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# The Options Greeks

The following document describes the 5 major variables related to options collectively known as "The Greeks."  Understanding Delta, Gamma, Vega, Theta, and Rho is the key to understanding how options prices are determined.

The Options Greeks

Delta (Δ)

Delta measures the change in the price of an option versus a change in the price of the underlying security.  Delta is the first derivative of the value of an option with respect to the price of the underlying security (δV/δS).  Delta can be perceived as the likelihood that an option will expire in the money.   Options that are at the money have a Delta of slightly greater than 0.5; an option with a delta of 0.9 approximates a 90% chance of the option expiring in the money.  Finally, Delta is used to calculate the equivalent number of shares in an options position.  For example, an investor holding 5 options contracts with a delta of 0.75 holds an equivalent of 375 shares of the underlying stock (position x Δ x 100).  At TFG, we often use the shares equivalent formula to calculate the firm’s exposure in our options portfolio.  For example, if one of the firm’s larger options positions also carries a high delta, we would use this calculation to determine the amount of risk inherent in that position.

Gamma (Γ)

Gamma is the rate of change of delta resulting from the movement in price of the underlying option.  Gamma is largest for options that are at the money and smallest for options that are deeply in or out of the money.  For example, when we buy options it is important to understand that a high gamma indicates that upward movement in the price of the stock can translate into a larger increase in delta.  This in turn will increase the shares equivalent for the position and make the option riskier.

Vega (ν)

Vega measures the change in the price of an option for a change in implied volatility; it is the derivative of the option value with respect to the volatility of the underlying security (δV/δσ).  Vega is typically expressed as the value that an option would gain or lose for a 1% change in implied volatility of the underlying security.  An investor who wishes to be long volatility can employ a long options straddle.  This involves purchasing both a put and a call with an equal strike price and expiration.  The owner of the straddle will profit if the price of the underlying moves away from the strike price in either direction.

Theta (Θ)

Theta measures the change in option price related to that option’s time to expiration.  For a long position, Theta is always negative and measures the rate of time premium decay.  As the expiration date approaches, Theta increases and the option price decreases more rapidly in relation to time.  This is because as expiration approaches there is a decreased probability that the option will expire in the money.  Theta can be expressed in terms of the dollar value that a stock option will lose on a daily basis if all other pricing variables do not change.  If a trader has written an option going into a long weekend and expects that the markets will remain relatively unchanged, he or she may choose to wait before covering their position.  By doing so, the options will lose value due to time decay and the trader will be able to buy back the options at a lower price.

Rho (P)

Rho is a measure of the change in option prices for a change in interest rates.  When interest rates rise, call prices will rise and put prices will fall.  For example, the price of an option with a Rho of 10 will increase roughly ten cents for a 1% rise in interest rates.  While we have not devoted extensive analysis to rho, this variable will become increasingly important when the Fed ultimately begins a tightening cycle.

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