An investor expects a volatility increase of Stock A by 2 years, but dont know when. He is also convinced that Stock A will never gain or lose more than 40% of its current price. Competing banks Rotmar and Ciks are proposing the following products:

Rotmar Bank: Naked European Put option ITM (105% of the Spot price) 1 year Ciks Bank: Delta hedged ATM European call option 2 year Stock A characteristics: No dividend, historical 10y return 15% per annum, implied 2y volatility 40% per annum, Spot price 200 euros, 1y correlation to the global markets (Beta factor) +0.85.

1) Which proposal is the best for the client? Which one is the most expensive? [0.5 pts]

2) What are all the factors, which justify a difference in both proposal prices. Precise for each its influence (increasing or decreasing the price) [1 pt]

3) Based on option combination (combining 4 options), propose an alternative solution than the ones of banks Rotmar and Ciks and explain why it is more suitable to the client. Provide the graphical representation of the payoff [1 pt].

4) Knowing your proposal, bank Rotmar revises its offer and proposes now a 2y European Straddle Strike 95%. Using a CRR approach with a two period tree, price the Straddle [2 pts]

5) Adopting the Black and Scholes framework, based on Itos result (see Annex), show that G = ln(S) also follows a normal law and details its drift and variance parameters [1 pt].

6) Based on (5) considering expected return value is equal to historical one, and considering lognormality property a. Provide the expected value interval of S (T ) (T = 2y) for Stock A with a 95% confidence level [1 pt] b. Based on (5.a.) refine your proposal (3) to improve the price you propose to the client without significantly degrading its risk [0.5 pt]

7) Calculate the price of (4) using BS formula [1 pt]

8) Based on (7), precise: a. What explains the difference with CRR calculation? How can we refine CRR approach to reach a better precision on price? [0.5 pt] b. Why dont we need expected return of Stock A to price an option on it? [0.5 pt]

9) What would be the hedging strategy to cover delta of (4)? its vega? [1 pt]

10) What is your trading management of Gamma risk all along the life of the product provided that the realized volatility is 5% with no price jump (very stable market)? [Optional 1pt]

Appendix

Black-Scholes formula:

CRR formula: Ito PDE. Let G be a regular function of S and t. Then:

Normal cumulative distribution: