Questions and Answers of An Introduction To Derivative Securities

What are the seven-year historical hazard rates that would be calculated from Table 24.1 for companies with different credit ratings? Assuming a 40% recovery rate, what credit spread is necessary to
In what is known as an asset swap, the promised coupons on a bond that is worth par are exchanged for a floating rate plus a spread. Show that the cost of default on the bond is the present value of
In Example 25.2, what is the tranche spread for the 6% to 9% tranche assuming a tranche correlation of 0.15?Example 25.2A 5-year credit default swap requires a quarterly payment at the rate of 60
The payoff from a forward contract is delayed by two years. Explain how the impact of the delay on value is affected by the correlation between the price of the underlying asset and interest rates.
On July 1, 2021, a company enters into a forward contract to buy 10 million Japanese yen on January 1, 2022. On September 1, 2021, it enters into a forward contract to sell 10 million Japanese yen on
On May 21, 2020, as indicated in Table 1.2, the spot ask price of Apple stock is $316.50 and the ask price of a call option with a strike price of $320 and a maturity date of September is $21.70. A
The price of gold is currently $$1$,200 per ounce. The forward price for delivery in 1 year is $$1$,300 per ounce. An arbitrageur can borrow money at 3% per annum. What should the arbitrageur do?
On May 21, 2020, an investor owns 100 Apple shares. As indicated in Table 1.3, the share price is about $$316$ and a December put option with a strike price of $$290$ costs $$21.30$.The
Suppose that in September 2021 a company takes a long position in a contract on May 2022 crude oil futures. It closes out its position in March 2022. The futures price (per barrel) is $$48.30$ when
Suppose that, on October 24, 2022, a company sells one April 2023 live cattle futures contract. It closes out its position on January 21, 2023. The futures price (per pound) is 121.20 cents when it
It is July 2021. A mining company has just discovered a small deposit of gold. It will take 6 months to construct the mine. The gold will then be extracted on a more or less continuous basis for 1
A company wishes to hedge its exposure to a new fuel whose price changes have a 0.6 correlation with gasoline futures price changes. The company will lose $1 million for each 1 cent increase in the
Between October 30, 2022, and November 1, 2022, you have a choice between owning a U.S. government bond paying a 12% coupon and a U.S. corporate bond paying a 12% coupon. Consider carefully the day
The 60-day SOFR rate has been estimated as 3%. The 3-month SOFR futures quote for the period between 60 and 150 days is 96.5. Estimate the 150-day SOFR rate.
It is March 10, 2022. The cheapest-to-deliver bond in a December 2022 Treasury bond futures contract is an 8% coupon bond, and delivery is expected to be made on December 31, 2022. Coupon payments on
OIS rates are 3.4% for all maturities. What is the value of an OIS swap with two years to maturity where 3% is received and the floating reference rate is paid. Assume annual compounding, annual
A financial institution has entered into a swap where it agreed to make quarterly payments at a rate of 3% per annum and receive the SOFR three-month reference rate on a notional principal of $100
A financial institution has entered into a swap where it agreed to receive quarterly payments at a rate of 2% per annum and pay the SOFR three-month reference rate on a notional principal of $100
Consider whether markets are efficient in each of the following two cases: (a) a stock price follows fractional Brownian motion and (b) a stock price volatility follows fractional Brownian motion.
Use a change of numeraire argument to show that the relationship between the futures rate and forward rate for the Ho-Lee model. Use the relationship to verify the expression for $\theta(t)$ given
Use a similar approach to that in Problem 30.13 to derive the relationship between the futures rate and the forward rate for the Hull-White model. Use the relationship to verify the expression for
Prove equations (30.25), (30.26), and (30.27). P(t,T) = (t, T)e B(T)R (30.25)
Prove equation (31.15). dFx(t) Fk(t) - i-m(t) . (1) 1 5 (1)5. (1) 1+8, F,(t) dt + 5k.q(t) dzq (31.15) q=1
Prove equation (31.19). To PN-IM To 10 q=1k=n m=1 Tk,mBk.m.q(t)Gk.m(0) (0) 1+Tk,m Gk.m (0) dt (31.19)
Calculate all the fixed cash flows and their exact timing for the swap in Business Snapshot 32.1.Assume that the day count conventions are applied using target payment dates rather than actual
Explain carefully why a bank might choose to discount cash flows on a currency swap at a rate slightly different from LIBOR.
Explain how a $5 \times 8$ option contract for May 2009 on electricity with daily exercise works. Explain how a $5 \times 8$ option contract for May 2009 on electricity with monthly exercise
Derive a relationship between the convenience yield of a commodity and its market price of risk.
A US Treasury bond pays a $7 \%$ coupon on January 7 and July 7 . How much interest accrues per $\$ 100$ of principal to the bondholder between July 7, 2011, and August 9, 2011? How would your
It is January 9,2013 . The price of a Treasury bond with a $12 \%$ coupon that matures on October 12, 2020, is quoted as 102-07. What is the cash price?
The 350-day LIBOR rate is $3 \%$ with continuous compounding and the forward rate calculated from a Eurodollar futures contract that matures in 350 days is $3.2 \%$ with continuous compounding.
It is May 5, 2011. The quoted price of a government bond with a $12 \%$ coupon that matures on July 27,2014 , is $110-17$. What is the cash price?
It is July 30, 2013. The cheapest-to-deliver bond in a September 2013 Treasury bond futures contract is a 13\% coupon bond, and delivery is expected to be made on September 30, 2013. Coupon payments
Between October 30, 2012, and November 1, 2012, you have a choice between owning a US government bond paying a $12 \%$ coupon and a US corporate bond paying a $12 \%$ coupon. Consider carefully
It is March 10, 2011. The cheapest-to-deliver bond in a December 2011 Treasury bond futures contract is an $8 \%$ coupon bond, and delivery is expected to be made on December 31, 2011. Coupon
The futures price for the June 2011 CBOT bond futures contract is 118-23.(a) Calculate the conversion factor for a bond maturing on January 1, 2027, paying a coupon of $10 \%$.(b) Calculate the
A $\$ 100$ million interest rate swap has a remaining life of 10 months. Under the terms of the swap, 6-month LIBOR is exchanged for $7 \%$ per annum (compounded semiannually). The average of the
Explain what a swap rate is. What is the relationship between swap rates and par yields?
A currency swap has a remaining life of 15 months. It involves exchanging interest at $10 \%$ on $£ 20$ million for interest at $6 \%$ on $\$ 30$ million once a year. The term structure of
Explain the difference between the credit risk and the market risk in a financial contract.
Prove the result in equation (10.7). So-K
Prove the result in equation (10.11). So-D-K
Call options on a stock are available with strike prices of $\$ 15, \$ 17 \frac{1}{2}$, and $\$ 20$, and expiration dates in 3 months. Their prices are $\$ 4, \$ 2$, and $\$ \frac{1}{2}$,
A stock price is currently $\$ 40$. It is known that at the end of 1 month it will be either $\$ 42$ or $\$ 38$. The risk-free interest rate is $8 \%$ per annum with continuous compounding.
A stock price is currently $\$ 50$. It is known that at the end of 6 months it will be either $\$ 45$ or $\$ 55$. The risk-free interest rate is $10 \%$ per annum with continuous compounding.
What are the formulas for $u$ and $d$ in terms of volatility?
Footnote 1 shows that the correct discount rate to use for the real-world expected payoff in the case of the call option considered in Figure 12.1 is $42.6 \%$. Show that if the option is a put
Consider a variable $S$ that follows the processFor the first three years, $\mu=2$ and $\sigma=3$; for the next three years, $\mu=3$ and $\sigma=4$. If the initial value of the variable is
Consider an American call option on a stock. The stock price is $\$ 50$, the time to maturity is 15 months, the risk-free rate of interest is $8 \%$ per annum, the exercise price is $\$ 55$,
Show that $S^{-2 r / \sigma^{2}}$ could be the price of a traded derivative security.
Explain how you would do the analysis to produce a chart such as the one in Figure 15.2. Figure 15.2 Erik Lie's results providing evidence of backdating. (Reproduced with permission, from
A company has granted $1,000,000$ options to its employees. The stock price and strike price are both $\$ 20$. The options last 10 years and vest after 3 years. The stock price volatility is \(30
Hedge funds earn a fixed fee plus a percentage of the profits, if any, that they generate. How is a fund manager motivated to behave with this type of arrangement?
Suppose you buy a put option contract on October gold futures with a strike price of $\$ 1,200$ per ounce. Each contract is for the delivery of 100 ounces. What happens if you exercise when the
How are recovery rates usually defined?
What is meant by a "haircut" in a collateralization agreement. A company offers to post its own equity as collateral. How would you respond?
A company enters into a 1-year forward contract to sell $\$ 100$ for AUD150. The contract is initially at the money. In other words, the forward exchange rate is 1.50 . The 1 -year dollar risk-free
Suppose that in Problem 23.15, the 6-month forward rate is also 1.50 and the 6 -month dollar risk-free interest rate is $5 \%$ per annum. Suppose further that the 6 -month dollar rate of interest
A credit default swap requires a semiannual payment at the rate of 60 basis points per year. The principal is $\$ 300$ million and the credit default swap is settled in cash. A default occurs after
Explain the two ways a credit default swap can be settled.
Explain how a cash $\mathrm{CDO}$ and a synthetic CDO are created.
In Example 24.2, what is the tranche spread for the $9 \%$ to $12 \%$ tranche?Data from Example 24.2Consider the mezzanine tranche of iTraxx Europe (5-year maturity) when the copula correlation
In Example 24.2, what is the tranche spread for the $6 \%$ to $9 \%$ tranche?Data from Example 24.2Consider the mezzanine tranche of iTraxx Europe (5-year maturity) when the copula correlation is
Calculate the price of a 1-year European option to give up 100 ounces of silver in exchange for 1 ounce of gold. The current prices of gold and silver are $\$ 380$ and $\$ 4$, respectively; the
Use Monte Carlo simulation to show that Merton's value for a European option is correct when $r=0.05, q=0, \lambda=0.3, k=0.5, \sigma=0.25$, and $S_{0}=30$.
Consider the case of Merton's jump-diffusion model where jumps always reduce the asset price to zero. Assume that the average number of jumps per year is $\lambda$. Show that the price of a
Consider the situation in Merton's jump-diffusion model where the underlying asset is a non-dividend-paying stock. The average frequency of jumps is one per year. The average percentage jump size is
A company caps 3 -month LIBOR at $10 \%$ per annum. The principal amount is $\$ 20$ million. On a reset date, 3-month LIBOR is $12 \%$ per annum. What payment would this lead to under the cap?
Use the Black's model to value a 1 -year European put option on a 10 -year bond. Assume that the current cash price of the bond is $\$ 125$, the strike price is $\$ 110$, the 1 -year interest
Calculate the value of a 4 -year European call option on bond that will mature 5 years from today using Black's model. The 5 -year cash bond price is $\$ 105$, the cash price of a 4 -year bond with
Derive a put-call parity relationship for European bond options.
Calculate the price of a cap on the 90-day LIBOR rate in 9 months' time when the principal amount is $\$ 1,000$. Use Black's model and the following information:(a) The quoted 9-month Eurodollar
An investor enters into a short forward contract to sell 100,000 British pounds for US dollars at an exchange rate of 1.4000 US dollars per pound. How much does the investor gain or lose if the
On July 1, 2011, a company enters into a forward contract to buy 10 million Japanese yen on January 1, 2012. On September 1, 2011, it enters into a forward contract to sell 10 million Japanese yen on
Suppose that USD/sterling spot and forward exchange rates are as follows:What opportunities are open to an arbitrageur in the following situations?(a) A 180-day European call option to buy $£ 1$
The price of gold is currently $\$ 1,000$ per ounce. The forward price for delivery in 1 year is $\$ 1,200$. An arbitrageur can borrow money at $10 \%$ per annum. What should the arbitrageur
On July 15, 2010, an investor owns 100 Google shares. As indicated in Table 1.3, the share price is about $\$ 497$ and a December put option with a strike price of $\$ 460$ costs $\$ 27.30$.
Suppose that in the situation of Table 1.1 a corporate treasurer said: "I will have $£ 1$ million to sell in 6 months. If the exchange rate is less than 1.41, I want you to give me 1.41.If it is
Describe how foreign currency options can be used for hedging in the situation so that(a) ImportCo is guaranteed that its exchange rate will be less than 1.4600, and(b) ExportCo is guaranteed that
Suppose that in September 2012 a company takes a long position in a contract on May 2013 crude oil futures. It closes out its position in March 2013. The futures price (per barrel) is $\$ 68.30$
On July 1, 2012, a Japanese company enters into a forward contract to buy $\$ 1$ million with yen on January 1, 2013. On September 1, 2012, it enters into a forward contract to sell \$1 million on
Live cattle futures trade with June, August, October, December, February, and April maturities. Why do you think the open interest for the June contract is less than that for the August contract in
Suppose that, on October 24, 2012, a company sells one April 2013 live cattle futures contract. It closes out its position on January 21, 2013. The futures price (per pound) is 91.20 cents when it
It is July 2011. A mining company has just discovered a small deposit of gold. It will take 6 months to construct the mine. The gold will then be extracted on a more or less continuous basis for 1
One orange juice futures contract is on 15,000 pounds of frozen concentrate. Suppose that in September 2011 a company sells a March 2013 orange juice futures contract for 120 cents per pound. In
A company enters into a short futures contract to sell 5,000 bushels of wheat for 450 cents per bushel. The initial margin is $\$ 3,000$ and the maintenance margin is $\$ 2,000$. What price
Suppose that there are no storage costs for crude oil and the interest rate for borrowing or lending is $5 \%$ per arnum. How could you make money on May 26, 2010, by trading July 2010 and December
A company wishes to hedge its exposure to a new fuel whose price changes have a 0.6 correlation with gasoline futures price changes. The company will lose $\$ 1$ million for each 1 cent increase in
It is now October 2010. A company anticipates that it will purchase 1 million pounds of copper in each of February 2011, August 2011, February 2012, and August 2012. The company has decided to use
A bank quotes an interest rate of $14 \%$ per annum with quarterly compounding. What is the equivalent rate with (a) continuous compounding and (b) annual compounding?
What is meant by LIBOR and LIBID. Which is higher?
The 6-month and 1 -year zero rates are both $10 \%$ per annum. For a bond that has a life of 18 months and pays a coupon of $8 \%$ per annum (with semiannual payments and one having just been
Suppose that zero interest rates with continuous compounding are as follows:Calculate forward interest rates for the second, third, fourth, fifth, and sixth quarters. Maturity (months) Rate (% per
Assuming that zero rates are as in Problem 4.5, what is the value of an FRA that enables the holder to earn 9.5% for a 3 -month period starting in 1 year on a principal of $1,000,000 ? The interest
The term structure of interest rates is upward-sloping. Put the following in order of magnitude:(a) The 5-year zero rate(b) The yield on a 5-year coupon-bearing bond(c) The forward rate corresponding
Suppose that zero interest rates with continuous compounding are as follows:Calculate forward interest rates for the second, third, fourth, and fifth years. Maturity (years) 1 1 12345 2 3 Rate (% per
A five-year bond provides a coupon of $5 \%$ per annum payable semiannually. Its price is 104. What is the bond's yield? You may find Excel's Solver useful.
Suppose that LIBOR rates for maturities of one, two, three, four, five, and six months are $2.6 \%, 2.9 \%, 3.1 \%, 3.2 \%, 3.25 \%$, and $3.3 \%$ with continuous compounding. What are the
Suppose that you enter into a 6-month forward contract on a non-dividend-paying stock when the stock price is $\$ 30$ and the risk-free interest rate (with continuous compounding) is $12 \%$ per
A stock index currently stands at 350 . The risk-free interest rate is $8 \%$ per annum (with continuous compounding) and the dividend yield on the index is $4 \%$ per annum. What should the

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