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1、Computational FinanceLecture 4Forward, Futures, and SwapsPart II: Swaps Main ContentsThe minimum-variance hedgeSwap basicsDefinition and relation to forwardsPhysical vs. financial settlementUses of swapsRisk managementBalance sheet transformationValuation of swaps4.7 The Minimum-Variance HedgeMinimu

2、m-Variance HedgeOne common method of hedging in the presence of basis risk is the minimum-variance hedge. From the previous discussion, if we want to hedge a cash flow x to occur at time T using h units of a futures, then the resulted cash flow is given byThe minimum-variance hedge is to find a prop

3、er h to minimize Minimum-Variance Hedging FormulaThe minimum-variance hedge and the resulting variance are ExampleAn airline expects to purchase 2 million gallons of jet fuel in 1 month and decides to use heating oil futures for hedging. Suppose that the data on the changes in the jet fuel price per

4、 gallon and in the heating oil futures are available for 15 successive months. (See the attached excel) Each heating oil contract traded on NYMEX is on 42,000 gallon. What is the optimal number of contracts? 4.8 Swap BasicsSwaps A swap (掉期) is an agreement to exchange cash flows at a sequence of spe

5、cified future times according to certain specified rules. In this sense, swaps are a multi-period extension of forward contracts. A forward contract can be viewed as a simple example of a swap. An Example of Commodity SwapAn industrial producer, IP Inc., is going to buy 100,000 barrels of oil 1 year

6、 from today and 2 years from today.It may separately long two forward contracts for delivery in 1 year and in 2 years. The forward prices are $110/barrel and $111/barrel respectively. Alternatively, it may enter a swap contract with another firm: they exchange cash for oil in 1 and 2 years. The amou

7、nt of each payment for IP Inc. is agreed upon $110.483/barrel. Physical SettlementPhysical settlement: Financial SettlementThe swap counterparties can also arrange for financial settlement. In the previous example, the oil buyer, IP Inc., can choose to pay the swap counterparty the difference betwee

8、n $110.483 and the spot price, and it then buys oil at the spot price from the market. Such operation will yield an identical economic e as the physical settlement. If the difference between $110.483 and the spot price is negative, then the swap counterparty pays the buyer. Physical vs. Financial Se

9、ttlementFinancial settlement: Interest Rate and Currency SwapsThe swap market is a large component of derivatives market, with well over US$200 trillion outstanding. Besides the commodity swaps, we have Interest rate swapsOne party agrees to pay the counterparty a fixed rate of interest in exchange

10、for paying a variable/floating rate of interest or vice versaForeign exchange swaps An exchange of currencies on several future dates 4.9 Uses of SwapsUses of SwapsSwaps are equivalent to a combination of forward contracts. Like forwards, they can be used to hedge risks. Another important economic i

11、ncentive for swap traders is that they will use swaps to change the nature of their assets or liabilities. Below I will use currency swaps as an example to illustrate this point. An Example of Currency SwapsConsider a hypothetical 5-year swap agreement between IBM and Qantas Airline entered into on

12、Nov 7, 2017. IBM agrees to pay QA a fixed rate of interest of 6.9% in Australian dollar and receives a fixed rate of 5% in US dollars from QA. The payments are done once a year and the principal amounts are US$18 million and AU$10 million. Cash Flows to IBMUSD DateAUDNov 7, 2017 Nov 7, 2018+0.90.69N

13、ov 7, 2019+0.90.69 Nov 7, 2020+0.90.69-18.00Nov 7, 2021+0.90.69 Nov 7, 2022+18.910.69+10.0Using the Swap to Transform LiabilitiesA swap as such can be used to convert a liability from US dollar to AUD for IBM.Consider a situation in which IBM wants to borrow AUD10 million to expand its business in A

14、ustralia. Meanwhile, QA wants to raise some amount of US dollars to maintain its operation in the US. Using the Swap to Transform Liabilities (Continued)Suppose that they are offered by the following borrowing rates: USDAUDIBM5.0%7.6%QA7.0%8.0%Using the Swap to Transform Liabilities(Continued)They m

15、ay do borrowing independently USD $18mAUD 7.6% AUD $10m USD 7%Total interest burdens 7.6%+7%=14.6% IBMQAUsing the Swap to Transform Liabilities(Continued)With a swap, they may borrow for each other: AUD 10m AUD 10m AUD 6.9% AUD 8%USD 5% USD 5%USD 18m USD 18m Effectively, IBM pays 6.9% (AUD)QA pays 1

16、.1% (AUD) + 5% (USD)The total interest burden is 13% (if we ignore the difference between AUD and USD)IBMQAUsing the Swap to Transform Liabilities(Continued)Alternatively, the arrangement may be AUD 10m AUD 10m AUD 8% AUD 8%USD 5% USD 6.1%USD 18m USD 18m Effectively, IBM pays 8% (AUD) 1.1% (USD)QA p

17、ays 6.1% (USD)IBM subjects to FX riskIBMQASwaps and Comparative AdvantagesExamine the reason why using swaps will lead to savings in interest payments.IBM has a comparative advantage in USD market. QA has a comparative advantage in AUD market. In other words, the extra amount that QA needs to pay ov

18、er the amount paid by IBM is less in AUD market than USD market. Role of Financial IntermediaryIn practice, two nonfinancial companies do not get in touch directly to arrange a swap. They each deal with a financial intermediary to swap their cash flows.Role of Financial IntermediaryIllustration: 5%

19、USD 5% USD 6.3% AUD 6.9% AUD 8% 8%Effectively, IBM pays 6.9% (AUD) QA pays 6.3% (USD)Financial intermediary pays 1.1% (AUD) 1.3% (USD), earning 0.2% premium and subject to FX risk. FinancialIntermediaryQAIBMUsing the Swap to Transform AssetsConverting an asset from pounds to US dollars. Consider a s

20、ituation in which IBM invested in Australia AUD 10 million to yield 6.9% per annum for the next 5 years, but it prefers a US-dollar dominated investment. The above swap has the effect of transforming for IBM the Australia investment into an $18 million investment in the US yielding 5%. 4.10 Valuatio

21、n of SwapsValuation of Currency Swaps To determine the value of swaps, we have two alternative ways. Take a currency swap as an example: A currency swap can be valued as the difference of 2 bonds.It can also be valued as a sequence of forward contracts. ExampleSuppose that the Japanese Yen interest

22、rate is 4% per year and the US dollar interest rate is 9% per year.A currency swap was entered into some time ago. The financial institution receives 5% per year in yen and pays 8% per year in USD. The principals are US$10 million and 1,200 million yen. The contract will last for another 3 years.The current exchange rate is 110 yen per dollar. Valuation in Terms of Bonds TimeCash Flows ($)PV ($)Cash flows (yen)PV (yen)10.80.73116057.6520.80.66826055.3930.80.61076053.22310.07.63381,2001,064.30Total9.64391,