BEA602 Assignment 1: Derivatives - Option Strategies and Valuation

This assignment solution for BEA602 explores various option strategies, including long straddles and short strangles, analyzing their profit and risk profiles. It delves into the binomial model for option pricing, discussing its applications and uses, particularly for American options. The assignment also includes a three-period stock tree analysis and calculates the fair price of an American call option using the Black-Scholes formula. Furthermore, it addresses delta hedging strategies, demonstrating how to adjust a portfolio to manage risk. This comprehensive analysis provides a strong understanding of options and their valuation.

2018 BEA602 Assignment 1

TABLE OF CONTENTS
Question 1........................................................................................................................................3
Long (or bottom) straddle strategy..............................................................................................3
Short (or top) strangle strategy....................................................................................................4
Question 2........................................................................................................................................4
Binomial model...........................................................................................................................4
Application and uses of Binomial model.....................................................................................5
Question 3........................................................................................................................................5
Part A...........................................................................................................................................5
Part B...........................................................................................................................................5
Part C...........................................................................................................................................6
Question 4........................................................................................................................................6
Calculation of fair price of the American call option by applying the Black-Scholes formula. .6
Question 5........................................................................................................................................8

QUESTION 1
Long (or bottom) straddle strategy
A long straddle is a simplest market strategy in an option strategy in which the trader buys the
call and put option simultaneously at the same strike price, and same expiration date. The strike
price is just near to at the money. The main objective of this strategy is to earn the profit from the
underlying asset from a movement in any direction that is either high or low because the trader
buys both the option call as well as put. Therefore either high or the low movement in the price
of the underlying asset will give the profit to the seller (Chu et al., 2017). In this strategy, the
maximum risk to the buyer is only up to the cost of purchasing the call and the put option. When
the price of the underlying increase then the trader will implement the call option and if the price
of the underlying asset decreased then the trader would implement the put option.
In the present at the strike price 22.5 for creating the long straddle trader purchase both the
option that is call option and the put option simultaneously. The current price of the underlying is
$ 22. Since the current price goes down from the strike price, therefore trader will exercise the
put option. Calculation of the profit by exercising the option is given below-
Call option premium $ 1.75
Put option premium $ 1.75
Total premium paid by the trader $ 3.5
Profit = strike price of put option- the price of an underlying asset- net premium paid
= 22.5-22-3.5
= -3
There is the loss to the trader because the movement of the price of the underlying was very less,
therefore, trader unable to cover the premium amount paid for buying the option.

Short (or top) strangle strategy
Another name of the short strangle strategy is to sell strangle. As the name suggests in this option
strategy, the trader sells the call and put option simultaneously at the same underlying stock and
the expiration time period. The trader adopts this strategy at that time when the movement of the
price very little of the underlying stock (Gordiaková and Lalić, 2014). The main objective of this
strategy is to earn the profit by way of the selling the call and put option, since in this strategy
trader thinks that the price movement either upward or the downward very less, therefore the
buyer of the option not exercise the option. In this strategy, there is a chance of the limited gain
to the trader, which means a trader can earn the maximum profit only up to the amount of the
premium received by selling the call option and the put option. However there is an unlimited
risk to the trader as is the price of the underlying move in a high or low direction significantly,
and then the buyer will exercise the option definitely.
At the strike price $ 22.5 selling the call, and at the strike price, 20 selling the put, the total
amount of premium received by the trader for selling the call and put option is-
Premium on selling the call option $ 1.75
The premium on selling the put option $ .75
Total premium received $ 2.5
The current price of the stock is $ 22, since the buyer of the call option at the strike price $ 22.5
will not exercise the option and the buyer of the put option at the strike price $ 20 also not
exercise the option.
Therefore the total gain to the trader is $ 2.5, which is the total premium received by him.
QUESTION 2
Binomial model
The Binomial option pricing method assists in valuation method through the numerical method,
in which the algorithm is used for analyzing the mathematical calculation. This method uses a

mathematical procedure, permitting for the requirement of the nodes, or the gap between the
valuation date and the expiration date of the option (Kim et al., 2016). Further, this method also
decreases the possibility of the arbitrage because this model reduces the volatility in price.
Therefore there is no chance of the purchase and sale of the asset together for generating the
profit due to an imbalance in the price. Therefore this model is a useful technique for
approximating the option price from the Black-Scholes model.
Application and uses of the Binomial model
Since the binomial pricing model applies the various factors for determining the fair value of the
option, therefore it is used for the valuation of the American option, which is exercisable at any
point of the time in a given time period (Pregibon and Hastie, 2017). Further, it is also applied to
those options which are exercisable at a specific occasion. Since this method is easy, therefore it
can be adopted in the computer software. It is also implemented in the application of the
computer such as a spreadsheet so that data can be analyzed and compared in a tabular format.
QUESTION 3
Part A
Three-period stock tree of
131.82
101.4 70.98
78
54.6 70.98
60 38.22
70.98
54.6 38.22
42
29.4 38.22
20.58
Part B
Tree price The strike price of a call or put option
Value of call
(tree price - strike price)
P1 131.82 45 86.82

P2 70.98 45 25.98
P3 70.98 45 25.98
P4 38.22 45 -6.78
P5 70.98 45 25.98
P6 38.22 45 -6.78
P7 38.22 45 -6.78
P8 20.58 45 -24.42
Part C
Evaluation of option
Tree price Value of option Probability Total value
P1 131.82 86.82 0.125 10.8525
P2 70.98 25.98 0.125 3.2475
P3 70.98 25.98 0.125 3.2475
P4 38.22 -6.78 0.125 -0.8475
P5 70.98 25.98 0.125 3.2475
P6 38.22 -6.78 0.125 -0.8475
P7 38.22 -6.78 0.125 -0.8475
P8 20.58 -24.42 0.125 -3.0525
Total value 15
Since the overall value of options analysis is positive, therefore option should be exercised.
QUESTION 4
Calculation of fair price of the American call option by applying the Black-Scholes formula
The formulas for d1 and d2 are:

Here the
So= underlying price
X= strike price
σ= volatility
r= continuously compounded risk-free interest rate (% p.a)
q= continuously compounded dividend yield (% p.a)
t= time to expiration
Here stock price= $ 120
Volatility= 20%
Risk fee interest rate= 5%
Time to expiration= 100days
Strike price= $ 150
Dividend= $ 10 per year

By applying the above formula, the fair price of the American call option is $ -0.0002.
QUESTION 5
Delta (Call) = Change in value of call / Change in value of strike price
=22.96-0/102.96-53.04
=.4595
Therefore for 46% of 2000 share call option will be taken delta and remaining will be invested in
Gama.
=78$ if increases by 32% $102.96 or if decreases by 32% 53.04
If increases than the value of the call
Increased price – Strike price
=$102.96-$80
=$22.96
If decreases than the value of the call
Increased price – Strike price
=$53.04-$75
=$0

REFERENCES
Chu, C.P., Hsiao, Y.L., Cho, C.M. and Chen, Y.C., 2017. Applying compound options in
logistics enterprise risk management. Journal of Industrial and Production Engineering, 34(2),
pp.135-146.
Gordiaková, Z. and Lalić, M., 2014. Long Strangle Strategy Using Barrier Options and its
Application in Hedging Against a Price Increase. Procedia Economics and Finance, 15,
pp.1438-1446.
Kim, Y.S., Stoyanov, S., Rachev, S. and Fabozzi, F., 2016. Multi-purpose binomial model:
Fitting all moments to the underlying geometric Brownian motion. Economics Letters, 145,
pp.225-229.
Pregibon, D. and Hastie, T.J., 2017. Generalized linear models. In Statistical Models in S (pp.
195-247). Routledge.