Financial Derivatives (N1559) – Spring 2024
Seminar Questions Week 6
Seminar Questions
Note: Create an Excel spreadsheet for each of the following questions! This will help you prepare for the mid-term test and the final exam as you can expect similar questions in the assessments.
1. (American Option)
• The stock’s price S is $100
• After three months, it either goes up and gets multiplied with the up factor U = 1.163287, or it goes down and gets multiplied with the down factor D = 0.861785
• Options mature after T = 0.5 years
• Options have a strike price of K = $110
• A dollar invested in the money market account grows to $1.012578 after three months (corresponding to an annual compounded interest rate of 5%)
(a) Set up a stock price tree that can price options maturing at T.
(b) Find the price of a put option.
(c) If the option was American, would it have been exercised early? If so, what is the price of the American option.
(d) Using PCP, what it the price of an European call option.
2. (Exotic Option)
Build a two-step stock price tree using the following information:
• maturity: 4 months
• annual volatility of the stock: 0.30
• annual, continuously compounded risk-free rate: 0.08
• stock price today: 100
• stock price moves up or down by: U = 1.1303 or D = 0.8847 respectively
(a) Calculate the price of an up-and-out put option (a barrier option) with strike 105, maturity 4 months and barrier level 110.
(b) What if it’s an up-and-in put instead of up-and-out? (Hint: think of a clever ‘model-free’ no arbitrage relationship.)
(c) Use the same stock price tree to instead calculate the price of a compound ‘call-on- call’ option with a maturity of 2 months and a strike of 5, where the underlying of the compound option is a vanilla call with maturity of 4 months and a strike price of 110. (i.e., this derivative is a call option written on a regular call option).
3. (JC 17.19) A stock’s price is $50. After 1 year, it either goes up to $61.0477 or down to $39.9405. The money market account has dollar return 1 + R = 1.01005017. An option with strike price K = 7 matures after 1 year. Consider the following exotic option whose payoff at maturity is given by the square root of the stock price less the strike price if is has a positive value, zero otherwise:
max(S(T) − K, 0)
Assuming that the strike price K is $7, determine the value of this exotic option under the assumption of no-arbitrage.
4. (JC 18)
• The stock’s price S is $100
• After three months, it either goes up and gets multiplied with the up factor U = 1.1282, or it goes down and gets multiplied with the down factor D = 0.9224
• Options mature after T = 1 year
• Options have a strike price of K = $110
• A dollar invested in the money market account grows to $1.0253 after six months (corresponding to an annual compounded interest rate of 5%)
• The price tree of the european put options is:
0
2.8942
• 8.7508 5.9345
15.0494
24.9246
(a) Find the price tree for an American put option