2 December 2018

“Option Greeks” is the common name for the sensitivities in price of options or, in general, of derivatives and portfolios of derivatives, with respect to different factors, such as spot price of the underlying, implied volatility, risk-free interest rate, etc. Each Greek represents the ratio between the change in price and a small change in one of the above-mentioned factors, all else equal. If an explicit formula for the price of the derivative exists, Greeks can be computed as partial derivatives with respect to the corresponding factor. For this reason, we can divide the Greeks into first-order Greeks, which are first order partial derivatives, and higher order Greeks.

Given that Greeks depends, in general, on the same factors as the price, higher order Greeks can also be considered as the sensitivities of the lower order Greeks with respect to a set of factors. With this perspective, higher order Greeks are a useful tool to understand the dynamics of the exposure of a derivative or portfolio.

In the first part of this article, we illustrate some of the most common and useful higher order Greeks. Moreover, we show their value for plain-vanilla call and put options as computed in the BSM framework for different times to expiry and underlying’s prices. In the second part, we show the Greeks we described before for a set of popular option strategies: a butterfly, a risk reversal, a bull call spread, a call ladder, a 1×2 ladder and a tripod.

**Vanna**

It is the change in Delta for a small change in IV or, equivalently, the change in Vega for a small change in spot price. It is positive for ITM puts and OTM calls and negative for OTM puts and ITM calls.

It is most interesting to consider the effect of Vanna in the context of non-constant IV across strikes. First, we consider a volatility smile where the IV is constant with respect to moneyness and minimum ATM. In this case, Vanna works against both long ATM puts and calls:

- as the spot price falls, the IV increases because we are moving along the volatility smile. Due to positive Vanna, this increase in IV makes the Delta increase towards 1 for calls and 0 for puts;
- as the spot price rises, the opposite happens: the IV increases because we are moving along the volatility smile and, due to negative Vanna, this increase in IV makes the Delta decrease towards 0 for calls and -1 for puts.

Thus, Vanna would decrease our Delta exposure when the spot price moves favorably for us and would increase it when the spot price is moving against us.

Second, we consider the effect of the negative correlation between returns and volatility. This relation, together with Vanna, has a positive effect for long OTM put and ITM call options: as the spot price rises (falls), volatility falls (rises). The negative Vanna makes the delta increase (decrease). Recalling that Vanna is positive for strikes greater than the spot price, it is trivial to see that the opposite holds for long ITM put and OTM call options.

**Charm**

Also known as “Delta bleed”, It is the change in Delta for a small change in time or, equivalently, the change in Theta for a small change in spot price.

It is useful to understand the evolution of Delta through time. Delta essentially tends towards the extremes with passage of time: for ITM puts and OTM calls it decreases respectively to -1 and 0, while for OTM puts and ITM calls it increases respectively to 0 and 1. The effect of Charm is higher for shorter term expiries.

**Speed**

It is the change in Gamma for a small change in spot price, the third partial derivative of the option price with respect to the spot price.

If the spot price is below the strike, you pick up gamma as the spot increases, while if the spot price is above the strike, you shed gamma as the spot increases. The Speed effect is higher for for shorter term expiries.

In a Delta-hedged strategy, as we profit from the realized volatility we want our Gamma exposure to increase. If we consider a Delta-hedged long option position, this means that:

- if the spot price is lower than the strike, we want realized volatility in the form of a rally;
- if the spot price is higher than the strike, we want realized volatility in the form of a sell-off.

**Zomma**

It is the change in Gamma for a small change in IV, or, equivalently, the change in Vanna for a small change in spot price. This Greek is crucial for dynamic Delta hedging under stochastic volatility and can be thought of as an exposure to the volatility of the realized volatility (given the typically high correlation between realized and implied volatility).

In general, when we are long a call or put option, we want positive Zomma, which is we want our gamma to increase as the IV increases. Both deep ITM and OTM options have positive Zomma and around ATM options have negative Zomma. This change in sign of Zomma is counterbalanced by the IV smile, as ITM and OTM options are relatively more expensive than ATM options.

**Color**

It is the change in Gamma for a small change in time or, equivalently, the change in Charm for a small change in spot price.

We need positive Color if we want Gamma to be handed to us with passage of time.

**Vega**

Vega is the change in option price for a small change in implied volatility. It is a first order Greek, but here we want to recall that even if the BSM IV is often considered the market’s best estimate of expected volatility for the duration of the option, it can also be interpreted as a basket of adjustments to for factors ignored by the BSM model, such us demand and supply for that particular strike and maturity, stochastic volatility, jumps, etc.

**Volga**

Also known as “Vega convexity” or Vomma, Volga is the change in Vega for a small change in IV.

If we are long options, we want a high and positive Volga such that we will earn more for every increase in IV and lose less if the IV is decreasing.

Volga can also be thought of as an exposure to the volatility of implied volatility.

Although the volatility of implied volatility and the volatility of actual volatility typically have high correlation, this is not always the case. Volga trading is a bet on changes on the price due to factors ignored by the BSM model: uncertainty in supply and demand, stochastic actual volatility, jumps, etc.

**Veta**

It is the change in Vega for a small change in time or, equivalently, the change in Theta for a small change in IV. It is almost always negative, with the exception of very long time to expiry with high risk-free interest rate.

**Option strategies**

Here, we chose to illustrate the Greeks of seven common options strategies: butterfly, a risk reversal, a bull call spread, a call ladder, a 1×2 ladder and a tripod. Many strategies are simple variations of these ones, so their Greeks can be derived easily from the one reported below.

**Butterfly**

A butterfly is a non-directional, short volatility strategy composed of:

- long one put (or call) with strike A
- short two puts (or calls) with higher strike B
- long one put (or call) at an even higher strike C

This short volatility strategy is less risky than a short straddle or strangle.

Our example strategy is composed of:

- short one put and one call with strike 120
- long one put with strike 108 (spread 10%)
- long one call with strike 132 (spread 10%)

**Risk reversal**

A risk reversal, or short combo, is a bullish, volatility neutral strategy composed of:

- short one put with strike A
- long one call with higher strike B

This strategy has a pay-off similar to that of a long future, with the exception of a plateau between the two strikes, which makes it more suitable if volatility expectations are uncertain.

Our example strategy is composed of:

- short one put with strike 108
- long one call with strike 132

**Bull Call Spread**

A bull call spread is a bullish, volatility neutral strategy composed of:

- long one call with strike A
- short one call with higher strike B

This bullish strategy has a lower cost and a capped profit potential compared to a simple long call.

Our example strategy is composed of:

- long one call with strike 108
- short one call with strike 132

**Call Ladder**

A call ladder is a bearish, short volatility strategy composed of:

- long one call with strike A
- short one call with higher strike B
- short one call with even higher strike C

This strategy delivers maximum profit when the underlying settles between strikes B and C.

Our example strategy is composed of:

- long one call with strike 108
- short one call with strike 132
- short one call with strike 144

**Double Ladder**

A double ladder, or short 2 by 1 ratio call spread, is a non-directional, short volatility strategy composed of:

- long one call with strike A
- short two calls with higher strike B

This strategy profits if the underlying settles at B.

Our example strategy is composed of:

- long one call with strike 108
- short two calls with strike 132

**Tripod**

A tripod, or call spread versus put, is a bullish, short volatility strategy composed of:

- short one put at strike A
- long one call at higher strike B
- short one call at even higher strike C

This bullish strategy is similar to a call spread, but the premium from short put position reduces its cost.

Our example strategy is composed of:

- short one put with strike 108
- long one call with strike 120
- short one call with strike 132

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