Question Three Option Greeks and Volatility Smiles

Part One

Consider European call option with the following parameters, S0 = 40, K = 40, = 0.3, r = 0.08 and T = 91 365. Your goal is to reproduce the price and Greek values of a call option in a data.frame variable in R, like the ones below.

Price Delta gamma Vega Rho Theta 2.78042 0.582402 0.06515618 0.07797232 0.051148889 -0.01734933

Using the above example write a short article on Option Greeks and hedging. Be as creative as possible [10 Marks]

Part Two

You are examining the reasons that the hedging program in your company has not been working as expected and would like to confirm that this smile does exist with the following market information: Strike Price 2,100 2,200 2,300 2,400 2,500 2,600 2,700 2,800 2,900 Market Price 600 500 410 325 260 215 190 175 180

Estimate the implied volatility of these options and plot the estimated results by strike price, which is supposed to render a volatility smile graph. Using the above example write a short article on volatility smile. [10 Marks]