options

1. Given Call option price at any K, C(K), what is the distributation of
stock price?

in Hull chap16 (6th ed) "volatility smile" Appendix. It's implied
distribution. g(K)=exp(rT)*(2nd order dericative of C wrt K). g(S_T) is the PDF.

3. Given call option price C(S, K, T, sigma, r), how to price option
with pay off (S_T^2-K^2)

the answer is
exp{(r+sigma^2)T}C(s^2, K^2, T, 2sigma, 2r+sigma^2)

4.I want to trade with you. A contract if in 3 months, if the
S&P 500 fell 90%, you give me $1000. Will you trade with me?
How to price. If I check in your account, you have $20,000.
Should I trade with you?

is a binary option