DER 9 Option Replication Using Put-Call Parity

Earlier lessons treated call and put payoffs, the upper and lower bounds on an option value, and the drivers of those values, contrasting the one-sided payoff of an option with the straight-line payoff of a forward or of the underlying itself. This lesson shows that options can be combined so that the package behaves exactly like the underlying plus a risk-free asset, and later like a forward commitment. The central result, called put-call parity, lets the price of a European put be recovered from the price of a European call on the same underlying, and the other way round, under a no-arbitrage condition. Throughout, the underlying is assumed to pay no income or benefit, and the options are European.

Consider an investor who wants to share in gains on an underlying while staying protected against a fall in its price below some level. Two portfolios, each formed at time t = 0, meet that goal:

1. A purchased call with exercise price X, together with a risk-free bond redeeming for X at time T. Since that bond is worth X(1+r)^-T today, the package costs c0 + X(1+r)^-T. A bought call plus a risk-free bond is called a fiduciary call.
2. A purchased unit of the underlying at S0, together with a purchased put on the same underlying at exercise price X. This package costs S0 + p0 and is called a protective put.

At expiration each portfolio is worth the greater of X or the terminal price ST. When ST finishes below X, the fiduciary call pays X from the bond, while the protective put also pays X, since the underlying worth ST combines with a put worth X minus ST. When ST finishes above X, both packages pay ST. The table walks through every scenario.

Because the two portfolios deliver the same cash flow in every state, the no-arbitrage condition, which says that two assets with identical future cash flows must trade at the same price, forces their prices to be equal at t = 0. That equality is put-call parity:

In words, a long underlying plus a long put must cost the same as a long call plus a risk-free bond that matures to the exercise price. The two sides therefore share the single kinked payoff drawn below.

If the two portfolios ever trade at different prices, an investor able to lend or borrow freely at the risk-free rate captures a riskless profit by buying the cheaper package and selling the dearer one.

Put-call parity is more than a pricing identity: rearranged, it is a recipe for building any one of the four positions out of the other three. Solving the parity equation for the put premium shows that a put is the same as a long call, a short position in the underlying, and a long risk-free bond.

That equation states both what the put should cost and how to replicate it with a call, a bond, and a short underlying. Solving the same relationship for the underlying instead shows that a long underlying equals a long call, a short put, and a long risk-free bond.

The full set of equivalences is collected below. Reading across a row gives the three positions that together replicate the position named at the left.

These building blocks do more than exploit mispricing; they construct familiar option strategies. A covered call, for instance, is a long underlying combined with a written call, or S0 minus c0 at inception. Rearranging parity isolates that combination.

So a covered call is equivalent to holding a risk-free bond and writing a put: the income from the sold call mirrors the premium collected on the sold put, and the capped upside mirrors the bond.

A long underlying position can itself be built synthetically. A long forward paired with a risk-free bond whose face value matches the forward price F0(T) reproduces the underlying, because the bond funds the forward purchase and the forward delivers the asset. Putting the present value of F0(T) in place of S0 in the parity equation yields put-call forward parity.

Replacing the cash underlying in a protective put with this synthetic underlying builds a synthetic protective put: a long forward, a risk-free bond worth F0(T)(1+r)^-T, and a purchased put. Its cash flows match the ordinary protective put in every state, as the table confirms.

Since the synthetic protective put and the fiduciary call share the same expiration cash flows, their costs must be equal, which is the forward parity equation itself. Rearranging it isolates a long put and a short call as a long risk-free bond and a short forward.

Put-call parity reaches beyond option desks: it describes how the value of a firm is split between the owners of its equity and the owners of its debt. At t = 0 a firm has asset value V0, funded by zero-coupon debt with face value D and by equity E0, so the assets equal the present value of the debt plus equity.

When the debt matures at T, one of two outcomes occurs. If the firm is solvent, meaning its value VT exceeds the face value D, debtholders are repaid D in full and shareholders keep the residual VT minus D. If the firm is insolvent, meaning VT is below D, debtholders hold the senior claim and receive the whole firm VT, which is less than D, while shareholders receive nothing. The firm carries risky debt here, not the risk-free bond of the earlier sections, because debtholders collect the full D only in the solvent case.

So at maturity the shareholder payoff is the greater of zero or VT minus D, and the debtholder payoff is the lesser of VT and D.

Read in option terms: shareholders hold the firm’s assets together with a put on those assets struck at D, and that combination behaves like a call on firm value. Debtholders hold a risk-free bond worth D and have written that put to shareholders, so their risky debt equals a risk-free bond minus a put. The figure sets the two payoffs side by side.

Substituting firm value V0 for the underlying and debt D for the risk-free bond in parity, then solving for V0, restates the same split in pricing terms.

Shareholders hold the call c0, the claim with unlimited upside and downside capped at losing the equity. Debtholders hold PV(D) minus p0, a risk-free bond less a written put, giving them limited upside and principal at risk on the downside. That written put is the credit spread on the debt, the extra yield above the risk-free rate that compensates debtholders for bearing insolvency risk. As the firm moves closer to insolvency the put becomes more valuable, which is exactly the same as saying the debt carries more credit risk.