Learning Module 9
Option Replication Using Put–Call Parity
Key Outcomes Summary & Practice Problems
What you must be able to do
Curriculum Year: 2026
Explain put–call parity for European options.
Explain put–call forward parity for European options.
1 · Put–Call Parity
S0 + p0 = c0 + X(1 + r)−T
Protective put (long underlying + long put) and fiduciary call (long call + long risk-free bond paying X) have identical payoffs at maturity under all scenarios — therefore they must have the same cost today.
If parity doesn't hold, a riskless arbitrage exists: sell the overpriced side, buy the underpriced side, pocket the difference today with zero net future cash flow.
Holds for European options with the same exercise price and expiration, on an underlying with no income/cost.
Scenario at T | Protective Put (S0+p0) | Fiduciary Call (c0+PV(X)) |
|---|---|---|
ST < X (put exercised) | X | X |
ST = X | ST (=X) | X (=ST) |
ST > X (call exercised) | ST | ST |
2 · Replicating Individual Positions from Parity
Rearranging the parity equation lets you solve for (and replicate) any one of the four positions using the other three.
To Replicate | Use |
|---|---|
Long underlying (S0) | Short put + long call + long risk-free bond |
Long call (c0) | Long underlying + long put + short risk-free bond |
Long put (p0) | Short underlying + long call + long risk-free bond |
Long risk-free bond | Long underlying + short call + long put |
Covered call = long underlying + short call (S0 − c0) — not long call.
Fiduciary call = long call + long risk-free bond (c0 + PV(X)).
3 · Put–Call Forward Parity
Since a long underlying can be replicated by a long forward + a risk-free bond paying F0(T), substituting this synthetic position into put–call parity gives put–call forward parity.
F0(T)(1 + r)−T + p0 = c0 + X(1 + r)−T
p0 − c0 = [X − F0(T)](1 + r)−T
Synthetic protective put = long forward + long risk-free bond (face value F0(T)) + long put — has the identical payoff to a (cash) protective put.
From the rearranged form: a long put + short call ≈ long risk-free bond + short forward; a short put + long call ≈ long forward + short risk-free bond.
4 · Firm Value Application — Equity and Debt as Options
Firm value at t=0: V0 = E0 + PV(D), where D is the face value of zero-coupon debt.
Solvency (VT > D): debtholders are repaid D in full; shareholders get the residual, VT − D.
Insolvency (VT < D): shareholders get nothing; debtholders receive only VT (less than their D claim).
takeholder | Payoff at T | Option Equivalent |
|---|---|---|
Shareholders | Max(0, VT − D) | Long call on firm value, X = D (equivalently: long firm assets + long put on firm value) |
Debtholders | Min(VT, D) | Long risk-free debt (D) + short put on firm value, X = D |
V0 = c0 + PV(D) − p0
The put option shareholders effectively hold (sold by debtholders) can be interpreted as the firm's credit spread — the premium debt-holders demand above the risk-free rate to bear insolvency risk.
As insolvency risk rises, this implicit put's value rises — benefiting shareholders' position and increasing debt-holders' effective credit risk exposure.