Corridor Implied Volatility and the Variance Risk Premium in the Italian Market

Corridor implied volatility introduced in Carr and Madan (1998) and recently implemented in Andersen and Bondarenko (2007) is obtained from model-free implied volatility by truncating the integration domain between two barriers. Corridor implied volatility is implicitly linked with the concept that the tails of the risk-neutral distribution are estimated with less precision than central values, due to the lack of liquid options for very high and very low strikes. However, there is no golden choice for the barriers levels’, which will probably change depending on the underlying asset risk neutral distribution. The latter feature renders its forecasting performance mainly an empirical question. The aim of the paper is twofold. First we investigate the forecasting performance of corridor implied volatility by choosing different corridors with symmetric and asymmetric cuts, and compare the results with the preliminary findings in Muzzioli (2010b). Second, we examine the nature of the variance risk premium and shed light on the information content of different parts of the risk neutral distribution of the stock price, by using a model-independent approach based on corridor measures. To this end we compute both realised and model-free variance measures which accounts for drops versus increases in the underlying asset price. The comparison is pursued by using intra-daily synchronous prices between the options and the underlying asset.


Introduction
Volatility modelling and forecasting is essential for asset pricing models, option pricing and hedging and risk management. In order to measure and forecast volatility we can resort either on the past history of the underlying asset (historical volatility) or on the information embedded in option prices which reflect the investors' opinion about the future underlying asset evolution (implied volatility). Focusing on implied volatility, how to extract the relevant information from the cross-section of option prices is still an open debate.
Black-Scholes implied volatility is a model dependent forecast, which relies on the strict assumptions of the Black-Scholes option pricing model about the asset price evolution (Brownian motion) and the constancy of the volatility parameter. Several papers have empirically shown the discrepancy between the assumptions of the model and the reality of financial markets (volatility varies with the strike price of the option, the time to maturity and the option type).
Model free implied volatility, introduced by Britten-Jones and Neuberger (2000), represents a valid alternative to Black-Scholes implied volatility, since it does not rely on a particular option pricing model, being consistent with several underlying asset price dynamics. A drawback of model free implied volatility is that it requires a continuum of option prices in strikes, ranging from zero to infinity, assumption which is not fulfilled in the reality of financial markets. Corridor implied volatility introduced in Carr and Madan (1998), and recently implemented in Andersen and Bondarenko (2007), is obtained from model-free implied volatility by truncating the integration domain between two barriers. Corridor implied volatility is implicitly linked with the concept that the tails of the riskneutral distribution are estimated with less precision than central values, due to the lack of liquid options for very high and very low strikes. For example, the 3 computation of market volatility indexes (see e.g. the VIX index for the Chicago Board Options Exchange, or the V-DAX New for the German stock market), which are closely followed by market participants, is done by operating a truncation of the domain of strike prices once two consecutive strikes with zero bid prices are observed.
The empirical literature about the forecasting performance of corridor implied volatility is very little and mixed: most studies are based on closing prices and investigate only symmetric corridor measures. Andersen and Bondarenko (2007), by using options on the S&P500 futures market, find that narrow corridor measures, closely related to Black Scholes implied volatility are more useful for volatility forecasting than broad corridor measures, which tend to model-free implied volatility as the corridor widens. A similar finding is obtained in Muzzioli (2010b) who finds that the best forecast for the Italian index options market is the one which operates a 50% cut of the risk neutral distribution. On the other hand, Tsiaras (2009), by using options on the 30 components of the DJIA index, concludes that CIV forecasts are increasingly better as long as the corridor width enlarges. To sum up, the empirical evidence suggests that there is no golden choice for the barriers levels', which will probably change depending on the underlying asset risk neutral distribution. The latter feature renders the forecasting performance of corridor implied volatility mainly an empirical question.
The investigation of corridor implied volatility has also important implications for the analysis of the variance risk premium. It is widely recognized that exposure to variance carries a negative risk premium: investors are willing to pay high prices in order to be insured against spikes in market variance. As noted by Carr and Wu (2006) investors are not only averse to increases in the variance level, but also to increases in variance of the return variance. Carr and Wu (2009) report evidence about five stock indexes and 35 stocks in the US market and find it to be strongly negative and highly significant. Andersen and Bondarenko (2009) are the first to exploit the concept of corridor implied variance in order to slice the risk neutral distribution of the stock price into different intervals and use the latter in order to investigate the pricing of market variance for different asset classes. In particular, for the SPX, they find a large negative risk premium which is very asymmetric since it is much larger in the downside part of the distribution than in the upside.
The aim of the paper is twofold. First we thoroughly investigate the forecasting performance of corridor implied volatility by pursuing a sensitivity analysis of corridor implied volatility with respect to the choice of the barrier. The analysis is motivated by the need to find an optimal cut for the Italian index options market, which can be found by analysing a grid of different corridors. To this end we investigate different corridors with both symmetric and asymmetric cuts and compare the results with the preliminary findings in Muzzioli (2010b), where only four cuts of the risk neutral distributions are explored (which correspond to a 5%, 10%, 20% and 50% overall cut). Second, we examine the nature of the variance risk premium and shed light on the information content of different parts of the risk neutral distribution of the stock price, by using a modelindependent approach based on corridor measures. To this end we compute both realised and model-free variance measures which accounts for drops versus increases in the underlying asset price. The comparison is pursued by using intradaily synchronous prices between the options and the underlying asset.
As for the sensitivity analysis to the choice of the barriers, the results substantially complement and corroborate the preliminary findings in Muzzioli (2010b), by finding an optimal cut around 50%-60% of the risk neutral distribution. Moreover, the use of asymmetric cuts highlights a weak evidence of superiority of the corridor measure which rely more on put prices on the one which relies more on call prices.
As for the analysis of the variance risk premium, we find it to be large and negative: investors are willing to pay sizable premiums and experience a loss on average in order to be hedged against peaks of variance. The results are consistent with previous empirical literature Wu (2006 and2009), Andersen and Bondarenko (2009)). Upside and Downside risk premia are negative and sizeable and both statistically significant. Downside risk premia are overwhelmingly higher than upside risk premia (more than two times higher). This means that investors heavily price downside risk: downside risk premium is the main component of the overall risk premium. The results are consistent with different computation methodologies of realised semi-variance.
The paper proceeds as follows. Sections 2 and 3 recall the basic features of variance swaps and corridor variance swaps. Section 4 presents the data set used.
Section 5 provides the details for the computation of the volatility and variance measures. Section 6 illustrates the computation of the variance risk premium and corridor variance risk premium. Sections 7 presents the results for the forecasting performance of corridor implied volatility and Section 8 the analysis of the variance risk premium. The last section concludes.

Variance swaps.
Variance swaps provide investors with a simple way in order to have a pure exposure to the future level of variance. Variance swaps are traded over the 6 counter. As they require a single payment at maturity, they are forward contracts on future realised variance. At maturity the long side pays a fixed rate (the variance swap rate) and receives a floating rate (the realised variance). A notional dollar amount is multiplied by the difference between the two rates. The payoff at maturity is: where N is a notional dollar amount, 2 R  is the realised variance, and VSR is the fix variance swap rate.
Assume that the stock price evolves as a diffusive process (no jumps allowed), as follows: Realized variance (also called integrated variance) in the period 0-T is given by: In practice, realised variance is monitored discretely (for example, the asset price could be observed each business day) and computed as: where n is the number of observations during the period 0-T and the day-count convention could be different depending on the term sheet conditions: business days/252 or actual/365.
At the time of entry the contract has zero value. If we assume absence of arbitrage opportunities and the existence of a unique risk-neutral measure we can write: where r is the risk-free discount rate corresponding to maturity T, and interest rates are assumed to be uncorrelated with realised variance. By no-arbitrage the variance swap rate should be equal to the risk-neutral expected value of realised variance over the life of the swap: Demeterfi et al. (1999) and Britten-Jones and Neuberger (2000) show how to replicate the risk-neutral expectation of variance with a portfolio of options with strike price ranging from zero to infinity, as follows: where M(K,T) is the minimum between a call or put option price, with strike price K and maturity T, i.e. only out-of-the-money options are used.
Equation (7) is also known as model-free implied variance, and its square root as model-free implied volatility, since, differently from Black-Scholes implied volatility it does not rely on any particular option pricing model. In fact, the definition is consistent with several underlying asset price dynamics: from diffusive to jump-diffusion process (Jiang and Tian (2005)). Relation (7) is exact if the underlying asset price evolves as a diffusive process and holds up to an approximation error when the underlying asset process displays jumps (Carr and Wu (2009)). Since in the reality of financial markets only a limited and discrete set of strike prices are quoted, both truncation and discretization errors arise in the computation of model free implied variance. Therefore, interpolation and extrapolation are needed in order to compute model-free implied variance (see Jiang and Tian (2005) and (2007)). The CBOE volatility index (VIX) and the 8 plethora of volatility indexes which have been introduced in various financial markets worldwide, are all based on the model free implied variance definition (which can be considered more precisely as a corridor implied variance because of the truncation of the domain of strike prices once two consecutive strikes with zero bid prices are observed). Therefore the volatility index squared represents a discretization of the 30-days variance swap rate, up to a discretization error and a jump induced error term.

Corridor variance swaps.
A corridor variance swap is a variant of variance swap which takes into account daily stock variations only when the underlying asset is in a specific corridor. At maturity the long side pays a fixed rate (the corridor variance swap rate) and receives a floating rate (the realised return variance which is accumulated only if the underlying lies in a pre-specified range). A notional dollar amount is multiplied by the difference between the two rates. The payoff at maturity is: where N is a notional dollar amount, 2

RC
 is the realised variance in the corridor, and CVSR is the corridor variance swap rate.
Corridor realised variance in the period 0-T, can be defined as follows: is the indicator function, which takes value 1 only if the underlying asset lies between the two barriers, n is the number of observations during the period 0-T and the day-count convention could be different depending on the term sheet conditions: business days/252 or actual/365. Carr and Madan (1998) and Andersen and Bondarenko (2007) show that it is possible to compute the expected value of corridor variance under the riskneutral probability measure (the corridor variance swap rate), by using a portfolio of options with strikes ranging from B 1 to B 2 , as follows: Equation (11) is known as corridor implied variance and its square root as corridor implied volatility. If B 1 =0 and B 2 =∞, then corridor variance degenerates into model-free implied variance. It follows that a corridor variance swap is cheaper than a variance swap, since it enables investors to bet on possible patterns of the stock price.
Upside and downside variance swaps are a variant of corridor variance swaps which have the following payoffs: where N is a notional dollar amount, VSRU and VSRD are the strike prices obtained by using formula (11) with barriers (B 1 =0 , B 2 =B) and (B 1 =B, B 2 =∞), respectively; upside (downside) corridor realised volatility is defined as: and accumulated only if the underlying lies above (below) the barrier B.
Note that equations (14) and (15) recognize the full square of the return that ends in the corridor, other alternative contract specifications may treat the movements across the barrier recognizing only a fraction of the total move (see e.g. Carr and Lewis (2004).
The sum of a downside corridor variance swap and an upside corridor variance swap yields a standard variance swap. An investor can be interested in an upside (downside) variance swap if she is bullish (bearish) on the underlying asset, or if the volatility skew is too steep making down-variance too expensive relative to up-variance.

The Data Set
The data set consists of intra-daily data on FTSE MIB-index options Therefore, the daily dividend yield is used in order to compute the appropriate value for the index, as follows: where S t is the FTSE MIB value at time t,  t is the dividend yield at time t and t is the time to maturity of the option.
As a proxy for the risk-free rate, Euribor rates with maturities one week, Finally, in order to reduce computational burden, we only retain options that are traded in the last hour of trade, from 16:40 to 17:40 (the choice is motivated by the high level of trading activity in this interval). Option prices and the underlying index prices are then matched in a one minute window in order to obtain implied volatilities from synchronous prices.

Measures
In the following volatility measures are taken as the square root of variance measures defined in Sections 2 and 3. We compute two volatility measures: corridor implied volatility ( CIV ) and realised volatility ( R ). As for corridor implied volatility, each day of the sample, we divide quoted option prices in two sets: near term and next term options and we follow the procedure described below, which consists of three steps (repeated for both near and next term options): fitting the smile function; obtaining the risk neutral distribution of 13 the underlying asset; computing corridor implied volatility. Last we interpolate between near and next term measures in order to have a 30-day measure.
As for the first step, we recover the Black-Scholes implied volatilities by using synchronous prices for the option and the underlying asset that are matched in one minute interval. These implied volatilities are averaged for each strike in the hour of trades resulting in a matrix of quoted strike prices and corresponding implied volatilities. Second, as only a discrete number of strikes are available, we need to interpolate and extrapolate option prices in order to generate the missing prices. As for the interpolation, following Campa et al. (1998), we use cubic splines to interpolate implied volatilities between strike prices. We extrapolate volatilities outside the listed strike price range by using a linear function that matches the slope of the smile function at K min and K max . This methodology has the advantage that the smile function remains smooth at K min and K max . As this latter methodology may generate implied volatilities that are artificially too high (in case the slope is positive) or too low (in case the slope is negative), we have imposed both a lower and an upper bound to implied volatilities equal to 0.001 and 0.999 respectively. Finally, we use the Black and Scholes formula in order to convert implied volatilities into call prices.
As for the second step, in order to obtain the risk neutral distribution of the underlying asset we resort to a non parametric method already tested in Muzzioli (2010b). In particular, we use the Derman and Kani (1994)  As for the third step, corridor implied volatility ( CIVi ) is computed both for near term (i=1) and for next term maturities (i=2), as a discrete version of the square root of equation (11): m is the number of strikes used, The barriers B 1 and B 2 are computed by looking at the risk-neutral distribution obtained by fitting an implied binomial tree with 100 levels to quoted option prices. As the implied tree yields a discrete cumulative distribution, term maturity and  2 for the next-term), as follows: where: T i = number of days to expiry of the i-th maturity index option, i=1,2.
where n is the number of index prices spaced by five minutes in the 30 days period. The choice of using five-minute frequency is made following Andersen and Bollerslev (1998) and Andersen et al. (2001) who showed the importance of using high frequency returns in order to measure realised volatility and point out that returns at a frequency higher than five minutes could be affected by serial  (2004) show that it is possible to separate two components in the quadratic variation process: the continuous evolution and the jump components. In this setting, realised semi-variance measures the variation of asset prices falls.
Downside realised variance is defined as: Symmetrically, upside realised variance is defined as: such that: Only negative (positive) returns are used in order to compute downside (upside) realised variance.
As upside and downside corridor measures are the risk neutral expectation of realised variance conditional on the underlying asset price being higher or lower than the barrier, we also compute the corresponding realised semi-variance corridor measures, as defined in equations (14) and (15). Corridor downside realised variance is defined as: and symmetrically, corridor upside realised variance is defined as: where F is the forward price.
The asymmetric treatment of the case S=F in the corridor upside and downside realised variance is necessary in order to have: Consistently with realised variance, semi-variance measures are annualised by means of the factor 365/30.

Variance risk premium and corridor variance risk premia.
The variance risk premium is the difference between the ex-post realised variance (over the lifetime of the swap) and the variance swap rate. Following

Carr and Wu (2009) the variance risk premium is measured both in Euro terms as
the Euro payoff of a long position in a variance swap with notional amount N=100 Euro, held up to expiry: and in log returns terms as the continuously compounded excess return given by: The average variance risk premium in Euro term and in log returns term is the sample average of (32) and (33) respectively.
The upside and downside variance risk premium is the difference between the ex-post upside and downside realised variance (over the lifetime of the swap) and the upside and downside corridor variance swap rate. As upside and downside realised variance is measured in two different ways (by separating either positive and negative returns or realisations higher or lower than a threshold), and the risk 19 premiums are computed either in Euro terms or in log-return terms, we have a total of eight different cases, as follows. The upside variance risk premium in Euro terms is the Euro payoff of a long position in an upside variance swap with notional amount N=100 Euro, held up to expiry: and in log returns terms it is measured as the continuously compounded excess return given by: Similarly are defined the downside variance risk premium in Euro terms: and in log returns terms:

The Results for the volatility measures.
Descriptive statistics for the volatility series are reported in In order to compare the results with Muzzioli (2010b) we gauge the forecasting performance of the different volatility measures, by resorting to the same metrics 1 widely used in the literature (see e.g. Poon and Granger (2003)). In particular, we use the MSE, the RMSE, the MAE, the MAPE and the QLIKE, defined as follows:  (1995)) by using the MSE function which is considered as robust to the presence of noise in the volatility proxy (Patton (2010)). The pair-wise comparisons are reported in Table 3  downside corridor measures obtain a poor performance, however CIVDW, which focus on put option prices performs better than CIVUP, which focus on call option prices.
In order to further investigate the robustness of the results, it is useful to look at the correlations between the implied volatility measures and subsequent realised volatility, which are reported in Figure 3. As we can see, CIV0.25 has the highest correlation with realised volatility, while CIV0.3 has a markedly lower correlation. Therefore the results point to an overall low degree of information of out-of-the-money options and substantially confirm the preliminary results of Muzzioli (2010b) which point to an overall 50% cut of the risk neutral distribution. Among corridor measures with asymmetric cuts, the one which uses more put prices than call prices (CIV0.3-0.1) has a strikingly higher correlation with future realised volatility than the one which uses more call prices than put prices. The result is the opposite than the one obtained by looking at the performance evaluation based on the ranking functions, where CIV0.1-0.3 obtains a little better performance than CIV0.3-0-1, which however is not statistically significant according to the Diebold and Mariano test. Looking at the different performance obtained by CIVUP and CIVDW, the results based both on the Diebold and Mariano test and on the correlation with realised volatility point to a better performance of CIVDW. Therefore we can say that overall, the asymmetric cut that trims the risk neutral distribution in the upper part more than in the lower part is more informative about future realised volatility. Put prices which carry information on the probability of a downturn move of the underlying asset convey better information about future realised volatility.

The results for the variance risk premium.
The investigation of the variance risk premium is pursued into two steps, first we investigate the overall variance risk premium; second we use upside and downside realised and implied variance measures in order to investigate the risk premium in the upper and lower part of the risk neutral distribution.
In Table 4 are reported the descriptive statistics for realised variance ( 2 R  ), the variance swap rate (VSR) and the risk premium measured both in Euro terms (EVRP) and in log returns terms (LVRP). As we can see the average swap rate is overwhelmingly higher than the average realised variance, reflecting the presence of a substantial variance risk premium. The variance swap rate is more volatile than realised variance, since it has been computed with the prices of different options series which differ in strike price and time to maturity. Overall, the variance risk premium is found to be large and negative: the average risk premium in Euro terms is equal to -2.472, and the average risk premium in log terms is -0.417. This means that investors do considerably price variance risk: the average Euro loss for each notional amount of 100 Euro invested in a long variance swap is -2.472 Euro. The average continuously compounded excess return of being long the variance swap and holding it to maturity is -41.7%. The negative average returns are both statistically different from zero (the t-statistics are adjusted for serial dependence, according to Newey-West). Therefore investors are willing to accept a strongly negative return being long in a variance swap, in order to be hedged against peaks in volatility.
In Table 5  It follows that downside risk premia are overwhelmingly higher than upside risk premia. If realised semi-variance is measured by positive and negative returns, the downside risk premium is almost twice than the upside risk premium, 25 if realised semi-variance is measured with returns higher or lower than the forward value of the underlying, then the downside risk premium is between two and three times the upside risk premium. This means that investors highly price downside risk (more than two times than upside risk).

Conclusions
In As for the sensitivity analysis to the choice of the barriers, the results substantially corroborates the preliminary findings in Muzzioli (2010b), by finding an optimal cut around 50%-60% of the risk neutral distribution. Moreover, the use of asymmetric cuts does not ameliorate the performance. There is a weak evidence of superiority of the corridor measure which rely more on put prices on the one which relies more on call prices, highlighting that the left part of the risk neutral distribution (the one which accounts for drops in the underlying asset price) is the most important in the derivation of the volatility measure.

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In the Italian market the variance risk premium is found to be large and negative. Investors are willing to pay sizable premiums and experience a loss on average, in order to be hedged against peaks of variance. Upside and Downside risk premia are negative and sizeable and both statistically significant. Downside risk premia are overwhelmingly higher than upside risk premia (more than two times higher). This means that investors heavily price downside risk. Downside