1. Chapter 7 Swaps
Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014

2. Nature of Swaps
A swap is an agreement to exchange cash flows at specified future times according to certain specified rules
Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014

3. An Example of a “Plain Vanilla” Interest Rate Swap
An agreement by Microsoft to receive 6-month LIBOR & pay a fixed rate of 5% per annum every 6 months for 3 years on a notional principal of $100 million
Next slide illustrates cash flows that could occur (Day count conventions are not considered)
Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014

4. One Possible Outcome for Cash Flows to Microsoft (Table 7.1, page 155)
Date
LIBOR
Floating Cash Flow
Fixed Cash Flow
Net Cash Flow
Mar 5, 2014
4.20%
Sep 5, 2014
4.80%
+2.10
−2.50
−0.40
Mar 5, 2015
5.30%
+2.40
−0.10
Sep 5, 2015
5.50%
+2.65
+ 0.15
Mar 5, 2016
5.60%
+2.75
+0.25
Sep 5, 2016
5.90%
+2.80
+0.30
Mar 5, 2017
+2.95
+0.45
Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014

5. Typical Uses of an Interest Rate Swap
Converting a liability from
fixed rate to floating rate
floating rate to fixed rate
Converting an investment from
Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014

6. Intel and Microsoft (MS) Transform a Liability (Figure 7.2, page 155)
LIBOR 5%
LIBOR+0.1%
5.2%
Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014

7. Financial Institution is Involved (Figure 7.4, page 157)
LIBOR
LIBOR+0.1%
4.985%
5.015%
5.2%
Intel MS
Financial Institution has two offsetting swaps
Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014

8. Intel and Microsoft (MS) Transform an Asset (Figure 7.3, page 156)
LIBOR 5%
LIBOR-0.2%
4.7%
Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014

9. Financial Institution is Involved (See Figure 7.5, page 157)
Intel
F.I. MS
LIBOR
4.7%
5.015%
4.985%
LIBOR-0.2%
Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014

10. Quotes By a Swap Market Maker (Table 7.3, page 158)
Maturity
Bid (%)
Offer (%)
Swap Rate (%)
2 years
6.03
6.06
6.045
3 years
6.21
6.24
6.225
4 years
6.35
6.39
6.370
5 years
6.47
6.51
6.490
7 years
6.65
6.68
6.665
10 years
6.83
6.87
6.850
Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014

11. Day Count
A day count convention is specified for for fixed and floating payment
For example, LIBOR is likely to be actual/360 in the US because LIBOR is a money market rate
Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014

12. Confirmations Confirmations specify the terms of a transaction
The International Swaps and Derivatives has developed Master Agreements that can be used to cover all agreements between two counterparties
Many interest rate swaps are now cleared through a CCP such as LCH Clearnet or the CME Group
Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014

13. The Nature of Swap Rates
Six-month LIBOR is a short-term AA borrowing rate
The 5-year swap rate has a risk corresponding to the situation where 10 six-month loans are made to AA borrowers at LIBOR
This is because the lender can enter into a swap where income from the LIBOR loans is exchanged for the 5-year swap rate
Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014

14. The Discount Rate
Pre-crisis derivatives cash flows were discounted at LIBOR
As Chapter 9 explains, this has changed
Here we illustrate the valuation methodology by assuming that LIBOR is the discount rate
Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014

15. Using Swap Rates to Bootstrap the LIBOR/Swap Zero Curve
Consider a new swap where the fixed rate is the swap rate
When principals are added to both sides on the final payment date the swap is the exchange of a fixed rate bond for a floating rate bond
The floating-rate rate bond is worth par assuming LIBOR discounting is used. The swap is worth zero. The fixed-rate bond must therefore also be worth par
This shows that swap rates define par yield bonds that can be used to bootstrap the LIBOR (or LIBOR/swap) zero curve
Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014

16. Example of Bootstrapping the LIBOR/Swap Curve (Example 7.1, page 164)
6-month, 12-month, and 18-month LIBOR/swap rates are 4%, 4.5%, and 4.8% with continuous compounding.
Two-year swap rate is 5% (semiannual)
The 2-year LIBOR/swap rate, R, is 4.953%
Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014

17. Valuation of an Interest Rate Swap
Initially interest rate swaps are worth close to zero
At later times they can be valued as the difference between the value of a fixed-rate bond and the value of a floating-rate bond
Alternatively, they can be valued as a portfolio of forward rate agreements (FRAs)
Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014

18. Valuation in Terms of Bonds
The fixed rate bond is valued in the usual way
The floating rate bond is valued by noting that it is worth par immediately after the next payment date
Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014

19. Valution of Floating-Rate Bondt*
Valuation Date
First Pmt
Date
Floating Pmt =k*
Second
Pmt Date
Maturity Date
Value = L
Value = L+k*
Value = PV of L+k* at t*
Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014

20. Example
Receive six-month LIBOR, pay 3% (s.a. compounding) on a principal of $100 million
Remaining life 1.25 years
LIBOR zero rates for 3-months, 9-months and 15-months are 2.8%, 3%, and 3.2% (cont comp)
6-month LIBOR on last payment date was 12.9% (s.a. compounding)
Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014

21. Valuation Using Bonds (page 166)
Time
Bfix cash flow
Bfl cash flow
Disc factor PV
Bfix
Bfl
0.25
1.5
0.9930
1.4895
0.75
0.9763
1.4644
1.25
101.5
0.9584
Total
Swap value = − =
Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014

22. Valuation in Terms of FRAs
Each exchange of payments in an interest rate swap is an FRA
The FRAs can be valued on the assumption that today’s forward rates are realized
Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014

23. Valuation of Example Using FRAs (page 167)
Time
Fixed cash flow
Floating cash flow
Net Cash Flow
Disc factor PV
Bfl
0.25
−1.5000
−0.0050
0.9930
−0.0497
0.75
0.9763
1.25
0.9584
Total
Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014

24. An Example of a Currency Swap
An agreement to pay 5% on a sterling principal of £10,000,000 & receive 6% on a US$ principal of $15,000,000 every year for 5 years
Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014

25. Exchange of Principal
In an interest rate swap the principal is not exchanged
In a currency swap the principal is usually exchanged at the beginning and the end of the swap’s life
Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014

26. The Cash Flows (Table 7.7, page 170)
Date
Dollar Cash Flows
(millions)
Sterling cash flow
Feb 1, 2014
−15.00
+10.0
Feb 1, 2015
+0.90
−0.50
Feb 1, 2016
Feb 1, 2017
Feb 1, 2018
Feb 1, 2019
+15.90
−10.50
Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014

27. Typical Uses of a Currency Swap
Convert a liability in one currency to a liability in another currency
Convert an investment in one currency to an investment in another currency
Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014

28. Valuation of Currency Swaps
Like interest rate swaps, currency swaps can be valued either as the difference between 2 bonds or as a portfolio of forward contracts
Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014

29. Example All Japanese LIBOR/swap rates are 4%
All USD LIBOR/swap rates are 9%
5% is received in yen; 8% is paid in dollars. Payments are made annually
Principals are $10 million and 1,200 million yen
Swap will last for 3 more years
Current exchange rate is 110 yen per dollar
Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014

30. Valuation in Terms of Bonds (Table 7.9, page 173)
Time
Cash Flows ($)
PV ($)
Cash flows (yen)
PV (yen) 1
0.8
0.7311 60
57.65 2
0.6682
55.39 3
0.6107
53.22
10.0
7.6338
1,200
1,064.30
Total
9.6439
1,230.55
Value of Swap = /110 − =
Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014

31. Valuation in Terms of Forwards (Table 7.10, page 174)
Time
$ cash
flow
Yen cash flow
Forward Exch rate
Yen cash flow in $
Net Cash Flow
Present value 1
-0.8 60
0.5734 2
0.6028 3
0.6337
-10.0
1200
2.0417
Total
1.5430
Options, Futures, and Other Derivatives, 9th Edition, Copyright © John C. Hull 2014