FRM Part I · FRM Exam Part I · Binomial Trees
A stock is at 50 and will be either 60 or 40 after one period. The per-period risk-free rate is 5% (simple, one period). Using a replicating portfolio of the stock and risk-free borrowing, what is the value of a European call with strike 50 at the start?
The call is worth about 5.95. Buy half a share for 25 and borrow the present value of 20, which is 19.05. The replicating cost is 25 minus 19.05. The risk-neutral approach, with p of 0.625 discounted at 5%, gives the same result.
- A5.95Correct
- B6.25
- C5.00
- D4.76
Explanation
Delta = 10/20 = 0.5. The down-state portfolio must pay 0, so the borrowing repayment is 0.5×40 = 20, and the amount borrowed is 20/1.05 = 19.05. The call value is 0.5×50 - 19.05 = 5.95. Check with the risk-neutral probability: p = (52.5 - 40)/20 = 0.625, and 0.625×10/1.05 = 5.95. The value of 5.00 comes from failing to discount the borrowing.
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