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options futures and other derivatives

Questions and Answers of Options Futures And Other Derivatives

Suppose that each day during July the minimum temperature is 68 Fahrenheit and the maximum temperature is 82" Fahrenheit. What is the payoff from a call option on the cumulative CDD during July with
Distinguish between the historical data and the risk-neutral approach to valuing a derivative. Under what circumstance do they give the same answer. P-968
How is a typical natural gas forward contract structured? P-968
What is meant by HDD and CDD? P-968
Suppose that the LIBOR zero rate is flat at 5% with annual compounding. In a 5-year swap, company X pays a fixed rate of 6% and receives LIBOR. The volatility of the 2-year swap rate in 3 years is
Suppose that you are trading a LIBOR-in-arrears swap with an unsophisticated counter- party who does not make convexity adjustments. To take advantage of the situation, should you be paying fixed or
Estimate the interest rate paid by P&G on the 5/30 swap in Section 32.7 if (a) the CP rate is 6.5% and the Treasury yield curve is flat at 6% and (b) the CP rate is 7.5% and the Treasury yield curve
LIBOR zero rates are flat at 5% in the United States and flat at 10% in Australia (both annually compounded). In a 4-year swap Australian LIBOR is received and 9% is paid with both being applied to a
In the accrual swap discussed in the text, the fixed side accrues only when the floating reference rate lies below a certain level. Discuss how the analysis can be extended to cope with a situation
Explain why a plain vanilla interest rate swap and the compounding swap in Section 32.2 can be valued using the "assume forward rates are realized" rule, but a LIBOR-in- arrears swap in Section 32.4
Calculate the total convexity/timing adjustment in Example 32.3 of Section 32.4 if all cap volatilities are 18% instead of 20% and volatilities for all options on 5-year swaps are 13% instead of 15%.
Explain carefully why a bank might choose to discount cash flows on a currency swap at a rate slightly different from LIBOR.P-987
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
What is the value of a 2-year fixed-for-floating compound swap where the principal is $100 million and payments are made semiannually. Fixed interest is received and floating is paid? The fixed rate
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?P-987
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
Prove equation (31.19).P-987
Prove the formula for the variance V(T) of the swap rate in equation (31.17).P-987
Prove equation (31.15).P-987
“An option adjusted spread is analogous to the yield on a bond.” Explain this statement.P-987
Explain why IOs and POs have opposite sensitivities to the rate of prepayments.P-987
Explain why a sticky cap is more expensive than a similar ratchet cap.P-987
Show that equation (31.10) reduces to (31.4) as the 8, tend to zero.P-987
Provide an intuitive explanation of why a ratchet cap increases in value as the number of factors increase.P-987
What is the advantage of LMM over HJM?P-987
“When the forwardrate volatility, s(t, T), in HJM is oe"’(T'”, the Hull-White model results.” Verify that this is true by showing that HJM gives a process for bond prices that is consistent
“When the forward rate volatility s(t, T) in HJM is constant, the Ho-Lee model results.”Verify that this is true by showing that HJM gives a process for bond prices that is consistent with the
Prove the relationship between the drift and volatility of the forward rate for the multifactor version of HJM in equation (31.6).P-987
Explain the difference between a Markov and a non-Markov model of the short rate.P-987
Suppose that the (CIR) process for short-rate movement in the risk-neutral world is dra(b-r)dt+ard: and the market price of interest rate risk is A. (a) What is the real world process for r? (b) What
Modify Sample Application G in the DerivaGem Application Builder software to test the convergence of the price of the trinomial tree when it is used to price a 2-year call option on a 5-year bond
Verify that the DerivaGem software gives Figure 30.11 for the example considered. Use the software to calculate the price of the American bond option for the lognormal and?P-987
Use the DerivaGem software to value 1x 4, 2 x 3, 3 x 2, and 4 x 1 European swap options to receive floating and pay fixed. Assume that the 1-, 2-, 3-, 3-, and 5-year interest rates are 3%, 3.5%,
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
Construct a trinomial tree for the Ho-Lee model where 0.02. Suppose that the the initial zero-coupon interest rate for a maturities of 0.5, 1.0, and 1.5 years are 7.5%, 8%, and 8.5%. Use two time
Suppose that short rate r is 4% and its real-world process is dr 0.110.05-rdt+0.01 dz?P-987
(a) What is the second partial derivative of P(t. T) with respect to r in the Vasicek and CIR models. (b) In Section 30.2, D is presented as an alternative to the standard duration measure D. What is
Prove equations (30.25), (30.26), and (30.27).P-987
Use the DerivaGem software to value 1x 4, 2 x 3, 3 x 2, and 4 x 1 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%,
What does the calibration of a one-factor term structure model involve?P-987
Calculate the price of an 18-month zero-coupon bond from the tree in Figure 30.10 and verify that it agrees with the initial term structure.P-987
Calculate the price of a 2-year zero-coupon bond from the tree in Figure 30.6. 30.17. Calculate the price of a 2-year zero-coupon bond from the tree in Figure 30.9 and verify that it agrees with the
Suppose a 0.05, 0.015, and the term structure is flat at 10%. Construct a trinomial tree for the Hull-White model where there are two time steps, each 1 year in length.P-987
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
Use a change of numeraire argument to show that the relationship between the futures rate and forward rate for the Ho-Lee model is as shown in Section 6.3. Use the relationship to verify the
Suppose that a 0.05 and 0.015 in the Hull-White model with the initial term structure being flat at 6% with semiannual compounding. Calculate the price of a 2.1-year European call option on a bond
In the Hull-White model, a 0.08 and 0.01. 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
Use the answer to Problem 30.9 and put-call parity arguments to calculate the price of a put option that has the same terms as the call option in Problem 30.9.P-987
Suppose that a=0.05, b=0.08, and =0.015 in Vasicek's model with the initial short-term interest rate being 6%. Calculate the price of a 2.1-year European call option on a bond that will mature in 3
Repeat Problem 30.7 valuing a European put option with a strike of $87. What is the put-call parity relationship between the prices of European call and put options? Show that the put and call option
Suppose that a 0.1, b 0.08, and a 0.015 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
Suppose that a 0.1 and b=0.1 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
Can the approach described in Section 30.4 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
Explain the difference between a one-factor and a two-factor interest rate model.P-987
If a stock price were mean reverting or followed a path-dependent process there would be market inefficiency. Why is there not a market inefficiency when the short-term interest rate does so?P-987
Suppose that the short rate is currently 4% and its standard deviation is 1% per annum. What happens to the standard deviation when the short rate increases to 8% in (a) Vasicek's model; (b)
What is the difference between an equilibrium model and a no-arbitrage model?P-987
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
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. The LIBOR
Suppose that the LIBOR yield curve is flat at 8% with annual compounding. 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
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
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
Describe how you would (a) calculate cap flat volatilities from cap spot volatilities and (b) calculate cap spot volatilities from cap flat volatilities.P-987
Suppose that zero rates are as in Problem 28.17. 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
Show that V += V2, where V is the value of a swaption to pay a fixed rate of 5K and receive LIBOR between times T1 and T2, f is the value of a forward swap to receive a fixed rate of sx and pay LIBOR
Carry out a manual calculation to verify the option prices in Example 28.2. 28.17. Suppose that the 1-year, 2-year, 3-year, 4-year, and 5-year zero rates are 6%, 6.4%, 6.7%, 6.9%, and 7%. The price
Suppose that the yield R on a zero-coupon bond follows the process dR=dt +dz where and are functions of R and t, and dz is a Wiener process. Use It's lemma to show that the volatility of the
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
When a bond's price is lognormal can the bond's yield be negative? Explain your answer. P-987
Explain why there is an arbitrage opportunity if the implied Black (flat) volatility of a cap is different from that of a floor. Do the broker quotes in Table 28.1 present an arbitrage opportunity?
Derive a put-call parity relationship for European swap options. P-987
Derive a put-call parity relationship for European bond options. P-987
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? P-987
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
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
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.
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
Explain carefully how you would use (a) spot volatilities and (b) flat volatilities to value a 5-year cap. P-987
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 rate is 10% per
Explain why a swap option can be regarded as a type of bond option. P-987
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? When would the
Consider a variable that is not an interest rate: (a) In what world is the futures price of the variable a martingale? (b) In what world is the forward price of the variable a martingale? (c)
Suppose that the price of a zero-coupon bond maturing at time T follows the process _ dP(t, T) = /,tPP(t, T) dt + oPP(t, T) dz and the price of a derivative dependent on the bond follows the process
A security’s price is positively dependent on two variables: the price of copper and the yen/dollar exchange rate. Suppose that the market price of risk for these variables is 0.5 and 0.1,
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
Prove the result in Section 27.5 that when df = [+]+[dz, dz i=1 and dg = with the dz, uncorrelated, f/g is a martingale for = (Hint: Start by using equation (13A.11) to get the processes for In f and
Explain the difference between the way a forward interest rate is defined and the way the forward values of other variables such as stock prices, commodity prices, and exchange rates are defined.
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
Prove the result in Section 27.5 that when df = [+]+[dz, dz i=1 and dg = with the dz, uncorrelated, f/g is a martingale for = (Hint: Start by using equation (13A.11) to get the processes for In f and
Explain the difference between the way a forward interest rate is defined and the way the forward values of other variables such as stock prices, commodity prices, and exchange rates are defined.
The variable S is an investment asset providing income at rate q measured in currency A. It follows the process dS = sSdt +S dz in the real world. Defining new variables as necessary, give the
The variable S is an investment asset providing income at rate q measured in currency A. It follows the process dS=sSdt +sSdz? P-987
"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
Show that when and g provide income at rates af and qg, respectively, equation (27.15) becomes fo = goes- E (Hint: Form new securities f* and g* that provide no income by assuming that all the income
Suppose that an interest rate x follows the process dx = a(x-x) dt +cxdz wherea, xo, and c are positive constants. Suppose further that the market price of risk for x is. What is the process for x in
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
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
Repeat the analysis in Section 26.8 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
Suppose that the volatilities used to price a 6-month currency option are as in Table 19.2. Assume that the domestic and foreign risk-free rates are 5% per annum and the current exchange rate is
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

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