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Does High Volatility Also Increase the Premium of the Sold Call Option?

Yes, high implied volatility increases the premium of both the purchased put and the sold call option. Since the call option also benefits from the increased chance of a large upward price movement, its value increases.

In a collar, the higher call premium is beneficial as it helps to offset the increased cost of the put.

Does High Implied Volatility Increase or Decrease the Delta-as-Probability Accuracy?
Does a High Volatility Environment Increase or Decrease the Time Value of an OTM Option?
Can a DAO Treasury Also Be Secured by a Multisig Wallet?
Does a Deep ITM Option Also Have Gamma near Zero?