Problem $2.$ Suppose an option has $=0\mathrm{.}2,=0\mathrm{.}4$ and Vega $=10.$ If the stock price increases by $$2$ and the $$ changes from $0\mathrm{.}2$ to $0\mathrm{.}17,$ how much dollar the option's price will increase by$?$

Problem $3.$ Which of the following are assumptions for the Black$-$Scholes Model? Select all that apply.

a$.$ Trade without transaction cost

b$.$ The movement of the underlying stock is continuous

c$.$ Trade continuously

d$.$ No bid$-$ask spread

Please explain your answer.

Problem $4.$ If we get a wrong $$ in the Black Scholes Model: $($Select all that apply.$)$

a$.$ We will get a wrong option price

b$.$ We will get a wrong delta$-$hedging strategy

c$.$ We will absolutely lose money because of the wrong hedging

Please explain your answer.

Problem $5.$ Which of the followings will cause problems in BSM$?$ Select all that apply.

a$.$ Security returns tend to have fatter tails than $log$ normal distribution

b$.$ The volatility surface is not flat

c$.$ Distribution of stock returns are not I.I.D$.$ in the practice

Please explain your answer.

Problem $6.$ According to the volatility surface:

a$.$ options with a lower strike tend to have high implied volatility

b$.$ the implied volatility must be lower for longer time$-$to$-$maturity

Please explain your answer.