Volatility smile or volatility skew refers to the relation between implied volatility and strike price (Hull chapter 19). Quotes for selected options on Facebook stock on April 26, 2019 are provided below. Determine implied volatility for call options with strike prices of 170, 180, 190, 200, and 210, and plot as a function of strike price. Use midpoint of bid and ask quotes for stock price and for option price. Facebook does not pay dividends. Assume the interest rate is 2.4% per annum with annual compounding (not continuous compounding). The expiration date of all options is July 19, 2019. To determine implied volatility, input all parameters in Black-Scholes formula and adjust volatility until the price given by formula matches the price quote. If you use Excel, Goal Seek function can be helpful. The Excel formula =DAYS(DATE(2019,7,19),DATE(2019,4,26)) returns the number of days to expiration. You may use the spreadsheet Black-Scholes Price.xlsx. (spreadsheet setup is shown in second picture below)

Dividend Yield Risk-free Rate Maturity (in years) European Call Price European Put Price 40% 12.0792 12.0792 12.0792 12.0792 12.0792 12.0792 12.0792 12.0792 12.0792 12.0792 9.6712 9.6712 9.6712 9.6712 9.6712 9.6712 9.6712 9.6712 9.6712 9.6712 Stock Price Strike Price Volatility (S.D 5% 5% 5% 5% 5% 5% 5% 5% 5% 5% 10% 10% 10% 10% 10% 10% 10% 10% 10% 10% 100 100 100 100 100 100 100 100 100 100 100 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 100 100 100 100 100 100 100 100 40% 40% 40% 40% 40% 10 Dividend Yield Risk-free Rate Maturity (in years) European Call Price European Put Price 40% 12.0792 12.0792 12.0792 12.0792 12.0792 12.0792 12.0792 12.0792 12.0792 12.0792 9.6712 9.6712 9.6712 9.6712 9.6712 9.6712 9.6712 9.6712 9.6712 9.6712 Stock Price Strike Price Volatility (S.D 5% 5% 5% 5% 5% 5% 5% 5% 5% 5% 10% 10% 10% 10% 10% 10% 10% 10% 10% 10% 100 100 100 100 100 100 100 100 100 100 100 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 100 100 100 100 100 100 100 100 40% 40% 40% 40% 40% 10