Show how a combination of options can reproduce the return profile of an underlying asset
The idea
Replication rests on put–call parity for an asset that pays no income: S + p = c + PV(X). Rearranged, S = c − p + PV(X), so a long call and a short put with the same strike and expiry, plus a risk-free bond paying X at expiry, behave like the asset. The call and put alone form a synthetic forward paying the terminal price minus X; the bond turns it into synthetic stock. The same algebra gives a synthetic put (long call, short asset, long bond) and a synthetic call (long asset, long put, borrowing PV(X)). Managers use these when options are cheaper, quicker or avoid a restriction on holding the asset. The usual slip is dropping the bond, which leaves only a forward.
Formulas
Put-call parity
c + X ÷ (1 + r)ᵗ = p + S₀
- c
- the price of a European call
- p
- the price of a European put, same strike and same expiry
- X
- the exercise price both options share
- r
- the risk-free rate
- t
- the years to expiry
- S₀
- the price of the underlying now
Rearranged, a protective put (S₀ + p) is worth a call plus the strike's present value, and a covered call (S₀ − c) is worth that bond less a put. That is why a covered call has the payoff shape of a short put.
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