In: Finance
Value of a stock is currently at $40. Volatility of that stock
is 30% per year and risk-free interest rate with
continuous compounding is at 5% per year. Suppose you are planning
to value a 3-month European call
option with strike price at $41 using a two-step binomial model.
Answer the following using this
information.
What is the value of the option at present?
S(0) = $40
Vol = 30%, therefore, size(up) = 1 + 0.3 = 1.3 and size(down) = 1 - 0.3 = 0.7
r(f) = 5%
Strike price = X = $41
each step time period = 3/2 = 1.5 month = 0.125 year
Prob(up) = (ert - size(down))/(size(up) - size(down))
Prob(up) = (e(0.05*0.125) - 0.7)/(1.3 - 0.7) = 0.5104 = 51.04%
Prob(down) = 1 - Prob(up) = 1 - 51.04% = 48.96%
At t = 0 | t = 1.5months | t = 3months | ||
52 * 1.3 = $67.6 |
max(67.6 - 41, 0) value of call = $26.6 |
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40 * 1.3 = $52 Value of call = $13.4920 |
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$40 Value of call = $6.8436 |
52 * 0.7 = $36.4 |
max(36.4 - 41, 0) value of call = $0 |
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40 * 0.7 = $28 Value of call = $0 |
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28 * 0.7 = $19.6 |
max(19.6 - 41, 0) value of call = $0 |
value of call (u,u) = 26.6
value of call (u,d) = 0
value of call (d,d) = 0
Therefore,
value of call (u) = ((0.5104 * 26.6) + (0.4896 * 0)) * e-(0.05 * 0.125)
Value of call (u) = 13.5766 * 0.9938 = 13.4920
Value of call (d) = 0
Value of call (t=0) = ((0.5104 * 13.4920) + (0.4896 * 0)) * e-(0.05 * 0.125)
Value of call (t=0) = 6.8863 * 0.9938 = 6.8436
Hence, value of call option at present = $6.84