Appendix - Shadow Greeks
Derivatives-based Portfolio Solutions


How a volatility surface changes over time can impact the profitability of a position. While the most important aspects have already been covered (and are relatively well understood by the market) there are ‘second order’ Greeks that are less well understood. Two of the most important are the impact of the passage of time on skew (volatility slide theta), and the impact of a movement in spot on OTM options (anchor delta). These Greeks are not mathematical Greeks, but are practical or ‘shadow’ Greeks.

As an option approaches expiry, its maturity decreases. As near-dated skew is larger than far-dated skew, the skew of a fixed maturity option will increase as time passes. This can be seen by assuming that skew by maturity (eg, three-month or one-year) is constant (ie, relative time, the maturity equivalent of sticky moneyness or sticky delta). We also assume that three-month skew is larger than the value of one year skew. If we buy a low strike one year option (ie, we are long skew) then, assuming spot and ATM volatility stay constant, when the option becomes a three-month option its implied will have risen (as three-month skew is larger than one-year skew and ATM volatility has not changed). We define ‘volatility slide theta’ as the change in price of an option due to skew increasing with the passage of time.

Given that skew increases as maturity decreases, this change in skew will increase the value of long skew positions (as in the example) and decrease the value of short skew positions. The effect of ‘volatility slide theta’ is negligible for medium- to far-dated maturities, but increases in importance as options approach expiry. If a volatility surface model does not take into account ‘volatility slide theta’, then its impact will be seen when a trader re-marks the volatility surface.

The constant smile rule (CSR) details how forward starting options should be priced. The impact of this rule on valuations is given by the ‘volatility slide theta’ as they both assume a fixed maturity smile is constant. The impact of this assumption is more important for forward starting options than for vanilla options.

Volatility surfaces are normally modelled via a parameterisation. One of the more popular parameterisations is to set the ATMf volatility from a certain level of spot, or ‘anchor’, and then define the skew (slope). While this builds a reasonable volatility surface for near ATM options, the wings will normally need to be slightly adjusted. Normally a fixed skew for both downside puts and upside calls will cause upside calls to be too cheap (as volatility will be floored) and downside puts to be too expensive (as volatility should be capped at some level, even for very low strikes). As the ‘anchor’ is raised, the implied volatility of OTM options declines (assuming the wing parameters for the volatility surface stay the same). We call this effect ‘anchor delta’.

Implied volatility has to be floored, and capped, for values to be realistic
There are many different ways a volatility surface parameterisation can let traders correct the wings, but the effect is usually similar. We shall simply assume that the very OTM call implied volatility is lifted by a call accelerator, and very OTM put implied volatility is lowered by a put decelerator. This is necessary to prevent call implieds going too low (ie, below minimum realised volatility), or put implieds going too high (ie, above maximum realised volatility).

Traders tend to refresh a surface by only changing the key parameters
For liquid underlyings such as indices, a volatility surface is likely to be updated several times a day (especially if markets are moving significantly). Usually only the key parameters will be changed, and the less key parameters such as the wing parameters are changed less frequently. We shall assume that there will be many occasions where there is a movement in spot along the skew (ie, static strike for near ATM strikes). In these cases a trader is likely to change the anchor (and volatility at the anchor, which has moved along the skew), but leave the remaining skew and wing parameters (which are defined relative to the anchor) unchanged. In order to have the same implied volatility for OTM options after changing the anchor, the call accelerator should be increased and the put decelerator decreased. In practice this does not always happen, as wing parameters are typically changed less frequently.

OTM options have a second order ‘anchor delta’

To simplify the example we shall assume the call wing parameter increases the implied volatility for strikes 110% and more, and the put wing parameter decreases the implied volatility for strikes 90% or lower. If spot rises 10%, the 120% call implied volatility will suffer when the anchor is re-marked 10% higher, because the call implied volatility is initially lifted by the call wing parameter (which no longer has an effect). OTM calls therefore have a negative ‘anchor delta’ as they lose value as anchor rises. Similarly, as anchor rises the effect of the put wing will increase, lowering the implied volatility of puts of strike 90% or less as anchor rises. So, under this scenario all options that are OTM have a negative ‘anchor delta’ that needs to be hedged.

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