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Financial Derivatives (N1559) – Spring 2024
Seminar Questions Week 5
Seminar Questions
1. (JC 17.8) A stock’s price is $100. After 1 year, it either goes up to $120.773898 or down to $89.47150422. The money market account has a continuous compounding interest rate of 5%. An option with strike price K = 105 matures after 1 year.
(a) Set up a perfect hedge and compute the call option’s value.
(b) What is the hedge ratio? What does it signify?
2. (JC 17.12) Consider the same market as the question above. Let a put option have the same strike K = $105.
(a) Set up a perfect hedge and compute the put option’s value.
(b) What is the hedge ratio? What does it signify?
3. (JC 17.14) Using the data from the two previous questions, verify that the put-call parity holds.
4. (JC 17.16) A stock’s price is $50. After six months, it either goes up by U = 1.22095341 or down by D = 0.79881010. Options mature after six months and has strike price K = 45. A dollar in the money market account earns continuously compounded risk-free interest rate at 2 percent per year. Compute the call option value.
5. (Binary Option:) A binary (or digital or cash-or-nothing) call option pays $X if the stock price exceeds the strike K on date T , and nothing otherwise. Consider a one-period binomial model with U = 1.25, D = 0.75. The interest rate r satisfies erT = 1.1, and suppose K = S0 , the current stock price.
1. What is the no arbitrage price of the digital at time zero?
2. Suppose a range of such options exist with any strike K ≥ 0 possible. Which (if any) of these options can be replicated without trading in the stock? Explain why this may be unrealistic in practice.