Forward, Swap,

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Forward, Swap, Option Pricing (the
very short story)Structured Notes
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What does this strategy replicate?
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A more general replication

• One can offset the risk of a forward by creating
  a synthetic forward to offset a position in the
  actual forward contract
• How can one do this? (assume continuous dividends
  at rate d)
• Recall the long forward payoff at expiration
  ST F0, T
• Borrow and purchase shares as follows
• Note that the total payoff at expiration is same
  as forward payoff

A tailed position
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Forward-Spot No-Arbitrage Condition

• Cash-and-carry arbitrage Buy the index, short
  the forward

Forward-Spot No-Arbitrage Condition
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Swaps
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Introduction to Swaps

• A swap is a contract calling for an exchange of
  payments, on one or more dates, determined by the
  difference in two prices
• A swap provides a means to hedge a stream of
  risky payments
• A single-payment swap is the same thing as a
  cash-settled forward contract

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Evolution of Swaps

• Increase in exchange rate volatility (1972)
• increase in earnings volatility
• fluctuation in asset value due to exchange rate
  volatility
• The Solution Parallel Loans
• two firms simultaneously make financial loans to
  each other
• increasing use in the 1970s
• difficult to find partners
• Swaps start being written 1981 by banks to help
  firms conduct parallel loan transactions

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Parallel loans

• Two firms with opposite exposure to something
• Foreign exchange rates, interest rates,
  volatility of an asset (equity, commodity,
  foreign currency)
• structure a loan to eliminate risk
• Currency parallel loan
• Firm B lends dollars to firm C
• Firm C lends foreign currency to firm B
• Interest parallel loan
• Firm B lends A to firm C, charges fixed rate
• Firm C lends A to firm B, charges floating rate

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Parallel Loan Example

• Suppose a US firm and a UK firm want to swap
  foreign currency exposure (US firm has UK
  exposure, and vice-versa). Suppose US firm is
  receiving 100,000 per year.
• Assume S1.562 /, rus .08 and ruk .10
• US firms perspective
• borrow 379,079 and repay 100,000 per year for
  5 years
• lend 592,121 and receive 148,300 per year for
  5 years
• Cash flow today 379,079 (1.562 /) -
  592,121 0
• Cash flow at date t1 through t5
• Ct 148,300 - 100,000(St /)
• Ct 100,000 (1.483/ - St /)
• which should look very familiar !!

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Parallel loan example continued

• What firm would enter such an agreement?
• A US exporter with 100,000 /year in revenue
• revenues at time period t parallel loan payoff
  hedged rev
• 100,000 (St/) 100,000 (1.483/ - St
  /) 148,300
• Problems with parallel loans
• default risk, impact on balance sheet, search
  costs
• Solution
• staple two contracts together to form a currency
  swap
• netting the payments on each of the dates
• First swap created in 1979 for IBM
• Swap market was largely developed by Chase
  Manhattan in 1981-82

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Swaps

• A swap is an agreement to exchange cash flows at
  a specified future times according to certain
  rules
• A swap may be viewed as a package of forward
  contracts
• Primary differences between swap and parallel
  loan

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A more formal treatment, tying swap prices
(rates) to forward prices (rates)
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An Example of a Commodity Swap

• An industrial producer, IP Inc., needs to buy
  100,000 barrels of oil 1 year from today and 2
  years from today
• The forward prices for delivery in 1 year and 2
  years are 20 and 21/barrel
• The 1- and 2-year zero-coupon bond yields are 6
  and 6.5

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An Example of a Commodity Swap (contd)

• IP can guarantee the cost of buying oil for the
  next 2 years by entering into two long forward
  contracts for 100,000 barrels in each of the next
  2 years.
• The PV of this cost per barrel is
• Thus, IP could pay an oil supplier 37.383, and
  the supplier would commit to delivering one
  barrel in each of the next two years
• If he did, it would consititute a prepaid swap.

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An Example of a Commodity Swap (contd)

• With a prepaid swap, the buyer might worry about
  the resulting credit risk. Therefore, a better
  solution is to defer payments until the oil is
  delivered, while still fixing the total price
• In theory, any payment stream with a PV of
  37.383 is acceptable. Typically, however, a swap
  will call for equal payments in each year
• For example, the payment per year per barrel, x,
  will have to be 20.483 to satisfy the following
  equation
• We then say that the 2-year swap price is 20.483

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Option Pricing(the short story)
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The problem

• Recall
• CT max0, ST - X
• implies that C is a function of S and X.
• The problem
• What is C0?
• From above,
• C0 Cte-rt
• C0 max0, Ste-rt - Xe-rt
• C0 max0, S0 - Xe-rt
• implies that C is a function of S, X, r, and t
• But at this point we have not captured the
  probability that S will be in the money
• this probability will depend upon volatility (?)

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The solution

• We will show that a solution is C SN(d1) -
  Xe-rtN(d2)
• where N(d)s can be interpreted as probabilities
• We will show this using
• 1. Binomial model
• 2. Continuous model (Black/Scholes)
• In both cases the math can be complex, but the
  intuition is fairly straightforward
• both are derived from no-arbitrage arguments

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The binomial world

• In a binomial world prices can only take one of
  two values per period (play).
• The binomial models are defined by the number of
  periods (plays)

One play Two
plays
110
100
80
Lets call the factor by which the stock rises,
u, and the factor by which the stock falls, d.
In this case u1.1, and d0.8
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Valuation/replication in a binomial world

• Arbitrage forces the price of a portfolio having
  the same payoff as a call option to have the same
  price as a call option
• call options can be replicated by borrowing money
  and buying shares
• Suppose S50, u2, d.5, r 22.31, and a call
  option with X50 is available for price C. What
  is C?
• Buy buying 2 shares and borrowing 40, we
  replicate 3 calls

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Valuation in a binomial world

• So a strategy that has a zero payoff at
  expiration better be worth zero.
• 3C - 60 0
• implies C 20
• How did I know that borrowing 40 and buying 2
  shares would replicate 3 calls?

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General Binomial Model

• Note that the value of stock next period will
  equal
• Su S0u or Sd S0d
• Value of call next period will equal
• cu max0, S0u -X or cd max0, S0d
  -X
• Now consider a portfolio of ? shares and B in
  bonds (Blt0borrowing), where we choose ? and B so
  as to replicate one call option.
• Value at t1
• ? S0u Bert cu
• ? S0d Bert cd

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General Binomial Model

• Solve for ? and B
• From our example, to replicate one call option
• ? (50-0) / (50(1.5)) 2/3
• B (20) - .5(50) / 1.5e.2231 -13 1/3
• So to replicate 3 call options
• 3 ? 2 shares
• 3B -40

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General Binomial Model

• Therefore, since ? and B are chosen to replicate
  C, to preclude arbitrage it must be that
• C ?S0 B
• Substitute the equations for ? and B, and we have

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General Binomial Model

• From the example
• p (e.2231 - .5) / (2-.5) .5
• c (.550) (.50)e-.2231 20
• Some intuition
• c is the present value of expected cash flows
  from holding the asset (option)
• p is the risk-neutral probability the option
  expires in the money
• p depends on u, d, and r
• u, d define a range of outcomes - they are
  measures of volatility

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Two-period binomial pricing model

• Call value possibilites
• Solve by backward iteration (letting time between
  periods t)
• See a pattern developing?

c2,uu
c1,u
c2,ud
c0
c1,d
c2,dd
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3-period binomial valuation

• 3-period valuation
• How many ways to get to each final payoff?
• 1, 3, 3, 1
• Solution

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N-period binomial model

• How do we keep counting the number of routes to
  the final payoffs?
• Use Pascals Triangle

1 1 1 2 1 1
3 3 1 1 4 6 4 1 1 5 10 10 5
1 1 6 15 20 15 6 1
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The N-period binomial option pricing model
The model can be written as follows where j
counts number of up moves
The number of price paths that lead to the final
node
The probability of taking any one of the
available price paths
The payoff of that node
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Binomial pricing in practice

• The models considered so far have been simple
  (few plays)
• We can make them more complex by considering more
  plays, where each play is over a shorter and
  shorter period of time
• typical to consider 1 play per day on options
  with life of 30 or greater days
• this implies that they are considering well over
  1 billion possible price paths (sum across the
  bottom row of Pascals Triangle)
• u and d are typically a function of the stocks
  return volatility
• letting ?t time period per play
• u e?(?t).5 and d e-?(?t).5
• everything else is solved following the usual
  steps
• computers, of course, do the heavy number
  crunching.

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Extending the Binomial Model

• Recall

Then, let PIM the first two terms, and sum over
only those nodes that have positive call payoff,
starting at ja
And does this look somewhat familiar?
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Black-Scholes Formula

• Binomial model fails to account where the stock
  price is at every point in time.
• Gets close when n gets real big
• Black-Scholes model solves the price of the
  option using continuous time or stochastic
  calculus.
• The model still relies on the simple idea that
  the price must preclude arbitrage.

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Black-Scholes set up

• Consider an European call (or put) option written
  on a stock, and assume that the stock pays
  dividend at the continuous rate d

Form a portfolio with stocks and the option that
is perfectly hedged at every second in time
(dynamic hedging). This portfolio must pay the
risk-free rate of return Solve for C.
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Black-Scholes Formula (contd)

• Call Option price
• Put Option price
• where
• and

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Black-Scholes (BS) Assumptions

• Assumptions about stock return distribution
• Continuously compounded returns on the stock are
  normally distributed and independent over time
  (no jumps)
• The volatility of continuously compounded returns
  is known and constant
• Future dividends are known, either as dollar
  amount or as a fixed dividend yield

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Black-Scholes (BS) Assumptions (contd)

• Assumptions about the economic environment
• The risk-free rate is known and constant
• There are no transaction costs or taxes
• It is possible to short-sell costlessly and to
  borrow at the risk-free rate

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Option price sensitivity to parameters
C fn(S, X, r, t, s) P fn(S, X, r, t, s)
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Structured Notes
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Start with a problem

• ExxonMobil is an oil producer, they have a
  natural long position in oil
• They need to hedge against a drop in oil prices
• Basic hedges
• Short oil futures/forwards (sell forward)
• Swap oil (sell gold through swap)
• Buy put options on oil (sell through options)

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Basic problems with these hedges

• Is timing good?
• Is it a direct hedge?
• Can the firm fund interim losses?

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Alternative solutions

• Engineer a product that also sells oil in the
  future.
• Widely used are structured notes
• for example, a commodity linked bond
• Get cash from investors and pay interest and
  principal in oil rather than in cash
• Selling gold forward through a debt instrument

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Pricing and Designing Structured Notes

• A structured note has interest or maturity
  payments that are not fixed in dollars, but are
  contingent in some way
• Structured notes can make payments based on stock
  prices, interest rates, commodities, or
  currencies
• Structured notes can have options embedded in them

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Structured Note Basics
What is the price of a bond that pays one barrel
of oil 1 year from now? We can contract today to
buy oil at 20.5 in one year, therefore, the
most Id pay for the bond is 20.5.9388
19.2454
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Profit loss on this bond
Buying a commodity linked bond is equivalent to a
long forward contract in oil Issuing a commodity
linked bond is equivalent to taking a short
forward position in the commodity
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Suppose the following

• If the oil company issued the bond, they raise
  19.245 on a per barrel basis
• Oil company might want to raise more per bond
• The bond buyers might want to receive cash during
  the life of the bond.

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New problem

• What quarterly cash coupon must the firm pay to
  make the bond worth 24.00?
• The promised barrel has a PV of 19.245
• The PV of 4 coupon payments must have a PV of
  4.755
• Coupon(.9852.9701.9546.9388)4.755
• Coupon(3.8487)4.755
• Coupon 1.235 each quarter

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Embed an option in the bond

• Suppose the corporation wants to buy a put and
  offers the following
• If oil lt 18, payment 24 (18-ST), otherwise
  payment 24
• Same as 24 max(0,18-ST)
• There are no coupon payments
• What is the price of this bond?
• Assume volatility of oil is 35
• 1-year interest rate 0.9388 1e-r(1), therefore
  r.0631
• Since F20.5 20.90e(.0631-?)1, therefore
  ?0.0824

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What does the bond buyer get?

• Bond price PV(24) - PV(put)
• First, value the put option component
• (S20.9, K18, r.0631, ?.0824, ?.20, t1)
• Bond Price (24)e-.0631 -1.5 21.03

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Bond buyers payoff
The bondholder keeps the premium if SgtK!
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Bond issuers payoff
The commodity-linked bond replicates a put option
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Another example embed call option in coupon
paying commodity-linked bond

• Suppose now that the bond promises a 1.235
  quarterly coupon, 1 barrel of oil at the end of
  the year, and 3max(0,ST-20.5), where 20.5 is
  the 1 year forward price of oil.
• What is the value of this bond if the volatility
  of oil is 18?

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Valuing the structured note

• Bond price
• PVquarterly coupon stream S1
  3max(0,ST-20.5),
• PV 4 coupon payments 4.755
• PV of S1 19.245
• Need to value a call option where the underlying
  asset pays a lease rate
• 1-year interest rate 0.9388 1e-r(1), therefore
  r.0631
• Since F20.5 20.90e(.0631-?)1, therefore
  ?0.0824
• Option price
• Bond value 4.75519.24531.38 28.14

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Payoff to the bondholder
The issuing firm has the opposite exposure Can
adjust the multiple to get required slope..