Verify that, when substituting the volatility parameter  with  the closed-form formula for  yields 

Closed-form formulas. The three questions are independent.

(a) Based on the lognormal model, what is the value of a 1-month European call on Kroger Co. struck at $25? Kroger Co. currently trades at $24, no dividend is scheduled, the risk-free rate is 1% p.a., and volatility is 20%.

(b) Based on the lognormal model, what is the value today of a European call and a European put struck at $50 and maturing in 6 months? The underlying asset is a stock with spot price $50, no dividend, and 30% volatility. The risk-free rate is 10% per annum.

(c) Verify that, when substituting the volatility parameter  with  the closed-form formula for  yields 

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