Lesson 3 of 5 · 12 min

Synthetic positions and option strategies from parity

Rearranging S0+p0=c0+X(1+r)−TS_0 + p_0 = c_0 + X(1+r)^{-T} shows how to build any one of the four instruments from the other three, which both prices it and gives its replicating portfolio.

In short

  • Move a term across the equals sign and its sign flips: a plus means long, a minus means short.
  • Synthetic put: p0=c0+X(1+r)−T−S0p_0 = c_0 + X(1+r)^{-T} - S_0 (long call, long bond, short share).
  • Synthetic call: c0=S0+p0−X(1+r)−Tc_0 = S_0 + p_0 - X(1+r)^{-T} (long share, long put, borrow PV(X)).
  • Synthetic share: S0=c0−p0+X(1+r)−TS_0 = c_0 - p_0 + X(1+r)^{-T} (long call, short put, long bond). The options' asymmetric payoffs cancel into a straight line.
  • A covered call (long share, short call) equals a long bond plus a short put: S0−c0=X(1+r)−T−p0S_0 - c_0 = X(1+r)^{-T} - p_0.

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Synthetic positions and option strategies from parity · Option Replication Using Put–Call Parity