Implied Volatility Smiles

Volatility smile refers to the U-shaped implied volatility estimates as a function of the exercise price. Previous option pricing studies have shown that both in-the-money and

out-the-money calls and puts have higher implied volatilities than at-the money calls and puts.

Moreover, short-maturity options, deep-in-the money calls, and deep-out-of-money puts have the highest estimated implied volatilities, giving rise to a skew-shaped implied volatility. We document a similar relation in table 4-1, table 4-2, table 4-3, and table 4-4. For the puts of ETF, Three ETF and SPX are existed implied volatility smiles whether past stock return are positive or negative. Furthermore, volatility smiles curve moved upward when the decline of stock prices increases volatility for puts . As observed in table 4-1, implied volatility of SPY put increases from 23.57% to 25.10% for out-the-money puts, increasing from 13.14% to 16.40% for at-the-money puts, and increasing from 21.74% to 24.66% for in-the-money puts.

For in-the-money puts, volatility smiles measure is 12.43% ( 25.57% minus 13.14% ) when past stock return are positive. For out-the-money puts, volatility smiles measure is 8.60% ( 21.74% minus 13.14% ). Also, For in-the-money puts, volatility smiles measure is 8.70% ( 25.10% minus 16.40% ) when past stock return are positive. For out-the-money puts, volatility smiles measure is 8.26% ( 24,66% minus 16.40% ).

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4-3 ETF Option Prices and Stock Market Momentum

This thesis is computed three weighting scheme of implied volatility. Past studies recommend that there is less mispricing and model error to use at-the-money option. Thus, we focus mainly on the volatility spread computed using vega-weighting. Nevertheless, we also replicated our tests using the two measures as well to examine the relation between past ETF returns and volatility spreads. The cross market momentum hypothesis predicts a negative relation between past ETF return and volatility spreads. Past ETF return are computed using the preceding 10-200 days (2-40 weeks) of close price. In order to guarantees that potential investors have the necessary information on hand to actually implement the tests conducted in this paper, we use a 1-day window between the ending day for computing stock returns distortions. Additionally, we take contemporary return into account to revise that current information from spot market may influences option market.

The properties of the volatility spread suggest a slow-moving time series. Hence, if daily average spreads are used as the dependent variable in ordinary least squares (OLS) regressions, the residuals will exhibit strong autocorrelations, leading to potential biases in the estimated regression coefficients. We need to consider autoregressive model and moving average simultaneously eliminating the correlation structure of the residuals, as judged by the Box-Pierce statistics. Selecting all appropriate model, we choose the best one by coefficient of determination, Akaike information, Schwartz Bayesian information criterion, and the likelihood Ratio test and replicated the tests to examine the effect of past ETF return from 10 days to 200 days. Table 4-5 is the results of the regression of the daily volatility spreads against SPY returns over the past 2 to 40 weeks for the entire sample period. Table 4-6 is the results of QQQQ , Table 4-7 is the results of DIA , and Table 4-8 is the results of S&P 500.

For 10-200 days, the relation between the volatility spread of SPY option and past SPY returns is negative and significant. In addition, that one period lagged is significant positive

accords with our observation in table 4-3. For 10-80 days, the relation between the volatility spread of QQQQ option and past QQQQ returns is negative and significant. For 10-120 days, the relation between the volatility spread of DIA option and past DIA returns is negative and significant. In term of our empirical results, for the overall ETFs, the relation between the volatility spread of ETF option and past ETF returns is negative and significant, especially in the underlying component of ETF are more active and the period of significant negative are longer. It also supported cross market momentum hypothesis and existed momentum trading.

On the other hand, when the ETF returns in past period are positive, this positive pressure, triggered by long call and short put trades, decrease the implied volatility for calls less than for puts, thus reducing the volatility spread at the end of period. The results is similar to what Tavakkol(2000) examined momentum trading. Namely, when the price of underlying asset increases, the positive feedback traders will trade in option market and expect to profit as a follower. Inversely, the downward movement in the stock market creates a negative pressure (resulting from short call and long put trades), increasing the volatility spread.

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TABLE 4-5 Regression of Daily Volatility Spread on Past Market Returns for SPY

This table reports the result of time series regressions of the vega-weighted volatility spread (put-implied volatility – call-implied volatility ) versus past 2-week to 40-week market returns. The numbers in parentheses are p-values.

10(2 weeks) 0.980216

(<0.001)

TABLE 4-6 Regression of Daily Volatility Spread on Past Market Returns for QQQQ

This table reports the result of time series regressions of the vega-weighted volatility spread (put-implied volatility – call-implied volatility) versus past 2-week to 40-week market returns. The numbers in parentheses are p-values.

, , 1 , 0 , 1 , , 1 10(2 weeks) 1.050745

(<0.001)

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TABLE 4-7 Regression of Daily Volatility Spread on Past Market Returns for DIA

This table reports the result of time series regressions of the vega-weighted volatility spread (put-implied volatility – call-implied volatility ) versus past 2-week to 40-week market returns. The numbers in parentheses are p-values.

, , 1 , 0 , 1 , , 1 10(2 weeks) 0.858171

(<0.001)

TABLE 4-8 Regression of Daily Volatility Spread on Past Market Returns for SPX

This table reports the result of time series regressions of the vega-weighted volatility spread (put-implied volatility – call-implied volatility) versus past 2-week to 40-week market returns. The numbers in parentheses are p-values.

, , 1 , 0 , 1 , , 1 10(2 weeks) 0.973096

(<0.001)

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