Question

A stock price is currently $60. Over each of the next two three-month periods it is expected to go up by 6% or down by 5%. The risk-free interest rate is 8% per annum with continuous compounding. What is the value of a six-month European call option with a strike price of $61?

Answer 1

Answer:
Value of one year European Call Option = $2.51 [Note 3.1]
Alternatively, Value will be $2.50 [Note 3.2], when calculations are not as accurately done, but acceptable.

Note 1: Basic Data
Strike price = $61
Spot price = $60
Maximum price after one move = $60 x 106% = $63.6
Minimum price after one move = $60 x 95% = $57
Maximum price after 2 moves (end of one year) = $67.42 [being 60 x 1.06 x 1.06]; profit = $6.42 [being 67.42 - 61]

Note 2: Calculation of Probability of Hike
u = Max/Spot = 1.06
d = Min/Spot = 0.95
Probability = (e^0.08x3/12 - 0.95) / (1.06 - 0.95) = (1.0202 - 0.95) / 0.11 = 63.82%
or Probability = (1.02-0.95) / (1.06-0.95) = 63.64%
The first one is more accurate. However, your teacher may allow you to use the easier one. I will work out both ways.
Other methods exist too, such as Risk Neutral Model. However I am following Binomial Model.

Note 3.1 [When probability is taken as 63.82%]: Option Valuation
At the end of 3 months: Value will be ($6.42 x 0.6382) / 1.0202 = $4.02
Now, the value will be ($4.02 x 0.6382) / 1.0202 = $2.51

Note 3.2 [When probability is taken as 63.64%]: Option Valuation
At the end of 3 months: Value will be ($6.42 x 0.6364) / 1.02 = $4.01
Now, the value will be ($4.01 x 0.6364) / 1.02 = $2.50

This is very logical. We are merely discounting the profit from the option at the end of 1 year, to the end of 6 months, and then that is discounted to the present moment. This is what it actually is.

To find e^0.08x3/12, we have two methods. Easy way is 1+(0.08 x 3/12) = 1.02. A more accurate method is, (1.08 + (0.08x0.08)/2)^3/12 = (1.08 + (0.08x0.08)/2)^1/4 = 1.0832^(1/4) = 1.0832^(1/2)x(1/2) = 1.0202

Hope this helps.
Feel free to ask for clarifications.
Consider leaving a thumbs up - that enables us to keep solving.
Good luck!