In this article, we will continue the exploration of different Greeks and examine the interesting second-order Greek known as “Charm”.

Charm is defined as the rate of change of Delta with respect to time, or equivalently, the rate of change of Theta with respect to the underlying spot.

Charm is important because it provides insight into the relationship between spot and time in option pricing. As for Theta, a positive direction signifies time advancing by one day or another unit, which is equivalent to a reduction in maturity.

When an option approaches maturity, its value can change quickly, making Charm especially important. The same applies to structured products with early termination opportunities, such as autocalls, since the determination of whether the product terminates can change significantly close to the relevant date.

The exact definition of Charm C(S,t,..), is given by:

C(S,t,..)=∂^2 f(S,t,..)/∂t ∂S

Where t is time elapsed, S is the spot price and f(..) is the option price function. This second-order derivative measures the change in price with respect to both volatility and time. As with all mathematical partial derivatives, the order of evaluation can be reversed. Thus:

This explains the two alternative definitions mentioned above, the rate of change of Theta with respect to spot and the rate of change of Delta with respect to time.

Charm is the last of the second-order Greeks to be covered in this series, as we have previously looked at Gamma and Vanna. These three are the most commonly measured and acted upon of the second-order Greeks by options and derivatives traders. Holding the right amount of Delta hedge is generally the most critical decision a trader has to make during the life of an option or derivative, since changes in the underlying directly translate to changes in its price. These second-order Greeks measure the change of Delta with respect to spot (Gamma), volatility (Vanna) and now time in the case of Charm. Hence, they represent quantities that anticipate changes in the required Delta hedge.

For an at-the-money call option with 2-year maturity, the chart below shows the values of Charm, Vanna and Delta rescaled as necessary.

Both Charm and Vanna exhibit curves that are positive and negative in different regions, this is unlike Delta and Gamma, which are always positive. Charm and Vanna are also opposite in sign and approximately of the same magnitude to each other right along the spot curve, as can be seen from the chart. This is to be expected for vanilla options, since increase in volatility and time to maturity both increase the variance of the final stock price and Charm moves in the direction of decreasing maturity.

Charm tends to be more pronounced as an option or product approaches expiration. This is because the rate of change in Delta accelerates as the option’s time value (Theta) decays more rapidly. This acceleration can create significant operational challenges for portfolios with short-dated options.

Although all the second-order Greeks analyse future changes in Delta hedge, Charm is unique in that time advancing is a deterministic process, unlike the variability of spot and volatility. For vanilla options, the main challenge arises near maturity when Delta can change quickly if the option is close to being at-the-money as ultimately it will end up as zero or one. If the option is far away from at-the-money, then Charm is generally less critical.

However, the position for Charm is more complex for structured products due to their path dependency and early maturity potential. Because the actual maturity can vary, the value of Charm may change significantly as an autocall date approaches.

Autocallable structured products present unique Charm challenges due to their multiple observation dates and barrier features. The Delta of an autocallable product can vary rapidly near the auto-call dates and levels, making Delta hedging costly to implement. As these products approach observation dates, Charm effects intensify around autocall levels, creating discontinuous risk profiles.

The multi-asset nature of many autocallables increases Charm sensitivity. This is particularly evident in the popular "worst-of" autocallables, where Charm behaviour becomes path-dependent based on which asset is performing worst.

Capital protection barriers can also cause changes in Charm. As the underlying approaches the barrier, particularly when there is little maturity left, Charm can spike dramatically, creating significant rebalancing costs and risks for issuers.

Unlike first-order Greeks that exhibit relatively stable behaviour, Charm can experience rapid non-linear changes, particularly in the final weeks before expiration. As with other second-order Greeks, this creates modelling challenges because of the need to calculate stable differences in price evaluations for second-order parameter changes.

The values of Charm for all the individual products are aggregated for the SRP Greeks service powered by FVC analytics by each underlying. These values will change quite quickly over time due to the sensitive nature of this Greek and its dependence on multiple variables. Charm is dominated by large notional products as they approach maturity or an autocall date and this data provides valuable insight into market exposure and potential overall Delta changes in the coming days or weeks.