DER 9 Option Replication Using Put-Call Parity

Combining options, the underlying, and a risk-free bond into portfolios that price one another with no arbitrage. Check-yourself items are study aids.

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:

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.
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.

Protective put versus fiduciary call at expiration

Portfolio position	Put exercised (ST < X)	No exercise (ST = X)	Call exercised (ST > X)
Protective put:
Underlying asset	ST	ST	ST
Put option	X − ST	0	0
Total	X	ST (= X)	ST
Fiduciary call:
Call option	0	0	ST − X
Risk-free asset	X	X	X
Total	X	X (= ST)	ST

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.

Figure 1: The shared payoff of a fiduciary call and a protective put

Below X the payoff is fixed at the exercise price; above X it climbs one-for-one with the terminal price. Both portfolios trace this same line, so they must cost the same today.

Example 1 · Worked

The Viswan Family Office (VFO) holds non-dividend-paying Biomian shares priced at INR295. VFO is weighing a six-month put on Biomian at an exercise price of INR265. A traded six-month call at the same INR265 strike is quoted at INR59, and the relevant risk-free rate is 4 percent.

What is the fair price of the put, per share?

Solution. Discount the exercise price: INR265 divided by (1.04) to the power 0.5 is INR259.85. Put-call parity gives INR295 plus the put equals INR59 plus INR259.85, so the put premium is INR59 plus INR259.85 minus INR295, which is INR23.85. The fair six-month put premium is therefore INR23.85 per share.

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.

Example 2 · Worked

Using the same Biomian figures, VFO finds that the put it valued at INR23.85 is actually quoted at INR30, with the risk-free rate still 4 percent.

Identify the arbitrage and the trades that lock in a riskless profit.

Solution. Substituting the traded put price into parity gives INR295 plus INR30, or INR325, on the left against INR59 plus INR259.85, or INR318.85, on the right. The protective put side is overpriced, so VFO sells it and buys the fiduciary call side: sell the shares and sell the put, buy the call and buy the risk-free bond. The opening cash flow is INR295 plus INR30 minus INR59 minus INR259.85, which is INR6.15. As the table shows, the two packages cancel in every future state, so that INR6.15 is a pure arbitrage profit.

Arbitrage cash flows when the put is overpriced

Arbitrage position	Cash flow at t = 0	Put exercised (ST < X)	No exercise (ST = X)	Call exercised (ST > X)
Sell underlying asset	S0	−ST	−ST	−ST
Sell put option	p0	−(X − ST)	0	0
Protective put total	S0 + p0	−X	−X	−ST
Buy call option	−c0	0	0	ST − X
Buy risk-free asset	−X(1+r)^-T	X	X	X
Fiduciary call total	−c0 − X(1+r)^-T	X	X	ST
Overall portfolio	S0 + p0 − c0 − X(1+r)^-T	0	0	0

Example 3 · Worked

South China Sprintwyck Investments (SCSI) is studying a protective put on ChinaWell Inc. (CWI), which trades at CNY127.50 and pays no dividends, with a risk-free rate of 4 percent. A six-month CWI call struck at CNY120 trades at CNY22.60.

What is the no-arbitrage price of the matching six-month put?

Solution. Discount the strike: CNY120 divided by (1.04) to the power 0.5 is CNY117.67. Rearranging parity for the put gives the call plus the discounted strike minus the spot, or CNY22.60 plus CNY117.67 minus CNY127.50, which is CNY12.77.

Check yourself

A fiduciary call combines which two positions, and why does its payoff never fall below the exercise price?

A purchased call plus a risk-free bond that matures to X. The bond guarantees X at expiration, and the call adds ST minus X only when that is positive, so the total is never less than X.

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.

Building each position from the other three

Position	Underlying (S0)	Risk-free bond	Call option (c0)	Put option (p0)
Underlying (S0)		Long	Long	Short
Risk-free bond	Long		Short	Long
Call option (c0)	Long	Short		Long
Put option (p0)	Short	Long	Long

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.

Example 4 · Worked

VFO still holds non-dividend-paying Biomian shares at INR295. Expecting the price to drift sideways for six months, the chief investment officer wants to write a six-month call struck at INR325 for income, a covered call. A put at the same terms trades at INR56, and the risk-free rate is 4 percent.

What is the no-arbitrage call premium?

Solution. Rearranged parity gives the covered call as a long bond and a short put: S0 minus c0 equals the discounted strike minus the put. The discounted strike is INR325 divided by (1.04) to the power 0.5, which is INR318.69, so INR295 minus the call equals INR318.69 minus INR56, or INR262.69. The call premium is INR295 minus INR262.69, which is INR32.31.

Why replication matters

Each equivalence is a synthetic position. If a desk cannot trade an instrument directly, or finds it cheaper to assemble the exposure from other contracts, parity says exactly which three positions reproduce the payoff of the fourth. The same logic tells an arbitrageur which side to buy and which to sell when a quoted price strays from its no-arbitrage value.

Check yourself

True or false: a covered call is a long underlying plus a long call, written as S0 + c0 at inception.

False. A covered call is a long underlying plus a short call, written as S0 minus c0. The investor already owns the shares and sells a call against them for income.

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.

Protective put versus synthetic protective put

Position	Cash flow at t = 0	Put exercised (ST < X)	No exercise (ST ≥ X)
Protective put:
Purchased put (p0)	p0	X − ST	0
Cash underlying (S0)	S0	ST	ST
Total	p0 + S0	X	ST
Synthetic protective put:
Purchased put (p0)	p0	X − ST	0
Forward purchase	0	ST − F0(T)	ST − F0(T)
Risk-free bond	F0(T)(1+r)^-T	F0(T)	F0(T)
Total	p0 + F0(T)(1+r)^-T	X	ST

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.

Example 5 · Worked

Return to the Biomian case, but assemble the underlying synthetically from a long forward plus a risk-free bond. Shares trade at INR295, the six-month put is struck at INR265, the matching six-month call trades at INR59, and the risk-free rate is 4 percent.

Using forward parity, what should the put cost?

Solution. First find the forward price: INR295 times (1.04) to the power 0.5 is INR300.84. Then the put minus the call equals the discounted difference between strike and forward: the put minus INR59 equals (INR265 minus INR300.84) divided by (1.04) to the power 0.5, which is minus INR35.14. The put premium is INR59 minus INR35.14, or INR23.86, matching the earlier parity answer up to rounding.

Example 6 · Worked

SCSI observes a three-month CWI call struck at CNY130 trading at a premium of CNY3, while a three-month CWI forward trades at a forward price of CNY128.76. The risk-free rate is 4 percent.

What is the no-arbitrage premium of a three-month CWI put struck at CNY130?

Solution. Discount both the forward and the strike over three months. The forward leg is CNY128.76 divided by (1.04) to the power 0.25, which is CNY127.50, and the strike leg is CNY130 divided by (1.04) to the power 0.25, which is CNY128.73. Forward parity reads CNY127.50 plus the put equals CNY3 plus CNY128.73, or CNY131.73, so the put premium is CNY4.23.

Check yourself

In a synthetic protective put, what replaces the cash holding of the underlying?

A synthetic underlying: a long forward on the asset plus a risk-free bond whose face value matches the forward price F0(T). Together they deliver the asset at T, just like owning it outright.

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.

Figure 2: Shareholder and debtholder payoffs at maturity

Below D the shareholder gets nothing while the debtholder takes the whole firm; above D the debtholder is capped at D and the shareholder keeps the surplus. Their payoffs always add to the firm value VT.

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.

A rise in leverage

If a firm takes on more debt, the put written by debtholders gains value, since insolvency grows more likely. The debtholder position, a risk-free bond minus that put, deteriorates relative to the shareholders, who effectively hold the appreciating call on firm value. This is why an increase in the leverage ratio supports a bearish view on the debt and, other things equal, shifts value toward equity.

Check yourself

Why is a debtholder said to have sold a put on firm value?

The debtholder payoff, the lesser of VT and D, equals D minus the greater of zero or D minus VT. That second term is a put on firm value struck at D, and the minus sign means the debtholder has written it. In insolvency the put is exercised against them, so they receive VT rather than the full D.

Overview

Put-Call Parity

Option Strategies Based on Put-Call Parity

Put-Call Forward Parity

Option Applications to Firm Value