Question

Consider an at-the-money European call option with one year left to maturity written on a non-dividend paying stock. Let today’s stock price be 80 kr and strike price be 40 krand the stock volatility be 30%. Furthermore let the risk free interest rate be 6%. Construct a one-year, two-step Binomial tree for the stock and calculate today’s price of the European call.

Answer 1

Current Stock price (S) = 80 kr

Strike Price (X) = 40 kr

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	65.80	0.60	39.48	38.20	0.40	15.28	54.76	0.97	53.17
2	38.20	0.60	22.92	17.80	0.40	7.12	30.04	0.97	29.17

At Node A:

Scenarios	Call price
1	52.00
2	28.00

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	53.17	0.97	51.58
2	29.17	0.97	28.3

Note: As the all the fluctuations leads to increase in spot price ( more than Strike Price ). There are 2 scenarios and hence 2 call prices for each scenario.