Computational Finance

1
Computational Finance

• Lecture 3
• Derivatives Forwards, Futures and Swaps

2
Derivatives

• A derivative security is a security whose payoff
  is explicitly tied to the value of some other
  variable.
• Forwards
• Futures
• Swaps
• Options

3
Forwards

• A forward contract is an agreement where one
  party promises to buy an asset from another party
  at some specified time in the future and at some
  specified price.
• Underlying asset
• Maturity or delivery date
• Delivery price or Forward Price
• Long position and short position

4
An Example of Forward Contract

• A US corporation enters a forward contract with
  an international bank on Jan. 23 to agree to buy
  1M British pounds at a price of 1.3 US dollars
  per pound in 1 month.
• Underlying asset British pounds
• Delivery date Feb. 23
• Delivery price 1.3/pound
• Long position

5
Payoffs from Forward Contracts

• To distinguish from the forward market, the open
  market for immediate delivery of the underlying
  asset is called the spot market.
• Forward contracts are obligations. Two positions
  must honor the agreement no matter what happens
  in the market.

6
Payoffs from Forward Contracts

• Suppose the exchange rate between US dollars and
  British pounds turns out to be
• US1.35 per pound. The corporation would be
  favored by the contract and it would pay
  (1.35-1.30.05M) less than the market price
• US 1.25 per pound. The forward contract would
  have a negative value to the corporation because
  it would pay (1.3-1.250.05M) more than the
  market price.

7
Payoffs from Forward Contracts

• In general the payoff from a long position in a
  forward contract should be
• where is the delivery price and is
  the spot price of the underlying asset at the
  maturity.
• Short position

8
Using Forwards to Manage Risks

• Risk hedger can reduce their risks with forward
  contracts.
• Consider the above example. If the company has 1M
  pounds obligation to pay on Feb. 23, then it can
  hedge its foreign exchange risk by fixing the
  price at 1.3, no matter what the market will be.
  

9
Speculating on Forwards

• Forwards also can be used as a tool of
  speculating.
• Consider the previous case again. Suppose that
  the company speculates that the pound will
  strengthen against USD in one month and want to
  back the hunch to the tune of 1M pounds. The
  current exchange rate is assumed to be
  US1.2/pound.

10
Speculating on Forwards

• One thing the speculator can do is simply
  purchase 1M pounds in the hope that the sterling
  can be sold later at a profit.
• For each one more dollar the British pound
  appreciates the speculator earns 1M. But he or
  she needs 1.2M upfront to invest.

11
Speculating on Forwards

• The alternative strategy is to long a forward
  contract such as the one in the previous slide.
• When on Feb. 23 the exchange rate is more than
  1.3, the speculator earns the difference.
  Meanwhile, it does not cost anything for him or
  her when entering a forward. The forward market
  allows the speculator to obtain leverage.

12
Determination of Delivery Price

• What is a fair delivery price for both
  positions?
• Information available , ,
• Idea risk hedging

13
A Copper Forward Example

• A manufacturer of heavy electrical equipment
  wishes to take a long position of a forward
  contract for delivery of copper in 9 months. The
  current price of copper is 84.85 cents per pound,
  and the interest rate for 9 month is 4 per year.
  What is the appropriate forward price?

14
A Copper Forward Example

• From the perspective of short position,
• Borrow 84.85 cents at risk free interest rate
• Buy 1 pound of copper now
• Deliver the asset to the long position at the
  maturity.
• Total loan at the maturity for the short position
  is . The forward price should not
  be larger than it, i.e.,

15
A Copper Forward Example

• From the perspective of long position
• Short the underlying asset to get 84.85 cents
• Deposit 84.85 cents at risk free interest
• Repurchase the asset back using the forward
  contract at the maturity.
• The total deposit will be .
• The forward price should not be less than it,
  i.e.,

16
Determination of Forward Price

• In general, consider a forward contract on a
  non-dividend stock. Suppose that the current
  stock price is , the time to the maturity is
  and the risk free interest rate is , then a
  fair delivery price should be

17
Cost of Carry

• Sometimes, the underlying assets incur storage
  costs for the holder and we should take it into
  account.
• For instance, a 5-month sugar forward. The
  current price of sugar is 12 cents per pound. The
  carrying cost of sugar is 0.1 cent per pound per
  month, to be paid at the beginning of every
  month. The interest rate is 9 per year.

18
Cost of Carry

• Short position
• Borrow 12 cents at interest rate 9
• Buy 1 pound sugar now
• (A hidden cost here! 0.1 cent for a pound each
  month)
• Deliver the asset to the long position at the
  maturity.
• Total net obligation at the maturity for the
  short position is

19
Known Income

• Sometimes, the underlying asset will provide a
  perfectly predictable cash income to the holder.
  The forward price should be adjusted accordingly.
• One example is cum dividend stock.

20
Cum Dividend Stocks

• Consider a 6-month forward contract on a stock
  with a price of 50. We assume that the risk-free
  interest rate is 8 per annum. A dividend of
  0.75 per share is expected in 4 months. What
  should the forward price be?

21
Cum Dividend Stocks

• Short position
• Borrow 50 at risk free interest rate 8
• Buy the underlying asset now
• (A hidden income here! A dividend of 0.75 is
  expected)
• Deliver the asset to the long position at the
  maturity.
• Total net obligation at the maturity for the
  short position is

22
Cum Dividend Stocks

• Long position
• Short 1 share of stock to get 50
• Deposit 50 at risk free interest rate 8
• Repurchase the asset back using the forward
  contract at the maturity.
• The total deposit will be . But
  it loses the dividends worth .
• The delivery price should be

23
The Value of Forward Contract

• The delivery price of a forward contract is
  chosen according to the information when the
  contract is entered.
• As time goes by, the delivery price will not be
  fair any longer.
• Stock Price Forward 1
  Forward 2
• Day 0
  -
• Day 1

24
The Value of Forward Contract

• A forward contract on a non-dividend-paying stock
  was entered into some time ago. It currently has
  6 months to maturity.
• The delivery price is 24 and the current stock
  price is 25. The interest rate is 10 per year.

25
The Value of Forward Contract

• Given the above data, the fair delivery price
  should be, if you enter the long position of a
  forward contract now,
• For the forward contract long position, it is
  being benefited because it pays 26.28-242.28
  less in 5 months.
• The forward contract has value

26
The Value of a Forward

• In general, a forward contract should have a
  positive/negative value after it is signed.
• Suppose that the delivery price is , the time
  to maturity is , and the current price of the
  underlying is . The value of this forward
  contract is given by

27
Futures

• The trading of forward contracts is usually
  over-the-counter. It involves risks. Sometimes
  one of the parties may not have enough financial
  resources, or may regret the deal, to honor the
  agreement.
• To avoid the risk, futures contracts are
  introduced.

28
Futures

• Futures are forward contracts traded in the
  exchanges.
• Exchanges provide standardized contracts and play
  the counterparty for both long and short traders.
  
• Convenience and security
• No need to find an appropriate counterparty
• No need to face default risks

29
Marking to Markets

• It is impossible for exchanges to track all
  delivery prices for every individual futures they
  trade.
• To avoid the difficulty, the mechanism of marking
  to market is introduced
• Margin account
• Adjust the amount in the account by market prices
• Margin call

30
The Operation of MarginsMargin Account and
Initial Margins

• An investor wants to buy two March gold futures
  contracts on the New York Commodity Exchange.
• Each contract size is 100 ounces. The current
  futures price is 400 per ounce and the initial
  margin is 2,000 per contract.
• Investor should deposit 4,000 in his margin
  account.

31
The Operation of MarginsMarking to Market

• On the next day, the March gold futures price
  drops down to 397 per ounce. The investor has a
  loss of
• 2 (400-397) 100 600.
• 600 is deducted from the margin of the investor
  and the total balance is reduced to
  4,000-6003,400.

32
The Operation of MarginsMaintenance Margin

• Usually the exchanges will set a lower bound for
  each margin account, known as the maintenance
  margin to prevent the balance in the margin
  account from being negative.
• Once the balance is below the maintenance margin,
  the investor will receive a margin call to top up
  the account to the initial level.

33
Swap

• Swaps are contracts to transform one cash flow
  into another. It can be viewed as a combination
  of several forwards.
• Whereas a forward contract leads to the exchange
  of cash flows on just one future date, swaps
  typically lead to cash flow exchanges taking
  place on several future dates.
• Commodity swap and interest rate swap

34
Commodity Swaps

• An electric power company purchases oil every
  month to generate power facility. But due to
  significant fluctuation in the energy market, it
  does not want to buy oil from the spot market. It
  wants to pay a constant price.
• Another company may want to speculate in the
  fluctuation of oil price.

35
Commodity Swap

• Illustration of a swap contract
• spot price
  spot price
• oil
  X
• Power company pays a fix amount X to the
  speculator
• Speculator pays the spot oil price to power
  company

Power Company
Speculator
Energy Market
36
How to Evaluate Commodity Swap Contract

• Such a swap contract can be viewed as a group of
  forwards
• Every month the power company pays X to buy some
  amount of oil from the speculator.

37
Interest Rate Swaps and Floating Rate Loans

• A floating interest rate, also known as a
  variable rate or adjustable rate, refers to any
  type debt instrument, such as a loan, bond,
  mortgage, or credit, that does not have a fixed
  rate of interest over the life of the instrument.
  
• Such debt typically uses an index or other base
  rate for establishing the interest rate for each
  relevant period.

38
Interest Rate Swaps and Floating Rate Loans

• A typical index used is the London Inter-bank
  Offered Rate, or LIBOR. Locally, we also have
  Hong Kong Inter-bank Offered Rate, HIBOR, as a
  reference for loans issued in Hong Kong dollars.

39
LIBOR

• LIBOR is a kind of interest rates at which a bank
  is prepared to deposit money with other banks in
  the Eurocurrency market.
• It is calculated by British Bankers Association
  (BBA) and released to the market shortly after
  11am London time every day.

40
LIBOR and Floating Interest Rate

• Consider a 5-year loan with a rate of interest
  specified as 6-month LIBOR plus 0.5 per annum.
• 0.5y 1.0y 1.5y
  4.0y 4.5y 5.0y

Interest rate in the period of 1.0y to 1.5y is
set at 0.256 month LIBOR at this time
Interest in the period of 1.0y to 1.5y is paid at
this time
41
Interest Rate Swaps

• Interest rate swaps are often used by firms to
  alter their exposure to interest-rate
  fluctuations, by swapping fixed-rate obligations
  for floating rate obligations, or vice versa. By
  swapping interest rates, a firm is able to alter
  its interest rate exposures and bring them in
  line with management's appetite for interest rate
  risk.

42
Interest Rate Swaps

• Consider two companies, A and B. Both of them
  wish to borrow 100M for 3 years. As credit
  quality is better than B. They are offered the
  following loan opportunities
• Fixed Floating
• A 4.0 6-month
  LIBOR0.3
• B 5.2 6-month
  LIBOR1.0
• (semiannually
  payment)

43
Comparative Advantage and Swaps

• Suppose that company A prefers a loan with
  floating rate and at the same time company B
  prefers a loan with fixed rate for some reason.
• If they borrows money directly from where they
  prefer, then
• A pays 6-month LIBOR0.3
• B pays 5.2
• Total interest payment LIBOR5.5

44
Comparative Advantage and Swaps

• However, A has a comparative advantage in the
  fixed loan market while B has one in the
  floating.
• Can they do what they prefer and meanwhile keep
  their respective comparative advantages?

45
Interest Rate Swaps

• Interest rate swaps are able to team these two
  companies up.
• Company A borrows money at fixed rate and Company
  B at floating rate
• They swap the interest payments, i.e., A pays
  floating interest for B and B pays fixed interest
  for A.

46
Interest Rate Swaps

• Illustration
• 3.95
• 4.0
• LIBOR
  LIBOR 1
• A pays LIBOR 0.05
• B pays 13.95 4.95.
• Total interest payment LIBOR 5

A
B
47
Interest Rate Swaps

• One possible cash flows for Company B
• Dates LIBOR Rate Floating cash flow
  Fixed cash flow Net
• 01/02/2009 4.2
  
• 01/08/2009 4.8 2.1M
  -1.975M 0.125M
• 01/02/2010 5.3 2.4M
  -1.975M 0.425M
• 01/08/2010 5.5 2.65M
  -1.975M 0.675M
• 01/02/2011 5.6 2.75M
  -1.975M 0.775M
• 01/08/2011 5.9 2.80M
  -1.975M 0.825M
• 01/02/2012 2.95M
  -1.975M 0.975M
• 
  (102.95M -101.975M 0.975M)

48
Interest Rate Swaps

• Cash flow analysis
• 0 0.5 1 1.5 2
  2.5 3
• 1.975M 1.975M
  1.975M

49
Interest Rate Swaps

• The value of the interest rate swap for company B
  should be