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CFA® Level I · Derivatives · free lesson
Option Replication Using Put-Call Parity
Puts, calls, the underlying and a risk-free bond are tied together by a single no-arbitrage relationship. This module derives put-call parity, uses it to build synthetic calls, puts, shares and bonds and to spot arbitrage, extends it to forwards with put-call forward parity, and applies it to corporate finance by viewing a firm's equity and debt as options on its assets.
What you will learn
1. Explain put-call parity for European options and use it to calculate the price of a call or put from the other instruments.
- Fiduciary call (call + bond paying X) = protective put (underlying + put) at expiration.
- Put-call parity: S_0 + p_0 = c_0 + X(1 + r)^-T.
- Any one of the four prices can be solved from the other three.
- Parity is model-free; it relies only on no arbitrage and European exercise.
- With dividends, use S_0 minus the present value of dividends.
2. Use put-call parity to construct synthetic calls, puts, underlying positions and risk-free bonds and to identify and exploit arbitrage opportunities.
- Synthetic call = underlying + put - bond; synthetic put = call - underlying + bond.
- Synthetic underlying = call - put + bond; synthetic bond = underlying + put - call.
- Arbitrage: sell the overpriced instrument and buy its synthetic replica.
- Quick test: compare c_0 - p_0 with S_0 - X(1 + r)^-T.
- Synthetic shorts (long put, short call) provide bearish exposure without borrowing shares.
3. Explain put-call forward parity and use it to relate option prices to the forward price of the underlying.
- Synthetic underlying = long forward + bond paying F_0(T).
- Put-call forward parity: F_0(T)(1 + r)^-T + p_0 = c_0 + X(1 + r)^-T.
- c_0 - p_0 = F_0(T) - X^-T: long call + short put = synthetic forward at X.
- When X = F_0(T), call and put prices are equal.
- The forward price embeds carry, so the formula handles income and costs directly.
4. Describe a firm's equity and debt as options on its assets and use put-call parity to relate the value of risky debt to a risk-free bond and a put.
- Equity = call on the firm's assets with exercise price = face value of debt.
- Risky debt = risk-free bond paying D minus a put on the firm's assets.
- The put's value measures the debt's credit risk and drives its yield spread.
- Equity can have value even when assets are below debt, because of time value.
- Higher asset volatility benefits shareholders (long call) at the expense of debtholders (short put).
Module at a glance
- Put-call parity: S_0 + p_0 = c_0 + X(1 + r)^-T; the protective put equals the fiduciary call.
- Any of the four instruments can be replicated with the other three.
- Arbitrage: sell the overpriced instrument, buy its synthetic replica; compare c_0 - p_0 with S_0 - PV(X).
- Put-call forward parity: F_0(T)(1 + r)^-T + p_0 = c_0 + X(1 + r)^-T.
- Long call + short put at X = synthetic long forward at X.
- Equity is a call on the firm's assets; risky debt is a risk-free bond minus a put.
- Higher asset volatility transfers value from debtholders to shareholders.
This free module has 4 lessons, 15 practice questions with full explanations.