Explain with reasons as to what happens to the following: 

a. A look back call as we increase the frequency with which we observe the asset price in calculating the minimum 

b. A down and out call as we increase the frequency with which we observe the asset price in determining whether the barrier has been crossed 

c. The price of an ``out'' barrier option when the volatility is increased 

d. The price of an American digital option when volatility is increased 

e. The relative prices of out - of - the - money American and European digital calls for an asset that follows the Brownian motion and there are no interest rates. 

f. The price of a derivative if it has a negative vega and the volatility increases