CHAPTER 29
Interest Rate Derivatives: The Standard Market Models
Practice Questions
Problem 29.1.
A company caps three-month LIBOR at 10% per annum. The principal amount is $20 million. On a reset date, three-month LIBOR is 12% per annum. What payment would this lead to under the cap? When would the payment be made?
An amount $20?000?000?0?02?0?25?$100?000 would be paid out 3 months later.
Problem 29.2.
Explain why a swap option can be regarded as a type of bond option.
A swap option (or swaption) is an option to enter into an interest rate swap at a certain time in the future with a certain fixed rate being used. An interest rate swap can be regarded as the exchange of a fixed-rate bond for a floating-rate bond. A swaption is therefore the option to exchange a fixed-rate bond for a floating-rate bond. The floating-rate bond will be worth its face value at the beginning of the life of the swap. The swaption is therefore an option on a fixed-rate bond with the strike price equal to the face value of the bond.
Problem 29.3.
Use the Black’s model to value a one-year European put option on a 10-year bond. Assume that the current value of the bond is $125, the strike price is $110, the one-year risk-free interest rate is 10% per annum, the bond’s forward price volatility is 8% per annum, and the present value of the coupons to be paid during the life of the option is $10.
In this case, F0?(125?10)e0?1?1?127?09, K?110, P(0?T)?e?0?1?1, ?B?0?08, and T?1?0.
ln(127?09?110)?(0?082?2)d1??1?8456 0?08d2?d1?0?08?1?7656From equation (29.2) the value of the put option is 110e?0?1?1N(?1?7656)?127?09e?0?1?1N(?1?8456)?0?12 or $0.12.
Problem 29.4.
Explain carefully how you would use (a) spot volatilities and (b) flat volatilities to value a five-year cap.
When spot volatilities are used to value a cap, a different volatility is used to value each
caplet. When flat volatilities are used, the same volatility is used to value each caplet within a given cap. Spot volatilities are a function of the maturity of the caplet. Flat volatilities are a
function of the maturity of the cap.
Problem 29.5.
Calculate the price of an option that caps the three-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 quarterly
compounding), the 18-month risk-free interest rate (continuously compounded) is 11.5% per annum, and the volatility of the forward rate is 12% per annum.
In this case L?1000, ?k?0?25, Fk?0?12, RK?0?13, r?0?115, ?k?0?12, tk?1?25, P(0?tk?1)?0?8416.
L?k?250
ln(0?12?0?13)?0?122?1?25?2d1???0?5295 0?121?25d2??0?5295?0?121?25??0?6637The value of the option is 250?0?8416?[0?12N(?0?5295)?0?13N(?0?6637)]
?0?59
or $0.59.
Problem 29.6.
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 low? Explain.
The implied volatility measures the standard deviation of the logarithm of the bond price at the maturity of the option divided by the square root of the time to maturity. In the case of a five year option on a ten year bond, the bond has five years left at option maturity. In the case of a nine year option on a ten year bond it has one year left. The standard deviation of a one year bond price observed in nine years can be normally be expected to be considerably less than that of a five year bond price observed in five years. (See Figure 29.1.) We would therefore expect the price to be too high.
Problem 29.7.
Calculate the value of a four-year European call option on bond that will mature five years from today using Black’s model. The five-year cash bond price is $105, the cash price of a four-year bond with the same coupon is $102, the strike price is $100, the four-year risk-free interest rate is 10% per annum with continuous compounding, and the volatility for the bond price in four years is 2% per annum.
The present value of the principal in the four year bond is 100e?4?0?1?67?032. The present value of the coupons is, therefore, 102?67?032?34?968. This means that the forward price of the five-year bond is
(105?34?968)e4?0?1?104?475
The parameters in Black’s model are therefore FB?104?475, K?100, r?0?1, T?4,
and ?B?0?02.
ln1?04475?0?5?0?022?4d1??1?1144 0?024d2?d1?0?024?1?0744The price of the European call is e?0?1?4[104?475N(1?1144)?100N(1?0744)]?3?19 or $3.19.
Problem 29.8.
If the yield volatility for a five-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 be 4.2 years and the forward yield on the bond is 7%.
The option should be valued using Black’s model in equation (29.2) with the bond price volatility being
4?2?0?07?0?22?0?0647
or 6.47%.
Problem 29.9.
What other instrument is the same as a five-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?
A 5-year zero-cost collar where the strike price of the cap equals the strike price of the floor is the same as an interest rate swap agreement to receive floating and pay a fixed rate equal to the strike price. The common strike price is the swap rate. Note that the swap is actually a forward swap that excludes the first exchange. (See Business Snapshot 29.1)
Problem 29.10.
Derive a put–call parity relationship for European bond options.
There are two way of expressing the put–call parity relationship for bond options. The first is in terms of bond prices: c?I?Ke?RT?p?B0
where c is the price of a European call option, p is the price of the corresponding
European put option, I is the present value of the bond coupon payments during the life of the option, K is the strike price, T is the time to maturity, B0 is the bond price, and R is the risk-free interest rate for a maturity equal to the life of the options. To prove this we can consider two portfolios. The first consists of a European put option plus the bond; the second consists of the European call option, and an amount of cash equal to the present value of the coupons plus the present value of the strike price. Both can be seen to be worth the same at the maturity of the options.
The second way of expressing the put–call parity relationship is c?Ke?RT?p?FBe?RT
where FB is the forward bond price. This can also be proved by considering two portfolios. The first consists of a European put option plus a forward contract on the bond plus the present value of the forward price; the second consists of a European call option plus the
present value of the strike price. Both can be seen to be worth the same at the maturity of the options.
Problem 29.11.
Derive a put–call parity relationship for European swap options.
The put–call parity relationship for European swap options is c?V?p
where c is the value of a call option to pay a fixed rate of sK and receive floating, p is the value of a put option to receive a fixed rate of sK and pay floating, and V is the value of the forward swap underlying the swap option where sK is received and floating is paid. This can be proved by considering two portfolios. The first consists of the put option; the second consists of the call option and the swap. Suppose that the actual swap rate at the maturity of the options is greater thansK. The call will be exercised and the put will not be exercised. Both portfolios are then worth zero. Suppose next that the actual swap rate at the maturity of the options is less than sK. The put option is exercised and the call option is not exercised. Both portfolios are equivalent to a swap where sK is received and floating is paid. In all states of the world the two portfolios are worth the same at time T. They must therefore be worth the same today. This proves the result.
Problem 29.12.
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 29.1 present an arbitrage opportunity?
Suppose that the cap and floor have the same strike price and the same time to maturity. The following put–call parity relationship must hold: cap?swap?floor
where the swap is an agreement to receive the cap rate and pay floating over the whole life of the cap/floor. If the implied Black volatilities for the cap equal those for the floor, the Black formulas show that this relationship holds. In other circumstances it does not hold and there is an arbitrage opportunity. The broker quotes in Table 29.1 do not present an arbitrage opportunity because the cap offer is always higher than the floor bid and the floor offer is always higher than the cap bid.
Problem 29.13.
When a bond’s price is lognormal can the bond’s yield be negative? Explain your answer.
Yes. If a zero-coupon bond price at some future time is lognormal, there is some chance that the price will be above par. This in turn implies that the yield to maturity on the bond is negative.
Problem 29.14.
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 four 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 curve is used for
discounting and is flat at 5% per annum with annual compounding and the volatility of the swap rate is 20%. Compare your answer to that given by DerivaGem. Now suppose that all
swap rates are 5% and all OIS rates are 4.7%. Use DerivaGem to calculate the LIBOR zero curve and the swap option value?
In equation (29.10), L?10?000?000, sK?0?05, s0?0?05, d1?0?24?2?0?2, d2??0.2, and
111 A????2?2404 5671?051?051?05The value of the swap option (in millions of dollars) is 10?2?2404[0?05N(0?2)?0?05N(?0?2)]?0?178
This is the same as the answer given by DerivaGem. (For the purposes of using the
DerivaGem software, note that the interest rate is 4.879% with continuous compounding for all maturities.)
When OIS discounting is used the LIBOR zero curve is unaffected because LIBOR swap rates are the same for all maturities. (This can be verified with the Zero Curve worksheet in DerivaGem). The only difference is that
111A????2.2790 5671.0471.0471.047
so that the value is changed to 0.181. This is also the value given by DerivaGem. (Note that the OIS rate is 4.593% with annual compounding.)
Problem 29.15.
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 Ito’s lemma to show that the volatility of the zero-coupon bond price declines to zero as it approaches maturity.
The price of the bond at time t is e?R(T?t) where T is the time when the bond matures. Using It?’s lemma the volatility of the bond price is
??R(T?t) ?e???(T?t)e?R(T?t)
?RThis tends to zero as t approaches T.
Problem 29.16.
Carry out a manual calculation to verify the option prices in Example 29.2.
The cash price of the bond is 4e?0?05?0?50?4e?0?05?1?00?卐?4?0?05?10?100e?0?05?10?122?82
As there is no accrued interest this is also the quoted price of the bond. The interest paid during the life of the option has a present value of 4e?0?05?0?5?4e?0?05?1?4e?0?05?1?5?4e?0?05?2?15?04 The forward price of the bond is therefore
(122?82?15?04)e0?05?2?25?120?61
The yield with semiannual compounding is 5.0630%.
期权期货与其他衍生产品第九版课后习题与答案Chapter
