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Learning Module 9 Option Replication Using Put–Call Parity

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Overview and Put-Call Parity Statement5 min

This chapter explains put–call parity for European options and its extension to forward contracts (put–call forward parity). Put–call parity: for European options on a non-dividend-paying underlying with the same strike X and maturity T, the no-arbitrage relationship is S0 + p0 = c0 + PV(X), where PV(X) = X(1 + r)^{-T}. This equality arises because a protective put (long underlying + long put) and a fiduciary call (long call + long risk-free bond paying X at T) deliver identical payoffs at expiration in all states. If prices depart from this equality, arbitrage strategies (buy the cheap portfolio, sell the expensive one, and finance appropriately) produce riskless profits. Rearranged forms provide synthetic positions: S0 = c0 - p0 + PV(X); p0 = c0 + PV(X) - S0; c0 = S0 + p0 - PV(X). Put–call forward parity substitutes a synthetic underlying (long forward + risk-free bond) for the cash underlying: F0(T)(1 + r)^{-T} + p0 = c0 + PV(X), where F0(T)(1 + r)^{-T} denotes the present value of the forward price. Rearranged, p0 - c0 = [X - F0(T)](1 + r)^{-T}, which shows equivalence between (long put + short call) and (long PV-bond + short forward). The chapter demonstrates how to use parity to price missing option premiums given traded primitives and to construct covered-call and synthetic-protective-put relationships (covered call = long bond - short put; fiduciary call = protective put).

Key Points

Put–call parity: S0 + p0 = c0 + PV(X) for European options on non-income underlyings.
Protective put and fiduciary call have identical payoffs at T, hence equal prices at t = 0.
Rearrangements produce synthetic positions and pricing formulas for missing option prices.
Parity violations imply arbitrage opportunities: buy the low-cost portfolio, sell the high-cost one.

Replication, Synthetic Positions, and Arbitrage Mechanics5 min

Practical replication uses parity to build equivalent portfolios: a covered call (long underlying, short call) is equivalent to long PV(X) bond minus long put (S0 - c0 = PV(X) - p0). Similarly, p0 = c0 + PV(X) - S0 shows a put as a portfolio of a long call, a long PV bond, and a short underlying. When parity fails in observed prices, one can perform arbitrage: if S0 + p0 > c0 + PV(X), sell S0 and p0, buy c0 and PV(X), pocket initial cash difference, and unwind at T for zero net liability but positive initial cash. Steps must include borrowing/lending at the risk-free rate and being long/short the correct instruments in the right quantities. Examples highlight numerical computations (solve for p0 given c0, S0, PV(X); compute covered call implied call price from put price).

Key Points

Replication maps options to combinations of calls, puts, underlying, and bonds.
Covered call equals long PV bond minus long put: S0 - c0 = PV(X) - p0.
Arbitrage procedure: identify inequality, construct opposite portfolios, lock in riskless profit.
Examples show step-by-step algebra and cash flows at t = 0 and T.

Put-Call Forward Parity and Forward-Based Replication5 min

Put–call forward parity replaces the cash underlying with a synthetic underlying (long forward plus PV of forward price), yielding F0(T)(1 + r)^{-T} + p0 = c0 + PV(X). This form is useful when forwards rather than cash positions are traded or when financing is structured via forward contracts. Rearranging demonstrates equivalence: p0 - c0 = [X - F0(T)](1 + r)^{-T}. Using these relationships, one can construct synthetic protective puts and fiduciary calls where the synthetic underlying substitutes for S0. The parity explains how option and forward prices interact in arbitrage-free markets. Applications include pricing puts from calls and forwards, and checking consistency between option markets and forward markets.

Key Points

Put–call forward parity: F0(T)(1 + r)^{-T} + p0 = c0 + PV(X).
Useful when forward prices are observed or used for synthetic underlying construction.
Leads to formulas connecting p0, c0, F0(T), and PV(X) for arbitrage tests.
Enables replication using forwards plus bonds and options.

Applications to Strategies and Firm Capital Structure5 min

Put–call parity applies beyond simple option trades. Covered-call and protective-put strategies are reinterpretations of parity equations. The chapter discusses an example where equity equals a call on firm assets with strike equal to debt face value; E0 approximates c0 when firm assets and debt map to option constructs. Debtholders effectively hold PV(D) minus a put on firm value, connecting credit risk to put values. Thus, the put price can be viewed as reflecting credit spreads; increases in default risk raise p0 and affect the decomposition V0 = c0 + PV(D) - p0. These mappings provide intuition for corporate finance applications: equity upside and limited downside mirror call option payoff; debt combines risk-free payoff and sold put exposure.

Key Points

Equity resembles a call option on firm assets; debt resembles risk-free bond minus a put.
Put values can be interpreted as credit-risk premia in firm valuation.
Parity links derivative pricing to corporate capital structure insights.
Practical examples show numerical derivation of put or call premiums given firm/debt parameters.

Worked Examples and Common Problem Types5 min

The chapter contains worked problems: computing missing option premiums using S0 + p0 = c0 + PV(X); computing lower and upper bounds for option prices (calls bounded above by S0 and below by Max(0, S0 - PV(X)); puts bounded above by X and below by Max(0, PV(X) - S0)); verifying arbitrage trades when parity violated; converting covered-call positions into bond-and-put positions numerically; and forward-parity calculations substituting observed forward prices. Students are guided to ensure correct discounting at risk-free rates and consistent use of European exercise assumptions. Emphasis is placed on frictionless markets and non-dividend-paying underlyings as core assumptions for the algebra to hold exactly.

Key Points

Typical problems: solve for p0 given c0, S0, PV(X); solve for c0 given p0 and PV(X).
Check parity before trading; small pricing frictions in real markets can prevent pure arbitrage.
Remember the assumptions: European options, no income on underlying, frictionless markets.
Use parity to derive synthetic strategies and to interpret market-implied credit risk.

Questions

Question 1

If S0 = 120, c0 = 6, PV(X) = 110, what is the put price p0 implied by put–call parity?

Learning Module 9 Option Replication Using Put–Call Parity

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