Practice for Final Exam

K Foster, Options & Futures, Eco 275, CCNY, Spring 2010

 

 

Exam will cover chapters 1-17, 20, 22, 23, 34 from the book as well as everything in the lecture notes.  (We might get to chapters 24, 28, & 30 but these will not be on the exam.)

 

Like I've indicated, great questions might ask for Bl-S-M valuations and tree valuations of options (or option combinations), then calculate deltas and gammas and VaR.  Of course some bond problems (finding zero rates and forward rates), since those have given everyone trouble but are very important to be able to calculate.

 

So go online and find prices of, say, the SPX and its derivatives (http://www.marketwatch.com/investing/index/SPX/options) or (given recent news) the euro and its derivatives (http://quotes.ino.com/).  Hint: use the prices with larger open interest – that means there's actual trading not just quotes offered.  Find implied volatilities from these option prices, then delta, gamma, and VaR.

 

Example (from class): euro September options with current spot price at 1.2833 (you can put in appropriate r and q),

call/put

Strike

option price

implied volatility

delta

gamma

VaR

c

1.32

0.0236

c

1.335

0.0183

c

1.35

0.0139

c

1.365

0.0108

c

1.4

0.0052

c

1.44

0.0028

c

1.45

0.0024

c

1.49

0.0009

c

1.5

0.0011

c

1.6

0.0003

c

1.38

0.0077

p

1.38

0.1123

p

1.08

0.0035

p

1.14

0.0082

p

1.2

0.0161

p

1.21

0.0144

p

1.22

0.0155

p

1.23

0.0171

p

1.25

0.0253

p

1.26

0.0243

p

1.27

0.0267

p

1.29

0.0332

p

1.3

0.0376

 

 

Past Exam (2005)

  1. (10 points) Using a binomial tree with a single step, assume that a stock with current price of 17 can either rise to 22 or fall to 12 in the next one month.  If I have a put with a strike price of 16, what is its current value?  If that put were American, would I exercise it early?  Assume that the risk-free rate is 4% (continuously compounded).
  2. (10 points) Two bond portfolios both have the same value and the same duration.  Will they have the same Value at Risk?  Explain (perhaps with examples).
  3. (15 points) A call option that matures in 4 months has a strike price of 15 and a volatility of 28% (annualized).  LIBOR is 0.045.  The value of the underlying security is presently 12.  What price would be implied by the Black-Scholes-Merton model?  What is the 1-day 1% VaR, calculated parametrically?  As I consider alternate models, what are the upper and lower bounds for this option price?
  4. (15 points) A stock trades at 100; its volatility is 0.2; the risk-free rate is 0.05.  What is the delta of a butterfly spread that is centered at the money with strike prices at 95 and 105?  Is delta the same, whether it is a butterfly call or butterfly put?  (Use the Black-Scholes-Merton formulas for the Greeks.)
  5. (10 points) Can I create a straddle that is delta-neutral?  Is it gamma-neutral?  Vega-neutral?  Give some examples and show the calculations.  (Use the Black-Scholes-Merton formulas for the Greeks.)
  6.  (15 points) Using a two-step binomial tree model, I am considering pricing a straddle that expires in 2 months.  Currently the stock trades at 100; it can move up or down by 10 in each time period (a month).  The straddle has both options with strike prices of 100.  Assume that the risk-free rate is 4.5% (continuously compounded).
  7. (15 points) A novice trader at your bank holds a portfolio of derivatives and shares of stock in ZZZ Corp, which currently trades at 144 although its volatility is 0.45 (annualized).  The trader has advised the bank to write (i.e. take a short position in) 25 calls with a strike price of 160 expiring in 6 months and to write (i.e. take a short position in) 25 puts with a strike price of 130 also expiring in 6 months.  Assume that the risk-free rate is 4%.  The trader says that the two positions (the calls and puts) form an excellent hedge since, as the price of the underlying stock goes up, the gains and losses balance each other.  Is this novice trader correct?  Can you figure out a better hedge?  Explain (show calculations).
  8. (15 points) I am considering an investment in the country of Cunystan, which uses currency called the "zing".  Since my home country offers interest rates of just 4.5%, I am looking to get higher returns.  You are my portfolio manager, whose expertise I depend upon.  The currency markets allow investors to buy and sell zing at a spot rate of $1 to 70 zing, and buy and sell forward (in one year) at a rate of $1 to 75 zing.  Cunystan government bonds pay a riskless interest rate of 5% in one year (the zero rate, assuming discrete quarterly compounding).
    1. What investment strategy would you recommend, in this situation? 
    2. Right now, at-the-money calls and puts are both priced in the market from Black-Scholes basics – given your knowledge from above, would you rather buy calls or puts?  Why?
  9. (15 points) A bond pays $10,000 in 6 months; its current price is $9811.79.  Another bond pays $1000 in 9 months and then $10,000 in 12 months; its current price is $10,544.77.  What is the 6-month zero rate (in continuous time)?  The 9-month zero rate (also continuous) is 20 basis points higher than the 6-month rate.  Calculate the 12-month zero rate (in continuous time).

 

Past Exam (2007)

 

  1. (25 points) A riskless bond pays 4% per year (compounded continuously).  A stock has 25% volatility over a year. The stock's current price is 80. Your portfolio is long with 2 puts with strike prices of 75 and 70 and long one call with a strike price of 82.  All these options have three months to expiry. 
    1. Draw a payoff graph for the portfolio.
    2. Next a short call position, with an at-the-money strike, is added to the portfolio (same expiry).  Draw the new payoff graph for the portfolio.
    3. Then a short put position is added, with a strike of 78 and 3 months to expiry.  Draw the new payoff graph.
    4. How does the portfolio's 1% VaR change with each additional option position?
    5. How does the portfolio's delta change with each additional option position?
  2. (25 points) Two riskless bonds for the same company are valued in the market (assume they are now riskless because the US government guarantees them!).  The first, with face of 10,000, pays this principal in six months.  It is currently worth 9875.78.  The second bond is a strip of the coupons, paying 500 in three months and then another 500 in six months.  It is currently worth 990.06.
    1. What is the six-month zero rate?
    2. What is the three-month zero rate?
    3. What is the three-to-six month forward rate?
  3. (25 points) Consider a one-step tree model, where each step is one month.  Consider a stock currently trading at 1.25 (which, without any particular realism, we can label "GM") but which has highly asymmetric possible returns: if the stock goes up it will quadruple in price to 5; if it goes down it will drop by one-quarter to 0.3125.  A put option has a strike price of 2.  Assume the risk-free interest rate is 2%.
    1. What does the model imply is the value of this put option?
    2. If the put option is actually priced at 1.25 (an odd coincidence that the stock value and option value are the same!), what values of "up" and "down" would produce this price?
  4. (25 points) Consider the Black-Scholes-Merton model for the same "GM" security as above:
    1. What is the formula for the put price?
    2. What is the implied value of a put option with strike of 2, if the volatility is 70%? 
    3. How much higher would volatility have to be, to imply a price as high as 1.25 for this put?
  5. (20 points) Finally consider a currency call option: currently the Japanese Yen/US Dollar rate is 95 ¥/$; assume that the riskless interest rates are 1% in Japan and 1.5% in the US.  Assume that volatility is 15%. 
    1. What is the implied value (from Black-Scholes-Merton model) of a three-month currency call option with strike of 96? 
    2. What is the implied value of the same option, if the current rate moves up to 96 ¥/$? 
    3. What is delta for this call?  Explain.

 

Other

 

1.        Economists such as Robert Shiller have suggested many more option contracts that might be written, such as options for house prices or for average wages.  For instance, suppose a student could sell a call option that would mature in 10 years (in 2015), that would at that point pay 1/10 of the average wage (in 2014) of an individual with a Bachelor's degree in finance.  The proceeds from this option sale could be used to pay college tuition now.  Under what circumstances would this be a good sale for the student?  How is it different from a loan?  (For now don't worry about the problem of guaranteed payment – assume that both transactions have the same rules preventing forfeiture.)

2.        What other option markets could help your life as a college student?  (A 4-month put for your textbook?)

 

Of course all of the midterm practice problems

 

Midterm Practice Problems

 

 

1.        Using a binomial tree with a single step, assume that a stock with current price of 17 can either rise to 22 or fall to 12 in the next one month.  If I have a put with a strike price of 16, what is its current value?  If that put were American, would I exercise it early?  Assume that the risk-free rate is 4%.

2.        Do the problem above to value a call with strike of 16.  Check if put-call parity holds.

3.        Do those problems for another month (so two-step models) and value options with 2 months to expiration.

4.        Using a two-step binomial tree model, consider pricing a straddle that expires in 2 months.  Currently the stock trades at 100; it can move up or down by 10 in each time period (a month).  The straddle has both options with strike prices of 100.  Assume that the risk-free rate is 4.5%.  What is the cost of the straddle?

5.        Do the problems above to value bear/bull/box spreads, strangles, strips & straps, butterflies, etc. (Pick interesting strike prices.)  What if you had calendar spreads on those?

6.        How do above prices change if risk-free rates are just 1% not 4%?

7.        A  stock currently trades at $5.  A call option has a strike price of $7.50 and a put has strike price of $2.50.  Both options have maturity of 3 months.  If I assume that the riskless interest rate is 5%, what are the upper and lower bounds for the call price?  What are the upper and lower bounds for the put price?  Using a one-step tree, what prices are implied?

8.        Consider a currency call option: currently the Japanese Yen/US Dollar rate is 89 ¥/$; assume that the relevant riskless interest rate is 1%.  Assume that after one month the yen will be worth either 85 or 93. 

a.        What is the implied value of an at-the-money one-month currency call option? 

b.        What is the implied value of the same option, if the current rate moves up to 91 ¥/$?

9.        Or currency forwards: the 'zing' trades at 20zing/$.  The US riskless 3-month rate is 1.5%; the zing rate is 3%.  Over what zing forward prices are there arbitrage opportunities?  What if the bid-ask spread is ½ zing per transaction?

10.     … trying to think of a cross-hedging problem…

11.      Bond valuation problems – continuous time:

a.        A bond paying $100,000 in a year is worth $96,079 now.  What is the implied zero rate?

b.        Another bond from the same issuer pays $10,000 in one year and $10,000 in two years; it is worth $18,747.81.  What is implied two-year zero rate?  Forward rate from 1 to 2 years?

c.        Another bond from the same issuer pays $10,000 in 1, 2, and 3 years.  It is worth $27,550.  What is the 3-year zero rate?  Forward rate from 2-3 years?

d.        Another bond from the same issuer pays $10,000 in 1, 2, 3, 4, and 5 years.  It is worth $43,995.  The 4-year zero rate is 10bps less than the 5-year zero rate.  What are these zero rates & forward rates?

12.     Bond valuation problems – discrete semiannual compounding:

a.        A bond paying $100,000 in a year is worth $97,066 now.  What is the implied zero rate?

b.        Another bond from the same issuer pays $10,000 in one year and $10,000 in two years; it is worth $19,082.  What is implied two-year zero rate?  Forward rate from 1 to 2 years?

c.        Another bond from the same issuer pays $10,000 in 1, 2, and 3 years.  It is worth 28,094.  What is the 3-year zero rate?  Forward rate from 2-3 years?

d.        Another bond from the same issuer pays $10,000 in 1, 2, 3, 4, and 5 years.  It is worth 45,014.  The 4-year zero rate is 10bps less than the 5-year zero rate.  What are these zero rates & forward rates?

 

Past Exams

 

Midterm • Econ 275

Kevin R Foster, CCNY Spring 2009

The questions are worth 75 points.  You have 75 minutes to do the exam, one point per minute.  All answers should be put into the blue books.

You may refer to your books, notes, calculator, computer, or astrology table.  The exam is "open book." However, you must not refer to anyone else, either in person or electronically!   You must do all work on your own.  Cheating is harshly penalized. Please silence all electronic noisemakers such as mobile phones. Good luck.  Stay cool.

 

1.        (20 points) A portfolio of bonds includes the following:

§  Bond ZZZ pays a $4000 coupon every six months, including 6 months from today, 12 months from today, and 18 months from today.  It also pays its principal of $200,000 in 18 months at the same time as its last coupon.  Its current market value is $201,359.31.

§  Bond YYY pays its $8500 coupon in 6 months and then that coupon again in 12 months along with its $500,000 principal.  Its current market value is $499,342.04.

§  Bond VVV pays its principal of $600,000 plus its $8000 semi-annual coupon in 6 months.  Its current market value is $595,960.79.

a.        Find the six month zero rate (continuously compounded).

b.        Find the six-to-twelve month forward rate and the twelve month zero rate (continuously compounded).

c.        Find the twelve-to-eighteen month forward rate and the eighteen month zero rate (continuously compounded).

d.        Find the par yield for each bond.

e.        Find the semiannually-compounded (discrete time every 6 months) zero rates.

f.         What is the duration of bond VVV?

2.        (20 points) A stock can be modeled with a one-step discrete-time tree so that after one month the stock, currently trading at $70 per share, will be worth either $75 or $60 (note that this is asymmetric).  Assume that the riskfree interest rate is 3½%.

a.        Find the value of an at-the-money call.

b.        Find the value of an at-the-money put.

c.        Verify that put-call parity holds.

d.        Find the risk-neutral probabilities of up and down movements.

e.        What value of V (shares of the stock) make a riskless portfolio when combined with one short call?  With one short put?

f.         Why are these values of V different?

3.        (20 points) Use the Black-Scholes-Merton formulas, find prices for the same call and put options as above, assuming a volatility of 0.45.  Does put-call parity hold now?  Explain which model price(s) you would use, if you had to give practical advice in real life.

4.        (15 points) A bank has written 1000 calls on stock ABCD.  These calls expire in 6 months and have a strike price of 15.  ABCD's stock currently trades at 16.  The portfolio of calls is currently worth 1956.88.

a.        What Black-Scholes volatility is implied by this call price (hint: 100s is divisible by 5)? (Riskfree rate is 2%.)

b.        Puts with the same strike trade at the same implied volatility.  What price is this?

c.        The bank believes that the stock will be less volatile in the future: five percentage points less volatile than current option prices imply.  What strategy of long/short positions in options, riskless bonds, or stock would allow the bank to profit from this knowledge?  Explain how risky each strategy is.

 

 

 

Midterm Exam

Econ 27500

fall 2005

 

 

 

The questions are worth 60 points.  You have 60 minutes to do the exam, one point per minute.  All answers should be put into the blue books. You may refer to your books, notes, calculator, computer, or astrology table.  The exam is open book. However, you must not refer to anyone else!  You must do all work on your own.   Cheating is harshly penalized. Please silence all electronic noisemakers such as cellphones. Good luck.  Stay cool.

 

1.        (5 points) A portfolio consists of two options on the same underlying stock, with the same expiration date in three months.  The stock currently trades at $85.  One option is a call with strike price of $87.  The other option is a put with strike price of $84.  Show the payoff graph for the portfolio.

2.         (5 points) A bond pays $3000 in six months and then $3000 in twelve months.  The current market value is $5740.41.  At semi-annual compounding, what is the annual interest rate?  (You can solve the quadratic or iterate; assume the interest rate is a whole number less than 10%.)

3.        (10 points) A bond with face value of $1,000,000 pays the $1,000,000 of principal in six months.  The current market value is $964,399.  What is the six-month zero rate (continuously compounded)?  What is the bond yield (continuously compounded)? 

4.        (10 points) A bond with face $10,000,000 pays that principal in one year and has current market value of $9,441,219.  What is the one-year zero rate?  Based on this answer and the answer from Question 3 above, what is the forward rate for the time period from 6 to 12 months ahead?

5.        (10 points) What is the duration of the bond in Question 3 above?  What is the duration of the bond in Question 4 above?  Calculate how much each bond changes in price when rates rise by 0.0025% after another meeting of Greenspan and the FOMC.

6.         (5 points) An insurance company wants to hedge its position against hurricanes.  The company believes that a Category-3 hurricane hitting a large city will cost it $500,000,000.  It can buy "catastrophe bonds" that each pay out $2,500,000 in the event of a Cat-3 hurricane hitting the city.  Each bond costs $10,000.  The company would like to hedge at least 50% of its exposure.  Should it buy or sell the catastrophe bonds?  How many?  What is the total cost?  Or, the company could hedge using the city's municipal bonds, which would decrease in price if a hurricane struck.  Explain the advantages and disadvantages to this alternate hedging strategy.

7.        (10 points) I am considering an investment in the country of Cunystan, which uses currency called the "zing".  Since my home country offers interest rates of just 1.5%, I am looking to get higher returns.  You are my portfolio manager, whose expertise I depend upon.  The currency markets allow investors to buy and sell zing at a spot rate of $1 to 30 zing, and buy and sell forward (in one year) at a rate of $1 to 35 zing.  Cunystan government bonds pay a riskless interest rate of 5% in one year (that's the continuous-time zero rate).  What investment strategy would you recommend, in this situation?

8.        (5 points) A stock currently trades at $5.  A call option has a strike price of $7.50 and a put has strike price of $2.50.  Both options have maturity of 3 months.  If I assume that the riskless interest rate is 5%, what are the upper and lower bounds for the call price?  What are the upper and lower bounds for the put price?