The instantaneous return and volatility of a covered call position

Abstract

This article derives and examines the instantaneous return and volatility of a covered call position under standard Black–Scholes dynamics and compares it with that of a long position in the underlying asset. It is demonstrated that the instantaneous volatility and instantaneous expected return of the covered call position are always less than or equal those of being long the underlying asset, while the instantaneous Sharpe Ratios of these two positions are equal.

Keywords:

• covered call
• equity options
• instantaneous risk
• investment management

JEL Classification:

• G11
• G12

Notes

¹ Black and Scholes (1973) demonstrated that under this asset price process assumption and given the existence of a riskless tradable bond under complete markets, the value of a European call option at time is , where is the standard normal distribution function, is the instantaneous return on the riskless tradable bond and and .

² Note that under Black–Scholes dynamics, the call option price formula satisfies the Euler Equation and is homogeneous of degree one in and ; that is if the current asset price and the strike price are both scale by the same amount, the value of the call option is scaled by that same amount.

³ Note that unlike the constant instantaneous return and volatility of the underlying stock ( and ), the return and volatility of the covered call in Equations 6 and 7 depend upon and which are not constant, but changing throughout the passage of time within a holding period.

⁴ However, these violations generally have not limited the implementation or usefulness of this common framework for numerous applications in pricing and hedging.