Quantitative Finance
Exam Answer Key
Course Introduction
Quantitative Finance explores the application of mathematical models, statistical techniques, and computational tools to analyze financial markets and manage financial risks. This course covers essential topics such as asset pricing, portfolio optimization, risk management, derivatives valuation, and financial econometrics. Students will develop a strong foundation in the use of programming languages and software relevant to quantitative analysis, gaining practical experience in modeling real-world financial scenarios and interpreting data-driven results to inform investment and risk management decisions.
Recommended Textbook
Derivatives Markets 3rd Edition by Robert L. McDonald
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Page 2
Chapter 1: Introduction to Derivatives
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Sample Questions
Q1) Who from the following list would be considered a speculator by entering into a futures or options contract on commodities?
A) Farmer
B) Corn delivery truck driver
C) Food manufacturer
D) None of the above
Answer: B
Q2) Assume that an investor lends 100 shares of Jiffy,Inc.common stock to a short seller.The bid-ask prices are $32.00 - $32.50.When the position is closed,the bid-ask prices are $32.50 - $33.00.The commission rate is 0.5%.The market interest rate is 5.0% and the short rebate rate is 3.0%.Calculate the gain or loss to the lender.Assume the lender is not subject to a bid-ask loss or commissions.
A) $164.00 gain
B) $164.00 loss
C) $100.00 gain
D) $100.00 loss
Answer: A
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Chapter 2: An Introduction to Forwards and Options
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Sample Questions
Q1) The spot price of the market index is $900.A 3-month forward contract on this index is priced at $930.The market index rises to $920 by the expiration date.The annual rate of interest on treasuries is 2.4% (0.2% per month).What is the difference in the payoffs between a long index investment and a long forward contract investment? (Assume monthly compounding.)
A) $10.84
B) $24.59
C) $26.40
D) $43.20
Answer: B
Q2) The spot price of the market index is $900.A 3-month forward contract on this index is priced at $930.What is the profit or loss to a short position if the spot price of the market index rises to $920 by the expiration date?
A) $20 gain
B) $20 loss
C) $10 gain
D) $10 loss
Answer: C
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4
Chapter 3: Insurance, collars, and Other Strategies
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Sample Questions
Q1) What is the maximum profit that an investor can obtain from a strategy employing a long 830 call and a short 850 call over 6 months? Interest rates are 0.5% per month.
A) $6.80
B) $7.68
C) $9.24
D) $12.32
Answer: B
Q2) Using option strategy concepts,what is the value of an insured home,if the value of the uninsured home is $220,000,the house was purchased for $180,000 and the house has a casualty policy costing $500 with a $2,000 deductible? Ignore interest costs.
A) $180,000
B) $217,500
C) $220,000
D) $222,500
Answer: B
Q3) Why is a straddle position considered a speculation on the asset's volatility?
Answer: The strategy loses money if prices stay constant,but benefits from large changes in prices,either up or down.
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Chapter 4: Introduction to Risk Management
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Sample Questions
Q1) Explain the relationship between options costs and profits under a put option insurance strategy.
Q2) A farmer expects to harvest 800,000 bushels of corn.To eliminate price risk,the farmer elects to short corn futures.What would cause the farmer to short only 720,000 bushels of corn?
A) Basis risk
B) Illiquid futures markets
C) Margin requirements
D) Quantity uncertain
Q3) A $1.75 strike call option has a $0.14 premium.The $1.75 strike put option premium is $0.12.What is the net cost for Farmer Jayne to create a synthetic short forward contract? (Assume 4.0% interest.)
A) $0.0208
B) -$0.0208
C) $0.000
D) -$0.0424
Q4) Why are synthetics created and/or calculated when the actual derivative is available?
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Chapter 5: Financial Forwards and Futures
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Sample Questions
Q1) The annualized dividend yield on the S&P 500 Index is 1.40%.The continuously compounded interest rate is 6.4%.If the 9-month forward price is $925.28 and the index is priced at $950.46,what is the profit/loss from a cash-and-carry strategy?
A) $25.18 loss
B) $25.18 gain
C) $61.50 loss
D) $61.50 gain
Q2) Explain the steps necessary to take advantage of an arbitrage opportunity,which may exist between the dollar and yen,when a future yen payment is required.
Q3) Interest rates on the U.S.dollar are 6.5% and euro rates are 5.5%.The dollar per euro spot rate is 0.950.What is the arbitrage profit on a required 1 million euro payment if the forward rate is 0.980 dollars per euro and the exchange occurs in one year?
A) $10,000
B) $21,000
C) $28,000
D) $34,000
Q4) Name some advantages that futures contracts have over forward contracts.
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Chapter 6: Commodity Forwards and Futures
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Q1) Refer to the table 6.1.Given a lease rate of 7.0% on the 24-month corn forward contract,what is the approximate potential arbitrage profit per contract?
A) 3.68 cents
B) 4.48 cents
C) 5.84 cents
D) 6.90 cents
Q2) The spot price of gasoline is 258 cents per gallon and the annualized risk free interest rate is 4.0%.Given a lease rate of 1.0%,a continuously paid storage rate of 0.5%,and a convenience yield of 0.75%,what is the no-arbitrage price range of a 1-year forward contract (in cents)?
A) 265.19 to 267.19
B) 258 to 265.19
C) 258 to 267.19
D) 247.16 to 265.19
Q3) Explain how a negative correlation between agricultural production and commodity prices creates a natural hedge.
Q4) What function does the convenience yield serve in setting forward prices and how does this influence arbitrage opportunities?
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Page 8
Chapter 7: Interest Rate Forwards and Futures
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Sample Questions
Q1) Explain the process of creating a synthetic Forward Rate Agreement.
Q2) A Forward Rate Agreement contains an agreed interest rate of 3.1% on a 6-month loan.If settled in arrears,what amount would the borrower pay or receive on an $800,000 loan if the prevailing 6-month interest rate is 3.6%?
A) $4,000 payment
B) $4,000 receipt
C) $1,729 payment
D) $1,729 receipt
Q3) The annual coupon rate on a 1-year treasury bond is 5.5%.The coupon on a 2-year treasury bond is 5.8%.What is the continuously compounded yield on a 2-year zero coupon bond?
A) 5.55%
B) 5.65%
C) 5.75%
D) 5.85%
Q4) How is duration calculated? What is the nature and use of duration? How does duration compare to the linear concept of the bond price and interest rate relationship? Is duration better than convexity or worse? Duration is considered common knowledge in the fixed income world and should be discussed at length.
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Chapter 8: Swaps
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Sample Questions
Q1) An investor enters into a 2-year swap agreement to purchase crude oil at $105.65 per barrel.Soon after the swap is created forward prices rise and the new swap price on a similar swap is $108.32.If interest rates are 3.0% per year,what is the gain to be made from unwrapping the original swap agreement?
A) $2.67
B) $5.11
C) $5.34
D) $5.67
Q2) Given zero-coupon bond yields are 2.0%,2.5%,and 2.8% in years 1,2,and 3,respectively,calculate the prepaid swap price for corn.Assume corn forward prices for the proceeding 3 years are $5.00,$5.20,and $5.35,respectively.
A) $14.87
B) $15.04
C) $16.12
D) $16.20
Q3) How would a market-maker hedge a swap involving variable price and quantity?
Q4) Describe briefly the nature of a swap and its primary component.
Q5) Why do arbitrage profits rarely exist in interest rate swap pricing?
Q6) Explain a "diff swap" as it relates to currency swaps.
Page 10
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Chapter 9: Parity and Other Option Relationships
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Sample Questions
Q1) Consider the case of an exchange option in which the underlying stock is Eli Lilly and Company with a current price of $56.00 per share.The strike asset is Merck,with a per share price of $52.00.Interest rates are 5% and the 3-month call option is trading for $7.00.What is the price of the put?
A) $3.00
B) $4.00
C) $7.00
D) $11.00
Q2) Put-call parity allows a discussion of option pricing relationships without actually pricing an option.Have the class list all the possible pricing relationships they can recall.Add to the list until reasonably complete.Follow up this exercise by listing puts and calls,while asking students to state if certain premiums are possible.
Q3) Jillo,Inc.stock is selling for $54.70 per share.Calls and puts with a $55.00 strike and 40 days until expiration are selling for $1.65 and $1.23,respectively.Draw a profit and loss graph illustrating the arbitrage.
Q4) Explain in simple terms why a call option on a non-dividend paying stock should never be exercised early.
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11
Chapter 10: Binomial Option Pricing: Basic Concepts
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Sample Questions
Q1) A stock is currently selling for $22.00 per share.Ignoring interest,determine the intrinsic value of a call option should there exist equally probable stock prices of $25.00 and $23.00.
A) $0.00
B) $1.00
C) $2.00
D) $3.00
Q2) A stock is selling for $32.70.The strike price on a call,maturing in 6 months,is $35.The possible stock prices at the end of 6 months are $39.50 and $28.40.If interest rates are 6.0%,what is the option price?
A) $1.90
B) $2.80
C) $3.40
D) $4.20
Q3) Discuss options on other assets.Ask students to define currency options,futures options,index options,and commodity options.Require that the students state which variables in the securities listed above correspond with the binomial pricing inputs used for stock options.
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Chapter 11: Binomial Option Pricing: Selected Topics
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Sample Questions
Q1) Ask the class to prove that option pricing is consistent with standard discounted cash flow calculations.Propose that students form groups and develop two binomial trees for the same set of data.One tree should use real probabilities as defined in chapter 11 and the other as defined in chapter 10.
Q2) Consider a two-period binomial model,where each period is 6 months.Assume the stock price is $60.00, = 0.30,r = 0.05.An American put option with a strike price of $65 would be exercised early at what dividend yield?
A) 5.0%
B) 6.0%
C) 11.0%
D) Never exercised early
Q3) Consider a one-period binomial model of 6 months.Assume the stock price is $63.00, = 0.28,r = 0.05 and the stock's expected return is 14.0%.What is the true probability of the stock going up?
A) 56.6%
B) 52.4%
C) 48.2%
D) 46.4%
Q4) Under what circumstances should an option be exercised early?
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Chapter 12: The Black-Scholes Formula
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Sample Questions
Q1) What is the difference between implied volatility and historical volatility?
Q2) As the date of expiration approaches,what change in theta might counteract or slow down the drop in the option price?
A) Decrease
B) Increase
C) Stay constant
D) Indifferent
Q3) Assume that an investor is currently holding a reverse straddle position (i.e.a short put and short call),which is currently a profitable investment.All else being equal,what would this investor like to happen to vega?
A) Decrease
B) Increase
C) Stay constant
D) Indifferent
Q4) Which Greek is also called time decay and why?
Q5) Draw a payoff diagram for a long put position,depicting options that expire at 0,30 and 60 days.
Q6) What unique feature about perpetual options makes it possible to derive a valuation formula?
Page 14
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Chapter 13: Market-Making and Delta-Hedging
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Sample Questions
Q1) Assume S = $62.50, = 0.20,r = 0.03,div = 0.0,on a $60 strike call and 81 days until expiration.Given a delta = 0.7092,gamma = 0.0582,and theta = -0.0158,what is the PREDICTED call price,using the delta,gamma,theta approach,after 1 day,assuming a $0.50 rise in the stock price?
A) $4.364
B) $4.376
C) $4.390
D) $4.392
Q2) What actions are required to both delta-hedge and gamma-hedge a written option position?
Q3) Assume that a $50 strike put pays a 2.0% continuous dividend,r = 0.07, = 0.25,and the stock price is $48.00.What is the profit or loss,per share,for a short put position if the option expires in 60 days and the price rises to $50.00 after 5 days?
A) $1.05 loss
B) $1.05 gain
C) $1.12 gain
D) $1.12 loss
Q4) What are the two methods by which insurance companies hedge their risk of extreme losses?
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Chapter 14: Exotic Options: I
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Sample Questions
Q1) Why is the premium on a standard option and down-and-in call the same when the barrier price exceeds the stock price?
Q2) Why might a down-and-in put option be more attractive than a standard option when hedging a foreign currency position?
Q3) Assume S = $55,K = $55,r = 0.07, = 0.27,div = 0.0,and 180 days until expiration.What is the premium on an Asian average price call,where N = 5?
A) $2.89
B) $2.99
C) $3.09
D) $3.19
Q4) When hedging a foreign currency position,what makes a down-and-out put unattractive?
Q5) Assume S = $60,K = $60,r = 0.07, = 0.24,div = 0.02,and 90 days until expiration.What is the premium on an Asian average price put where N = 4?
A) $1.74
B) $1.84
C) $1.94
D) $2.04
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Chapter 15: Financial Engineering and Security Design
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Q1) Assume the price of Mary,Inc.stock is $56.00,interest rates are 4.8%,div yield = 0,and = 0.35.What is the price of a $1,000 par value 2-year price-participation note paying a 5.0% annual coupon and receiving 50.0% of all price appreciation above $65.00?
A) $896.44
B) $996.44
C) $1006.44
D) $1106.44
Q2) Which of the following financially engineered products is NOT used to defer the payment of capital gains taxes on securities that have appreciated?
A) Commodity Linked Options
B) DECS
C) Equity Linked Notes
D) PEPS
Q3) What possible tax advantage exists in equity-linked notes?
Q4) What is the primary difference between an equity-linked bond and a currency-linked bond?
Q5) How does a coupon bond differ from an equity-linked bond?
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Page 17
Chapter 16: Corporate Applications
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Q1) A company issues an option grant with an outperformance feature,against the S&P 500.Assume S&P 500 = 950,S = 22,k = 25, = 0.25,r = 0.06,and 5 years until expiration.The S&P 500 has a dividend yield of 2%,standard deviation of 18.0% and a 0.30 correlation coefficient with the stock.What is the value of the outperformance feature?
A) $0.99
B) $1.31
C) $1.59
D) $1.72
Q2) Daniels,Inc.has assets valued at $2 million and 50,000 outstanding shares.A 5-year zero-coupon bond exists,which pays $400,000 at maturity.The bond is convertible into 10,000 shares.Assume = 0.30,r = 0.055,and no dividend is paid.What is the value of the bond?
A) $402,672
B) $452,172
C) $415,022
D) $385,172
Q3) What three components exist in the value of an "outperform stock option"?
Q4) Why does a company sell a put when issuing compensation options?
Q5) What feature of reload options prevents the use of a Black-Scholes valuation?
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Chapter 17: Real Options
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Sample Questions
Q1) The price of oil is $115 per barrel.The effective lease rate and risk free rate are 3.0% and 4.0%,respectively.The constant cost of extraction is $85 per barrel and the volatility of prices is 15.0%.If an untapped well costs $2,100 to open and can produce indefinitely,at what price per barrel should the well be opened?
A) $349
B) $423
C) $454
D) $484
Q2) What is the relationship,in general,between volatility and trigger prices,assuming constant costs?
Q3) The current price per ton of iron ore is $145.00.The effective lease rate is 3.0% and the risk free rate is 4.5%.The cost to mine one ton of iron ore is $110.00 and constant.What is the trigger price at which we will mine the iron ore?
A) $163.80
B) $180.40
C) $210.50
D) $205.70
Q4) What two components go into valuing an infinite commodity reserve?
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Chapter 18: The Lognormal Distribution
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Q1) A stock is valued at $28.00.The annual expected return is 9.0% and the standard deviation of annualized returns is 19.0%.If the stock is lognormally distributed,what is the price of the stock given a one standard deviation move up after 4 years?
A) $28.00
B) $32.33
C) $40.13
D) $54.60
Q2) Why do we assume a lognormal distribution in option pricing? Ask the class to explain the pluses and minuses to this assumption.Once the downfalls are established,probe students to find out if a better alternative exists.
Q3) A stock is valued at $55.00.The annual expected return is 12.0% and the standard deviation of annualized returns is 22.0%.If the stock is lognormally distributed,what is the price of the stock given a one standard deviation move up after 3 years?
A) $64.41
B) $74.41
C) $84.41
D) $94.41
Q4) What assumption is made in the Black-Scholes model concerning volatility?
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Chapter 19: Monte Carlo Valuation
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Q1) What type of random variable is necessary for a Monte Carlo valuation?
A) Standard normal distribution
B) Normal distribution
C) Lognormal distribution
D) All of the above
Q2) A stock owned by a portfolio has a bankruptcy probability of 1% per year.Using a Poisson distribution,what is the probability that this firm will not declare bankruptcy over the upcoming 10 years?
A) 60%
B) 70%
C) 80%
D) 90%
Q3) Which distribution is a discrete probability distribution that counts the number of events,such as large stock price moves,that occur over a period of time?
A) Latin hypercube
B) Normal
C) Lognormal
D) Poisson
Q4) How does the number of draws impact the validity of a Monte Carlo simulation?
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Chapter 20: Brownian Motion and Itos Lemma
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Q1) Assume the following: LN(S)and LN(Q)have a correlation coefficient of -0.65,S(0)= 55,Q(0)= 60,r = 0.04, s = 0.22 Q = 0.15,and dividends = 0.Using formula 20.39,what is the price of a claim that pays Q/ \(\sqrt{S}\) ?
A) $8.16
B) $9.16
C) $10.16
D) $11.16
Q2) What is the relationship of the Sharpe ratios and risk premiums between stocks and options?
Q3) Define the term drift.
Q4) Assume a stock price of S(0)= $83.00,r = 0.045, = 0.25,and dividend = 0.02.What is the price of a claim that pays S³? Use formula 20.29.
A) $423,323
B) $710,695
C) $624,165
D) $818,123
Q5) Provide a definition of Brownian motion.
Q6) What are two important implications of assuming that prices follow a geometric Brownian motion?
Page 22
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Chapter 21: The Black-Scholes-Merton Equation
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Q1) What is the boundary condition for a European call option?
A) Max [0,S(T)-K]
B) Max [0,K-S(T)]
C) Min [0,S(T)-K]
D) Min [0,K-S(T)]
Q2) Lapel Inc.stock price is $32.00.Joe bets Sarah that the price will be above $35.00 in 6 months (180 days).The standard deviation of the stock is 0.25 and the risk free interest rate is 5.0%.If Joe wins the bet,he wishes to be paid with one share of stock.What is the value of the wager to Joe?
A) $3.00
B) $9.65
C) $12.44
D) $19.58
Q3) Briefly define a terminal boundary condition.
Q4) What is the boundary condition for a European put option?
A) Max [0,S(T)-K]
B) Max [0,K-S(T)]
C) Min [0,S(T)-K]
D) Min [0,K-S(T)]
Q5) Give an example of currency translation that is a change in numeraire.
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Chapter 22: Risk-Neutral and Martingale Pricing
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Q1) The pricing of derivatives is linked to the decisions investors make relative to:
A) Black-Scholes variables
B) Money markets
C) Portfolios
D) T-bills
Q2) How do probabilities change with a change of measure?
Q3) Which of the following is not commonly used as a numeraire?
A) Futures contract
B) Money market account
C) Risky asset
D) Zero coupon bond
Q4) In martingale pricing,the observed price of a stock follows a process which substitutes what variable for alpha?
A) Delta
B) Epsilon
C) Money market rate
D) Risk-free rate
Q5) What aspect of risk-neutral pricing valuation links it to portfolio selection?
Q6) What does Girsanov's theorem tell us about drift and Brownian motion?
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Chapter 23: Exotic Options: 2
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Q1) The Buckingham Casino offers to give every gambler one share of Buckingham Casino Corp.stock if the price drops below $40.00,as an incentive to spur business.If S = $45.25, = 0.15,r = 0.05 and div = 0,how much profit or loss is Buckingham incurring if they charge $0.25 to participate in this wager?
A) $0.31 loss
B) $0.31 profit
C) $0.19 loss
D) $0.19 profit
Q2) A multivariate option that has a claim with a payoff determined by the average of two or more asset prices is known as:
A) Basket options
B) Multioptions
C) Quantos options
D) Rainbow options
Q3) What purpose do currency linked options serve?
Q4) Donald Trump offers to give you a partnership share in his casinos if the price of his shares drops below a certain level.He charges a nominal fee for this right.What is he offering you and is he wise?
Q5) What is the characteristic that makes options,like quantos,multivariate options?
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Chapter 24: Volatility
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Q1) The sum of the squared,continuously compounded returns used to calculate a volatility is referred to as:
A) ARCH
B) EWMA
C) GARCH
D) Realized quadratic variation
Q2) The process of emphasizing more recent observations of data in calculating volatility is commonly known as:
A) ARCH
B) EWMA
C) GARCH
D) Realized quadratic variation
Q3) Plotting the volatility of a security in a three dimensional graph,using time to maturity on one axis and strike price on another,is referred to as volatility:
A) Skew
B) Smile
C) Smirk
D) Surface
Q4) What is the primary difference between ARCH models and GARCH models?
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Chapter 25: Interest Rate and Bond Derivatives
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Q1) Assume a = 0.15,b = 0.08,r = 0.05,and 0.30.Using the CIR model,calculate the delta of a zero coupon bond maturing in 5 years.
A) -4.08
B) -3.08
C) -2.08
D) -1.08
Q2) What are the various models in bond pricing and behavior? Ask students to describe the various models along with an explanation of each model's strengths and weaknesses.
Q3) Under what conditions does delta-gamma-theta approximate the exact bond price change?
Q4) If next year's bond prices for 3-year zero coupon bonds may be either 0.8923 or 0.8644,what is the yield volatility?
A) 12.7%
B) 13.7%
C) 14.7%
D) 15.7%
Q5) What is calibration?
Q6) How does the node configuration in interest rates and bonds differ from stocks?
Page 27
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Chapter 26: Value at Risk
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21 Verified Questions
21 Flashcards
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Sample Questions
Q1) Why is recent data more relevant than older data when calculating volatility?
Q2) A bond maturing in 5 years has a YTM = 0.065 and an annual yield volatility of 2.0%.Given a $15 million portfolio,what is the value at risk over 2 weeks at a 95% confidence level?
A) $283,917
B) $383,917
C) $483,917
D) $583,917
Q3) You own $4 million of Jacko Corp.The expected return is 14.0% and = 0.20.What is the value at risk over 4 weeks at a 99% confidence level?
A) $383,000
B) $413,000
C) $453,000
D) $473,000
Q4) What is implied volatility?
Q5) Why is VaR an important tool in measuring risk? What are some of its shortcomings? Ask the class to explain the rationale for a company to rely heavily on VaR in the absence of other measurement tools.
Q6) How is VaR used in credit risk scenarios?
Page 28
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Chapter 27: Credit Risk
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Sample Questions
Q1) The chance that a counter party may fail to meet a contractual obligation on a debt instrument is referred to as:
A) Credit risk
B) Credit spread
C) Loss given default
D) Recovery rate
Q2) What is the recovery rate?
Q3) Suppose that B = $500 and A = $470, = 9%,r = 4%, = 17%,and = 0.If T = 8,what is the risk neutral default probability?
A) 13.0%
B) 20.5%
C) 38.3%
D) 44.4%
Q4) What is meant by the phrase "tranche" when referring to collateralized debt obligations?
Q5) What does a transition matrix indicate about a bond's future credit risk?
Q6) What are the two ways that the payoff conditional on default can be expressed?
Q7) What is a credit default swap and what function does it serve?
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