Appendix - The Greeks & Their Meaning
Derivatives-based Portfolio Solutions


 

Greeks is the name given to the (usually) Greek letters used to measure risk. We give the Black-Scholes formula for the key Greeks and describe which risk they measure. As a reminder, the variables for the below formulae are identical to the formulae in the Black-Scholes Formula section. In addition:
N'(z) is the normal density function, ( (e(-x2/2) ) / √(2π) )
N(z) is the cumulative normal distribution, i.e. N(0) = 0.5


DELTA MEASURES EQUITY EXPOSURE
The most commonly examined Greek is delta, as it gives the equity sensitivity of the option (change of option price due to change in underlying price). Delta is normally quoted in percent. For calls it lies between 0% (no equity sensitivity) and 100% (trades like a stock). The delta of puts lies between -100% (trades like short stock) and 0%. If a call option has a delta of 50% and the underlying rises USD1, the call option increases in value USD0.50 (= USD1 * 50%). Note the values of the call and put delta in the formula below give the equity sensitivity of a forward of the same maturity as the option expiry. The equity sensitivity to spot is slightly different. Please note that there is a (small) difference between the probability that an option expires ITM and delta.
Call Delta = N(d1)
Put Delta = -N(-d1) = N(d1)-1


GAMMA MEASURES CONVEXITY (AMOUNT EARNED DELTA HEDGING)
Gamma measures the change in delta due to the change in underlying price. The higher the gamma, the more convex is the theoretical payout. Gamma should not be considered a measure of value (low or high gamma does not mean the option is expensive or cheap); implied volatility is the measure of an option’s value. Options are most convex, and hence have the highest gamma, when they are ATM and also about to expire. This can be seen intuitively as the delta of an option on the day of expiry will change from approx. 0% if spot is just below expiry to approx. 100% if spot is just above expiry (a small change in spot causes a large change in delta; hence, the gamma is very high).
Gamma = - ( N'(d1) ) / ( S√Tσ )


THETA MEASURES TIME DECAY (COST OF BEING LONG GAMMA)
Theta is the change in the price of an option for a change in time to maturity; hence, it measures time decay. In order to find the daily impact of the passage of time, its value is normally divided by 252 (trading days in the year). If the second term in the formula below is ignored, the theta for calls and puts are identical and proportional to gamma. Theta can therefore be considered the cost of being long gamma.
Call Theta = - ( Sσ N'(d1) ) / ( 2√T ) - rKe(-rT) N(d2)
Put Theta = - ( Sσ N'(d1) ) / ( 2√T ) + rKe(-rT) N(-d2)


VEGA MEASURES VOLATILITY EXPOSURE (AVERAGE OF GAMMAS)
Vega gives the sensitivity to volatility of the option price. Vega is normally divided by 100 to give the price change for a 1 volatility point (ie, 1%) move in implied volatility. Vega can be considered to be the average gamma (or non-linearity) over the life of the option. As vega has a √T in the formula power vega (vega divided by square root of time) is often used as a risk measure (to compensate for the fact that near dated implieds move more than far-dated implieds).
Vega = ( S√T ) N'(d1)


RHO MEASURES INTEREST RATE RISK (RELATIVELY SMALL)
Rho measures the change in the value of the option due to a move in the risk-free rate. The profile of rho vs spot is similar to delta, as the risk-free rate is more important for more equity sensitive options (as these are the options where there is the most benefit in selling stock and replacing it with an option and putting the difference in value on deposit). Rho is normally divided by 10,000 to give the change in price for a 1bp move.
Call Rho = KTe(-rT) N(d2)
Put Rho = - KTe(-rT) N(-d2)


VOLGA MEASURES VOLATILITY OF VOLATILITY EXPOSURE
Volga is short for VOLatility GAmma, and is the rate of change of vega due to a change in volatility. Volga (or Vomma/vega convexity) is highest for OTM options (approximately 10% delta), as these are the options where the probability of moving from OTM to ITM has the greatest effect on its value. For more detail on Volga, see the section Measuring Skew.
Volga = ( S √T d1 d2 N'(d1) ) / σ


VANNA MEASURES SKEW EXPOSURE
Vanna has two definitions as it measures the change in vega given a change in spot and the change in delta due to a change in volatility. The change in vega for a change in spot can be considered to measure the skew position, as this will lead to profits on a long skew trade if there is an increase in volatility as spot declines. The extreme values for vanna occur for approx. 15 delta options, similar to volga’s approx. 10 delta peaks. For more detail on vanna, see the section Measuring Skew.
Vanna = ( -d2 N'(d1) ) / σ
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