Corollaries of Structured Products Hedging
Derivatives-based Portfolio Solutions


 

The sale of structured products leaves investment banks with a short skew position (eg, short an OTM put in order to provide capital-protected products). Whenever there is a large decline in equities, this short skew position causes the investment bank to be short volatility (eg, as the short OTM put becomes more ATM, the vega increases). Thecovering of this short vega position lifts implied volatility further than would be expected. As investment banks are also short vega convexity, this increase in volatility causes the short vega position to increase in size. This can lead to a ‘structured products vicious circle’ as the covering of short vega causes the size of the short position to increase. Similarly, if equity markets rise and implied volatility falls, investment banks become long implied volatility and have to sell. Structured products can therefore cause implied volatility to undershoot in a recovery, as well as overshoot in a crisis.


IMPLIED VOL OVERSHOOTS IN CRISIS, UNDERSHOOTS IN RECOVERY
The sale of structured products causes investment banks to have a short skew and short vega convexity position. Whenever there is a significant decline in equities and a spike in implied volatility, or a recovery in equities and a collapse in implied volatility, the position of structured product sellers can exaggerate the movement in implied volatility. This can cause implied volatility to overshoot (in a crisis) or undershoot (in a recovery post-crisis). There are four parts to the ‘structured products vicious circle’ effect on implied volatilities.


           1. Equity markets decline
While implied volatility moves – in both directions – are exaggerated, for this example we shall assume that there is a decline in the markets and a rise in implied volatility. If this decline occurs within a short period of time, trading desks have less time to hedge positions, and imbalances in the market become more significant.

2. Desks become short implied volatility due to short skew
Investment banks are typically short skew from the sale of structured products. This position causes trading desks to become short implied volatility following declines in the equity market. To demonstrate how this occurs, we shall examine a short skew position through a vega flat risk reversal (short 90% put, long 110% call).

Short skew & equity markets decline = short vega (i.e. short implied volatility)
If there is a 10% decline in equity markets, the 90% put becomes ATM and increases in vega. As the risk reversal is short the 90% put, the position becomes short vega (or short implied volatility). In addition, the 110% call option becomes more OTM and further decreases the vega of the position (increasing the value of the short implied volatility position).

Even if skew was flat, markets declines cause short skew position to become short vega
The above example demonstrates that it is the fact options become more or less ATM that causes the change in vega. It is not the fact downside put options have a higher implied than upside call options. If skew was flat (or even if puts traded at a lower implied than calls), the above argument would still hold. We therefore need a measure of the rate of change of vega for a given change in spot, and this measure is called vanna.
Vanna = dVega/dSpot

Vanna measures size of skew position, skew measures value of skew position
Vanna can be thought of as the size of the skew position (in a similar way that vega is the size of a volatility position), while skew (i.e. 90%-100% skew) measures the value of skew (in a similar way that implied volatility measures the value of a volatility position). For more details on different Greeks, including vanna, see the section Greeks and Their Meaning.


3. Short covering of short vega position lifts implied volatilities
As the size of trading desks’ short vega position increases during equity market declines, this position is likely to be covered. As all trading desks have similar positions, this buying pressure causes an increase in implied volatility. This flow is in addition to any buying pressure due to an increase in realized volatility and hence can cause an overshoot in implied volatility.

4. Short vega position increases due to vega convexity
Options have their peak vega when they are (approximately) ATM. As implied volatility increases, the vega of OTM options increases and converges with the vega of the peak ATM option. Therefore, as implied volatility increases, the vega of OTM options increases. The rate of change of vega given a change in volatility is called volga (VOLGAmma) or vomma, and is known as vega convexity.
Volga = dVega/dVol

Vega convexity causes short volatility position to increase
As the vega of options rises as volatility increases, this increases the size of the short volatility position that needs to be hedged. As trading desks’ volatility short position has now increased, they have to buy volatility to cover the increased short position, which leads to further gains in implied volatility. This starts a vicious circle of ever increasing volatility.


VEGA CONVEXITY IS HIGHEST FOR LOW-TO-MEDIUM IMPLIED VOLATILITIES
The slope of vega against volatility is steepest (i.e. vega convexity is highest) for low-to-medium implied volatilities. This effect of vega convexity is therefore more important in volatility regimes of approx. 20% or less; hence, the effect of structured products can have a similar effect when markets rise and volatilities decline. In this case, trading desks become long vega, due to skew, and have to sell volatility. Vega convexity causes this long position to increase as volatility declines, causing further pressure on implied volatilities. This is typically seen when a market recovers after a volatile decline such as in 2003 following the end of the tech bubble and in 2009 after the credit crunch.


IMPACT GREATEST FOR FAR-DATED IMPLIEDS
While this position has the greatest impact at the far end of volatility surfaces, a rise in far-dated term volatility and skew tends to be mirrored to a lesser extent for nearer-dated expiries. If there is a disconnect between near- and far-dated implied volatilities, this can cause a significant change in term structure.


STRUCTURED PRODUCT CAPITAL GUARANTEE IS LONG AN OTM PUT
The capital guarantee of many structured products leaves the seller of the product effectively short an OTM put. A short OTM put is short skew and short vega convexity (or volga). This is a simplification, as structured products tend to buy visually cheap options (ie, OTM options) and sell visually expensive options (ie, ATM options), leaving the seller with a long ATM and short OTM volatility position. As OTM options have more volga (or vega convexity) than ATM options (see the section Greeks and Their Meaning) the seller is short volga. The embedded option in structured products is floored, which causes the seller to be short skew (as the position is similar to being short an OTM put).
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