <¿=$ 22.50 , Payoff =0
If S > $ 22.50 , Payoff =( 22.50−S)
Net Profit at different Expiration price is shownbelow :
S A B C=A+B+2.55
Price at
Expiration
Payoff
Short Put
Strike=20
Payoff Short
Call
Strike=22.5
Net Profit from Short
Strangle
$15 ($5.00) $0 ($2.45)
$16 ($4.00) $0 ($1.45)
$17 ($3.00) $0 ($0.45)
$18 ($2.00) $0 $0.55
$19 ($1.00) $0 $1.55
$20 $0.00 $0 $2.55
$21 $0.00 $0 $2.55
$22 $0.00 $0 $2.55
$22.5 $0.00 $0 $2.55
$23 $0.00 ($0.50) $2.05
$24 $0.00 ($1.50) $1.05

2019 BEA602 ASSIGNMENT 1 5
$25 $0.00 ($2.50) $0.05
$26 $0.00 ($3.50) ($0.95)
$27 $0.00 ($4.50) ($1.95)
$28 $0.00 ($5.50) ($2.95)
$29 $0.00 ($6.50) ($3.95)
$30 $0.00 ($7.50) ($4.95)
Question 2
a) Although both Binomial model and Black Scholes model both are very good model and
provide the accurate figures and used for options pricing, Binomial Model is preferred in
the approximation of option prices due to its flexibility and simplicity of application
(Dastranj & Latifi, 2013). Binomial Model breaks down the time to expiration of option
hence not only simplifying the process of determination but also enabling determination
of value at each node or at the end of the year or period. Moreover, Binomial Model,
unlike Black Scholes Model, allows for determination of prices in both directions (UP

2019 BEA602 ASSIGNMENT 1 6
and DOWN), side by side (Krznaric, 2016). Besides, given that Black Scholes Model
assumes a constant volatility, it fails to recognise large variation in the prices. Contrarily,
Binomial recognises volatility effect on the prices (Krznaric, 2016).
b) The Binomial Model may be applied in the modeling of price of options based on the
underlying price volatility. This is because the model allows control of complexity based
on the requirement. Therefore, the model may be of use of changes or simplification of
the model is required to fit a case. Besides, a simple one-step binomial model could be
used to compute the prices of call options (Dastranj & Latifi, 2013).
Question 3
a) ¿ Up Move Factor :
U =er √ t=eσ √ t=e0.25∗ √ 60
365 =1.11
¿ down Move Factor :
D= 1
U = 1
1.11 =0.9
Probability of Up Move , π= e
rt
n −d
U −D = e
0.06∗60
365 −0.9
1.11−0.9 =1.01−0.9
0.21 = 0.11
0.21 =0.52
Probaility of Down Move=1− p=1−0.52=0.48
For put option,
P=π∗p+¿+ ( 1−π ) ∗p−¿
1 +r ¿
¿
Put Option Payoff at Day 120=max ( X−u∗uS ,0 )

2019 BEA602 ASSIGNMENT 1 7
( T h e formula is Adjusted for ot h er sections ) .
Given that the stock price is greater than the put exercise price at all levels, the value of the put
option is 0 at all level. Hence the following is the tree for the put.
C++¿=110.89−65=45.89 ¿
C+−¿∨C−+ ¿=89.91−65=$ 24.91¿ ¿
C−−¿=72.90−65=$ 7.90 ¿
C
+¿at Year 1= 0.52∗45.89+0.48∗24.91
1+ 0.06∗60
365
= 35.8196
1.009863 =$ 35.47 ¿
C−¿= 0.52∗24.91+ 0.48∗7.9
1.009863 = 16.7452
1.009863 =$ 16.58 ¿
C=π∗C
+¿+ (1−π )∗C−¿
1+r = 0.52∗35.47+0.48∗16.58
1 + 0.06∗60
365
= 26.4028
1.009863 =$ 26.14 ¿
¿
Using this information, the following is the tree for call option:

2019 BEA602 ASSIGNMENT 1 8
b) C=π∗C
+¿+ (1−π )∗C−¿
1+r = 0.52∗35.47+0.48∗16.58
1 + 0.06∗60
365
= 26.4028
1.009863 =$ 26.14 ¿
¿
Hence, the price of the call option is $26.14.
Given that the stock price is greater than the put exercise price at all levels, the price of
put option is $0.
c) The call option should be exercised at 60 days because in addition to the value of stock
being high, dividend will be paid out.
Question 4
To determine the value of the call option via Black-Scholes model, we adjust the current
stock price of $120 downward by the PV of the dividends to be acquired before expiration of the
option. Since the dividend of $5 will be paid in 40 days,
D1 X e−r (T −t )=5∗e−0.06∗( 40
365 )=$ 4.97
S' =$ 120−$ 4.97=$ 115.03

2019 BEA602 ASSIGNMENT 1 9
For option that expires in 100 days:
d1=ln ( $ 115.03
150 )+ ¿ ¿
d2=−1.50722−0.3∗ √ 100
365 =−1.664247177
N ( d1 ) =0.065877
N ( d2 ) =0.048031
c=115.03∗0.065877−150∗e−( 0.06∗100
365 )
∗0.048031=$ 0.49
Particulars: 100 days
Std Deviation 0.3
Time to Expiration
Yrs 0.273973
Risk Free Rate 0.06
Stock Price 115.03
Strike Price 150
Dividend Yield
Output
d1 -1.50722
d2 -1.66425
N(d1) 0.065877
N(d2) 0.048031
c $0.49

2019 BEA602 ASSIGNMENT 1 10
For the option that expires in 40 days, the strike price is decreased by the amount of the
dividend that will be received in 40 days, just before expiration. Accordingly, the current strike
price will be reduced to $145. That is,
Strike Price Reduction=Initial Strike Price−Divident =150−5=$ 145
X' =150−5=$ 145
d1=ln ( $ 115.03
145 )+ ¿ ¿
d2=−2.21557−0.3∗
√ 40
365 =−2.31488
N ( d1 ) =0.013361
N ( d2 )=0.01031
c=115.03∗0.013361−150∗e− ( 0.06∗40
365 )∗0.01031=$ 0.05
The following table summarises the findings:
Particulars: 100 days 40 Days
Std Deviation 0.3 0.3
Time to Expiration
Yrs 0.273973 0.109589
Risk Free Rate 0.06 0.06

2019 BEA602 ASSIGNMENT 1 11
Stock Price $ 115.03 $ 115.03
Strike Price $ 150.00 $ 145.00
Dividend Yield $ - $ -
Output
d1 -1.50722 -2.21557
d2 -1.66425 -2.31488
N(d1) 0.065877 0.013361
N(d2) 0.048031 0.01031
c
$
0.49 $ 0.05
Given that the value of the American call option is the largest between the two values, the
price of the American call option using the Black-Scholes Model is $0.49.
Question 5
T h e Gammaof t h e underlying stock is always 0 w h ile Delatais always 1.
Assuming t h at t h e number of call options for x=75 is a
w hile t h e number of call options for x=80 isb ,if neutral portfolio is ¿ be gamma∧delta neutral ,
t h e following equations hold :
( Quantity of Stock )∗( Delta of Stock )+ a∗ ( Delta of Option 1 )+b∗( Delta of Option 2 )=0
( Quantity of Stock )∗( Gamma of Stock )+ a∗ (Gamma f Option 1 ) +b∗( Gamma of Option 2 )=0

2019 BEA602 ASSIGNMENT 1 12
From the information given,
0.6674 a+0.574 b+5000=0
0.6674 a+0.574 b=−5000 … … … … … … … … … … …( i)
0.0176 a+ 0.019 b=0 … … … … … … … … … … … … … …(ii)
Solving equation (i) and (ii),
b=29083.69 ≈ 29084
a=−32505.30 ≈ 32505(absolute value)
Therefore, to construct a portfolio that is DELTA and GAMMA neutral using the call options,
5000 shares of ABC as follows:
29084 calls having a strike price of 75 and 32505 calls having a stroke price of 80.
Accordingly, the portfolio can be summarized as shown below:
Particulars Call Call Stock
X 75 80 77
Delta 0.6674 0.574 1
Gamma 0.0176 0.019 0
Quantity 29084 32505 5000

2019 BEA602 ASSIGNMENT 1 13
References
BIBLIOGRAPHY Dastranj, E., & Latifi, R. (2013). A comparison of option pricing models. Retrieved from
World Scientific: https://www.worldscientific.com/doi/abs/10.1142/S2424786317500244
Krznaric, M. J. (2016). Comparison of Option Price from Black-ScholesModel to Actual Values.
Ohio: The University of Akron.