Option Pricing Strategies And Binomial Model Essay.

Long (or bottom) straddle strategy

Current stock price S is $22. Time to maturity T is six months. Continuously compounded, risk-free interest rate r is 5 percent per annum. European options
prices are given in the following table:
Strike Price Call Price Put Price
K1=$17.50 $5.00 $0.05
K2=$20.00 $3.00 $0.75
K3=$22.50 $1.75 $1.75
K4=$25.00 $0.75 $3.50
(a) What is the aim of a long (or bottom) straddle strategy? Create a long straddle by buying a call and put with strike price K3=$22.50 
(b) What is the aim of a short (or top) strangle strategy? Create a short strangle by writing a call with strike price K3=$22.50 and a put with strike price K2=$20.

Question 2 
(a) Why is the binomial model a useful technique for approximating options prices from the Black–Scholes model?

(b) Describe some applications and uses of this model. 

Question 3 
Consider the binomial model for an American call and put on a stock whose price is $60. The exercise price for both the put and the call is $45. The standard deviation of the stock returns is 30 percent per annum, and the risk-free rate is 5 percent per annum. The options expire in 90 days. The stock will pay a dividend equal to 3 percent of its value in 50 days.
(a) Draw the three-period stock tree and the corresponding trees for the call and the put. 
(b) Compute the price of these options using the three-period trees. 
(c) Explain when, if ever, each option should be exercised. 

Question 4 
Consider a stock with a price of $120 and a standard deviation of 20 percent. The stock will pay a dividend of $5 in 40 days and a second dividend of $5 and 130 days. The current risk-free rate is 5 percent per annum. An American call on this stock has an exercise price of $150 and expires in 100 days.What is the price of the American call? Show all calculations.

Question 5 
ABC is currently trading at $78 per share. Your previous calculation of the historical volatility for ABC indicated an annual standard deviation of return of 27 percent, but examining the implied volatility of several ABC options reveals an increase in annual volatility to 32 percent.There are two traded options series that expire in 245 days as show in the
following table:
X = 75 X = 80
Call Put Call Put
DELTA 0.6674 -0.3326 0.574 -0.426
GAMMA 0.0176 0.0176 0.019 0.019
The options have $75 and $80 strike prices respectively. The current 245-day riskfree interest rate is 4.75 percent per annum, and you hold 2,000 shares of ABC.Construct a portfolio that is DELTA - and GAMMA- neutral using the call options written on ABC. Show all calculations.

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 Lali?, 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.

Short (or top) strangle strategy

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 Lali?, 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.