Asset pricing / Finance / Trading: options implied volatilities. options expires in one year.

The question:

The following is a snapshot of S&P 500 index options implied volatilities. The options expire in one year. Underlying future is trading at 2000 and lets assume zero interest rate & zero dividend yield.

given hint for Q3:

Questions:

3. Use the fitted volatility curve to calculate the implied probability density function (Hint: Use Butterfly Spread). Please indicate the formula you use in this step. 4. Based on step 3, does the implied distribution have positive or negative skewness? Why?

Many Thanks!

The following is a snapshot of S&P 500 index options implied volatilities. The options expire in one year. Underlying future is trading at 2000 and let's assume zero interest rate & zero dividend yield. Strike 2900 2800 2700 2600 2500 2400 2300 2200 2100 0.05 0.07 0.08 0.1 0.12 0.13 0.15 0.16 0.18 Volatility Strike Volatility 2000 0.19 1000 1900 1800 1700 1600 1500 1400 1300 1200 1100 0.21 0.23 0.24 0.26 0.28 0.29 0.31 0.33 0.34 0.35 Profit or Loss LONG BUTTERFLY $600 30.00 40.00 50.00 $0 Stock Price at Expiration -$400 The following is a snapshot of S&P 500 index options implied volatilities. The options expire in one year. Underlying future is trading at 2000 and let's assume zero interest rate & zero dividend yield. Strike 2900 2800 2700 2600 2500 2400 2300 2200 2100 0.05 0.07 0.08 0.1 0.12 0.13 0.15 0.16 0.18 Volatility Strike Volatility 2000 0.19 1000 1900 1800 1700 1600 1500 1400 1300 1200 1100 0.21 0.23 0.24 0.26 0.28 0.29 0.31 0.33 0.34 0.35 Profit or Loss LONG BUTTERFLY $600 30.00 40.00 50.00 $0 Stock Price at Expiration -$400