Second Order Greeks

Look for a new TTT today.
We will follow a debit spread when the long option is in the money.


As option traders, we want to know about ‘the Greeks’ and how they influence our trading. Delta, vega, theta, and gamma become our allies in the risk management business. The 2nd order Greeks are less important, less is written about them, and I don’t know any service that supplies values for those Greeks. Nevertheless, it cannot hurt to learn more about them. We may even discover an explanation for something that has been puzzling us.

Sometimes we seek education for its own sake. It feels good to learn about ‘stuff’ that does not affect our everyday lives. But please remember that a general idea about these 2nd order Greeks may be all that anyone needs. Do not make an effort to understand everything in this post in complete detail. For now just knowing that these risk parameters are available – should you need them – is enough.

Introduction

The Greeks are important tools in risk management. Each measures a specific sensitivity in the value of an options portfolio to a small change in a given underlying parameter (time, volatility, stock price etc). Each component risk can be measured and the portfolio rebalanced if necessary, to reduce that risk to an acceptable level.

Using an options-pricing model (Black-Scholes, for example), the Greeks are easy to calculate. Traders can use them to hedge their portfolios from adverse changes in market conditions. Information on the Greeks which are particularly useful for hedging – delta, theta, vega, and gamma – is readily available. Of these, only gamma is a 2nd order Greek.

The term ‘order’ refers to how many derivative levels are needed to calculate the specific Greek. For example, delta is a first order Greek because it represents the sensitivity of the options price with respect to changes in the stock price.

Gamma is a 2nd order Greek because it represents the rate of change of the delta (itself a 1st order Greek) with respect to a change in the stock price.

The other 2nd and 3rd order Greeks

Charm describes the rate at which delta changes as time passes. It represents ‘delta decay,’ a term that is unfamiliar to many option traders. Charm allows us to understand why time plays a big role when we look at the delta risk of a position.

    Example: Looking at a stock, price $54, IV 35, no dividend, 42 days before expiration. The Dec 50 put is worth $0.96 and has a delta of -24. As time passes, the danger (‘delta threat’) of being short that option declines. Two weeks prior to expiration, the value has decreased to $0.23 and the delta has also declined to -12. Two days before expiration the put becomes worth less than one penny with a delta of -0.1. Note that the stock remained unchanged, but theta had an affect on delta.

    And what if the stock dips and time passes? If that stock declines from 54 to 52, and if time to expiration declines from 42 to 12, then the option loses value. It’s worth only $0.53, with a -26 delta. If the stock moves to 51 and only 5 days remain, then .

    That’s charm.

For anyone trying to maintain a delta-neutral portfolio, paying attention to charm makes it more efficient to manage a portfolio over a weekend.

Note: This is not important for traders who manage ‘income generating’ strategies. Charm works in their favor, and it’s not crucial to crunch the numbers. Charm becomes important for a portfolio manager who cannot afford to lose money over the weekend.

For premium sellers, when a significant price change occurs soon after the trade, money is lost. When time has passed, that price change is not as costly. Charm quantifies the idea that when an option has a high chance of expiring worthless, delta for OTM options decreases.

Please note that if the option becomes ATM, that is a very different situation. Above, we discussed OTM options. When an option has a 50-delta, it is ATM. When the stock moves, the option gains or loses value quickly, depending on the direction of the move. [I often refer to the risk of owning short option positions near expiration. This is the problematic situation: Close to the money (CTM) options have high gamma, and delta changes quickly when the stock moves.]

Mathematically, charm is the derivative of theta with respect to the stock price.

Color measures the rate of change of gamma as time changes. Color is a third-order derivative.

The manager of a Greek ‘neutral’ portfolio uses color to help anticipate the effectiveness of a hedge as time passes. When we own options (kite spread, for example) color allows us to see how long the options will be effective. I’ve often stated that time is not friendly to a kite spread.

DvegaDtime measures the rate of change in the vega with respect to the passage of time. It’s the second derivative; once to volatility and once to time.
Apparently, there is no common name for this Greek.

DvegaDtime tells us that our vega risk diminishes as time passes.

Speed measures how Gamma changes as the stock price changes. Speed is sometimes referred to as the gamma of the gamma.

Speed is a third-order derivative and can be important to monitor when seeking delta neutrality. Gamma may be under control, but if speed is too high, that gamma gets explosive at the wrong time for premium sellers.

Vomma is the rate at which vega changes as the volatility of the underlying changes.

It is the second derivative of the option value with respect to volatility, and is useful for volatility-based option trades.

Positive vomma positions gain vega as implied volatility increases, and lose vega as it decreases. Vomma is positive for options away from the money, and initially increases with distance from the money (but declines as vega declines).

Ultima measures the sensitivity of the option vomma with respect to change in volatility. It’s a third-order Greek.

Vanna is a second order derivative of the option value, once to the underlying price and once to volatility. It is mathematically equivalent to the sensitivity of the option delta with respect to change in volatility.

Vanna can be a useful sensitivity measure to monitor when maintaining a delta- or vega-hedged portfolio because vanna helps the trader anticipate changes in the effectiveness of a delta-hedge as volatility changes or the effectiveness of a vega-hedge against change in the underlying price.

Thesse 3rd order Greeks seem trivial, but if you can get the values (see end of post), then you can understand ore about the chances that an adjustment will be effective. Not all adjustments work as we thought they would – and vanna may offer an explanation as to why that is true.

Zomma measures the rate of change of gamma with respect to changes in volatility. Zomma is a 3rd order Greek.

Zomma can be a useful sensitivity monitor when maintaining a gamma-hedged portfolio because zomma assists the trader in anticipating the effectiveness of a hedge as volatility changes.

Calculations

Each of the Greeks mentioned here can be calculated directly from the Black-Scholes equation.

Calculator. I found XXXX online, but do not know if the site is trustworthy (link no longer active, so it was removed: 121204).

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