Forward and Futures Prices
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Buy 1 unit of asset Take a short position in the forward contract
Forward Contracts
At time T:
Sold the asset for ST Pay back the loan, -St e r(T-t) Realize a profit, F – St e r(T-t) > 0
Forward Contracts: Example 1
A forward contract is written on a stock. The maturity of the contract is 6 months. The stock price is $50 today and the risk-free rate is 10% per year
3. Forward and Futures Prices
Summary
Short Selling The Repo Rate Forward Contracts Forward Price vs. Futures Price Stock Index Futures Forward and Futures Contracts on Currencies
Short Selling
Selling securities that you do not own and buying them back later
Yield a profit if the price of a security goes down
Yield a loss if the price of a security goes up
the repo rate, repurchase agreement
You sell securities to another investor Buy them back at a higher price The difference between the two prices is
Sell them in the open market at $50 per share
Deposit the cash in your account
Short Selling: Example
You call your broker and instruct him to short 100 shares of ATT
accounts Dividends and interest received by the
investor will be transferred to the client from whom the shares were borrowed
The Repo Rate
The risk-free rate available to investors is
the interest earned by the counterparty
The Repo Rate
Repo is slightly higher than the T-bill rate
Overnight repo, term repo
Forward Contracts
Notations:
F ST
Borrow at rate r
St I
St I er(T t )
0
ST F ST (St I )er(T t )
Charles Cao
25
Forward Contracts
Therefore, the price of the forward
contract at t is
Real world application
Short Selling: Example
You call your broker and instruct him to short 100 shares of ATT
He will borrow 100 shares of ATT from another client
He will borrow 100 shares of ATT from another client
Sell them in the open market at $50 per share
Deposit the cash in your account
Short Selling: Example
Recall
F ( St I )er(T t )
Forward Contracts : Example
T-t = 0.83 year (10/12) r = 0.08 St = $50
I 2e0.08(6 / 12) 2e0.08(9 / 12) 3.81
Thus, the forward price is:
F ( St I )er(T t )
Forward Contracts: Example
A 10-month contract is written on a dividend-paying stock. The stock price is $50 today. The risk free rate is 8% per year. The dividend is $2.00 per share and will be paid after 6 months and 9 months
Forward Contracts
Forward contracts on a security that provides no income
Determine today’s forward price using noarbitrage argument
Forward Contracts
or
Charles Cao
17
Forward Contracts
If F > St e r(T-t) , you can earn an
arbitrage profit by taking the following
positions at t (cash is not required): Borrow St dollars at rate r
T : expiration day t : current tTime T-t : the life of the contract (in years) St : price of asset underlying forward
contract at t
Forward Contracts
Notations: (cont.)
13
Forward Contracts
Therefore, the price of the forward contract
at t is:
F Ster(T t )
Intuition
You walk in a local Mercedes-Benz dealer’s office
You want to buy a white color CL500 model
ST : price of asset underlying forward contract at T
F : forward price at t r : risk-free rate per year K: delivery price in forward contract f : value of a long forward contract
A forward contract is written on a discount bond. The maturity of the contract is 4 months. The bond price is $950 today and the risk-free rate is 8% per year
Forward contracts on a security that provides a known cash income
Dividends-paying stock with known (discrete) dividends
Coupon bonds Determine today’s forward price using no-
arbitrage argument
I: present value of all known cash income
Forward Contracts
Action
Cash Flow at t
Cash Flow at T
Buy Underlying Asset
St I
ST
Short 1 Forward Contract
You agree to pay when the car is ready to be delivered
How much should the salesman charge you 30 days later?
Interest rate is 5%Charles Cao
16
Intuition
Action
Cash Flow at t
Cash Flow at T
Buy 1 Unit of Asset
St
ST
Short 1 Forward Contract
Borrow $ St
at rate r
F ST
St
Ster(T t )
0
ST F ST Ster(T t ) 0
Charles Cao
Today, the price is $90,000 However, only black and silver colors
are available
Charles Cao
15
Intuition
The salesman agrees to get a white color CL500 from Florida in 30 days
F (50 3.81)e0.080.83 $49.36
Short Selling: Example
You instruct your broker to close out the position 10 days later. He will buy 100 shares of ATT, and return them to the client
If the price is $48, the profit is $200 If the price is $51, the loss is $100
Short Selling
Shares can only be sold on an uptick (price increases)
Margin requirement Some brokers pay interest on margin
Recall F Ster(T t )
Forward Contracts: Example 1
T-t = 0.5 year r = 0.1 St = $50
Thus, the forward price is:
F 50e0.10.5 $52.56
Forward Contracts: Example 2
Recall
F Ster(T t )
Forward Contracts : Example 2
T-t = 0.33 year (4/12) r = 0.08 St = $950
Thus, the forward price is:
F 950e0.080.33 $975.41
Forward Contracts
You instruct your broker to close out the position 10 days later. He will buy 100 shares of ATT, and return them to the client
If the price is $48, the profit is $200 If the price is $51, the loss is $100
Forward Contracts
At time T:
Sold the asset for ST Pay back the loan, -St e r(T-t) Realize a profit, F – St e r(T-t) > 0
Forward Contracts: Example 1
A forward contract is written on a stock. The maturity of the contract is 6 months. The stock price is $50 today and the risk-free rate is 10% per year
3. Forward and Futures Prices
Summary
Short Selling The Repo Rate Forward Contracts Forward Price vs. Futures Price Stock Index Futures Forward and Futures Contracts on Currencies
Short Selling
Selling securities that you do not own and buying them back later
Yield a profit if the price of a security goes down
Yield a loss if the price of a security goes up
the repo rate, repurchase agreement
You sell securities to another investor Buy them back at a higher price The difference between the two prices is
Sell them in the open market at $50 per share
Deposit the cash in your account
Short Selling: Example
You call your broker and instruct him to short 100 shares of ATT
accounts Dividends and interest received by the
investor will be transferred to the client from whom the shares were borrowed
The Repo Rate
The risk-free rate available to investors is
the interest earned by the counterparty
The Repo Rate
Repo is slightly higher than the T-bill rate
Overnight repo, term repo
Forward Contracts
Notations:
F ST
Borrow at rate r
St I
St I er(T t )
0
ST F ST (St I )er(T t )
Charles Cao
25
Forward Contracts
Therefore, the price of the forward
contract at t is
Real world application
Short Selling: Example
You call your broker and instruct him to short 100 shares of ATT
He will borrow 100 shares of ATT from another client
He will borrow 100 shares of ATT from another client
Sell them in the open market at $50 per share
Deposit the cash in your account
Short Selling: Example
Recall
F ( St I )er(T t )
Forward Contracts : Example
T-t = 0.83 year (10/12) r = 0.08 St = $50
I 2e0.08(6 / 12) 2e0.08(9 / 12) 3.81
Thus, the forward price is:
F ( St I )er(T t )
Forward Contracts: Example
A 10-month contract is written on a dividend-paying stock. The stock price is $50 today. The risk free rate is 8% per year. The dividend is $2.00 per share and will be paid after 6 months and 9 months
Forward Contracts
Forward contracts on a security that provides no income
Determine today’s forward price using noarbitrage argument
Forward Contracts
or
Charles Cao
17
Forward Contracts
If F > St e r(T-t) , you can earn an
arbitrage profit by taking the following
positions at t (cash is not required): Borrow St dollars at rate r
T : expiration day t : current tTime T-t : the life of the contract (in years) St : price of asset underlying forward
contract at t
Forward Contracts
Notations: (cont.)
13
Forward Contracts
Therefore, the price of the forward contract
at t is:
F Ster(T t )
Intuition
You walk in a local Mercedes-Benz dealer’s office
You want to buy a white color CL500 model
ST : price of asset underlying forward contract at T
F : forward price at t r : risk-free rate per year K: delivery price in forward contract f : value of a long forward contract
A forward contract is written on a discount bond. The maturity of the contract is 4 months. The bond price is $950 today and the risk-free rate is 8% per year
Forward contracts on a security that provides a known cash income
Dividends-paying stock with known (discrete) dividends
Coupon bonds Determine today’s forward price using no-
arbitrage argument
I: present value of all known cash income
Forward Contracts
Action
Cash Flow at t
Cash Flow at T
Buy Underlying Asset
St I
ST
Short 1 Forward Contract
You agree to pay when the car is ready to be delivered
How much should the salesman charge you 30 days later?
Interest rate is 5%Charles Cao
16
Intuition
Action
Cash Flow at t
Cash Flow at T
Buy 1 Unit of Asset
St
ST
Short 1 Forward Contract
Borrow $ St
at rate r
F ST
St
Ster(T t )
0
ST F ST Ster(T t ) 0
Charles Cao
Today, the price is $90,000 However, only black and silver colors
are available
Charles Cao
15
Intuition
The salesman agrees to get a white color CL500 from Florida in 30 days
F (50 3.81)e0.080.83 $49.36
Short Selling: Example
You instruct your broker to close out the position 10 days later. He will buy 100 shares of ATT, and return them to the client
If the price is $48, the profit is $200 If the price is $51, the loss is $100
Short Selling
Shares can only be sold on an uptick (price increases)
Margin requirement Some brokers pay interest on margin
Recall F Ster(T t )
Forward Contracts: Example 1
T-t = 0.5 year r = 0.1 St = $50
Thus, the forward price is:
F 50e0.10.5 $52.56
Forward Contracts: Example 2
Recall
F Ster(T t )
Forward Contracts : Example 2
T-t = 0.33 year (4/12) r = 0.08 St = $950
Thus, the forward price is:
F 950e0.080.33 $975.41
Forward Contracts
You instruct your broker to close out the position 10 days later. He will buy 100 shares of ATT, and return them to the client
If the price is $48, the profit is $200 If the price is $51, the loss is $100