• Goal: to neutralize the risk as far as possible.
I. Derivatives
A. Option: contract that gives its holder the right to buy (or sell) an asset at a predetermined price within a specified period of time.

•
• A-1) Call option: option to buy an underlying asset at a certain price within a specific period
• A-2) Put option: option to sell an underlying asset at a certain price within a specific period
Call
Put


PN ( d
1
)


PN (
 d
1
)
Xe
 rt
N ( d
2
)

Xe
 rt
N (
 d
2
) d
1

(ln( P / X )

( r
 
2
/ 2 )
 t ) /(
 d
2
 d
1
  t
 t )

• call put
• Underlying asset + -
• Exercise price +
• Time to expiration + +
• Risk free rate + -
• Variance of return + +

• B. Forward Contracts:
- agreements where one party agree to buy a commodity at a specific price on a specific future date and other party agrees to sell.
- Physical delivery occurs

• C. Future contracts:
• - similar to forward contracts
• - marked to market on a daily basis
• (margin account with minimum requirement, reducing default risk)
• - settled with cash
• - standardized

• Ex) marking to market
• Setting up a margin account for futures contract with initial margin – e.g)$4000
• Rebalanced daily to reflect investors’ gains or losses, using daily futures prices.
• If the balance below maintenance margin, there will be a margin call.
• The investor is entitled to withdraw any balance in the margin account in excess of the initial margin.

• D. Swap: two parties agree to exchange obligations to make specified payment streams.
• Ex) Floating rate bond & Fixed rate bond
• E. Structured notes: a debt obligation derived from another debt obligation
• Ex) Stripping long term debts (30 years) to create a series of zero coupon bonds
• Ex) CMO with mortgages loans

F. Inverse Floaters
• A note in which the interest paid moves counter to market rates
Ex) note at prime plus 1%
II. Hedging with futures
• Short hedge: a hedge that involves a short position in futures contract. Here the hedger owns an asset and expects to sell it in the future.

• Ex-1) suppose that an oil producer sell a
August 15 futures contract at $18.75 per barrel. Now it is April.
• Suppose that the spot price on August 15 prove to be $17.50. And August 15 futures price will be close to $17.50.
• gain = Sales of oil + difference of futures price = 17.50
+(18.7517.50
) = 18.75
• If the spot price goes up to $19, gain =
19 + (18.75 19 ) =18.75

• Long hedges: hedges taking a long position in a futures contract. Here hedgers want to purchase a certain assets
• Ex) A copper fabricator buys a May futures contract at 120 cents per pound. He or she needs 10,000 pound in May. Now, It is January.
• If spot price goes up to 125 cents, costs =
10000* 1.25
- ( 1.25
-1.20)*10000=120000
• If spot price goes down to 105 cents, costs =
10000* 1.05
+(1.201.05
)*10000=120000

1) Basis Risk
As shown in the previous examples, to achieve hedging, spot price and future contract prices should converge around the expiration date. If not,……
• Basis = spot price of asset to be hedged
– futures price of contract used.
• Reasons of basis:
- the asset for hedging is not the same as the asset underlying the futures contract

- uncertain date when the asset will be bought or sold
- closed out before its expiration date
• Strengthening of the basis: increasing basis
• Weakening of the basis: decreasing basis

• 2) Minimum variance hedge ratio
• Ratio that minimize the variance of the hedger’s position h
*  


F s

S

F
: change
: change in in spot price futures during price the during

: S tan dard s


F
: S tan dard
: correlatio n deviation deviation of between

S of


S
F and

F life the of life hedge of hedge

• 3) Optimal Number of Contracts
• N: optimal number of contracts
• N
A
: Size of position hedged
• Q
F
: Size of one contract
N
 h
*
N
A
Q
F