Explain the difference between a forward start option and a chooser option.
> Suppose that in Example 29.3 of Section 29.2 the payoff occurs after 1 year (i.e., when the interest rate is observed) rather than in 15 months. What difference does this make to the inputs to Black’s model? //
> Explain how you would value a derivative that pays off 100R in 5 years, where R is the 1-year interest rate (annually compounded) observed in 4 years. What difference would it make if the payoff were in (a) 4 years and (b) 6 years?
> What other instrument is the same as a 5-year zero-cost collar where the strike price of the cap equals the strike price of the floor? What does the common strike price equal?
> 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 the same coupon is $102 and both bonds have a principal of $100
> A bank uses Black’s model to price European bond options. Suppose that an implied price volatility for a 5-year option on a bond maturing in 10 years is used to price a 9-year option on the bond. Would you expect the resultant price to be too high or too
> LIBOR zero rates are flat at 5% in the United States and flat at 10% in Australia (both annually compounded). In a 4-year diff swap Australian LIBOR is received and 9% is paid with both being applied to a USD principal of $10 million. Payments are exchan
> Explain carefully how you would use (a) spot volatilities and (b) flat volatilities to value a 5-year cap.
> Describe how you would (a) calculate cap flat volatilities from cap spot volatilities and (b) calculate cap spot volatilities from cap flat volatilities.
> Explain why a swap option can be regarded as a type of bond option.
> Suppose that risk-free zero rates and LIBOR forward rates are as in Problem 29.17. Use DerivaGem to determine the value of an option to pay a fixed rate of 6% and receive LIBOR on a 5-year swap starting in 1 year. Assume that the principal is $100 millio
> Show that , where V1 is the value of a swaption to pay a fixed rate of and receive LIBOR between times T1 and T2, f is the value of a forward swap to receive a fixed rate of and pay LIBOR between times T1 and T2, and V2 is the value of a swaption to
> Suppose that all risk-free (OIS) zero rates are 6.5% (continuously compounded). The price of a 5-year semiannual cap with a principal of $100 and a cap rate of 8% (semiannually compounded) is $3. Use DerivaGem to determine: (a) The implied 5-year flat vo
> Carry out a manual calculation to verify the option prices in Example 29.2. Example 29.2 Consider a European put option on a 10-year bond with a principal of 100. The coupon is 8% per year payable semiannually. The life of the option is 2.25 years a
> What is the value of a European swap option that gives the holder the right to enter into a 3-year annual-pay swap in 4 years where a fixed rate of 5% is paid and LIBOR is received? The swap principal is $10 million. Assume that the LIBOR/swap yield curv
> Explain why there is an arbitrage opportunity if the implied Black (flat) volatility of a cap is different from that of a floor.
> Derive a put–call parity relationship for European swap options.
> Suppose that the spot price, 6-month futures price, and 12-month futures price for wheat are 250, 260, and 270 cents per bushel, respectively. Suppose that the price of wheat follows the process in equation (36.3) with a = 0:05 and . Construct a two-tim
> Derive a put–call parity relationship for European bond options.
> ‘‘The expected future value of an interest rate in a risk-neutral world is greater than it is in the real world.’’ What does this statement imply about the market price of risk for (a) an interest rate and (b) a bond price. Do you think the statement
> Show that when f and g provide income at rates and , respectively, equation (28.15) becomes Hint: Form new securities and that provide no income by assuming that all the income from f is reinvested in f and all the income in g is reinvested in g.)
> Prove that, when the security f provides income at rate q, equation (28.9) becomes (Hint: Form a new security f that provides no income by assuming that all the income from f is reinvested in f.)
> Deduce the differential equation for a derivative dependent on the prices of two non-dividend-paying traded securities by forming a riskless portfolio consisting of the derivative and the two traded securities.
> An oil company is set up solely for the purpose of exploring for oil in a certain small area of Texas. Its value depends primarily on two stochastic variables: the price of oil and the quantity of proven oil reserves. Discuss whether the market price of
> In the example considered in Section 36.5: (a) What is the value of the abandonment option if it costs $3 million rather than zero? (b) What is the value of the expansion option if it costs $5 million rather than $2 million?
> How is the tree in Figure 35.2 modified if the 1- and 2-year futures prices are $21 and $22 instead of $22 and $23, respectively. How does this affect the value of the American option in Example 35.3. Figure 35.2 Tree for spot price of a commodity:
> An insurance company’s losses of a particular type are to a reasonable approximation normally distributed with a mean of $150 million and a standard deviation of $50 million. (Assume no difference between losses in a risk-neutral world and losses in the
> How would you calculate the initial value of the equity swap in Business Snapshot 34.3 if OIS discounting were used? Business Snapshot 34.3 Hypothetical Confirmation for an Equity Swap Trade date: 4-January, 2016 11-January, 2016 Following business
> A company uses the GARCH(1,1) model for updating volatility. The three parameters are , , and . Describe the impact of making a small increase in each of the parameters while keeping the others fixed.
> Suppose that all 12-month LIBOR forward rates are 5% with annual compounding. The OIS zero curve is flat at 4.8% with continuous compounding. In a 5-year swap, company X pays a fixed rate of 6% and receives LIBOR. The volatility of the 2-yearswap rate in
> Suppose that you are trading a LIBOR-in-arrears swap with an unsophisticated counterparty who does not make convexity adjustments. To take advantage of the situation, should you be paying fixed or receiving fixed? How should you try to structure the swap
> In the flexi cap considered in Section 33.2 the holder is obligated to exercise the first N in-the-money caplets. After that no further caplets can be exercised. (In the example, N = 5.) Two other ways that flexi caps are sometimes defined are: (a) The h
> Verify that the DerivaGem software gives Figure 32.9 for the example considered. Use the software to calculate the price of the American bond option for the lognormal and normal models when the strike price is 95, 100, and 105. In the case of the normal
> Use the DerivaGem software to value, , , and European swap options to receive floating and pay fixed. Assume that the 1-, 2-, 3-, 4-, and 5-year interest rates are 3%, 3.5%, 3.8%, 4.0%, and 4.1%, respectively. The payment frequency on the swap is sem
> A trader wishes to compute the price of a 1-year American call option on a 5-year bond with a face value of 100. The bond pays a coupon of 6% semiannually and the (quoted) strike price of the option is $100. The continuously compounded zero rates for mat
> Construct a trinomial tree for the Ho–Lee model where . Suppose that the initial zero-coupon interest rate for a maturity of 0.5, 1.0, and 1.5 years are 7.5%, 8%, and 8.5%. Use two-time steps, each 6 months long. Calculate the value of a zero-coupon bon
> What is the result corresponding to that given in Problem 31.7. for the CIR model. Use maximum likelhood methods to estimate the a, b, and  parameters for the CIR model using the same data as that used for the Vasicek model in Section 3
> Use the DerivaGem software to value a European swaption that gives you the right in 2 years to enter into a 5-year swap in which you pay a fixed rate of 6% and receive floating. Cash flows are exchanged semiannually on the swap. The continuously compound
> Use the DerivaGem software to value a 5-year collar that guarantees that the maximum and minimum interest rates on a LIBOR-based loan (with quarterly resets) are 7% and 5%, respectively. All 3-month LIBOR forward rates are 6% per annum (with quarterly co
> What is a first-to-default credit default swap? Does its value increase or decrease as the default correlation between the companies in the basket increases? Explain your answer.
> A swaption gives the holder the right to receive 7.6% in a 5-year swap starting in 4 years. Payments are made annually. The forward swap rate is 8% with annual compounding and its volatility is 25% per annum. The principal is $1 million and risk-free (OI
> Consider an 8-month European put option on a Treasury bond that currently has 14.25 years to maturity. The current cash bond price is $910, the exercise price is $900, and the volatility for the bond price is 10% per annum. A coupon of $35 will be paid b
> A 3-year convertible bond with a face value of $100 has been issued by company ABC. It pays a coupon of $5 at the end of each year. It can be converted into ABC’s equity at the end of the first year or at the end of the second year. At the end of the fir
> A European call option on a non-dividend-paying stock has a time to maturity of 6 months and a strike price of $100. The stock price is $100 and the risk-free rate is 5%. Use DerivaGem to answer the following questions: (a) What is the Black–Scholes–Mer
> Repeat the analysis in Section 27.8 for the put option example on the assumption that the strike price is 1.13. Use both the least squares approach and the exercise boundary parameterization approach.
> Suppose that the volatilities used to price a 6-month currency option are as in Table 20.2. Assume that the domestic and foreign risk-free rates are 5% per annum and the current exchange rate is 1.00. Consider a bull spread that consists of a long positi
> Outperformance certificates (also called ‘‘sprint certificates,’’ ‘‘accelerator certificates,’’ or ‘‘speeders’’) are offered to investors by many European banks as a way of investing in a company’s stock. The initial investment equals the stock price, S0
> In the DerivaGem Application Builder Software modify Sample Application D to test the effectiveness of delta and gamma hedging for a call on call compound option on a 100,000 units of a foreign currency where the exchange rate is 0.67, the domestic risk-
> Use the DerivaGem Application Builder software to compare the effectiveness of daily delta hedging for (a) the option considered in Tables 19.2 and 19.3 and // (b) an average price call with the same parameters. Use Sample Application C. For the avera
> Suppose that a stock index is currently 900. The dividend yield is 2%, the risk-free rate is 5%, and the volatility is 40%. Use the results in Technical Note 27 on the author’s website to calculate the value of a 1-year average price call where the strik
> Consider a down-and-out call option on a foreign currency. The initial exchange rate is 0.90, the time to maturity is 2 years, the strike price is 1.00, the barrier is 0.80, the domestic risk-free interest rate is 5%, the foreign risk-free interest rate
> Sample Application F in the DerivaGem Application Builder Software considers the static options replication example in Section 26.17. It shows the way a hedge can be constructed using four options (as in Section 26.17) and two ways a hedge can be constru
> Consider an up-and-out barrier call option on a non-dividend-paying stock when the stock price is 50, the strike price is 50, the volatility is 30%, the risk-free rate is 5%, the time to maturity is 1 year, and the barrier at $80. Use the DerivaGem softw
> Produce a formula for valuing a cliquet option where an amount Q is invested to produce a payoff at the end of n periods. The return earned each period is the greater of the return on an index (excluding dividends) and zero.
> What is the value in dollars of a derivative that pays off £10,000 in 1 year provided that the dollar/sterling exchange rate is greater than 1.5000 at that time? The current exchange rate is 1.4800. The dollar and sterling interest rates are 4% and 8% pe
> In Example 25.3, what is the spread for (a) a first-to-default CDS and (b) a second-to default CDS? //
> Suppose that: (a) The yield on a 5-year risk-free bond is 7%. (b) The yield on a 5-year corporate bond issued by company X is 9.5%. (c) A 5-year credit default swap providing insurance against company X defaulting costs 150 basis points per year. What ar
> Explain how you would expect the returns offered on the various tranches in a synthetic CDO to change when the correlation between the bonds in the portfolio increases.
> Table 25.6 shows the 5-year iTraxx index was 77 basis points on January 31, 2008. Assume the risk-free rate is 5% for all maturities, the recovery rate is 40%, and payments are quarterly. Assume also that the spread of 77 basis points applies to all matu
> Assume that the hazard rate for a company is and the recovery rate is R. The risk-free interest rate is 5% per annum. Default always occurs halfway through a year. The spread for a 5-year plain vanilla CDS where payments are made annually is 120 basis
> A 5-year credit default swap requires a quarterly 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 4 years and 2 months, and the price of the cheapest de
> The value of a company’s equity is $4 million and the volatility of its equity is 60%. The debt that will have to be repaid in 2 years is $15 million. The risk-free interest rate is 6% per annum. Use Merton’s model to estimate the expected loss from defa
> Use DerivaGem to calculate the value of: (a) A regular European call option on a non-dividend-paying stock where the stock price is $50, the strike price is $50, the risk-free rate is 5% per annum, the volatility is 30%, and the time to maturity is one y
> Estimate the value of a new 6-month European-style average price call option on a non-dividend- paying stock. The initial stock price is $30, the strike price is $30, the risk-free interest rate is 5%, and the stock price volatility is 30%.
> A new European-style floating lookback call option on a stock index has a maturity of 9 months. The current level of the index is 400, the risk-free rate is 6% per annum, the dividend yield on the index is 4% per annum, and the volatility of the index is
> Describe the payoff from a portfolio consisting of a floating lookback call and a floating lookback put with the same maturity.
> What is the value of a derivative that pays off $100 in 6 months if an index is greater than 1,000 and zero otherwise? Assume that the current level of the index is 960, the risk-free rate is 8% per annum, the dividend yield on the index is 3% per annum,
> Does a floating lookback call become more valuable or less valuable as we increase the frequency with which we observe the asset price in calculating the minimum?
> Is a European down-and-out option on an asset worth the same as a European down and out option on the asset’s futures price for a futures contract maturing at the same time as the option?
> What is the value of the swap in Problem 25.8 per dollar of notional principal to the protection buyer if the credit default swap spread is 150 basis points? 25.8. Suppose that the risk-free zero curve is flat at 7% per annum with continuous compounding
> Explain the term ‘‘single-tranche trading.’’
> Suppose that the spread between the yield on a 3-year zero-coupon riskless bond and a 3-year zero-coupon bond issued by a corporation is 1%. By how much does Black– Scholes–Merton overstate the value of a 3-year European option sold by the corporation.
> Explain carefully the distinction between real-world and risk-neutral default probabilities. Which is higher? A bank enters into a credit derivative where it agrees to pay $100 at the end of 1 year if a certain company’s credit rating f
> Show that under Merton’s model in Section 24.6 the credit spread on a T-year zero coupon bond is, where.
> What is the difference between a total return swap and an asset swap?
> Why does the credit exposure on a matched pair of forward contracts resemble a straddle?
> A company enters into a total return swap where it receives the return on a corporate bond paying a coupon of 5% and pays LIBOR. Explain the difference between this and a regular swap where 5% is exchanged for LIBOR.
> Suppose that a financial institution has entered into a swap dependent on the sterling interest rate with counterparty X and an exactly offsetting swap with counterparty Y. Which of the following statements are true and which are false? Explain your answ
> Explain the difference between a cash CDO and a synthetic CDO.
> What is the formula relating the payoff on a CDS to the notional principal and the recovery rate?
> A company can buy an option for the delivery of 1 million units of a commodity in 3 years at $25 per unit. The 3-year futures price is $24. The risk-free interest rate is 5% per annum with continuous compounding and the volatility of the futures price is
> The correlation between a company’s gross revenue and the market index is 0.2. The excess return of the market over the risk-free rate is 6% and the volatility of the market index is 18%. What is the market price of risk for the company’s revenue?
> Would you expect the volatility of the 1-year forward price of oil to be greater than or less than the volatility of the spot price? Explain your answer.
> A company has 1- and 2-year bonds outstanding, each providing a coupon of 8% per year payable annually. The yields on the bonds (expressed with continuous compounding) are 6.0% and 6.6%, respectively. Risk-free rates are 4.5% for all maturities. The reco
> ‘‘HDD and CDD can be regarded as payoffs from options on temperature.’’ Explain this statement.
> Why is the historical data approach appropriate for pricing a weather derivatives contract and a CAT bond?
> Why is the price of electricity more volatile than that of other energy sources?
> Suppose that each day during July the minimum temperature is Fahrenheit and the maximum temperature is Fahrenheit. What is the payoff from a call option on the cumulative CDD during July with a strike of 250 and a payment rate of $5,000 per degree-day
> How is a typical natural gas forward contract structured?
> Consider a commodity with constant volatility and an expected growth rate that is a function solely of time. Show that, in the traditional risk-neutral world, where is the value of the commodity at time T, is the futures price at time 0 for a contr
> How can an energy producer use derivatives markets to hedge risks?
> What are the characteristics of an energy source where the price has a very high volatility and a very high rate of mean reversion? Give an example of such an energy source.
> What is meant by HDD and CDD?
> Explain why a plain vanilla interest rate swap and the compounding swap in Section 34.2 can be valued using the ‘‘assume forward rates are realized’’ rule, but a LIBOR-in-arrears swap in Section 34.4 cannot.
> Suppose a 3-year corporate bond provides a coupon of 7% per year payable semiannually and has a yield of 5% (expressed with semiannual compounding). The yields for all maturities on risk-free bonds is 4% per annum (expressed with semiannual compounding).
> What is the value of a 5-year swap where LIBOR is paid in the usual way and in return LIBOR compounded at LIBOR is received on the other side? The principal on both sides is $100 million. Payment dates on the pay side and compounding dates on the receive
> What is the value of a 2-year fixed-for-floating compounding swap where the principal is $100 million and payments are made semiannually? Fixed interest is received and floating is paid. The fixed rate is 8% and it is compounded at 8.3% (both semiannuall
> Suppose that a swap specifies that a fixed rate is exchanged for twice the LIBOR rate. Can the swap be valued using the ‘‘assume forward rates are realized’’ rule?
> Explain why IOs and POs have opposite sensitivities to the rate of prepayments.
> Show that equation (33.10) reduces to (33.4) as the tend to zero. dFt) F(t) (33.10) 1+ 8, F;(t)
> What is the advantage of LMM over HJM?
