Lesson 2 of 5 · 11 min

Arbitrage when put–call parity fails

If the protective put and the fiduciary call trade at different prices, sell the expensive side, buy the cheap one, and keep the difference today with zero net payoff at expiration.

In short

  • Compare the two sides: S0+p0S_0 + p_0 versus c0+X(1+r)−Tc_0 + X(1+r)^{-T}. Any gap is a riskless profit for someone who can borrow and lend at r.
  • If S0+p0>c0+PV(X)S_0 + p_0 > c_0 + PV(X): sell the put, short the share, buy the call and buy the bond (lend).
  • If S0+p0<c0+PV(X)S_0 + p_0 < c_0 + PV(X): buy the put, buy the share, sell the call and borrow PV(X).
  • At expiration the long and short portfolios have equal values in every state, so the combined payoff is zero.
  • The profit is the initial net cash inflow: the size of the mispricing.

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Arbitrage when put–call parity fails · Option Replication Using Put–Call Parity