Trading the Skew
Derivatives-based Portfolio Solutions
FOUR IDEALISED REGIMES DESCRIBE MOVEMENT OF VOL SURFACEThere are four idealised regimes for a volatility surface. While sticky delta, sticky strike and (sticky) local volatility are well known and widely accepted names, we have added ‘jumpy volatility’ to define volatility with a high negative correlation with spot. **Sticky delta (or sticky moneyness).**Sticky delta assumes a constant volatility for options of the same strike as a percentage of spot. For example, ATM or 100% strike volatility has constant volatility. As this model implies there is a positive correlation between volatility and spot, the opposite of what is usually seen in the market, it is not a particularly realistic model (except over a very long time horizon).**Sticky strike.**A sticky strike volatility surface has a constant volatility for options with the same fixed currency strike. Sticky strike is usually thought of as a stable (or unmoving) volatility surface as real-life options (with a fixed currency strike) do not change their implied volatility.**Sticky local volatility.**Local volatility is the instantaneous volatility of stock at a certain stock price. When local volatility is static, implied volatility rises when markets fall (i.e. there is a negative correlation between stock prices and volatility). Of all the four volatility regimes, it is arguably the most realistic and fairly prices skew.**Jumpy volatility.**We define a jumpy volatility regime as one in which there is an excessive jump in implied volatility for a given movement in spot. There is a very high negative correlation between spot and volatility. This regime usually occurs over a very short time horizon in panicked markets (or a crash).
PUTTING ON LONG SKEW TRADES HAS A COST (SKEW THETA)If an investor initiates a long skew position by buying an OTM put and selling an OTM call, the implied volatility of the put purchased has a higher implied volatility than the implied volatility sold through the call. The long skew position therefore has a cost associated with it, which we shall define as ‘skew theta’. Skew theta is the difference between the cost of gamma (theta per unit of dollar gamma) of an OTM option compared to an ATM option. If skew is flat, then all strikes have an identical cost of gamma, but as OTM puts have a higher implied volatility than ATM ones they pay more per unit of gamma. Skew theta is explained in greater depth at the end of this section. If the long skew position does not give the investor enough additional profit to compensate for the skew theta paid, then skew can be sold at a profit. Skew trades profit from negative spot volatility correlationIf there is a negative correlation between the movement of a volatility surface and spot (as is usually seen in practice), then this movement will give a long skew position a profit when the volatility surface is re-marked. For example, let us assume an investor is long skew via a risk reversal (long an OTM put and short an OTM call). If equity markets decline, the put becomes ATM and is the primary driver of value for the position (as the OTM call becomes further OTM it is far less significant). The rise in the volatility surface (due to negative correlation between spot and volatility) boosts the value of the (now ATM) put and, hence, the value of the risk reversal. SKEW IS USUALLY OVERPRICED DUE TO HEDGINGAs volatility markets tend to trade between a static strike and static local volatility regime, long skew trades are usually unprofitable (usually there is negative spot volatility correlation, but not enough to compensate for the skew theta). As long skew trades break even during static local volatility regimes, they are only profitable in periods of jumpy volatility. This overpricing of skew can be considered to be a result of excessive demand for downside put options, potentially caused by hedging. Another reason for the overpricing of skew could be the popularity of short volatility long (downside) skew trades (traders often hedge a short volatility position with a long skew (OTM put) position, in order to protect themselves should markets suddenly decline). The profits from shorting expensive volatility are likely to more than compensate for paying an excessive amount for the long skew position (OTM put). VOL REGIME DETERMINED BY TIME AND SENTIMENTImplied volatility can be thought of as the equity derivative market’s estimate of future volatility. It is therefore investor sentiment that determines which implied volatility regime the market trades in, and this choice is largely determined by how much profit (or loss) a long skew position is expected to reveal over a certain time period (ie, investor sentiment). The choice of regime is also determined by the time horizon chosen. Sticky delta regimes occur over long time horizon or trending marketsA sticky delta regime is typically one in which markets are trending in a stable manner (either up or down, with ATM volatility staying approximately constant) or over a very long time horizon of months or years (as over the long term the implied volatility mean reverts as it cannot go below zero or rise to infinity).Jumpy volatility regimes occur over very short time horizons and panicked marketsIt is rare to find a jumpy volatility regime that occurs over a long time horizon, as they tend to last for periods of only a few days or weeks. Markets tend to react in a jumpy volatility manner after a sudden and unexpected drop in equity markets (large increase in implied volatility given a decline in spot) or after a correction from such a decline (a bounce in the markets causing implied volatility to collapse).Markets tend to trade between a sticky strike and sticky local volatility regimeSticky delta and jumpy volatility are the two extremes of volatility regimes. Sticky strike and sticky local volatility are far more common volatility regimes. Sticky strike is normally associated with calmer markets than sticky local volatility (as it is closer to a sticky delta model than jumpy volatility).DELTA OF OPTION DEPENDS ON VOLATILITY REGIMEHow a volatility surface reacts to a change in spot changes the value of the delta of the option. For sticky strike, as implied volatilities do not change, the delta is equal to the Black-Scholes delta. However, if we assume a sticky delta volatility regime if an investor is long a call option, then the implied volatility of that option will decline if there is a fall in the market. The value of the call is therefore lower than expected for falls in the market. The reverse is also true as implied volatility increases if equities rise. As the value of the call is lower for declines and higher for rises (as volatility is positively correlated to spot), the delta is higher than that calculated by Black-Scholes (which is equal to the sticky strike delta). A similar argument can be made for sticky local volatility (as volatility is negatively correlated to spot, the delta is less than that calculated by Black-Scholes). VOL CAN BE EXAMINED IN RELATIVE OR ABSOLUTE DIMENSIONSTo evaluate the profit – or loss – from a skew trade, assumptions have to be made regarding the movement of volatility surfaces over time (as we assume a skew trader always delta hedges, we are not concerned with the change in premium only the change in volatility). Typically, traders use two main ways to examine implied volatility surfaces. Absolute dimensions tend to be used when examining individual options, a snapshot of volatilities, or plotting implied volatilities over a relatively short period of time. Relative dimensions tend to be used when examining implied volatilities over relatively long periods of time. **Absolute dimensions.**In absolute dimensions, implied volatility surfaces are examined in terms of fixed maturity (eg, Dec14 expiry) and fixed strike (eg, €4,000). This surface is a useful way of examining how the implied volatility of actual traded options changes.**Relative dimensions.**An implied volatility surface is examined in terms of relative dimensions when it is given in terms of relative maturity (eg, three months or one year) and relative strike (eg, ATM, 90% or 110%). Volatility surfaces tend to move in relative dimensions over a very long period of time, whereas absolute dimensions are more suitable for shorter periods of time.
Care must be taken when examining implieds in relative dimensionsAs the options (and variance swaps) investors buy or sell are in fixed dimensions with fixed expiries and strikes, the change in implied volatility in absolute dimensions is the key driver of volatility profits (or losses). However, investors often use ATM volatility to determine when to enter (or exit) volatility positions, which can be misleading. For example, if there is a skew (downside implieds higher than ATM) and equity markets decline, ATM implieds will rise even though volatility surfaces remain stable. A plot of ATM implieds will imply buying volatility was profitable over the decline in equity markets; however, in practice this is not the case. Absolute implied volatility is the key driver for equity derivative profitsAs options that are traded have a fixed strike and expiry, it is absolute implied volatility that is the driver for equity derivative profits and skew trades. However, we accept that relative implied volatility is useful when looking at long-term trends. For the volatility regimes (1) sticky delta and (2) sticky strike, we shall plot implieds using both absolute and relative dimensions in order to explain the difference. For the remaining two volatility regimes (sticky local volatility and jumpy volatility), we shall only plot implied volatility using absolute dimensions (as that is the driver of profits for traded options and variance swaps).STICKY DELTA ASSUMES ATM VOL NEVER CHANGESA sticky delta model assumes a constant implied volatility for strikes as a percentage of spot (eg, ATM stays constant). RANGE-BOUND VOLATILITIES SUPPORT A STICKY DELTA MODELAs implied volatility cannot be negative, it is therefore usually floored close to the lowest levels of realised volatility. Although an infinite volatility is theoretically possible, in practice implied volatility is typically capped close to the all-time highs of realised volatility. Over a long period of time, ATM implied volatility can be thought of as being range bound and likely to trend towards an average value (although this average value will change over time as the macro environment varies). As the trend towards this average value is independent of spot, the implied volatility surface in absolute dimensions (fixed currency strike) has to move to keep implied volatility surface in relative dimensions (strike as percentage of spot) constant. Thinking of implied volatility in this way is a sticky delta (or sticky moneyness) implied volatility surface model. Sticky delta most appropriate over long term (many months or years)While over the long term implied volatility tends to return to an average value, in the short term volatility can trade away from this value for a significant period of time. Typically, when there is a spike in volatility it takes a few months for volatility to revert back to more normal levels. This suggests a sticky delta model is most appropriate for examining implied volatilities for periods of time of a year or more. As a sticky delta model implies a positive correlation between (fixed strike) implied volatility and spot, the opposite of what is normally seen, it is not usually a realistic model for short periods of time. Trending markets (calmly rising or declining) are usually the only situation when a sticky delta model is appropriate for short periods of time. In this case, the volatility surface tends to reset to keep ATM volatility constant, as this implied volatility level is in line with the realised volatility of the market. LONG SKEW IS UNPROFITABLE IN STICKY DELTA VOLATILITY REGIMEIn a sticky delta volatility regime the fixed strike implied volatility (and, therefore, the implied volatility of traded options) has to be re-marked when spot moves. The direction of this remark for long skew positions causes a loss, as skew should be flat if ATM volatility is going to remain unchanged as markets move (we assume the investor has bought skew at a worse level than flat). Additionally, the long skew position carries the additional cost of skew theta, the combination of which causes long skew positions to be very unprofitable. STICKY STRIKE HAS ZERO SPOT VOL CORRELATIONA sticky strike model assumes that options of a fixed currency strike are fixed (absolute dimensions). The diagrams below show how a volatility surface moves in both absolute/fixed strike and relative strike due to a change in spot. TRADER’S SYSTEMS CAN GIVE ILLUSION OF STICKY STRIKEWhile Figure 120 above describes which volatility regime normally applies in any given environment, there are many exceptions. A particular exception is that for very small time horizons volatility surfaces can seem to trade in a sticky strike regime. We believe this is due to many trading systems assuming a static strike volatility surface, which then has to be remarked by traders (especially for less liquid instruments, as risk managers are likely to insist on volatilities being marked to their last known traded implied volatility). As the effect of these trading systems on pricing is either an
illusion (as traders will re-mark their surface when asked to provide a
firm quote) or well within the bid-offer arbitrage channel, we believe
this effect should be ignored. LONG SKEW UNPROFITABLE WITH STICKY STRIKEWhile there is no profit or loss from re-marking a surface in a sticky strike model, a long skew position still has to pay skew theta. Overall, a long skew position is still unprofitable in sticky strike regimes, but it is less unprofitable than for a sticky delta regime. STICKY LOCAL VOLATILITY PRICES SKEW FAIRLYAs a sticky local volatility causes a negative correlation between spot and Black-Scholes volatility (shown below), this re-mark is profitable for long skew positions. As the value of this re-mark is exactly equal to the cost of skew theta, skew trades break even in a sticky local volatility regime. If volatility surfaces move as predicted by sticky local volatility, then skew is priced fairly (as skew trades do not make a loss or profit). BLACK-SCHOLES VOL IS AVERAGE OF INSTANTANEOUS LOCAL VOL Local volatility is the name given for the instantaneous volatility of an underlying (ie, the exact volatility it has at a certain point). The Black-Scholes volatility of an option with strike K is equal to the average local (or instantaneous) volatility of all possible paths of the underlying from spot to strike K. This can be approximated by the average of the local volatility at spot and the local volatility at strike K. This approximation gives two results. - The ATM Black-Scholes volatility is equal to the ATM local volatility.
- Black-Scholes skew is half the local volatility skew (due to averaging).
Example of local volatility skew = 2x Black-Scholes skewThe second point can be seen if we assume the local volatility for the 90% strike is 22% and the ATM local volatility is 20%. The 90%-100% local volatility skew is therefore 2%. As the Black-Scholes 90% strike option will have an implied volatility of 21% (the average of 22% and 20%), it has a 90%-100% skew of 1% (as the ATM Black-Scholes volatility is equal to the 20% ATM local volatility). STICKY LOCAL VOL IMPLIES NEGATIVE SPOT VOL CORRELATIONAs local volatility skew is twice the Black-Scholes skew, and ATM volatilities are the same, a sticky local volatility surface implies a negative correlation between spot and implied volatility. This can be seen by the ATM Black-Scholes volatility resetting higher if spot declines and is shown in the diagrams above. Example of negative correlation between spot and Black-Scholes volatilityWe shall use the values from the previous example, with the local volatility for the 90% strike = 22%, Black-Scholes of the 90% strike = 21% and the ATM volatility for both (local and Black-Scholes) = 20%. If markets decline 10%, then the 90% strike option Black-Scholes volatility will rise 1% from 21% to 22% (as ATM for both local and Black-Scholes volatility must be equal). This 1% move will occur in parallel over the entire surface (as the Black- Scholes skew has not changed). Similarly, should markets rise 10%, the Black-Scholes volatility surface will fall 1% (assuming constant skew). LONG SKEW PROFITS FROM VOLATILITY SURFACE RE-MARKIn order to demonstrate how the negative correlation between spot and (Black-Scholes) implied volatility causes long skew positions to profit from volatility surfaces re-mark, we shall assume an investor is long a risk reversal (long OTM put, short OTM call). When markets fall, the primary driver of the risk reversal’s value is the put (which is now more ATM than the call), and the put value will increase due to the rise in implied volatility (due to negative correlation with spot). Similarly, the theoretical value of the risk reversal will rise (as the call is now more ATM – and therefore the primary driver of value – and, as implied volatilities decline as markets rise, the value of the short call will rise as well). The long skew position therefore profits from both a movement up or down in equity markets, as can be seen in the diagram below as both the long call and short put position increase in value. VOLATILITY RE-MARK WITH STICKY LOCAL VOL = SKEW THETAWhile a sticky local volatility regime causes long skew positions to profit from (Black-Scholes) implied volatility changes, the position still suffers from skew theta. The combination of these two cancel exactly, causing a long (or short) skew trade to break even. As skew trades break even under a static local volatility model, and as there is a negative spot vol correlation, it is arguably the most realistic volatility model JUMPY VOLATILITY IS ONLY REGIME WHERE LONGSKEW IS PROFITABLE During very panicked markets, or immediately after a crash, there is typically a very high correlation between spot and volatility. During this volatility regime (which we define as jumpy volatility) volatility surfaces move in excess of that implied by sticky local volatility. As the implied volatility surface re-mark for a long skew position is in excess of skew theta, long skew positions are profitable. A jumpy volatility regime tends to last for a relatively short period of time. Example of volatility regimes and skew tradingIf one-year 90%-100% skew is 25bp per 1% (ie, 2.5% for 90-100%) and markets fall 1%, volatility surfaces have to rise by 25bp for the profit from realised skew to compensate for the cost of skew theta. If surfaces move by more than 25bp, surfaces are moving in a jumpy volatility way and skew trades are profitable. If surfaces move by less than 25bp then skew trades suffer a loss. SKEW IS FAIRLY PRICED ONLY IF ATM MOVES BY TWICE THE SKEWFor a given movement in spot from S _{0} to S_{1}, we shall define the movement of the (Black-Scholes) implied volatility surface divided by the skew (implied volatility of strike S_{1} – implied volatility of strike S_{0}) to be the realised skew. The realised skew can be thought of as the profit due to re-marking the volatility surface. Defining realised skew to be the movement in the volatility surface is similar to the definition of realised volatility, which is the movement in spot. realised skew = movement of surface/skew where: movement of surface = movement of surface when spot moves from S _{0} to S_{1}skew = difference in implied volatility between S _{1} and S_{0}The ATM volatility can then be determined by the below equation: - ATM
_{time 1}= ATM_{time 0}+ skew + movement of surface
- ATM
_{time 1}= ATM_{time 0}+ skew + (skew × realised skew)
- ATM
_{time 1}= ATM_{time 0}+ skew × (1 + realised skew)
The realised skew for sticky delta is therefore -1 in order to keep ATM constant (and hence skew flat) for all movements in spot. A sticky strike regime has a realised skew of 0, as there is no movement of the volatility surface and skew is fixed. A local volatility model has a realised skew of 1, which causes ATM to move by twice the value implied by a fixed skew. As local volatility prices skew fairly, skew is only fairly priced if ATM moves by twice the skew. We shall assume the volatility surface for jumpy volatility moves more than it does for sticky local volatility, hence has a realised skew of more than 1. Skew profit is proportional to realised skew – 1 (due to skew theta)In order to calculate the relative profit (or loss) of trading skew, the value of skew theta needs to be taken away, and this value can be thought of as -1. Skew profit is then given by the formula below: Skew profit α realised skew - 1 |

### Trading the Skew

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