The Greeks are the quantities representing the sensitivity of the price of derivatives such as options to a change in underlying parameters on which the value of an instrument or portfolio of financial instruments is dependent. Each risk variable of the (major) Greeks is defined below:

- Δ (Delta) represents the sensitivity in the options value to a change in the underlying price.
- Γ (Gamma) represents the rate of change between an options delta and the underlying price changes, (second-order derivative).
- ϒ (Vega) represents the sensitivity of an options value to the underlying’s volatility.
- Θ (Theta) represents the sensitivity of an options value to time.
- ρ (Rho) represents the sensitivity of an options value to interest rate changes.

For options traders the Greeks are essential, as they provide a guide to analysing the current and potential exposures of a position. If the option holder is planning to keep the option to maturity the Greeks become potentially irrelevant. However if the trader wants to manage the exposures of options then the sensitivities are fundamentally important. If buying an option gives a trader a positive delta exposure, then that trader needs to hedge by generating a short position. Similarly, estimates of the change in position will guide a trader as to which future trades may be necessary to rebalance the exposure.

For a portfolio of positions, the sensitivities are broadly additive, and care should be taken when the maturities of instruments differ. For example a single vega exposure may be misleading since the volatilities of instruments with different maturities/strikes may change at different rates. As time passes the sensitivities of a portfolio change irrespective of movements in other market factors and option prices will change along with the delta and gamma of the options (bleed). Combinations of options give rise to variability in the gamma of a portfolio as the underlying price changes. In this instance the amount of gamma may rise and fall but additionally the gamma may change sign.

**Delta **

Delta (Δ) is the sensitivity of the option price to movements in the price of the underlying asset. Delta is the most important measure of option sensitivity. Delta can be represented as (where C = security price, and S = asset price):

For an individual option, delta is usually expressed as a percentage. For instance, if an option had a 50% delta, then if a ‘small change in the price of the underlying asset’ was USD 1, the option price should increase by about USD 0.50. Note that while the delta for a simple call option is positive, the delta of a put will be negative. As the price of the underlying asset rises, the value of a put option will decrease. When talking about a portfolio of positions, the idea of delta as a percentage becomes less meaningful. Instead, delta is better considered as the amount of market exposure to some underlying asset. The concept of delta as positional exposure extends beyond the option market. If a trader owns a security, this can be represented as having 100% delta in the amount of that security. Many non-option trading desks therefore use the term delta in describing position risks.

Delta neutrality can be achieved in a multitude of ways. The delta of an option can be eliminated not only through the purchase/sale of an underlying asset, but also by buying/selling an option that has (or options that have) equivalent delta. Similarly, a trader looking to be ‘long’ some market can achieve this either by buying the underlying security, by buying call options, or by selling put options. Although the payoff from each of the strategies may be quite different, in each case the trader is better off if the market rises. The position is additive; as long as a trader is consistent about the units of measurement, the instantaneous position risk of a portfolio will be the sum of the deltas from the different instruments. Note that the delta of a position will change with time. This is especially the case with options, where the delta must theoretically be rebalanced continuously. The correct hedge for today will not be appropriate for tomorrow.

**Gamma**

Gamma (Γ) is the rate of change of the delta of an option, with respect to a change in the underlying price. An option always has a positive gamma, although gamma may tend to zero. A positive gamma means that as the price of the underlying asset rises, the effective position thrown up by ownership of the option also increases; for a call option the positive delta goes up, for a put option the negative delta is reduced. Positive gamma generates the ‘super-trader’ effect; the position gets longer as the market rises, shorter as it falls. There is a flipside to the ‘super-trader’ effect of being long gamma – the option has to be paid for first. The profit that a long option trader makes from rebalancing may well be less than the premium outlay. Indeed, one-way of thinking about the premium price of an option is that it is ‘a fair price’ equivalent to the amount of gamma profit that a buyer should generate over time. An increase in volatility will increase the gamma of options that have strikes away from ‘the money’, while decreasing the gamma of at-the-money options. Gamma can be represented as (where C = security price, and S = asset price):

The delta exposure generated by an option can be eliminated by trading in linear securities, such as an underlying asset or a forward contract on that asset. The delta for these securities is constant and the gamma is zero. In order to hedge the gamma of a portfolio, it is necessary to trade in securities that also generate gamma positions – essentially an option trader must hedge options with other options. However, just as with delta hedging, gamma hedging will itself require rebalancing, especially since the rate of change of gamma with respect to time will itself differ between securities. If the perfect continuous-time/price securities market idealized in the Black-Scholes formulas actually existed, then there would be no need to worry about the gamma of a portfolio. Option positions could be hedged by continuous trading in the underlying asset. However, in reality rebalancing costs money and involves some risk; market conditions will not perfectly equate to the idealized situation. If a portfolio can be constructed so that delta rebalancing occurs less frequently, then this may be advantageous. Since the gamma position of an option trader represents the amount of rebalancing required, then that position needs to be diminished in order to reduce the requirement to rebalance.

Gamma is almost at a maximum for ‘at-the-money’ options. This makes sense as deeply ‘in’ or ‘out’ of the money options will reach extreme values, which will not change particularly if the underlying price changes. Gamma is at a maximum when delta equals 0.5, i.e. when the option is close to the money, but not at the money.

**Vega**

Vega represents the sensitivity of a security price with respect to the change in volatility. It is also known as kappa or zeta. Vega can be represented as (where C = security price, and σ = volatility):

The volatility used in these calculations will generally be the implied volatility that is currently associated in the market with options of this type (think of volatility as a market price, not an estimation of market movements). Conventionally, a vega value is multiplied by the underlying position to express a cash amount that would be made or lost if the implied volatility used to price an option increased by 1%. An operator who is long vega (typically the owner of an option/s) will make money if implied volatility rises. Simple European-style puts and calls with identical strikes/maturities will have identical vega exposure. For simple options vega is always positive, though it may be close to zero.

**Theta**

Theta (θ) measures the rate of change in an option value with respect to a change in the remaining maturity (time) of the option. If all the parameters (asset price, risk-free rate, volatility, and so on) remain constant, the value of the option will be constantly reduced as time passes. This is because the option’s time value is lost. Therefore, theta is a measure of time decay. Theta is nearly always negative as, if nothing else changes, an option price declines as it approaches expiry and the ‘optionality’ disappears. As an option reaches expiry, the absolute level of theta for ATM options increases, whilst for OTM/ITM it decreases. In many respects theta is a simple reverse of gamma with one key exception: the passage of time is a certainty whereas changes in the underlying price are unpredictable. Theta can be represented as (where C = security price, and T = time period):

**Rho **

Rho measures the sensitivity of a security price to the change in risk-free rate. The price of a simple call option on a non-futures contract increases with rises in rates because the forward price rises – the converse is true for a put. The amount of increase/decrease will rise with time. Deeply ITM calls and puts will have the largest (absolute) amount of rho, since these options are almost certain to be exercised (they have a delta of approximately +/-100%), the value will move in step with the change in the forward prices. Rho can be represented as (where C = security price, and r = risk-free rate):

**Table of Greeks**