Q&A: What is the appropriate arbitrage strategy given the following involving...

...call options with an exercise price of $100 expiring in 6 months. The risk-free rate is 10%. The call is priced at $7.5, and the put is priced at $4.25. The underlying price is $99.

CppQuant Answer

According to the put-call parity, c(0) + X/(1 + r)^T = p(0) + S(0)

  • the fiduciary call (lhs) = 7.5 + 100/(1.10)^0.5 = 102.85.
  • the protective put (rhs) = 4.25 + 99 = 103.25
  • Since protective put > fiduciary call, that means the protective put is overpriced.
  • We could sell the protective put: sell the put and sell short the underlying. Doing so will generate a cash inflow of $103.25. The we buy fiduciary call, paying out $102,85, netting a cash inflow of $0.4.
    • At expiration, if the price of the underlying is above 100:
      • the bond matures, paying $100. use the $100 to exercise call, receiving the underlying.
      • deliver the underlying to cover the short sale.
      • the put expires with no value.
      • net effect: no money in or out.
    • Similarily If the price of the underlying is below 100 at expiration.

  • So we receive $0.4 up front and do not have to pay anything out.

Bonus Points

  • Put-call parity: c(0) + X/(1 + r)^T = p(0) + S(0) : the lhs is known as fiduciary call, rhs protective put
  • If the underlying assets make cash payments: c0 + X/(1 + r)T = P0 + [S0 - PV(CF, 0, T)] where PV(CF, 0, T) represents the present value of these cash flows.
  • The position is perfectly hedged and represents an arbitrage profit.

Category: C++ Quant > Derivatives > Options

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