Option Replication Using Put-Call Parity

Put-call parity is a fundamental no-arbitrage relationship that links the prices of European call and put options that have the same underlying asset, strike price (X), and expiration date (T). It is built on the principle that two different portfolios with identical payoff profiles must have the same initial cost.

The Core Equation: Protective Put = Fiduciary Call

The relationship is derived from the fact that a "protective put" portfolio has the exact same payoff as a "fiduciary call" portfolio. To avoid arbitrage, their values at inception must be equal. This leads to the put-call parity equation:

Where:

• p₀: The price (premium) of the put option at time 0
• S₀: The price of the underlying asset at time 0
• c₀: The price (premium) of the call option at time 0
• X: The strike price of the options
• r: The risk-free interest rate
• T: The time to expiration

Portfolio 1: The Protective Put

A protective put is a strategy that combines a long position in an underlying asset with a long put option on that asset.

• Initial Value: p₀ + S₀
• Payoff at Expiration:
  • If S_T ≥ X, the put expires worthless (out-of-the-money), and the payoff is simply the value of the asset, S_T
  • If S_T < X, the put is exercised (in-the-money), and its payoff is X – S_T. The total payoff is (X – S_T) + S_T = X
• Summary: The payoff of the protective put is max(X, S_T). It provides downside protection at the strike price X while retaining upside potential.

Portfolio 2: The Fiduciary Call

A fiduciary call is a strategy that combines a long call option with a long position in a risk-free, pure-discount bond that pays the strike price X at maturity.

• Initial Value: c₀ + X (1 + r)ᵀ
• Payoff at Expiration:
  • If S_T ≤ X, the call expires worthless (out-of-the-money), and the payoff is the bond maturing at a value of X
  • If S_T > X, the call is exercised (in-the-money), and its payoff is S_T – X. The total payoff is (S_T – X) + X = S_T
• Summary: The payoff of the fiduciary call is also max(X, S_T). It provides a minimum payoff of X while also providing exposure to the asset's upside.

By rearranging the put-call parity equation, we can create a "synthetic" position for any of the four components (call, put, underlying asset, or bond) using the other three. This is useful for arbitrage and for creating custom risk exposures.

Synthetic Position	Replicating Portfolio	Formula
Synthetic Call	Long Put + Long Underlying + Short Bond	c₀ = p₀ + S₀ – X (1 + r)ᵀ
Synthetic Put	Long Call + Short Underlying + Long Bond	p₀ = c₀ – S₀ + X (1 + r)ᵀ
Synthetic Underlying	Long Call + Short Put + Long Bond	S₀ = c₀ – p₀ + X (1 + r)ᵀ
Synthetic Bond	Long Put + Long Underlying + Short Call	X (1 + r)ᵀ = p₀ + S₀ – c₀

A variation of the parity relationship exists for options priced using a forward contract instead of the spot asset. This is derived by replacing the spot price S₀ in the standard parity equation with the present value of the forward price, F₀(T) (1 + r)ᵀ .

The put-call-forward parity equation is:

Option theory can be used to model the claims of a firm's shareholders and debtholders on its total assets.

For Shareholders

Shareholders have a claim on the firm's assets after all debts are paid. Their upside potential is unlimited, while their downside potential is limited to their investment (the stock price can't go below zero). This payoff profile is identical to a long call option on the firm's assets (V_T) with a strike price equal to the face value of the firm's debt (D).

For Debtholders

Debtholders have a senior claim on the firm's assets. Their upward potential is restricted to receiving their principal and interest (D), while their downside includes receiving less than face value or nothing in the case of insolvency.