Business 4119 Derivative Securities

1
Business 4119 Derivative Securities

• Prof. Nikola Gradojevic, Ph. 343-8419
  (nikola.gradojevic_at_lakeheadu.ca)
• Web http//foba.lakeheadu.ca/gradojevic/4119/
• Office Ryan Building RB-1035(Hours Thursdays
  1-230PM)
• Lectures M/W 530-7 PM at AT-1005 (ATAC
  Building)
• Textbook Risk Management and Derivatives, 1/e
  by René M. Stulz, Thomson, South-Western, 2003.

2
Derivative Securities

Weeks 1,2.3 - Introduction to Derivative
Securities and Forward (Futures) Contracts
1,23 ?
3
Overview

• Risk regarding the possibility of loss can be
  especially problematic
• If a loss is certain to occur
• It may be planned for in advance and treated as a
  definite, known expense
• When there is uncertainty about the occurrence of
  a loss (e.g., weather, political factors)
• Risk becomes an important problem

4
Overview

• Some risks involve only the possibility of loss
• Risks surrounding potential losses create
  significant economic burdens for businesses,
  government, and individuals
• Billions of dollars are spent each year to
  finance potential losses
• But when losses are not planned for in advance
  they may cost even more
• Risk of loss may deprive society of services
  judged to be too risky

5
Overview

• Businesses may try to either avoid risk of loss
  or to reduce its negative consequences
• Risk management
• Process used to systematically manage risk
  exposures
• Integrated risk management and enterprise risk
  management
• Intent to manage all forms of risk, regardless of
  type

6
Overview

• Many businesses have special departments charged
  with overseeing the firms risk management
  activities
• The head of such a department often is called a
  risk manager
• Some firms have formed risk management committees
  
• Some firms have created the position of chief
  risk officer to coordinate the firms risk
  management activities

7
Overview

• Identify risks
• Evaluate risks
• Select risk management techniques
• Implement and review decisions

8
Risk Management Techniques

• It has become possible to be protected against
  the financial consequences of e.g., bad weather
  financial engineers produce financial instruments
  (derivatives) that can help us deal with
  uncertainty
• Derivatives financial contracts that derive
  their value from the underlying, cash market
  instrument (stocks, bonds, currencies,
  commodities, etc.).

9
Risk Management Techniques

• Derivatives
• Futures and forwards
• Options
• Swaps
• Underlying instruments (securities)
• Stocks
• Currencies
• Interest rates (bonds, notes, T-bills)
• Indexes (SP-500, CRB index, etc.)
• Commodities, etc.

10
The Changing Financial Landscape

• Some changes have been gradual.
• The smaller role played by the United States and
  the U.S. dollar (in favor of DM, Yen, Euro).
• The growing importance of new financial products
  (options, futures, swaps), new financial
  institutions (investment banks), and emerging
  markets around the world.
• Other changes were more abrupt.
• The collapse of the pegged exchange rate system
  in the early 1970s (Bretton Woods) and in the
  latter half of the 1990s (Mexico, Thailand,
  Korea, etc.).

11
The Changing Financial Landscape

• The menu of financial choices is expanding.
• Financial engineering Corporate issuers and
  private investors have at their disposal
  innumerable futures and option contracts to
  acquire or lay off risks associated with
  economy-wide shocks or with firm-specific events.
  
• Financial markets have greater volatility now.
• Large price swings over the past three decades
  led to increased demands for financial
  forecasting, as well as a greater emphasis on
  risk management.

12
The Changing Financial Landscape

• New instruments can help, but they carry their
  own risks
• Counterparty risk default risk
• Liquidity risk position cannot be sold
• Delivery risk buyer does not pay
• Rollover risk credit rating risk
• Systemic risk cascade effect

13
Accidents along the International Financial
Superhighway
14
Accidents along the International Financial
Superhighway
15
Accidents along the International Financial
Superhighway
16
The Changing Financial Landscape
17
FX market

• Foreign exchange market the market in which
  currencies are traded.
• Major players in the FX market
• -dealers (e.g., commercial banks)
• -customers (e.g., corporations, central banks,
  etc.)
• -brokers (mediators between dealers and dealers
  or customers).
• Spot FX market decentralized multiple-dealer
  market accounts for 40 across all FX
  instruments categories (in 1998). It was 59 in
  1989 (BIS survey).
• -Of the 900 billion of daily volume that is not
  from the spot market, 734 billion of this is FX
  swaps (BIS 1999) the derivatives market have
  grown up around the spot market!

18
Risk Management basic ideas

• Options
• Definition a contract between two parties that
  gives one party, the buyer, the right to buy or
  sell something from or to the other party, the
  seller, at a later date at a price agreed upon
  today
• Option terminology
• price (premium)
• call/put
• American/European
• exchange-listed vs. over-the-counter options

19
Risk Management basic ideas

• Gains and losses from buying shares and a call
  option on Risky Upside Inc.

20
Risk Management basic ideas
21
Risk Management basic ideas

• Forward Contracts
• Definition a contract between two parties for
  one party to buy something from the other at a
  later date at a price agreed upon today
• Exclusively over-the-counter
• Futures Contracts
• Definition a contract between two parties for
  one party to buy something from the other at a
  later date at a price agreed upon today subject
  to a daily settlement of gains and losses and
  guaranteed against the risk that either party
  might default
• Exclusively traded on a futures exchange

22
Risk Management basic ideas
23
Risk Management basic ideas

• Payoff of a short forward position (sell EURO
  forward 6 months _at_ 1/EURO).

24
Risk Management basic ideas

• Hedged firm income perfect hedge. Garmans cash
  flow is the same regardless of the exchange rate
  movements

25
Risk Management basic ideas

• Comparison of income with put contract (pays the
  premium) and income with forward contract.

26
Forward and Futures Contracts

• The possibility of default poses a potentially
  serious problem for counterparties in a forward
  contract.
• One method of dealing with this risk is to make
  forward contracts only with people of high
  character, reputation, and credit quality.
• Another method, associated with futures
  contracts, calls for both counterparties to post
  a good-faith bond that is held in escrow by a
  reputable and disinterested third party.

27
Forward and Futures Contracts

• Futures exchanges require each counterparty to
  post a bond in the form of a margin requirement
  that is marked to market.
• Until 1972, futures contracts were associated
  exclusively with physical commodities. The
  multiplication of the demand for financial
  hedging instruments set the stage for financial
  futures contracts.

28
Distinctions between Futures and Forwards
29
Distinctions between Futures and Forwards
30
Distinctions between Futures and Forwards
31
Prices and the Margin Account
32
Barings Bank

• Example Collapse of Barings BankIn January
  1995, a trader named Nick Leeson excessively
  bought index futures contracts in Singapore
  International Monetary Exchange (SIMEX) hoping
  the index had bottomed out.
• Unfortunately, it had not and continued to fall.
• Daily adjustment at the close of each trading
  day, dt Ft - Ft-1 is added to the account of
  the buyer of the contract, where Ft is the
  closing futures price and Ft-1 is that of the
  previous trading day. Thus entering a futures
  contract at time t entitles the buyer the cash
  flows
• dt1, . . . ,dT.
• -Nick Lessons dt-s were mostly negative, i.e.
  losses, which eventually accumulated to 1
  billion in March 1995 when the bank collapsed.

33
Example

• Why would you engage in forward contracts
  (transactions)?
• Answer To avoid the risk of unexpected future
  movements of the underlying security (exchange
  rate).
• Canadian company imports radios from the US and
  knows that in 30 days must pay US dollars to a US
  supplier for a shipment.
• The importer can sell each radio for 100 and
  must pay 68 US per radio.
• The current spot exchange rate is 1.3918 per US
  dollar.
• Thus, the profit per radio is 100 - (1.3918
  per US dollar) ? (68 US per radio) 94.64
  5.36

34
Example-contd.

• However, suppose the importer will not have funds
  to pay the supplier until the radios arrive and
  are sold.
• If over the next 30 days the Canadian dollar
  unexpectedly depreciates to 1.4918 per US
  dollar, the importer will have to pay (1.4918
  per US dollar) ? (68 US per radio) 101.44
  per radio and will take a loss of 1.44 per
  radio!
• To avoid this risk, the importer can make a
  30-day forward exchange rate contract with his
  bank the bank would agree to sell US dollars to
  the importer at a rate of 1.4518 per US dollar.
• The importer is guaranteed a profit of 100 -
  (1.4518 per US dollar) ? (68 US per radio)
  98.72 1.28 per radio!

35
Pricing Forwards and Futures

• The Concept of Price Versus Value
• Normally in an efficient market, price value.
• For futures or forward, price is the contracted
  rate of future purchase. Value is something
  different.
• At the beginning of a contract, value 0 for
  both futures and forwards.
• Notation
• Vt(0,T), Ft(0,T), vt(T), ft(T) are values and
  prices of forward and futures contracts created
  at time 0 and expiring at time T.

36
Pricing Forwards and Futures

• The Value of a Forward Contract
• Forward price at expiration
• F(T,T) ST.
• That is, the price of an expiring forward
  contract is the spot price.
• Value of forward contract at expiration
• VT(0,T) ST - F(0,T).
• An expiring forward contract allows you to buy
  the asset, worth ST, at the forward price F(0,T).
  The value to the short party is -1 times this.

37
Pricing Forwards and Futures

• Pricing forwards by arbitrage (on T-bills)
• buy T-bills (maturing Aug 30) on June 1 or borrow
  now (short T-billsPVftJune 1(T)), repay June
  1 and buy T-bills (maturing Aug 30)?

38
Pricing Forwards and Futures

• Replicating portfolio 3-month loan and long
  position in T-bills
• If value of the replicating portfolio 0, there
  are no arbitrage opportunities
• PtMarch 1(Aug 30) - PtMarch 1(June 1) x
  ftMarch 1(Aug 30) 0
• gt
• or more generally, for T-bills

39
Pricing Forwards and Futures

• Value of a forward position/dollar of face value
• Long
• Short

40
Pricing Forwards and Futures

• Generalizing to stocks
• where Pt(ti) is the price of a zero-coupon bond
  that pays 1 at ti and r is the continuously
  compounded interest rate for that bond

41
Pricing Forwards and Futures

• Generalizing to multiple payouts before maturity
• where the asset has N intermediate payouts
  (dividends) of DtDh, h1,...,N (example pp.123)

42
Pricing Forwards and Futures

• Foreign currency forwards (/units of foreign
  currency)
• where St is the spot exchange rate (price of the
  foreign currency), rFX is the continuously
  compounded foreign interest rate for the
  zero-coupon bond maturing at ti.

43
Pricing Forwards and Futures

• Commodity forwards (/one unit of commodity)
• where c is the convenience yield (benefit of
  holding the commodity), accrued continuously
  (percent/year).

44
Pricing Forwards and Futures

• General formula for the forward price
• where d denotes the payout rate (/year), v
  denotes storage costs (/year e.g., fraction of
  the holdings of oil we have to sell to pay for
  storage)

45
Pricing Forwards and Futures

• Properties of the forward price
• S? (increase the cost of buying today) gt f?
• r? (more expensive to borrow) gt f?
• v? (more expensive to buy and hold) gt f?
• c? (decrease cost of holding) gt f?
• d? (decrease cost of holding) gt f?
• NB keep in mind the replicating portfolio buy
  today and finance it through borrowing (the
  present value of the forward price) until the
  maturity of the forward

46
Pricing Forwards and Futures

• fgtS gt contango
• fltS gt backwardation
• cash delivery delivery takes place in cash
• physical delivery e.g., apples
• usually no physical delivery takes place in
  futures markets (positions are closed out before
  that)

47
Pricing Forwards and Futures

• Forwards, futures, spot prices
• The theoretical relationship between futures and
  forward prices is ambiguous.
• Empirical studies also show that futures and
  forward prices for foreign exchange are not
  statistically different from each other.
• Forwards on a currency with high r are on average
  profitable
• Can we use forward prices to forecast spot
  exchange rates?

48
Pricing Forwards and Futures

• where denotes the spot rate realized at time
  t30 (Canada/U.S.), ft,30 denotes the 30-day
  forward rate of the Canadian dollar, settled at
  time t30, and et is a random error term. The
  data for this exercise were obtained from the
  Statistics Canada CANSIM database (data range
  January 1994 May 2002).

49
Hedging with Forwards and Futures

• Suppose t0March 1, 1999 and we will receive 1
  mil Swiss Francs on June 1, 1999
• i?i(t)const., DSt??(E(DSt),Var(DSt))
• What will be the dollar value of the Francs?
• First, based on the past T 131 monthly
  observations, find

50
Hedging with Forwards and Futures
51
Hedging with Forwards and Futures

• Square root rule for volatility
• For N1, VolatilityV1 (1)1/2(0.000631)1/2
• For N3, VolatilityV1x(3)1/20.0435V3
• Volatility of the payoffV3x1M43,508.6
• E(DStN3 months)3x(-0.000115)-0.000345

52
Hedging with Forwards and Futures

• Thus, there is a 5 probability that we will
  receive an amount of US that is at least
  71,789.2 below the expected amount
• Suppose that S00.75/SFrgtE(DST)(0.75-0.000345)
  
• 0.749655 E(CF)749,655
• There is 5 chance CF677,865.8
• CaR1.65x0.0435x1M 71,789.2gt39,655 (target)

53
Hedging with Forwards and Futures

• Now suppose that we compute the value of our long
  SFr position and analyze its change over the next
  day
• What is the dollar value (V) of this position
  now?
• Replicating portfolio SFr 3-month zero coupon
  bond is 0.96 SFrgt960,000 SFrV
• gtV0.75/SFr x 0.96 x 1M SFr720,000
• -assume rconst., ignore DST over one day

54
Hedging with Forwards and Futures

• need daily volatility now, assume daily DSt??(.),
  assume 21 trading days per month
• Volatility (position)V1D x SFrV5,260.41
• There is a 5 chance that the position will lose
  at least 1.65x5,260.41 over the next day
• VaR(Value-at-Risk)1,65x ? x portfolio value

55
Hedging with Forwards and Futures

• Eliminate risk using forward, money and futures
  markets
• Forward contract (sell 1M SFr forward maturing on
  June 1) hedge and money market
• Value of the position on June 1
• This makes CaR0 (s0)
• PV1(T-bill mat. on June 1)
• On March 2-PV2
• One-day VaR1.65 x volatilityPV1-PV20
• Assume there is no uncertainty about the price
  of T-bill

56
Hedging with Forwards and Futures

• Same can be achieved using money market
  instruments long position in SFr T-bill and
  short position in US (domestic) T-bill.
• Hedge with futures (transform it into a forward
  by reinvesting the gains (until June 1) and
  borrowing the losses (repay June 1) daily at the
  risk-free rate (assume constant)

57
Hedging with Forwards and Futures

• This changing of the short position (tailing the
  hedge) requires forecasting of the next days
  futures price (T-bills price). We can simply
  use todays price.
• Assume now that maturity is in 10 years
• Tailing factor is based on PVT-bill price
• Suppose it is 0.5, but we decide not to use it

58
Hedging with Forwards and Futures

• Also, assume that f increases tomorrow by 10
  cents and stays like that for 10 years
• No tailing sell 1M SFr on the futures market
• In 10 years we do not use tailing, but sell spot
  for 0.85/SFr and make 850k, but lose from
  marking-to-market
• 0.10x1Mx(1/0.5)200k, total650klt750k
  (forward hedge)
• This is overhedging!

59
Hedging with Forwards and Futures

• The tailed hedge using
• solves the problem and generates
• 850k (cash position)-100k(loss on futures
  position)750k
• loss on futures50k immediately which amounts to
  100k in 10 years

60
Hedging with Forwards and Futures

• Basis risk there is no futures contract that
  matures on June 1 available
• Thus, on June 1, f and S are not equal
• Payoff of the hedged position on June 1

61
Hedging with Forwards and Futures
Figure 6.2. Relation between cash position and
futures price when there is a deterministicrelatio
n between the futures price and the spot price
(assume slope 0.9).
62
Hedging with Forwards and Futures

• For our example, cash market position on June 1
  is 1M x SJune 1
• Using the assumed slope of 0.9, an unexpected
  change in the f of D units changes the cash
  position by 1M x 0.9 x D
• Suppose we go short h SFr in futures then

63
Hedging with Forwards and Futures
Figure 6.3. Change in value of the hedged
position as a function of the exchange rate
change and the size of the hedge h
64
Hedging with Forwards and Futures
Figure 6.3. Change in value of the hedged
position as a function of the exchange rate
change and the size of the hedge h
65
Hedging with Forwards and Futures

• Notice that the hedged position gains when
  hlt900kSFr and the SFr appreciates (DSgt0), and
  incurs a loss when hgt900k and DSgt0 (error in the
  book!)
• The position loses when hlt900k and DSlt0, and
  gains when hgt900k and DSlt0
• Thus, the only point when there is no loss is
  h900k (solve 0.9x1MxD-hxD0)
• Hedging strategy sell value of 900K SFr futures.

66
Hedging with Forwards and Futures

• What if basis risk is random?

Figure 6.4. Relation between cash position and
futures price changes when the futures price
changes are imperfectly correlated with the cash
position changes
67
Hedging with Forwards and Futures

• What is the optimal h?
• We regress (OLS) DS on Df to obtain an estimate
  of h
• That gives

68
Hedging with Forwards and Futures
Figure 6.5. Regression line obtained using data
from Figure 6.4
69
Hedging with Forwards and Futures

• We can generalize formula to
• This is the classic formula for the
  volatility-minimizing hedge of arbitrary cash
  position
• To be able to use the OLS method we assumed that
  classical assumptions hold (Box 6.4, pp. 172)
• We also assume that the choice of T (sample size)
  does not have any impact on the estimate of h

70
Hedging with Forwards and Futures

• Using weekly data from September 1, 1997
  February 22, 1999 we estimate h
• With the use daily data the estimate of h is 0.91
  (t52.48) more precise
• R2 tells us how much of the variance in DS is
  explained by Df. Through hedging we eliminated
• of the volatility of the unhedged position.
• (higher R2 is preferred)

71
Hedging with Forwards and Futures

• What if returns rather than levels are IID?
• Volatility minimizing hedge of cash position
• r(cash)-return on the cash position
• r(hedge)-the rate of change of the price of the
  hedging instrument