Given,

Stock price = $80

Strike price = $75

Risk free rate = 10% = 0.1

time period in years = 6/12 = 0.5

A call option gives right not obligation to purchase the stock at a specific price known as strike price on a specific date i.e., expiration day.

The lower bound is given by the formula,

Lower Bound = Stock Price -Strike Price*(e^(-r*t))

= 80-(75*(e^(-0.1*0.5)))

= 80-(75*(e^(-0.05))

= 80-(75*(0.95)

  = 80-(71.34)

= 8.66

Therefore the lower bound on a 6 month call option is $8.66

b)Given,

Stock price = $58

Strike price = $65

Risk free rate = 5% = 0.05

time period in years = 2/12 = 0.167

A European put option is the right to sell the option at a predetermined price on a specific day.

The lower bound is given by the formula,

lower bound = (strike price*e^(-rt)) - stock price

= (65*(e^(-0.05*0.167))) - 58

= 65*e(-0.0083) - 58

   = 65*0.9917 - 58

= 64.4606 - 58

= 6.46

Therefore the lower bound on a 2 month European put option is $6.46