A European call option on a non-dividend-paying stock has a time to maturity of 6 months and a strike price of $$\$ 100$$. The stock price is $$\$ 100$$ and the risk-free rate is $5 \%$. Use DerivaGem to answer the following questions:
(a) What is the Black-Scholes-Merton price of the option if the volatility is $30 \%$ ?
(b) What is the CEV volatility parameter that gives the same price for the option as you calculated in (a) when $\alpha=0.5$ ?
(c) In Merton's mixed jump-diffusion model, the average frequency of jumps is 1 per year, the average percentage jump size is $2 \%$, and the standard deviation of the logarithm of 1 plus the percentage jump size is $20 \%$. What is the volatility of the diffusion part of the process that gives the same price for the option as you calculated in (a)?
(d) In the variance-gamma model, $\theta=0$ and $v=40 \%$. What value of the volatility gives the same price for the option as you calculated in (a)?
(e) For the models you have developed in (b), (c), and (d), calculate the volatility smile by considering European call options with strike prices between 80 and 120 . Describe the nature of the probability distributions implied by the smiles.