Question: Why is the historical data approach appropriate

> 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

> Explain the difference between a forward start option and a chooser option.

> 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 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?

> Prove the relationship between the drift and volatility of the forward rate for the multifactor version of HJM in equation (33.6). m(t, T,2.) = Esa{t, T, 2.) salt, 7, 2,) dr (33.6)

> Show that the swap volatility expression (33.19) in Section 33.2 is correct. To rN-1 M dt (33.19) To Jt-0 1+ TEmGkm (0)

> Prove the formula for the variance of the swap rate in equation (33.17). PrN-1 (33.17) 1+ G¿(t) where +

> Prove equation (33.15). dFt) Σ dt + (33.15) Filt) 1+ 8, F;(t) (t)

> The LIBOR/swap curve is flat at 3% with continuous compounding and a 4-year bond with a coupon of 4% per annum (paid semiannually) sells for 101. How would an asset swap on the bond be structured? What is the asset swap spread?

> Explain the difference between a Markov and a non-Markov model of the short rate.

> In the Hull–White model, a = 0:08 and
. Calculate the price of a 1-year European call option on a zero-coupon bond that will mature in 5 years when the term structure is flat at 10%, the principal of the bond is $100, and the strike price is $68.

> Use the answer to Problem 32.5 and put–call parity arguments to calculate the price of a put option that has the same terms as the call option in Problem 32.5.

> Suppose that a =0:1, b =0:08, and
in Vasicek’s model, with the initial value of the short rate being 5%. Calculate the price of a 1-year European call option on a zero-coupon bond with a principal of $100 that matures in 3 years when the strike price is

> Can the approach described in Section 32.2 for decomposing an option on a coupon-bearing bond into a portfolio of options on zero-coupon bonds be used in conjunction with a two-factor model? Explain your answer.

> Prove equations (32.15), (32.16), and (32.17). P(t, T') = Â(t, T)e- -t-T)R (32.15) where In Â(t, T) = In- P(0, T) B(t, T) P(0, t + At) -In- P(0, t) B(t, t +At) P(0, t) (1— е 2а) в(t, Т)В(, Т) — В(t, г + д)) (32.16) 4a and B(t, T') = - B(t, T) At (32.

> Use the DerivaGem software to value
,
,
, and
European swap options to receive fixed and pay floating. Assume that the 1-, 2-, 3-, 4-, and 5-year interest rates are 6%, 5.5%, 6%, 6.5%, and 7%, respectively. The payment frequency on the swap is semian

> What does the calibration of a one-factor term structure model involve?

> Calculate the price of a 2-year zero-coupon bond from the tree in Figure 32.4. Figure 32.4 Example of the use of trinomial interest rate trees. Upper number at each node is rate; lower number is value of instrument. E 14% 3 12% F 12% B 1.11 10% 10%

> Suppose that a =0:1 and b=0:1 and in both the Vasicek and the Cox, Ingersoll, Ross model. In both models, the initial short rate is 10% and the initial standard deviation of the short-rate change in a short time
is
. Compare the prices given by the mo

> Suppose that the risk-free zero curve is flat at 6% per annum with continuous compounding and that defaults can occur at times 0.25 years, 0.75 years, 1.25 years, and 1.75 years in a 2-year plain vanilla credit default swap with semiannual payments. Supp

> Observations spaced at intervals t are taken on the short rate. The ith observation is ri (1 ≤ i ≤ m). Show that the maximum-likelihood estimates of a, b*, and  in Vasicek’s model are given by maximizing

> a) What is the second partial derivative of P(t,T) with respect to r in the Vasicek and CIR models? (b) In Section 31.2,
is presented as an alternative to the usual duration measure, D. What is a similar alternative,
, to the convexity measure in Secti

> Explain whether any convexity or timing adjustments are necessary when: (a) We wish to value a spread option that pays off every quarter the excess (if any) of the 5-year swap rate over the 3-month LIBOR rate applied to a principal of $100. The payoff oc

> If the yield volatility for a 5-year put option on a bond maturing in 10 years’ time is specified as 22%, how should the option be valued? Assume that, based on today’s interest rates the modified duration of the bond at the maturity of the option will

> Calculate the price of an option that caps the 3-month rate, starting in 15 months’ time, at 13% (quoted with quarterly compounding) on a principal amount of $1,000. The forward interest rate for the period in question is 12% per annum (quoted with quart

> 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 risk-free interest rate is 10% per annum, the bond’s forward price volatility is 8

> Suppose that the yield R on a zero-coupon bond follows the process
where and are functions of R and t, and dz is a Wiener process. Use Itoˆ ’s lemma to show that the volatility of the zero-coupon bond price declines to zero as it approaches maturity.

> When a bond’s price is lognormal can the bond’s yield be negative? Explain your answer.

> A company caps 3-month LIBOR at 2% per annum. The principal amount is $20 million. On a reset date, 3-month LIBOR is 4% per annum. What payment would this lead to under the cap? When would the payment be made?

> Suppose that an interest rate x follows the process
where a, x0, and c are positive constants. Suppose further that the market price of risk for x is
. What is the process for x in the traditional risk-neutral world?

> 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

> Consider two securities both of which are dependent on the same market variable. The expected returns from the securities are 8% and 12%. The volatility of the first security is 15%. The instantaneous risk-free rate is 4%. What is the volatility of the s

> Show that when w = h/g and h and g are each dependent on n Wiener processes, the ith component of the volatility of w is the ith component of the volatility of h minus the ith component of the volatility of g. Use this to prove the result that if
is t

> Prove the result in Section 28.5 that when
and
with the dzi uncorrelated, f/g is a martingale for
(Hint: Start by using equation (14A.11) to get the processes for ln f and ln g.)

> How is the market price of risk defined for a variable that is not the price of an investment asset?

> ‘‘The IVF model does not necessarily get the evolution of the volatility surface correct.’’ Explain this statement.

> Write down the equations for simulating the path followed by the asset price in the stochastic volatility model in equations (27.2) and (27.3). ds/s = (r - q) dt + Vī dzs dV = a(V1 – V)dt +§Vª dzv (27.2) (27.3) %3D

> Suppose that the volatility of an asset will be 20% from month 0 to month 6, 22% from month 6 to month 12, and 24% from month 12 to month 24. What volatility should be used in Black–Scholes–Merton to value a 2-year option?

> Confirm that the CEV model formulas satisfy put–call parity.

> Explain why a down-and-out put is worth zero when the barrier is greater than the strike price.