# Secrets of a smile

#### Published in Automated Trader Magazine Issue 19 Q4 2010

It’s hard to interpret a smile, whether you’re looking at a human face or a set of equity returns. Here, we present an exclusive summary of a recent report1 from Societe Generale that lifts the veil on the causes, behaviour, and consequences of a smile – skew - in the derivatives world.

Skew is foremost an implied concept: on the Equity derivatives market, out-of-the-money calls are cheaper, and out-of-the-money puts are more expensive than established by the original Black & Scholes theory, which assumes that equity returns are log-normally distributed. There are basically two explanations for this phenomenon.

Firstly, the market price of vanilla options is mostly driven by flows: the more buyers of an option, the more expensive its price, and conversely. And flows are actually one way as institutional investors buy downside puts to protect their portfolio, that they finance through the sale of calls. In his preliminary behavioural analysis of skew, SG Strategist Dylan Grice calls this asymmetry as the "life-dinner principle": a human investor does not consider losing an opportunity and fronting actual losses on his existing assets in the same way.

Secondly, this correction fits with reality. As shown by the graph below, volatility does increase when markets drop. The main consequence of skew is to increase the mathematical probability of sharp downward moves (the "fat tails") and to explain why so-called "centennial market crashes" occur more often than once in a century.

#### Figure 1: 1-year performance and realised volatility on EuroStoxx50. (source: SG)

### From implied to realized : the relationship between market performance and realized volatility

The main question in the equity market is whether skew is correctly priced or not. In a first attempt to assess it, one could consider skew as a forecast, by the market, of the level of realization of the volatility, depending on the performance of the market.

In order to appreciate the accuracy of this "prediction", one can, for any 1-year period, take the final performance P of the market and compare the realized volatility over the period to the implied volatility of vanilla option, as priced at the start of the period, choosing options whose strike was P% out-of-the-money.

The graph (figure 2) plots on the X-axis the performance of the market and on the Y-axis the spread between realized and implied volatilities. The graph spots different regimes of spot-volatility relationships:

• Two bullish periods (Jan 04 to Jun 07 and Jul 09 to Jun 10), where volatility realizes below the implied level

• Two bearish periods (Jul 07 to Aug 08 and Sep 08 to Jun 09) where volatility realizes over the implied level.

#### Figure 2: 1-year difference between realised (RV) and implied (IV) volatility with respect to the final spot on S&P 500 (source: SG)

One would deduce from this comparison that skew is under-priced by the market: volatility at higher strikes should be cheaper and volatility at lower strikes should be lower.

This is however a quite naïve approach: options premiums are foremost not used to predict volatility but to cover the hedging costs of traders. A more rationale approach would rely on this "hedging" definition of skew. One can define a realized volatility at a given strike, as the volatility which breakevens the delta-hedging of an option with such a strike.

Proceeding this way, one can calculate realized volatilities at different strikes and compare them with the implied levels. The graph (figure 4) shows the result of such a computation on S&P500 for a short period of last year. The conclusion is that there is actually a realized skew, with a shape very similar to the implied one. Hedging downside puts actually costs more than hedging upside calls!

#### Figure 3: 3-months theoretical and implied volatility on S&P 500 (Nov09-jan10 period) (source: SG)

Going further, skew is often characterized as the spread between implied volatilities at strikes 90% and 110%. The graph (figure 3) compares both values over the past 8 years for the S&P500 and leads to a conclusion totally opposite to the previous one. Skew realizes below the implied level almost all the time, in a very erratic way. But when it spikes it can reach 3 times the implied level… so that its behavior is very similar to the one of volatility, realizing often below the implied level but spiking sometimes much above.

#### Figure 4: The 1-year theoretical and implied skew spread (90% - 110%) on S&P 500 (source SG)

### From implied to realized: the relationship between market performance and implied volatility

In a quite complementary approach, skew can designate the relationship between equity performance and implied volatility - realized volatility being let apart.

There are three main models which describe this behaviour:

#### Figure 5: Change of volatility as spot increases in the three simple regimes

The sticky-strike model, mostly used to update volatility matrix for intraday moves of the market, considers that the price of an option is mostly changed by delta. The implied volatility per strike (expressed in points of the index) remains unchanged when the index moves. With such a model, the strike of variance swap also simply needs to be delta-adjusted.

The sticky-delta model assumes that the at-the-money volatility remains unchanged when the market moves. The price of options have to be both delta and vega-adjusted.

Finally, the "sticky local-volatility" model assumes that implied volatility over-reacts to market movements - as it is the case in violent drops.

The volatility regimes described above imply very different correlations between market performance and implied volatility. A sticky-delta model means that the at-the-money volatility is independent from the performance of the market, whereas a sticky-strike regime means that the correlation between implied volatility and spot is equal to -100%. In order to observe how this correlation evolves in the Equity markets, we can firstly plot the 1-month realised correlation between S&P 500 and VIX (the VIX is representing the short-term implied volatility). As shown below, this correlation is constantly negative, never reaches -100% and realizes on average at -80%. (see "Correlation" chart below)

In order to further investigate the relationship between implied volatility and market performance, one can expect that the changes of the ATM implied volatility are proportional to the skew multiplied by the daily returns of the spot:

Indeed, this equation holds for most classical regimes of volatility, though with different values of ß:

• In sticky strike regime, the volatility surface is a fixed function of strike ∑=f(K), therefore ßT=1

#### Figure 6: The 3-months realised correlation between S&P 500 1-month returns and VIX (source: SG)

• In sticky delta regime the volatility is a fixed function of K/S, therefore the ATM volatility remains constant and ßT =0.

• In sticky-local volatility, one uses the current volatility surface to calibrate a local vol model, i.e. to construct a local vol function (S,t) such that current option prices are matched. Then, when the spot moves this function should be unchanged and match the new option prices. The resulting behaviour of the ATM implied volatility is not straightforward, however it can be proven that ßT = 2 for short expirations.

The report estimates ßT for different times to maturity for the S&P 500 using data between 2003 and 2010. The skew is computed by considering the spread between the 95% and 105% implied volatilities. The values obtained are all very close to 1.5

In order to study these changes, the report cuts the period into 4 separate ones according to the level of volatility, as shown on the following chart below:

• January 2005 to June 07: low volatility (around 10%), few big spot moves. ß is close to 1, meaning that volatility follows approximately the sticky strike regime.

• July 07 to August 08: volatility around 20%, ß is close to 1.5, value which is consistent with a stochastic volatility model. The linear relationship seems to hold with a good precision (R2 of 0.87)

• September 08 to June 09: because of the very high volatility and the big moves of the spots, the points are much more scattered. Nevertheless, the values of ß and R2 don't change significantly compared to the previous period, showing that at the heart of the Lehman crisis the volatility is still driven by the same factors as before.

• July 09 to June 10: The volatility is almost back to its pre-crisis level. However, the beta increases significantly to 1.8, coming close to a sticky-local-volatility regime.

#### Figure 7

Surprisingly, the different regimes of skew are not linked to any other market regime, being performance of volatility. Skew has its own dynamic - and ironically the different regimes of betas are consistent with distinct clusters of plots in the first graph.

Going further, the report studies the dynamics of the volatility surface and shows how it evolved during the recent vol spikes, in September 08 and May 10.

### Trading skew

Besides this complex dynamic, skew is currently at historically high levels.

#### Figure 8: Evolution of EuroStoxx50 3-month skew from July 2008 to July 2010. (source: SG)

This steepness explains the current popularity of skew trades. The purest skew trades are the conditional variance trade.

The up-variance swap is a variance swap where observations are taken into account only if the spot is above a certain level, and weighted by the number of days where the spot is above this level. For an equity index such as S&P 500 or Eurostoxx50, the presence of a downward sloping skew means that a higher level of the spot is associated with a lower volatility. As a consequence, the fair strike of an Up Var is lower than that of a standard Varswap. The buyer of the Up Var gets a discount due to the implied skew and is betting on the fact that the skew won't realize as much as implied. As seen in the study of the theoretical skew, this is the case most of the time.

Similarly, the down-variance swap is defined as a variance swap where observations are taken into account only where spot is below a certain level. Because of the skew, the fair strike is usually higher than that of a standard Varswap. Therefore, a short position in a Down Var is favored if the skew does not realize as much as implied.

Combining the two positions, the skew lock trade consists in being long an up-variance swap at a certain level L ("Low Barrier", let's say 90%) and short a down-variance swap at level above H ("High Barrier", let's say 110%). This trade "locks" the spread between both the UpVar and the DownVar strikes, and the investor gets this positive carry as long as the spot stays between both barriers. The following graph plots the dependency of the Up Var Strike and the Down Var Strike with respect to the strike for a 1-year maturity trade on the EuroStoxx50.

Besides direct skew trades, a few products benefit, in their pricing, from the steepness of skew. This is for instance the case for puts down and out.

#### Figure 9: UpVar and DownVar Strikes with respect to the barrier on EuroStoxx50. (source: SG)

The buyer of a Put Down and Out ("PDO") with strike K and Barrier B receives a payoff equal to that of a vanilla put with same strike, provided that the underlying price stayed above the barrier level during the whole period.

If the underlying reaches the barrier then the option knocks out and the buyer receives nothing (alternatively, a rebate usually taken equal to the premium, may be offered if the option knocks out).

Payoff formula

This option allows the investor to bet on a down movement while paying a lower premium than for a standard put. The PDO buyer is short skew and usually short vega (depending on the level of the barrier).

In the following, the base example will be an at-the-money PDO on S&P 500 with a maturity of 6 months and a barrier at 75%. In this case, with a premium around 2%, leverage up to 12 can be achieved.

#### Figure 10: Premium and delta of the PDO with respect to the level of the skew, for different values of the barrier. (source: SG)

The price of the down and out put is very sensitive to the skew, which makes it very cheap when the skew is pronounced. The figure below shows the premium of the option for a linear shift of the skew, varying between 0 and 150% of a reference skew.