Option Replication Using Put–Call Parity

CFA Level I · CFA · 6 min read

Welcome to the World of Put–Call Parity!

Hello there! Today, we are diving into one of the most elegant and useful concepts in the CFA Level I Derivatives curriculum: Put–Call Parity. If you’ve ever wondered how traders know if an option is "fairly priced" or how they can create a "synthetic" stock using options, this is the key. Don't worry if derivatives feel a bit abstract right now—we’re going to break this down using simple logic, a classic formula, and some easy-to-remember analogies.

By the end of these notes, you’ll see that options, stocks, and bonds are all part of one big family, and Put-Call Parity is the "DNA" that connects them.

1. The Foundation: European Options Only

Before we look at the math, there is one very important rule to remember: Put–Call Parity applies specifically to European-style options.

As a quick refresher:

European options can only be exercised at the very end (maturity).
American options can be exercised at any time before they expire.

Because the timing of the exercise is fixed for European options, we can create a perfect relationship between calls, puts, stocks, and bonds. If they have the same underlying asset, the same strike price (\(X\)), and the same expiration date (\(T\)), their prices must follow a specific balance.

2. The Two Famous Portfolios

To understand Put–Call Parity, we compare two different ways of ending up with the exact same amount of money in the future. These are called the Fiduciary Call and the Protective Put.

A. The Fiduciary Call

Imagine you want to own a stock in the future. You decide to buy a Call option (giving you the right to buy the stock at strike price \(X\)). To make sure you actually have the money to buy that stock if you exercise the option, you also invest some money in a risk-free zero-coupon bond that will grow to exactly the strike price \(X\) by the expiration date.

Value of Fiduciary Call = \(C_0 + \frac{X}{(1+r)^T}\)

(Where \(C_0\) is the call price and the second part is the Present Value of the strike price).

B. The Protective Put

Now imagine a different strategy. You buy the Stock today, but you are worried the price might drop. To protect yourself, you also buy a Put option (giving you the right to sell the stock at strike price \(X\)). This "protects" your downside.

Value of Protective Put = \(S_0 + P_0\)

(Where \(S_0\) is the current stock price and \(P_0\) is the put price).

Why are they equal?

Whether the stock price goes up or down, both portfolios will be worth the same amount at expiration (the higher of the Stock Price or the Strike Price). Because they have the same payoff at the end, they must cost the same at the beginning! If they didn't, there would be an "arbitrage" opportunity (free money), and traders would jump on it until the prices balanced out again.

Quick Review:
Fiduciary Call: Long Call + Long Risk-Free Bond
Protective Put: Long Stock + Long Put

3. The Put–Call Parity Formula

Here is the "Golden Equation" you need to memorize:

\[ S_0 + P_0 = C_0 + \frac{X}{(1+r)^T} \]

Mnemonic Aid: "Sip Coke"
Think of it as: S + P = C + K (where K is the discounted strike price).
S (Stock) + P (Put) = C (Call) + K (Kash/Bond).

Did you know?
This formula is like a scale. If one side gets "heavier" (more expensive) than the other, the scale tips, and arbitrageurs will step in to profit from the imbalance!

4. Option Replication (Synthetics)

This is where the magic happens for the CFA exam. If you know three of the components, you can "synthetically" create the fourth one by rearranging the algebra. To "replicate" something, just isolate it on one side of the equals sign.

1. Want to replicate a Stock? (Long Stock)
Move the Put to the other side: \(S_0 = C_0 + \frac{X}{(1+r)^T} - P_0\)
Translation: Buy a Call, Buy a Bond, and Sell (Short) a Put.

2. Want to replicate a Call Option? (Long Call)
Move the Bond to the other side: \(C_0 = S_0 + P_0 - \frac{X}{(1+r)^T}\)
Translation: Buy the Stock, Buy a Put, and Borrow money (Short a Bond).

3. Want to replicate a Put Option? (Long Put)
Move the Stock to the other side: \(P_0 = C_0 + \frac{X}{(1+r)^T} - S_0\)
Translation: Buy a Call, Buy a Bond, and Sell (Short) the Stock.

Common Mistake to Avoid:
When you see a minus sign in the formula (e.g., \(- S_0\)), it means you are shorting or selling that asset. When you see a plus sign, it means you are buying or long that asset.

5. Arbitrage: When the Scale Tips

If the equation does not equal, an arbitrage opportunity exists. The rule is simple: Buy the undervalued side and Sell the overvalued side.

Example:
Suppose \(S + P > C + PV(X)\).
The Protective Put is too expensive, and the Fiduciary Call is too cheap.
Action: Sell the Stock, Sell the Put (the expensive side), and Buy the Call, Buy the Bond (the cheap side). You pocket the difference instantly with zero risk!

Step-by-Step for Arbitrage Questions:
1. Calculate the left side (\(S+P\)).
2. Calculate the right side (\(C + PV(X)\)).
3. Identify which side is cheaper.
4. "Buy Low, Sell High."

6. Put–Call Forward Parity

The curriculum sometimes replaces the spot price of the stock (\(S_0\)) with the forward price (\(F_0(T)\)). Because of the relationship between spot and forward prices, the formula looks slightly different:

\[ \frac{F_0(T) + P_0}{(1+r)^T} = \frac{C_0 + X}{(1+r)^T} \]

Or more simply: The present value of the forward price plus the put equals the present value of the call plus the strike. This is just a variation of the same balance—don't let the "Forward" terminology scare you!

Key Takeaways for Exam Day

• The Formula: \(S + P = C + PV(X)\). This is your best friend.
• Replication: If the question asks for a "Synthetic Put," just solve the equation for \(P\).
• Components: Remember that "Bond" or "\(PV(X)\)" refers to a risk-free zero-coupon bond maturing at strike \(X\).
• European Only: Parity doesn't strictly hold for American options because of the possibility of early exercise.
• Don't Panic: If you forget which way to rearrange the formula, just write down \(S + P = C + PV(X)\) and use basic algebra to move things around!

You've got this! Derivatives can be intimidating, but Put-Call Parity is just a simple balance. Keep practicing the algebra, and it will become second nature.

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

To which specific type of options does the Put–Call Parity relationship strictly apply?

Put–Call Parity applies strictly to European-style options.

What are the components of a Fiduciary Call portfolio?

A Fiduciary Call consists of a long call option and a risk-free zero-coupon bond with a face value equal to the strike price (X).

What are the components of a Protective Put portfolio?

A Protective Put consists of a long position in the underlying stock and a long put option on that same stock.

Why must the Fiduciary Call and the Protective Put have the same initial cost?

They must have the same cost because they produce identical payoffs at the expiration date, regardless of the stock's price.

What is the fundamental Put–Call Parity formula?

\( S_0 + P_0 = C_0 + \frac{X}{(1+r)^T} \)

How do you create a synthetic long stock position using options and bonds?

By buying a call, buying a risk-free bond, and selling (shorting) a put: \(S_0 = C_0 + \frac{X}{(1+r)^T} - P_0\).

How is a call option replicated synthetically using Put–Call Parity?

A call is replicated by buying the stock, buying a put, and borrowing the present value of the strike price: \(C_0 = S_0 + P_0 - \frac{X}{(1+r)^T}\).

How is a put option replicated synthetically using Put–Call Parity?

A put is replicated by buying a call, buying a risk-free bond, and selling (shorting) the underlying stock: \(P_0 = C_0 + \frac{X}{(1+r)^T} - S_0\).

In option replication, what do the plus (+) and minus (-) signs signify?

A plus sign represents a long position (buying), while a minus sign represents a short position (selling or borrowing).

What action should be taken if the Protective Put is more expensive than the Fiduciary Call?

Sell the components of the Protective Put (Stock and Put) and buy the components of the Fiduciary Call (Call and Bond).

What is the Put–Call Forward Parity formula?

\( \frac{F_0(T) + P_0}{(1+r)^T} = \frac{C_0 + X}{(1+r)^T} \)

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