A company is currently awaiting the outcome of a major lawsuit. This is expected to be known within 1 month. The stock price is currently $$\$ 20$$. If the outcome is positive, the stock price is expected to be $$\$ 24$$ at the end of 1 month. If the outcome is negative, it is expected to be $$\$ 18$$ at this time. The 1 -month risk-free interest rate is $8 \%$ per annum.
(a) What is the risk-neutral probability of a positive outcome?
(b) What are the values of 1 -month call options with strike prices of $$\$ 19$$, $$\$ 20$$, $$\$ 21$$, $$\$ 22$$, and $$\$ 23$$ ?
(c) Use DerivaGem to calculate a volatility smile for 1-month call options.
(d) Verify that the same volatility smile is obtained for 1-month put options.