风险管理案例 This project score will constitute 50% of the semester

Individual project

EF4321 Derivatives and Risk Management


Notes:

This is NOT a group project. Every student should work INDIVDUALLY and submit INDIVDU-
This problem set is to be turned in by Tuesday, 10 December 11:00 pm. Please present your work using MS Word or PDF and submit online on You may use Excel for calculation but the final solution should be presented in MS Word orPDF.
This project score will constitute 50% of the semester
Allof questions are from lectures 6 though
风险管理案例风险管理案例


True/False Questions [30 points – 3 points for each]
For each of the following statements, mark (a) if the statement is true or (b) if it is false.





The profit for a short position in a call option is always negative or
True
False




As the time to expiration becomes longer, an American call option always becomes more
True
False




Consider two European put options with the same expiration dates and the same strike prices. The underlyingassets of the two options are  One option is for stock A whose current price is $50 and has the volatility of 30%. The other is for stock B whose current price is $45 and has the volatility of 25%. Both stock A and B will pay no dividends. The price of put A is always higher than the price of put B.
True
False




The put-call parity holds only when future stock price changes as described in a binomial
True
False




To find the option price in a binomial tree, we need an assumption regarding the probability of the increase/decrease in the stock
True
False






Suppose that risk-averse investors expect the return on a stock to be µ per annum and the risk-free rate is r per annum. In a binomial tree, if µ < r, the real probability of an increase in the stock price is lower than the risk-neutral probability of the
True
False




Consider an American put option on a non-dividend paying stock. The option will expire on date T . On date t(< T ), the option payoff from the immediate exercise is always lower than the value that results from not exercising and holding the
True
False




In risk-neutral valuation, we recognize that investors are risk-averse and thus modify the probability of an increase in a stock price from the real
True
False




In the Discounted Cash Flow, the required return on a European option should be higher than the risk-free rate. Otherwise, an arbitrage
True
False




An investor wants to construct a bull spread using put options with the same expiration dates. The investorneeds to long the put with strike price K1 and short the put with strike price K2(> K1).
True
False




2. Short-Answer Questions
Consider a three-year European call option with the strike price of $150.  The underlying stock will     pay $10-dividend two years later from now.  The current stock price is $170.  The risk-free rate is 3%  per Find the range of the call prices that do not allow any arbitrage. [5 points]


An investor wants to construct a bear spread using two put options with the same expiration T . One put has the strike price of $20 and currently sells for $2. The other put has the strike price of $30  and currently sells for $7. Find the range of stock price on T that results in a positive profit for the investor. [5points]


In time 0, an investor takes a calendar spread by selling two-year European call option and buying three-year European call option. These two options have the same strike price of $80 and are for the same stock that pays no dividends. The two-year option sells for $5 and the three-year option sellsfor
$7. Two years later, the stock price turns out to be $90. The risk-free rate is 2% per annum. What is the minimum of the profit from this strategy? (We assume that we sell the longer-term option in year two) [5 points]







Consider a European call and a European put on a non-dividend-paying stock. Both the call and the putwill expire in one year and have the same strike prices of $120. The stock currently sells for $115. The risk-free rate is 5% per  The price of the call is $7 and the price of the put is $5. Is there an arbitrage? If so, show an arbitrage strategy. (To show the arbitrage, present the table listing actions and resulting cash flows) [6 points]




Consider a two-year European put on Canadian Dollar (CAD). The strike price of the put is 6.50 HKD(HongKong Dollar)/CAD. The risk-free rate is 2% per annum in Hong Kong and 3% per annum in Canada. The current exchange rate is 5.90 HKD/CAD. The put currently sells for $0.4 in Hong Kong. Is there an arbitrage for Hong Kong investors? If so, show an arbitrage strategy. (To show the arbitrage, present the table listing actions and resulting cash flows) [6 points]


Supposethat there are two possible states of the economy, A and B, in year T . A stock’s price in year T will depend on the economic state as in the table below. A risk-free bond currently sells for $8 and it will pay $10 in year T in every
state A state B
stock $150 $80
bond $10 $10
In addition, we also see a year-T European call option on the stock above.  The call has the strike price  of $100 and currently sells for $20. What is the current stock price? [6 points]


An one-year European call option has the strike price of $60. The underlying stock pays no dividend and currently sells for $70. One time step is six months long, and the stock price may move up or down by 10% in each step. The risk-free rate is 3% per


What is the risk-neutral probability of an increase in the stock price in each step? [5points]




What is the time-0 current price of the call? [6points]




Find the replicating portfolio that we construct in time 0 to generate the same value as the call  six months later. [6points]




Suppose that risk-averse investors require the stock return to be 12% per annum. In the approach of Discounted Cash Flow, what is the discount rate for the call per annum? [7points]
(Hint: Use the replicating portfolio found in (c).)







The current term-structure of risk-free rate is as


Term-structure in year 0 maturity (years) zero-rate (%)

1 2.5

2 3.0



A risk-free bond will pay $1,000 two years from now. The price of the bond one year later depends on  the term structure then. There are two possible scenarios in year 1:



Term-structure in year 1

Scenario A Scenario B

maturity (years) zero-rate (%) maturity (years) zero-rate (%)
1 1.0 1 4.0
2 2.0 2 5.0


An investor considers buying an one-year European call option on the bond with the strike price of

$970.



Whatare the payoffs in scenario A and B in year one for a long position in the call? [6 points]


What is the present value (in year 0) of the call?[7 points]
(Hint: Find a replicating portfolio using the two-year bond and an one-year bond. For an one-year bond, you can choose any face value as you like.)