Derivatives

1
Derivatives Risk Management

• Lecture 4
• a) Swaps
• b) Options properties and non-parametric bounds

2
Nature of Swaps

• A swap is an agreement to exchange cash flows at
  specified future times according to certain
  specified rules

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Example a Plain Vanilla Interest Rate Swap

• An agreement by Company B 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

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Cash Flows to Company B
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Typical Uses for an Interest Rate Swap

• Converting a liability from
• fixed rate to floating rate
• floating rate to fixed rate

• Converting an investment from
• fixed rate to floating rate
• floating rate to fixed rate

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A and B Transform a Liability
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Financial Institution is Involved

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A and B Transform an Asset

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Financial Institution is Involved

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The Comparative Advantage Argument

• Company A wants to borrow floating
• Company B wants to borrow fixed

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The Swap

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The Swap when a Financial Institution is Involved

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Criticism of the Comparative Advantage Argument

• The 10.0 and 11.2 rates available to A and B in
  fixed rate markets are 5-year
• The LIBOR0.3 and LIBOR1 rates available in
  the floating rate market are six-month rates
• Bs fixed rate depends on the spread above LIBOR
  it borrows at in the future i.e. it is fixed only
  as long as its creditworthiness stays the same

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Alternatives

• Information asymmetry
• Flexible and liquid instruments
• Tax and regulatory arbitrage

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Valuation of an Interest Rate Swap

• Interest rate swaps can be valued as the
  difference between the value of a fixed-rate
  bond the value of a floating-rate bond

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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

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Valuation as bonds
K is the floating rate know from at the
beginning of the period
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Example

• A financial institution pays 6 month LIBOR and
  receives 8 (semi-annually) on 100 million
  notional principal.
• The FI has sold a floater and bought a fixed rate
  bond
• remaining life 1.25 years
• market rates for 3, 9, 15 months to go are 10,
  10.5 and 11

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Example II

• The 6 month LIBOR when the swap was set up 3
  months ago was 10.2.

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Example III
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Forward Rate Agreement

• A forward rate agreement (FRA) is an agreement
  that a certain rate will apply to a certain
  principal during a certain future time period
• An FRA is equivalent to an agreement where
  interest at a predetermined rate, RK is exchanged
  for interest at the market rate

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Forward Rate Agreementcontinued (Page 100)

• Capital R is the rate measured with compounding
  rate reflecting maturity, i.e. if the T2 T1 is
  three months the rate is compounded quarterly
  etc.
• The agreed cash flows are
• T1 - L
• T2 L 1 Rk (T2-T1)

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FRA

• Note if Rf Rk the FRA is worth 0. Why?
• To value the FRA, we can compare now two payments
  at time T2
• One that pays Rk and one that pays Rf
• Note we are not assuming anything more than no
  arbitrage

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Valuing future cash flows
Hypothetical cash flow
Cash settlement
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Alternative Valuation of Interest Rate Swap
portfolio of FRA

• Swaps can be valued as a portfolio of forward
  rate agreements (FRAs)
• Each exchange of payments in an interest rate
  swap can be analyzed as an FRA
• The relevant interest rates are the fixed for one
  leg, and the forward associated with the period
  to be valued for the other leg

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Swaps as FRAs

• Suppose an interest rate swap promises fixed rate
  payments C and receives floating payments P1fl at
  regular intervals
• We have seen that this could be valued as a
  portfolio of bonds
• What about valuating it as a package of FRAs?

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Swaps as FRAs II
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Swaps as FRAs III

• Consider the second exchange of payments (the
  first is known)

• The floating rate payment is computed based on
  the prevailing spot rate at T1

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Swaps as FRAs IV
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Swaps as FRAs V

• So we want to compute the PV of

• Which can be written as

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Swaps V

• The value of the fixed part of this payment is
  obvious

• The value of the floating part less so because it
  involves a random interest rate

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Swaps as FRAs VI

• We know that at T2, the floating rate payment
  will be worth

• And thus T1, it must be worth

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Swaps as FRAs VII

• And thus at time t it must be worth

• Recall that by no arbitrage
• So that

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Swaps as FRAs VIII

• Hence

Changing compounding convention
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Swaps as FRA
So to value a fixed for floating exchange,
compute the present value of the exchange between
the forward rate and the fixed rate
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An Example of a Currency Swap

• An agreement to pay 11 on a sterling
  principal of 10,000,000 receive 8 on a US
  principal of 15,000,000 every year for 5 years

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Exchange of Principal

• In an interest rate swap the principal is not
  exchanged
• In a currency swap the principal is exchanged at
  the beginning the end of the swap

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The Cash Flows
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Typical Uses of a Currency Swap

• Conversion from an investment in one currency to
  an investment in another currency

• Conversion from a liability in one currency to a
  liability in another currency

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Comparative Advantage Arguments for Currency
Swaps

• Company A wants to borrow AUD
• Company B wants to borrow USD

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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

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Swaps Forwards(continued)

• The value of the swap is the sum of the values
  of the forward contracts underlying the swap
• Swaps are normally at the money initially
• This means that it costs NOTHING to enter into
  a swap
• It does NOT mean that each forward contract
  underlying a swap is at the money initially

43
Credit Risk

• A swap is worth zero to a company initially
• At a future time its value is liable to be either
  positive or negative
• The company has credit risk exposure only when
  its value is positive