## NAKED & COVERED CALL OPTIONS #

A **naked position** occurs when one party sells a call option without owning the underlying asset.

A **covered position** occurs when the party selling a call option owns the underlying asset.

## EXPECTED VALUE #

Stop-loss strategies with call options are designed to limit the losses associated with short option positions (i.e., those taken by call writers). The strategy requires purchasing the underlying asset for a naked call position when the asset rises above the option’s strike price. The asset is then sold as soon as it goes below the strike price. The objective here is to hold a naked position when the option is out-of-the-money and a covered position when the option is in-the-money.

## DELTA HEDGING #

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. Mathematically:

Delta = Δ = δc/δs

**OPTION DELTA**

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.**

Delta can also be calculated as the N(d_{1}) in the Black-Scholes-Merton option pricing model.

**FORWARD DELTA**

A forward contract position can easily be hedged with an offsetting underlying asset position with the same number of securities.

**FUTURES DELTA**

The delta of a futures position is not ordinarily one because of the spot-futures parity relationship. An investor would hedge short futures positions by going long the amount of the deliverable asset.

## DELTA OF A PORTFOLIO #

The delta of a portfolio of options on a single underlying asset can be calculated as the weighted average delta of each option position in the portfolio:

**Portfolio delta = ****Δ**_{p}** =****Ʃ****w _{i}**

**Δ**

_{i}Where: w_{i} = the portfolio weight of each option position

Δ_{i} = the delta of each option position

## DYNAMIC ASPECTS OF DELTA HEDGING #

When the delta changes, the portfolio will no longer be hedged, and the investor will need to either purchase or sell the underlying asset. This rebalancing must be done on a continual basis to maintain the delta-neutral hedged position.

The goal of a delta-neutral portfolio (or delta-neutral hedge) is to combine a position in an asset with a position in an option so that the value of the portfolio does not change with changes in the value of the asset.

Number of options needed to delta hedge = (number of shares hedged)/ delta of call options.

**MAINTAINING THE HEDGE**

The delta-neutral position only holds for very small changes in the value of the underlying stock. Hence, the delta-neutral portfolio must be frequently (continuously) rebalanced to maintain the hedge. Adjusting the hedge on a frequently basis is known as **dynamic hedging. **If, instead, the hedge is initially set-up but never adjusted, it is referred to as **static hedging**. This type of hedge is also known as a **hedge-and forget strategy.**

**OTHER PORTFOLIO HEDGING APPROACHES**

** **A delta-neutral position can also be created by purchasing the correct number of put options so that:

Δ value of puts = -Δ value of long stock position

When using puts in constructing a delta-neutral portfolio, purchase [1 / (call delta — 1)] put options to protect a share of stock held long. When using calls you would sell (1 / call delta) call options for each long share of stock.

## THETA, GAMMA, VEGA, AND RHO #

**THETA**

Theta,θ,measures the option’s sensitivity to a decrease in time to expiration. Theta is also termed the “time decay” of an option. Theta varies as a function of both time and the price of the underlying asset. Theta for a call option is calculated using the following equation:

θ = δc/δt

Where: δc = change in the call price.

δt = change in time

The specific characteristics of theta are as follows:

- Theta affects the value of put and call options in a similar way .
- Theta varies with changes in stock prices and as time passes.
- Theta is most pronounced when the option is at-the-money, especially nearer to expiration.
- Theta values are usually negative, which means the value of the option decreases as it gets closer to expiration
- Theta usually increases in absolute value as expiration approaches.
- It is possible for a European put option that is in-the-money to have a positive theta value.

**GAMMA**

Gamma, Γ, represents the expected change in the delta of an option. It measures the curvature of the option price function not captured by delta. The specific mathematical relationship for gamma is:

Γ = δ^{2}c/δs^{2 }

Where: δ^{2}c & δs^{2 }= the second partial derivatives of the call and stock prices, respectively.

- When gamma is large, delta will be changing rapidly. On the other hand, when gamma is small, delta will be changing slowly.
- Delta-neutral positions can hedge the portfolio against small changes in stock price, while gamma can help hedge against relatively large changes in stock price. Therefore, it is not only desirable to create a delta-neutral position but also to create one that is gamma- neutral.

**VEGA**

Vega measures the sensitivity of the option’s price to changes in the volatility of the underlying stock. Vega for a call option is calculated using the following equation:

Vega=δc/δσ

Where: δc= change in call price

Δσ = change in volatility

Options are most sensitive to changes in volatility when they are at-the-money. Deep out- of-the-money or deep in-the-money options have little sensitivity to changes in volatility (i.e., vega is close to zero).

**RHO**

Rho, ρ, measures an option’s sensitivity to changes in the risk-free rate. Large changes in rates have only small effects on equity option prices. Rho is a much more important risk factor for fixed-income derivatives.

Rho for a call option is calculated using the following equation:

Rho= δc/δr

Where: δc = change in the call price

δr = change in interest rate

In-the-money calls and puts are more sensitive to changes in rates than out-of-the-money options. Increases in rates cause larger increases for in-the-money call prices (versus out-of- the-money calls) and larger decreases for in-the-money puts (versus out-of-the-money puts).

**RELATIONSHIP AMONG DELTA, THETA, AND GAMMA**

Stock option prices are affected by delta, theta, and gamma as indicated in the following relationship:

This Equation shows that the change in the value of an option position is directly affected by its sensitivities to the Greeks.

## HEDGING IN PRACTICE #

Large financial institutions usually adjust to a delta-neutral position and then monitor exposure to the other Greeks. Two offsetting situations assist in this monitoring activity. First, institutions that have sold options to their clients are exposed to negative gamma and vega, which tend to become more negative as time passes. In contrast, when the options are initially sold at-the-money, the level of sensitivity to gamma and vega is highest, but as time passes, the options tend to go either in-the-money or out-of-the-money. The farther in- or out-of-the-money an option becomes, the less the impact of gamma and vega on the delta-neutral position.

Scenario analysis involves calculating expected portfolio gains or losses over desired periods using different inputs for underlying asset price and volatility. In this way, traders can assess the impact of changing various factors individually, or simultaneously, on their overall position.

## PORTFOLIO INSURANCE #

Portfolio insurance is the combination of (1) an underlying instrument and (2) either cash or a derivative that generates a floor value for the portfolio in the event that market values decline, while still allowing for upside potential in the event that market values rise.

The simplest way to create portfolio insurance is to buy put options on an underlying portfolio. In this case, any loss on the portfolio may be offset with gains on the long put position.