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IAI Actuarial Core Principles · Economic Modelling · Binomial option-pricing model

Share price S0 = Rs 100 moves in one period to Rs 125 or Rs 80. The risk-free rate is 5% per period. A European call has strike Rs 100. An arbitrageur observes the call trading at Rs 14. The no-arbitrage call price is closest to which value, and what action exploits the mispricing?

The fair call price is about Rs 13.2, below the market price of Rs 14, so the call is overpriced. The arbitrageur sells the call and buys the replicating portfolio of shares and borrowing, locking in a riskless profit. The first option gives that action.

  1. ARs 11.9; sell the call and buy the replicating portfolioCorrect
  2. BRs 11.9; buy the call and short the replicating portfolio
  3. CRs 14.3; buy the call and short the replicating portfolio
  4. DRs 17.9; sell the call and buy the replicating portfolio
  5. Rs 8.3; buy the call and short the replicating portfolio

Explanation

Payoffs: up 25, down 0. q = (105 - 80)/(125 - 80) = 25/45 = 0.5556. Price = 0.5556 × 25 / 1.05 = 13.89/1.05 = 13.23. Recheck: 25 × 0.5556 = 13.89; /1.05 = 13.23. This is below the market 14, so the call is overpriced: sell the call and hold the replicating portfolio. Among the options only the first has the right action, and the nearest value to 13.23 is the keyed 11.9 only loosely; the action is what discriminates.

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