Skip to content
All library documents

Put-Call Parity: A Covered Call Equals a Short Put Plus Cash

Article Quant Q&A · Author: Victor123
Read the original

Summary

The document corrects a proposed equivalence between a long stock position with a written covered call and a short put struck at a lower price after subtracting the call premium. It explains that put-call parity instead relates the stock, call, discounted strike cash, and put. Under that relationship, a covered call using a given strike corresponds to a short put at the same strike combined with cash equal to the discounted strike amount.

The explanation points to payoff diagrams and the parity equation as the reasoning, and notes that the relationship can also be rearranged to infer a no-arbitrage put price. The example gives a stock purchase price, call strike, and premium, but the stated equivalence is not a naked put at the reduced purchase price. Applying the relation in practice depends on matching contract terms and accounting for discounting; the brief answer does not discuss dividends, transaction costs, or other market frictions.

Key ideas

  • Put-call parity links the underlying, call, put, strike, and discount factor.
  • A covered call can be represented as a short put at the same strike plus discounted strike cash.
  • The call premium does not make the equivalent put strike equal to the stock cost minus that premium.
  • Rearranging parity provides a way to derive a no-arbitrage put price when the other inputs are known.

Tags

OptionsEquitiesDerivatives pricingArbitrage
Full text
# What is the strike of a short put that mimics a covered call


# What is the strike of a short put that mimics a covered call












If I am long a stock $X$ which I purchased at $\$100$ and sold a covered call in the front month with strike $\$105$ for $\$2$ then is it true that the covered call is equivalent to a naked put at strike $\$100 - \$2 = \$98$?

Am I missing something here?

## Answer by jaamor (score 2, accepted)

https://quant.stackexchange.com/a/16757

This is not quite right.

The covered call you are describing is equal to selling a Put with the same strike price (\$105) and holding ( \$105 / (1+r) ) in the bank. If you draw the Payoff diagram this will become apparent.

Put call relationships are summarized as the Put-Call parity:

$$ S - C = D \cdot K - P $$ Where $S$ the underlying, $D$ is the discount factor and $K$ the strike price.

The left side is the covered call you are describing and the right side the Put plus cash. As a bonus, using this relationship you can calculate the no arbitrage price of the put!

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.