Question

A stock price is currently $200. Over each of the next two six-month periods it is expected to go up by 10% or down by 10%. The risk-free interest rate is 6% per annum with continuous compounding. What is the value of a one-year European call option with a strike price of $200?

Answer 1

Answer:
Value of one year European Call Option = $16.82 [Note 3.1]
Alternatively, Value will be $16.72 [Note 3.2], when calculations are not as accurately done, but acceptable.

Note 1: Basic Data
Strike price = $200
Spot price = $200
Maximum price after one move = $200 x 110% = $220
Minimum price after one move = $200 x 90% = $180
Maximum price after 2 moves (end of one year) = $242; profit = $42

Note 2: Calculation of Probability of Hike
u = Max/Spot = 1.1
d = Min/Spot = 0.9
Probability = (e^0.06x0.5 - 0.9) / (1.1 - 0.9) = (1.0304 - 0.9) / 0.2 = 65.22%
or Probability = (1.03-0.9) / (1.1-0.9) = 65%
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 65.22%]: Option Valuation
At the end of 6 months: Value will be ($42 x 0.6522) / 1.0304 = $26.58
Now, the value will be ($26.58 x 0.6522) / 1.0304 = $16.82

Note 3.2 [When probability is taken as 65%]: Option Valuation
At the end of 6 months: Value will be ($42 x 0.65) / 1.03 = $26.50
Now, the value will be ($26.50 x 0.65) / 1.03 = $16.72

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^0.06x6/12, we have two methods. Easy way is 1+(0.06 x 6/12) = 1.03. A more accurate method is, (1.06 + (0.06x0.06)/2)^6/12 = (1.06 + (0.06x0.06)/2)^1/2 = 1.0618^1/2 = 1.0304. There are other methods too.

Hope this helps.
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