Hedging volatility skew in equity derivative markets is non-trivial in that the delta calculation is subject to assumptions made on volatility surface dynamics. These assumptions represent various volatility regimes, including "sticky strike," "sticky delta" and "sticky local volatility." The deltas calculated under different dynamics can be substantially different.
Using the mathematical chain rule, the theoretical option delta ( ) is:
Where * is the implied volatility and is the volatility delta that specifies the volatility change due to the spot move. If the volatility delta is calculated using an implied volatility surface without any adjustments, one has implicitly assumed that the implied forward volatility and skew will prevail in the future. In reality, however, as time moves forward the volatility skew persists. So does the volatility surface in general. The implied forward volatility and skew from today's volatility surface is not a good prediction for the future volatility evolution. This raises an important question regarding how delta positions should take volatility dynamics into account.
Delta Hedging Volatility Dynamics
To include volatility dynamics in the option delta, one could adjust the volatility delta term and rewrite equation 1 as:
The term XXXX is is the adjusted volatility delta. The value of the L-factor represents a particular view of volatility dynamics. For example, for an option with a fixed strike:
L=-1 Sticky Delta (ATM vol moves with spot);
L=0 Sticky Strike (BlackScholes);
L=1 Sticky Local Vol (Static local vol).
In the "sticky delta" case, we have used the approximation:
Figure 1 plots the delta versus maturity for at-the-money European calls using different L-factors. The market data is a typical FTSE-100 set with skew. With L = -1 ("sticky delta"), long-dated deltas are much larger than those calculated with L = 0 (Black-Scholes). As spot shifts up, the fixed strike becomes lower than the shifted spot. Given that in the "sticky delta" case, the ATM volatility moves with the spot, skew will result in an increase in volatility for the given strike and hence a larger option delta. In contrast, with L = 1 ("sticky local vol"), when spot shifts up, given a static skewed volatility surface, a smaller volatility is used for calculating the option delta. The long-dated option deltas are smaller comparing to those calculated when L = 0 (Black-Scholes).
To put Figure 1 in perspective, Table 1 summarises long-dated cash delta positions for a notional of USD100 million for various L-factors. For a five-year ATM European call, the cash delta positions range from USD49-80 million, with L = 1 being the smallest and L = -1 the largest.
Figure 2 plots the delta versus maturity for ATM European puts using different L-factors. The differences among these curves can be explained similarly as in Figure 1. Table 2 summarises long-dated cash delta positions for a notional of USD100 million.
An L-Factor Term Structure
Different L-factors represent different volatility dynamics. A term structure of L-factor determines an overall delta hedging strategy across all option maturities. In the following, we examine a particular L-factor term structure and the corresponding delta hedges.
Figure 3 shows one L-factor term structure inclined towards the sticky delta view. The L-factor is -29% at the six-month point and decays as the maturity gets longer. This indicates that for longer-dated option, the sticky delta view becomes more dominant. For ATM European calls, the absolute cash delta positions calculated using this particular L-factor term structure are larger than those of Black-Scholes and static local vol. For ATM European puts, the absolute cash delta positions calculated off this L-factor term structure are less than those of Black-Scholes and static local vol.
This particular shape of L-factor term structure is qualitatively observable on the market. Figure 4 plots the FTSE-100 average ATM implied volatility for various maturities. The data were taken in 1999 and the average is done daily over the whole year. The error bars show the magnitude of the variation of the averaged volatility. As can be seen in the figure, there is larger variation for short-dated volatility than long-dated. If we assume the implied volatility surface was mainly driven by the spot and given that FTSE-100 daily spots were different, the short-end ATM volatility was more susceptible to spot movement, indicating a less sticky delta situation. The long-end ATM volatility, however, was relatively more stable with the changing spot, indicating a more sticky delta dynamic.
Conclusion
The delta hedge positions can be substantially different under different volatility dynamics. The hedging P&L in the long run will be different as a result. This has implications in the risk management of equity derivatives products and hedging strategies aiming at achieving relatively stable long term P&L. Using an L-factor term structure one can apply a consistent hedging strategy across all option maturities. Typically, for long dated options, L-factors reflecting the sticky delta view seem to work well.
This week's Learning Curve was written by Dong Qu, head of equity derivatives quantitative research and applications at Abbey National Treasury Servicesin London.
