Naked And Covered Positions
A naked call option is written without owning the underlying asset, whereas a covered call is a short call option where the writer owns the underlying asset. Neither of these positions is a hedged position.
A Stop-Loss Strategy
Stop-loss trading strategies are designed to minimize losses in the event the price of the underlying exceeds the strike price of a short call-option position.
Delta Hedging
To completely hedge a long stock or short call position, an investor must purchase the number of shares of stock equal to delta times the number of options sold. Another term for being completely hedged is delta-neutral.
A forward/futures contract position can easily be hedged with an offsetting underlying asset position with the same number of securities.
Delta
The delta of an option, Δ, is the ratio of the change in price of the call option, c, to the change in price of the underlying asset, s, for small changes in s.
Dynamic Aspects of Delta Hedging
Delta-neutral hedges are sophisticated hedging methods that minimize changes in a portfolio’s position due to changes in the underlying security.
Delta-neutral hedges are only appropriate for small changes in the underlying asset and need to be rebalanced when large changes in the asset’s value occur.
Portfolio Delta
The delta of a portfolio is a weighted average of the deltas of each portfolio position.
Theta, Gamma, Vega, And Rho
Theta, also referred to as the time decay of an option, measures the sensitivity of an option’s price to decreases in time to expiration.
Gamma measures the sensitivity of an option’s price to changes in the option’s delta.
Vega measures the sensitivity of an option’s price to changes in the underlying asset’s volatility.
Rho measures the sensitivity of an option’s price to changes in the level of interest rates.
Gamma-neutral
Gamma is used to correct the hedging error associated with delta-neutral positions by providing added protection against large movements in the underlying asset’s price.
Gamma-neutral positions are created by matching the gamma of the portfolio with an offsetting option position.
The Relationship Between Delta, Theta, Gamma, And Vega
Theta, delta, and gamma directly affect the rate of return of an option portfolio.
Stock option prices are affected by delta, theta, and gamma as indicated in the following relationship:
rΠ = Θ + rSΔ + 0.5σ2S2Γ
where:
r = the risk-neutral risk-free rate of interest
Π = the price of the option
Θ = the option theta
S = the price of the underlying stock
Δ = the option delta
σ2 = the variance of the underlying stock
Γ = the option gamma
Hedging Activities In Practice
Hedging usually entails actively managing a delta-neutral position while monitoring the other option Greek sensitivities.
Portfolio Insurance
Portfolio insurance is the combination of (1) an underlying instrument and (2) either cash or a derivative that generates a floor value of the portfolio in the event that market valuations decline, while allowing for upside potential in the event that market valuations rise.
