# Unlocking the Mystery of Option Replication: Put-Call Parity | CFA Level I Derivatives

Ever wondered how to price options based on their relationship to other options? Today, we’ll dive into the fascinating world of put-call parity and learn how it can be applied to pricing options. We’ll also explore protective puts and fiduciary calls, which form the basis of the put-call parity relationship.

## Understanding Put-Call Parity

Put-call parity is an essential relationship in options pricing, and it involves European options. To derive this relationship, we need to understand two strategies: protective puts and fiduciary calls.

### Protective Puts: Protecting Your Asset

A protective put strategy involves holding an asset while also buying a put option with the same exercise price. This strategy is designed to protect against a decline in the value of the asset. Here’s how it works:

An investor owns an asset with a current price of S0. The investor buys a put option with an exercise price of X and a premium of p0. If the asset’s value increases at expiration, the investor lets the put option expire and keeps the asset. If the asset’s value decreases, the investor exercises the put option, selling the asset for X dollars.

A protective put strategy ensures the investor benefits from unlimited upside potential while limiting downside risk to X.

### Fiduciary Calls: A Different Approach

A fiduciary call strategy involves buying a call option and a risk-free zero-coupon bond simultaneously. The strategy works like this:

The investor buys a call option with an exercise price of X and a premium of c0. The investor also buys a risk-free zero-coupon bond with a face value of X. If the asset’s value increases at expiration, the investor exercises the call option and buys the asset. If the asset’s value decreases, the call option expires worthless, and the bond is worth X dollars.

Both the protective put and fiduciary call strategies produce the same result, leading to the put-call parity relationship:

S0 + p0 = c0 + PV(X)

Where PV(X) is the present value of the exercise price.

## Applying Put-Call Parity to Pricing Options

Put-call parity can be used to price options by identifying the relationship between the put and call prices. If the relationship doesn’t hold, an arbitrage opportunity exists.

### EXAMPLE

Suppose an asset is trading at \$90, with a call and put option at an exercise price of \$100. The risk-free rate is 4%, and the options expire in 3 months. What is the relationship between the put and call prices?

Following the put-call parity formula,

S0 + p0 = c0 + PV(X)
-> p0 – c0 = X/(1+Rf)T – S0
-> p0 – c0 = 100/1.04^0.25 – 90
-> p0 – c0 = 9.02

we find the difference between the put and call prices must be \$9.02. If the put is trading at \$15, the call price should be \$5.98.

## Expanding to Put-Call-Forward Parity

Put-call-forward parity is an extension of the put-call parity concept to incorporate forward contracts. It helps us understand the relationship between put options, call options, and forward contracts.

The put-call-forward parity formula is:

PV(F0) + p0 = c0 + PV(X)

Where F0 is the current forward contract price.

## Conclusion

Understanding put-call parity is essential for options pricing and risk management. The concept demonstrates the relationship between European call and put options, allowing investors to identify potential arbitrage opportunities and ensure fair pricing in the market. By extending the concept to include forward contracts, investors can further understand the relationship between options and other derivatives.

Remember, put-call parity is a crucial concept for CFA Level I candidates in the Fixed Income section. To succeed in your exam, ensure you have a solid grasp of protective puts, fiduciary calls, and the put-call parity relationship. Good luck!

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