Level 1 CFA® Exam:
Put–Call-Forward Parity

Last updated: January 09, 2023

Put-Call-Forward Parity: Assumptions & CFA Exam Formulas

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We have (A) an underlying asset, (B) a forward contract, and (C) two options on the underlying: a call option and a put option. Both options:

  • are European-style options,
  • have the same expiration date,
  • have the same exercise price, and
  • cover the same quantity of the underlying.

Forward Price

There is a relationship between an asset and a forward contract. It is given by the following formula that you can use in your level 1 CFA exam:

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Forward Price
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\(F = S_{0}\times (1+r)^{T}\)

  • \(F\) - forward price
  • \(S_{0}\) - underlying price at contract initiation
  • \(r\) - risk-free interest rate
  • \(T\) - time until contract expiration
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Put-Call Parity

The relationship between a put option, a call option, and the underlying asset is given by the following formula:

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Put-Call Parity
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\(c_{0}+\frac{X}{(1+r)^{T}}=p_{0}+S_{0}\)

  • \(c_{0}\) - current price of a call option
  • \(X\) - exercise price
  • \(r\) - risk-free interest rate
  • \(T\) - time to expiration (number of days to expiration divided by 365)
  • \(p_{0}\) - current price of a put option
  • \(S_{0}\) - current price of the underlying asset less the PV of any future benefits & plus the PV of any future costs expected to be earned or incurred on the underlying before the expiration of the option
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Put-Call-Forward Parity

Taking both formulas into account we can derive the so-called put–call-forward parity, which takes the following form:

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Put-Call-Forward Parity
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\(p_0-c_0=\frac{X-F}{(1+r)^T}\)

  • \(c_{0}\) - current price of a call option
  • \(X\) - exercise price
  • \(r\) - risk-free interest rate
  • \(T\) - time to expiration (number of days to expiration divided by 365)
  • \(p_{0}\) - current price of a put option
  • \(F\) - forward price
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Thanks to arbitrage the left-hand side of the equation must equal the right-hand side.

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Put-Call-Forward Parity: Calculations

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Example 1

Suppose we know that the price of a 1-year put on a stock of Hearts Inc. with an exercise price of USD 70 is USD 5 and the forward price of the contract expiring in 1 year is USD 81. Knowing that the annual risk-free interest rate is 10%, determine the price of a call option on a stock of Hearts Inc., with an exercise price of USD 70 that expires in one year.

Because we are given the forward price, we will use the put-call-forward parity formula:

\(p_0-c_0=\frac{X-F}{(1+r)^T}\)

\( c_0=p_0-\frac{X-F}{(1+r)^T}\)

\( c_0=5-\frac{70-81}{1+10\%}=5-\frac{-11}{1.1}=5+10=15\)

\( c_0=5-\frac{70-81}{1+10\%}=\\=5-\frac{-11}{1.1}=5+10=15\)

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Example 2

Let’s assume that the prices of a call option and a put option on the same asset and expiring at the same date and having the same exercise price are equal. What conclusions can we reach from this assumption?

(...)

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Level 1 CFA Exam Takeaways for Put–Call-Forward Parity

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  1. The relationship between a put option, a call option, a forward contract, and the exercise price is called put-call-forward parity.
  2. Both options included in the put-call-forward parity are European-style options, have the same expiration date, have the same exercise price, and cover the same quantity of the underlying.
  3. Forward price is equal to the options’ exercise price if only the prices of the options are the same.
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