# Put-Call Parity for European Options

Although parity means equivalence, puts and calls are not equivalent. However, there is a relationship between the price of a call and its corresponding put option. This is referred to as put-call parity.

## Protective Puts and Fiduciary Calls

First, let’s consider a protective put strategy where an investor who holds the underlying asset purchases a protective put option. At inception, the investor commits S0, the underlying asset cost, and p0, the put premium.

$$\text{Value at inception} = p_0 + S_0$$

A protective put gives the holder a limited downside but reduces his upside a bit because of the cost paid to buy the put option. Graphically, it can  be represented as:

A strategy where an investor enters into a fiduciary call means an investor buys a call option on an underlying asset and a risk-free zero-coupon bond with a face value equal to the exercise price.

$$\text{Value at inception} = c_0 + \frac{X}{(1+r)^T}$$

The payoff graph from this strategy will be the exact same as that of the protective put strategy.

## Put-Call Parity Arbitrage

By examing the payoff profiles of a protective put and a fiduciary call, we note that they are identical. Therefore, if two combinations of assets or portfolios of assets have the same payoff, their acquisition cost must be identical. In any other case, there is an arbitrage opportunity. In other words, someone can make a risk-free profit by buying the cheaper one and selling the most expensive one.

As a result of the no-arbitrage principle, we can set the value at the inception of the protection call equal to the value at the inception of the protective put as follows:

$$c_0 + \frac{X}{(1+r)^T} = p_0 + S_0$$

This equation is a key concept in derivatives pricing called put-call parity. This formula equates the value of calls and puts through equivalent portfolios. It must be assumed that since these are European options, they have the same strike, same expiry date, and the same underlying asset.

By re-arranging the prior equation:

$$c_0 = p_0 + S_0 – \frac{X}{(1+r)^T}$$

The right side of this equation is equivalent to a call option and is referred to as a synthetic call. It consists of a long put, a long position in the underlying asset, and a short position in the risk-free bond.

Another re-arrangement can be made to isolate the put option:

$$p_0 = c_0 +\frac{X}{(1+r)^T} – S_0$$

The right side of this equation is now referred to as a synthetic put which consists of a long call, a short position in the underlying, and a long position in the risk-free bond.

## Question

European put and call options both have an exercise price of $50 that expires in 120 days. The underlying asset is priced at$52 and makes no cash payments during the life of the option. The risk-free rate is 4.5% and the put is selling for $3.80. According to the put-call parity, the price of the call option should be closest to: A.$6.52

B. $6.32 C.$7.12

Solution

c0 = p0 + S0 – X/(1+r)T

c0 = 3.80 + 52 – (50/1.045120/365) = 6.52

Shop CFA® Exam Prep

Offered by AnalystPrep

Featured Shop FRM® Exam Prep Learn with Us

Subscribe to our newsletter and keep up with the latest and greatest tips for success

Sergio Torrico
2021-07-23
Excelente para el FRM 2 Escribo esta revisión en español para los hispanohablantes, soy de Bolivia, y utilicé AnalystPrep para dudas y consultas sobre mi preparación para el FRM nivel 2 (lo tomé una sola vez y aprobé muy bien), siempre tuve un soporte claro, directo y rápido, el material sale rápido cuando hay cambios en el temario de GARP, y los ejercicios y exámenes son muy útiles para practicar.
diana
2021-07-17
So helpful. I have been using the videos to prepare for the CFA Level II exam. The videos signpost the reading contents, explain the concepts and provide additional context for specific concepts. The fun light-hearted analogies are also a welcome break to some very dry content. I usually watch the videos before going into more in-depth reading and they are a good way to avoid being overwhelmed by the sheer volume of content when you look at the readings.
Kriti Dhawan
2021-07-16
A great curriculum provider. James sir explains the concept so well that rather than memorising it, you tend to intuitively understand and absorb them. Thank you ! Grateful I saw this at the right time for my CFA prep.
nikhil kumar
2021-06-28
Very well explained and gives a great insight about topics in a very short time. Glad to have found Professor Forjan's lectures.
Marwan
2021-06-22
Great support throughout the course by the team, did not feel neglected
Benjamin anonymous
2021-05-10
I loved using AnalystPrep for FRM. QBank is huge, videos are great. Would recommend to a friend
Daniel Glyn
2021-03-24
I have finished my FRM1 thanks to AnalystPrep. And now using AnalystPrep for my FRM2 preparation. Professor Forjan is brilliant. He gives such good explanations and analogies. And more than anything makes learning fun. A big thank you to Analystprep and Professor Forjan. 5 stars all the way!
michael walshe
2021-03-18
Professor James' videos are excellent for understanding the underlying theories behind financial engineering / financial analysis. The AnalystPrep videos were better than any of the others that I searched through on YouTube for providing a clear explanation of some concepts, such as Portfolio theory, CAPM, and Arbitrage Pricing theory. Watching these cleared up many of the unclarities I had in my head. Highly recommended.