Appendix - Skew & Term Structure Derivatives-based Portfolio Solutions
SKEW AND TERM STRUCTURE CUT SURFACE IN DIFFERENT DIMENSIONS A volatility surface has three dimensions (strike, expiry and implied volatility), which is difficult to show on a two dimensional page. For simplicity, a volatility surface is often plotted as two separate two dimensional graphs. The first plots implied volatility vs expiry (similar to the way in which a yield curve plots credit spread against expiry) in order to show term structure (the difference in implied volatility for options with different maturities and the same strike). The second plots implied volatility vs strike to show skew (the difference in implied volatility for options with different strikes and the same maturity). We examine a volatility surface in both these ways (ie, term structure and skew) and show how they are related. TERM STRUCTURE IS NORMALLY UPWARD SLOPINGWhen there is a spike in realized volatility, near-dated implied volatility tends to spike in a similar way (unless the spike is due to a specific event such as earnings). This is because the high realized volatility is expected to continue in the short term. Realized volatility can be expected to mean revert over an approx. 8-month period, on average. Hence far-dated implied volatilities tend to rise by a smaller amount than near-dated implied volatilities (as the increased volatility of the underlying will only last a fraction of the life of a far-dated option). Near-dated implieds are therefore more volatile than far-dated implieds. If equity markets decline, term structure becomes invertedTypically, an increase in volatility tends to be accompanied by a decline in equity markets, while a decline in volatility tends to occur in periods of calm or rising markets. If volatility surfaces are assumed not to move as spot moves (ie, sticky strike), then this explains why the term structure of low strike implied volatility is normally downward sloping (as the 80% strike term structure will be the ATM term structure when equities fall 20%). Similarly, this explains why the term structure of high strike implieds is normally upward sloping (as the 120% strike term structure will become the ATM term structure when equities rise 20%). Slope of rising term structure is shallower than slope of inverted term structure In practice, the slope of rising term structure is shallower than the slope of inverted term structure. This can be seen by looking at a volatility cone based on historical time-series. Despite the fact that the inverted term structure is steeper, the more frequent case of upward sloping term structure means the average term structure is slightly upward sloping. Implied volatility is usually greater than realised volatility and less volatileWhile historic and realised volatility are linked, there are important differences which can be seen when looking at empirical volatility cones. Average implied volatility lies slightly above average realised volatility, as implieds are on average slightly expensive. Implied volatility is also less volatile (it has a smaller min-max range) than realized volatility for near-dated maturities. This is because implieds are forward looking (ie, similar to an average of possible outcomes) and there is never 100% probability of the maximum or minimum possible realized. This effect fades for longer maturities, potentially due to the additional volatility caused by supply-demand imbalances (i.e. varying demand for put protection). This causes inverted implied volatility term structure to be less steep than realized volatility term structure. SKEW IS INVERTED AND IS HIGHER FOR NEAR-DATED EXPIRIESAssuming volatility surfaces stay constant (ie, sticky strike), the effect of near-dated ATM implieds moving further than far-dated implieds for a given change in spot is priced into volatility surfaces by having a larger near-dated skew. The more term structure changes for a given change in spot, the steeper the skew is. As near-dated ATM volatility is more volatile than far-dated ATM volatility, near-dated implied volatility has higher skew. Skew for equities is normally inverted Unless there is a high likelihood of a significant jump upwards (i.e. if there were a potential takeover event), equities normally have negative skew (low strike implied greater than high strike implied). There are many possible explanations for this, some of which are listed below. **Big jumps in spot tend to be down, rather than up.**If there is a jump in the stock price, this is normally downwards as it is more common for an unexpectedly bad event to occur (bankruptcy, tsunami, terrorist attack, accident, loss or death of key personnel, etc.) than an unexpectedly good event to occur (positive drivers are normally planned for).**Volatility is a measure of risk and leverage (hence risk) increases as equities decline.**If we assume no change in the number of shares in issue or amount of debt, then as a company’s stock price declines its leverage (debt/equity) increases. Both leverage and volatility are a measure of risk and, hence, they are correlated, with volatility rising as equities fall.**Demand for protection and call overwriting.**Typically, investors are interested in buying puts for protection, rather than selling them. This lifts low strike implieds. Additionally, some investors like to call overwrite their positions, which weighs on higher strike implieds.
REASONS WHY SKEW AND TERM STRUCTURE ARE CORRELATEDThere are three reasons why skew and term structure are correlated: - Credit events, such as bankruptcy, lift both skew and term structure
- Implied volatility is ‘sticky’ for low strikes and long maturities
- Implied correlation is ‘sticky’ for low strikes and long maturities (only applies to index)
1. Bankruptcy lifts both skew and term structureThere are various models that show the effect of bankruptcy (or credit risk) lifting both skew and term structure. As implieds with lower strikes have a greater sensitivity to credit risk (as most of the value of low strike puts is due to the risk of bankruptcy), their implieds rise more, which causes higher skew. Similarly, options with longer maturity are more sensitive to credit risk (causing higher term structure, as far-dated implieds rise more). Longer-dated options have a higher sensitivity to credit risk as the probability of entering default increases with time (hence a greater proportion of an option’s value will be associated with credit events as maturity increases). More detail on the link between volatility and credit can be seen in section Capital Structure Arbitrage.2. Implied volatilities are 'sticky' for low strikes and long maturitiesIf there is a sudden decline in equity markets, it is reasonable to assume realized volatility will jump to a level in line with the peak of realized volatility. Therefore, low-strike, near-dated implieds should be relatively constant (as they should trade near the all-time highs of realized volatility). If a low-strike implied is constant, the difference between a low-strike implied and ATM implied increases as ATM implieds falls. This means near-dated skew should rise if near-dated ATM implieds decline. For this reason, we do not view skew as a reliable risk indicator, as it can be inversely correlated to ATM volatility Similarly, term structure should also rise if near-dated ATM implieds fall, as far-dated ATM implieds are relatively constant (as they tend to include complete economic cycles). Hence skew and term structure should be correlated as a fall in near-dated ATM implied lifts both of them. 3. Correlation surface causes index skew and term structure to be correlatedIn the same way implied volatility is ‘sticky’ for low strikes and long maturities, so is implied correlation. This can be an additional reason why index skew and index term structure are correlated. CORRELATION LIFTS INDEX SKEW ABOVE SINGLE-STOCK SKEWAn approximation for implied correlation is the index volatility squared divided by the average single-stock volatility squared [ ρ = σ _{Index}² / average( σ_{( Single stock)} )² ]. Implied correlation is assumed to tend towards 100% for low strikes, as all stocks can be expected to decline in a crisis. This causes index skew to be greater than single stock skew. Index skew can be thought of as being caused by both the skew of the single stock implied volatility surface, and the skew of the implied correlation surface.Example of how index skew can be positive with flat single-stock skewWe shall assume all single stocks in an index have the same (flat) implied volatility and single-stock skew is flat. Low strike index volatility will be roughly equal to the constant single-stock volatility (as implied correlation is close to 100%), but ATM index volatility will be less than this figure due to diversity (as implied correlation ρ for ATM strikes is less than 100% and σ _{Index}² = ρ × average( σ_{( Single stock)} )². Despite single-stocks having no skew, the index has a skew (as low strike index implieds > ATM index implieds) due to the change in correlation. For this reason, index skew is always greater than the average single-stock skew. Implied correlation is likely to be sticky for low strikes and long maturities A correlation surface can be constructed for options of all strikes and expiries, and this surface is likely to be close to 100% for very low strikes. The surface is likely to be relatively constant for far maturities; hence, implied correlation term structure and skew will be correlated (as both rise when near-dated ATM implied correlation falls, similar to volatility surfaces). This also causes skew and term structure to be correlated for indices. DIVERSE INDICES HAVE HIGHER SKEW THAN LESS DIVERSE INDICESAs index skew is caused by both single-stock skew and implied correlation skew, a more diverse index should have a higher skew than a less diverse index (assuming there is no significant difference in the skew of the single-stock members). This is due to the fact that diverse indices have a lower ATM implied, but low strike implieds are in line with (higher) average single-stock implieds for both diverse and non-diverse indices. |

### Skew & Term Str.

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