Question

Develop a model and a chart to show how the value of a European QQQ call option varies with the price of the underlying stock. Make the chart dynamic by allowing the user to change all of the other variables that affect the price of the option. On the same graph, draw the option price and option payoff versus the stock price. What do you observe on this graph? Please, explain.

Answer 1

The relation in option price and the underlying asset can be depicted through Binomial Pricing Method.

Current Stock price (S) = $ 80

Strike Price (X) = $ 70

Premium = $2

Risk free rate = 6%

Change % = 30% (Both rise and decline)

Calculation of Probability: (r-d) / (u-d)

r = interest received = 1 + 6%/2 = 1.03

u = increase percentage = 1.15

d = decline percentage = 0.85

Probability of increase in the price = (1.03 - 0.85) / (1.15 - 0.85)

= 0.6

Probability of decrease in the price = 1- 0.6 = 0.4

	Period 1 (Node A)	Period 2 (Node B)
		105.8
	92
80		78.2
	68
		57.8

At Node B:

Scenarios	Increase	Probabilty	Probable Increase	Decrease	Probabilty	Probable Decrease	Total	DCF @ 3%	Call Price
1	35.80	0.60	21.48	8.20	0.40	3.28	24.76	0.97	24
2	8.20	0.60	4.92	0	0.40	0	4.92	0.97	4.77

At Node A:

Scenarios	Call price
1	22
2	0

It can be seen that its better to exercise at Node B i.e. after 1 year.

PV of Call option:

Scenarios	Call Price after 6 months	DCF @ 3%	PV of Call Price
1	24.00	0.97	23.28
2	4.77	0.97	4.63

It can be seen that the change in the prices of the call option, so much that in the 2nd scenario the price is just $ 4.63 as compared to scenario 1, where it is $ 23.28.

Comparison in Stock Price and Pay-off

Holder of the call option will only exercise the option when the S>X. In the above example,

Stock Price	Strike Price	Payoff
105.8	70	35.8
78.2	70	8.2
57.8	70	0

Note: As the answering mechanism doesn't allow me to upload active graphs and charts, I am providing detailed textual working and calculations for thorough understanding. Please let me know if any further information is required.