期权、期货及其他衍生产品课件3金融工程学
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5.3
Short Selling
(continued)
If at any time while the contract is open the broker is not able to borrow shares, the investor is forced to close out the position, even if not ready to do so, called shortsqueezed(挤空，挟仓). You must pay dividends and other benefits the owner of the securities receives
(Page 107, equation 5.3)
F0 = S0 e(r–q )T
where q is the average yield during the life of the contract (expressed with continuous compounding)
5.13
Example
5.22
Index Arbitrage
(continued)
Index arbitrage involves simultaneous trades in futures and many different stocks Very often a computer is used to generate the trades Occasionally (e.g., on Black Monday) simultaneous trades are not possible and the theoretical no-arbitrage relationship between F0 and S0 does not hold
5.6
Notation for Valuing Futures and Forward Contracts
S0: Spot price today F0: Futures or forward price today
T: Time until delivery date(in years) r: Risk-free interest rate for maturity T
源自文库
F0 = S0 e(r–q )T
where q is the average dividend yield on the portfolio represented by the index during life of contract
5.20
Stock Index
(continued)
For the formula to be true it is important that the index represent an investment asset In other words, changes in the index must correspond to changes in the value of a tradable portfolio
5.17
The value of a long forward contract
No income income
Known yield
5.18
Forward vs Futures Prices
Forward and futures prices are usually assumed to be the same. When interest rates are uncertain they are, in theory, slightly different A strong positive correlation between interest rates and the asset price implies the futures price is slightly higher than the forward price A strong negative correlation implies the reverse
5.19
Stock Index (Page 110-112)
Can be viewed as an investment asset paying a dividend yield The futures price and spot price relationship is therefore
5.4
Example
The investor is required to maintain a margin account with the broker.
5.5
Assumptions
1.
2.
3.
4.
The market participants are subject to no transaction costs when they trade. The market participants are subjects to the same tax rate on all net trading profits. The market participants can borrow money at the same risk-free rate of interest as they can lend money. The market participants take advantage of arbitrage opportunities as they occur.
F0 = (S0 – I )erT
where I is the present value of the income during life of forward contract. In our example, So that,
5.12
When an Investment Asset Provides a Known Yield
When an Investment Asset Provides a Known Dollar Income
5.10
套利者净收益：910.00-886.60=23.40
5.11
When an Investment Asset Provides a Known Dollar Income
(page 105, equation 5.2)
The Nikkei index viewed as a dollar number does not represent an investment asset
(See Business Snapshot 5.3, page 111)
5.21
Index Arbitrage
When F0 > S0e(r-q)T an arbitrageur buys the stocks underlying the index and sells futures When F0 < S0e(r-q)T an arbitrageur buys futures and shorts or sells the stocks underlying the index
5.7
Forward price for an investment asset
5.8
When Interest Rates are Measured with Continuous Compounding
F0 = S0erT
if
远期价格大于即期价格
F0 > S0erT ,arbitrageurs can buy the asset and short forward contracts on the asset. if F0 < S0erT , they can short the asset and enter into long forward contracts on it. This equation relates the forward price and the spot price for any investment asset that provides no income and has no storage costs. 5.9
Determination of Forward and Futures Prices
Chapter 5
5.1
Consumption vs Investment Assets
Investment assets are assets held by significant numbers of people purely for investment purposes (Examples: stock, bond, gold, silver) Consumption assets are assets held primarily for consumption (Examples: copper, oil)
Consider a 6-month forward contract on an asset that is expected to provide income equal to 2% of the asset price once during a 6-month period. The risk free rate of interest is 10% per annum. The asset price is $25. The yield is 4% per annum, it follows that q=0.0396(=2ln(1+4%/2)).
5.15
5.16
Example
A long forward contract on a not-dividend paying stock was entered into some time ago. It currently has 6 months to maturity. The risk-free rate of interest is 10% per annum, the stock price is $25, and the delivery price is $24.
5.23
Futures and Forwards on Currencies (Page 112-115)
A foreign currency is analogous to a security providing a dividend yield The continuous dividend yield is the foreign risk-free interest rate It follows that if rf is the foreign risk-free interest rate ( r r f )T 0 0
5.14
Valuing a Forward Contract
Page 108
Suppose that K is delivery price in a forward contract and F0 is forward price that would apply to the contract today The value of a long forward contract, ƒ, is ƒ = (F0 – K )e–rT Similarly, the value of a short forward contract is (K – F0 )e–rT
5.2
Short Selling (Page 99-101)
Short selling involves selling securities you do not own Your broker borrows the securities from another client and sells them in the market in the usual way At some stage you must buy the securities back so they can be replaced in the account of the client
Short Selling
(continued)
If at any time while the contract is open the broker is not able to borrow shares, the investor is forced to close out the position, even if not ready to do so, called shortsqueezed(挤空，挟仓). You must pay dividends and other benefits the owner of the securities receives
(Page 107, equation 5.3)
F0 = S0 e(r–q )T
where q is the average yield during the life of the contract (expressed with continuous compounding)
5.13
Example
5.22
Index Arbitrage
(continued)
Index arbitrage involves simultaneous trades in futures and many different stocks Very often a computer is used to generate the trades Occasionally (e.g., on Black Monday) simultaneous trades are not possible and the theoretical no-arbitrage relationship between F0 and S0 does not hold
5.6
Notation for Valuing Futures and Forward Contracts
S0: Spot price today F0: Futures or forward price today
T: Time until delivery date(in years) r: Risk-free interest rate for maturity T
源自文库
F0 = S0 e(r–q )T
where q is the average dividend yield on the portfolio represented by the index during life of contract
5.20
Stock Index
(continued)
For the formula to be true it is important that the index represent an investment asset In other words, changes in the index must correspond to changes in the value of a tradable portfolio
5.17
The value of a long forward contract
No income income
Known yield
5.18
Forward vs Futures Prices
Forward and futures prices are usually assumed to be the same. When interest rates are uncertain they are, in theory, slightly different A strong positive correlation between interest rates and the asset price implies the futures price is slightly higher than the forward price A strong negative correlation implies the reverse
5.19
Stock Index (Page 110-112)
Can be viewed as an investment asset paying a dividend yield The futures price and spot price relationship is therefore
5.4
Example
The investor is required to maintain a margin account with the broker.
5.5
Assumptions
1.
2.
3.
4.
The market participants are subject to no transaction costs when they trade. The market participants are subjects to the same tax rate on all net trading profits. The market participants can borrow money at the same risk-free rate of interest as they can lend money. The market participants take advantage of arbitrage opportunities as they occur.
F0 = (S0 – I )erT
where I is the present value of the income during life of forward contract. In our example, So that,
5.12
When an Investment Asset Provides a Known Yield
When an Investment Asset Provides a Known Dollar Income
5.10
套利者净收益：910.00-886.60=23.40
5.11
When an Investment Asset Provides a Known Dollar Income
(page 105, equation 5.2)
The Nikkei index viewed as a dollar number does not represent an investment asset
(See Business Snapshot 5.3, page 111)
5.21
Index Arbitrage
When F0 > S0e(r-q)T an arbitrageur buys the stocks underlying the index and sells futures When F0 < S0e(r-q)T an arbitrageur buys futures and shorts or sells the stocks underlying the index
5.7
Forward price for an investment asset
5.8
When Interest Rates are Measured with Continuous Compounding
F0 = S0erT
if
远期价格大于即期价格
F0 > S0erT ,arbitrageurs can buy the asset and short forward contracts on the asset. if F0 < S0erT , they can short the asset and enter into long forward contracts on it. This equation relates the forward price and the spot price for any investment asset that provides no income and has no storage costs. 5.9
Determination of Forward and Futures Prices
Chapter 5
5.1
Consumption vs Investment Assets
Investment assets are assets held by significant numbers of people purely for investment purposes (Examples: stock, bond, gold, silver) Consumption assets are assets held primarily for consumption (Examples: copper, oil)
Consider a 6-month forward contract on an asset that is expected to provide income equal to 2% of the asset price once during a 6-month period. The risk free rate of interest is 10% per annum. The asset price is $25. The yield is 4% per annum, it follows that q=0.0396(=2ln(1+4%/2)).
5.15
5.16
Example
A long forward contract on a not-dividend paying stock was entered into some time ago. It currently has 6 months to maturity. The risk-free rate of interest is 10% per annum, the stock price is $25, and the delivery price is $24.
5.23
Futures and Forwards on Currencies (Page 112-115)
A foreign currency is analogous to a security providing a dividend yield The continuous dividend yield is the foreign risk-free interest rate It follows that if rf is the foreign risk-free interest rate ( r r f )T 0 0
5.14
Valuing a Forward Contract
Page 108
Suppose that K is delivery price in a forward contract and F0 is forward price that would apply to the contract today The value of a long forward contract, ƒ, is ƒ = (F0 – K )e–rT Similarly, the value of a short forward contract is (K – F0 )e–rT
5.2
Short Selling (Page 99-101)
Short selling involves selling securities you do not own Your broker borrows the securities from another client and sells them in the market in the usual way At some stage you must buy the securities back so they can be replaced in the account of the client