Let r = 4 % , A ( 0 ) = 100 r = 4 % ,

Let $r=4\%,A\left(0\right)=100$, and let the price of a 3 -month $(T=0.25)$ ATMF ( $K=100e^{0.04/4})$ call option be 2.50 .

(a) Using your favorite solver, calculate the implied volatility of the call option.

(b) Calculate the delta, gamma, and vega of the call option.

(c) What is the price of an ATMF straddle (ATMF call + ATMF put)?

(d) Holding volatility constant, how much does the price of the above straddle with $K=100e^{0.04/4}$ change if the asset price $A\left(0\right)$ changes to 101 ?

(e) Holding $A\left(0\right)=100$ constant, how much does the price of the above straddle with $K=100e^{0.04/4}$ change if volatility increases by $1\%$, ${\sigma }_{\text{new}}={\sigma }_{\text{old}}+0.01$ ?

(f) Starting with the equation for the price of an ATMF straddle, use the Taylor series expansion of $N\left(x\right)$ at 0 to approximate the price of the ATMF straddle as a linear function of $\sigma$.