Let C = BS(S;X; ; r; ) denote the price of a European call option on this stock, where BS(:) is the

Black-Scholes formula derived in class.

(a) Suppose that we want to estimate the implied volatility but at each time point we have a large

cross-section of option prices with dierent exercise prices (and hence moneyness dened as

[erX] =S 1) and maturities. We would like to use options that contain the most information

about volatility, i.e. options that are most sensitive to changes in volatility. Find, in terms of

moneyness and maturities, the options that are most informative about volatility and discuss

the result.

(b) Find the moneyness and the maturity that make the Black-Scholes formula linear in volatility.

Discuss the result.

(c) Finally, nd the maturity at which the \vega" of the call option is maximized for at-the-money

options (i.e., erX=S 1).