### Options Greeks: Vanna, Charm, Vomma, DvegaDtime

The present article deals with second order Options Greeks and it constitutes the second part of a previously published article entitled **“Options Greeks: Delta,Gamma,Vega,Theta,Rho”**. Before getting started it is important to highlight the great contribution that Liying Zhao (Options Analyst at **HyperVolatility**) gave to this report. All the calculations and numerical simulations that will be shown and commented are entirely provided by Mr Zhao.

Second-order Greeks are sensitivities of first-order Greeks to small changes in different parameters. Mathematically, second-order Greeks are nothing else but the second-order partial derivatives of option prices with respect to different variables. In practical terms, they measure how fast first order options Greeks (Delta, Vega, Theta, Rho) are going to change with respect to underlying price fluctuations, volatility, interest rate changes and time decay. Specifically, we will go through Vanna, Charm (otherwise known as Delta Bleed), Vomma and DvegaDtime. It is important to point out that all charts have been produced by assuming that the underlying asset is a futures contract on WTI crude oil, the ATM strike (X) is 100, risk-free interest rate (r) is 0.5%, implied volatility is 10% while the cost of carry (b) is 0 (which is the case when dealing with commodity options).

**Vanna: **Vanna measures the movements of the delta with respect to small changes in implied volatility (1% change in implied volatility to be precise). Alternatively, it can also be interpreted as the fluctuations of vega with respect to small changes in the underlying price. The following chart shows how vanna oscillates with respect to changes in the underlying asset S:

The above reported chart clearly shows that vanna has positive values when the underlying price is higher than strike (in our case S>$100) and it has negative values when the underlying moves just below it (S<$100). What does that imply? The graph highlights the fact that vega moves much more when the underlying asset approaches the ATM strike ($100 in our case) but it tends to approximate 0 for OTM options. Consequently, the delta is very sensitive to changes in implied volatility when the ATM area is approached. However, it is important to point out that delta will not always increase if the underlying moves from, say, $80 to $100 because in many risky assets (stocks, equity indices, some currencies and commodities) the implied volatility is inversely correlated to the price action. As a result, if WTI futures go from $80 to $100 the implied volatility will probably head south and such a phenomenon would decrease vanna which, in turn, would diminish the value of delta.

**Charm (or Delta Bleed): **Charm measures delta’s sensitivity to a small movement in time to maturity (T). In practical terms, it shows how the delta is going to change with the passage of time. The next chart displays graphically the relationship between the aforementioned variables:

The chart suggests that, like in the case of vanna, the charm achieves its highest absolute values when the options are around the ATM area. Therefore, slightly in-the-money or out-of-the-money options will have the highest charm values. This makes sense because the greatest impact of time decay is precisely on options “floating” around the ATM zone. In fact, deep ITM options will behave almost like the underlying asset while OTM options with the passage of time will approach 0. Consequently, the deltas of slightly ITM or OTM options will be the most eroded by time. Charm is very important to options traders because if today the delta of your position or portfolio is 0.2 and charm is, for instance, 0.05 tomorrow your position will have a delta equal to 0.25. As we can clearly see, knowing the value of charm is crucial when hedging a position in order to keep it delta – neutral or minimize portfolio risk.

**Vomma**: Vomma measures how Vega is going to change with respect to implied volatility and it is normally expressed in order to quantify the influence on vega should the volatility oscillate by 1 point. The fluctuations of vomma with respect to S are shown in the next chart:

As displayed in the above reported chart out-of-the-money options have the highest vomma, while at-the-money options have a low vomma which means that vega remains almost constant with respect to volatility. The shape of vomma is something that every options trader should bear in mind while trading because it clearly confirms that the vega that will be influenced the most by a change in volatility will be the one of OTM options while the relationship with ATM options will be almost constant. This makes sense because a change in implied volatility would increase the probability of an OTM options to expire in-the-money and this is precisely why vomma is the highest around the OTM area.

**DvegaDtime: **DvegaDtime is the negative value of the partial derivative of vega in terms of time to maturity and it measures how fast vega is going to change with respect to the time decay. The next chart is a visual representation of its fluctuations with respect to the underlying asset S:

The above reported graph clearly displays that the influence of time decay on volatility exposure measured by vega is mostly felt in the ATM area especially for options with short time to maturity. The fact that DvegaDtime is mathematically expressed as negative derivatives makes sense because time decay is clearly a price that every options holder has to pay. In order to make things easier have a look at the **plots of vega and theta** because you will immediately realize that both volatility and time decay have their highest and lowest values in the ATM area. It goes without saying that ATM options have the highest volatility potential and therefore vega will be effected the most by the passage of time when the strike of our hypothetical options and the underlying price gets very close.

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