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Futures and Forwards
Financial Engineering and Computations
Dai, Tian-Shyr
Outline
Brief Introduction of Derivatives
Forwards
Relationships Between Forward Price and Spot Price
Futures
Relationships between Forward and Futures Prices
Cost of Carry
Hedging and Futures
Minimum Variance Hedge Ratio
Derivatives
Payoffs depend on other more fundamental assets:
Underlying assets
Commodity
Index
Interest rate
Other derivatives
Four types of derivatives
Forwards(远期契约)
Futures(期货)
Swaps(交换合约)
Options(选择权)
远期契约(Forwards)
定义:远期契约双方同意在未来某一天以某一价
格(履约价格)交易某一特定资产或商品.
Terms:
Maturity date:到期日
Strike price:履约价格
Underlying asset: 标的物
Why we Need Forwards
A Simple Story
考虑XYZ公司欲投资x元生产棉花,生产期为三个月
三个月后:
棉花价格上升:公司大赚一笔
棉花价格狂泄:公司亏本甚至倒闭
如何避免棉花的价格风险
A Simple Story on Forwards
假定有另一纺布公司ABC,三个月后需要棉花纺布
棉花不能今日进货:仓储问题
三个月后进货
棉花价格上升:公司有亏本甚至破产之虞
棉花价格狂泄:公司大赚一笔
如何规避棉花的价格风险
A Simple Story on Forwards
XYZ和ABC之间可签订合约
使用预定好的金额K於三个月后交易棉花
价格风险消失,但也无法得额外利润
XYZ (Forward的报酬为K-P)
P2
P1
市场价格远期合约
K-P2P2(亏损)价格下降至P2
K-P1P1(额外利润)价格上升至P1
签合约不签合约
总报酬锁定为K
A Simple Story on Forwards
上述为一远期合约
标的物:棉花
到期日:三个月后(假定棉花价格为P)
履约价格:K
买方:ABC (报酬为P-K)
卖方:XYZ (报酬为K-P)
交易方式:
实物交易
现金清算
从Buyer (ABC)来看Forward的报酬
1. 报酬构面(payoff profile)是直线的
2. 远期合约期初价值为零,期中无任何支付,
仅有期末会有支付产生.
V
K
V
Forward的缺点
当棉花价格上升,XYZ失去大赚一笔的机会
当棉花价格下降,ABC失去减少成本的机会
-Options
Default risk(违约风险)
-XYZ可能失火,导致无法履约
-当棉花价格上升,XYZ有违约的可能
-当棉花价格下降,ABC有违约的可能
采用期货合约每日清算的制度,减少违约风险
Forward的缺点
合约未标准化
-采用标准化的期货合约
-以台指期货为例
交易标的物:加权指数
规格:近三个月和两个季月到期的合约
Ex: Now: Oct.,
交易合约到期日: Oct., Nov., Dec., Mar., Jun.
市场撮合和流通
-期货合约可在交易所流通
(Introduced later)
Reasonable Strike Price
在理想的假定下( no tax, transaction cost, etc.)远期
的履约价格和标的物的现货价格有一定的关系
当该关系不满足时,存在套利机会
凯因斯(Keynes)使用Interest parity指出该状况
考虑远期汇率合约
汇率和本国利率,外国利率有连动的关系
Interest Rate Parity
Terms:
S: Spot exchange rate (Domestic/Foreign)
F: Forward exchange rate (Maturity: T year)
Rf: Foreign interest rate
Rl: Local interest rate
Then we have
Otherwise, there exists arbitrage opportunity.
()TlfRRF
e
S
=
Arbitrage Opportunity
Assume
Now:
Borrow 1 domestic dollar, change to 1/S foreign
dollar, save it at a interest rate of Rf
Sign a exchange rate forward with price F.
t year later:
You get foreign dollars.
Convert them at the rate F (Forward contract)
Since get free lunch
()TlfRRF
e
S
>
T1
fRe
S
TTfl
RRF
ee
S
>
In Class Exercise
Show that arbitrage opportunity exists as ()TlfRRF
e
S
还ST-K
(K=F0)
Short Forward
F0-S0erT得ST还时间T
0Buy the stocks借S0时间0
0
rTSe
00()0 !rT
TTSe S S F + >,期初不用支付任何成本,期末可得正的报酬
Cash flow
In Class Exercise
考虑 的套利机制00
rTFSe
0)(0000> = +++ rTrTrT
TTeISFeSIeSFS
Example
Consider a 10-month forward contract on a stock
with a price of $50. We assume that the risk-free rate
interest is 8% per annum for all maturity, and that
dividends of $0.75 per share are expected after three
months, six months and nine months. Please
calculate the forward price.
()
dividendstheofPVtheisIwhere
eF
eeeI
14.51$162.250
162.275.075.075.0
12/1008.0
12/908.012/608.012/308.0
= =
=++=
×
× × × ∵
Homework
Show that arbitrage opportunity exist if
Assume that the underlying stock pays a continuous
dividend yield at rate q. Show that the forward price
is
rTeISF)(00 <
TqreSF)(
00
=
期货合约和远期合约很类似,主要差别在於:
1.每日现金结算损益(marked-to-market)
2.买卖双方都必须提供保证金(margin)
-交易所藉由marked-to-market和margin account来降低
违约风险
期货合约(Futures)
保证金保证金
BA
双方交易的共同保证人
结算所
期货合约(Futures)
3. 合约标准化(期货均在集中交易所交易) .
4. 透过集中市场竞价来决定交易价格.
5.可以反向契约来抵销原有契约.
Black: Futures可视为一系列的远期合约.在每
日将前一天的Forward清算掉,并重新订约.
有关期货相关资讯可至台湾期交所http://www.taifex.com.tw/
台湾的期货市场
标的物: 股票指数
Check: http://tw.stock.yahoo.com/future/
走势图(台指期10)
5985
5991
6000
6020
6028
10:0009:00时间
价格
昨收盘价
10:07:51时,成交价6000
报价表(名词解释)
名称:台指期10,指交割月份为10月份的台指期货.
(期货报价通常会揭露近三个连续近月与二个接续季月共五个资讯)
时间:10:07:51,指系统揭露成交的时间.
涨跌:△9,指成交价减去昨天收盘价的价差(6000-5991=9).
总量:10806,指09:00~10:07期间市场成交的总数量.
基差:-33.72,指现货价格减去期货的价格,由此报价表可得知台股指
数约为5966点.(基差反应的是持有现货的成本,收益与市场对
未来走势的预期心理)
最高:6028,指09:00~10:07期间内最高价.
平仓:以等量但相反买卖方向冲销原有的部位.
未平仓量(open interest):买(卖)的期货契约在未平仓以前称为未平仓
量,代表等量的多头部位与空头部位.(报价表中常见的名词)
保证金清算
替代原则:结算公司介入每笔交易,成为买方的卖方,也
成为卖方的买方,承担双方之信用风险
保证金低至触及维持保证金时,投资人会收到保证金催
缴通知,必须将保证金补足至原始保证金水准,否则期
货商可迳自予以平仓
Initial margin
Maintenance margin
Example:期货保证金清算
假设某投资人於2006年11月23日买进一口台股指
数期货契约,成交价格是台股指数5600点,原始
保证金为105,000元,维持保证金为81,000元,此
契约的总价值为$1,120,000元(大台指契约乘数每
点200元) .在11月24,25,26日时,台股指数收
盘价为分别为5500,5450,5700点,期货保证金
清算过程如下:
Example:期货保证金清算
155,00050,000570011/26
75,000-10,000545011/25
85,000-20,000550011/24
105,000-560011/23
保证金余额当日损益台股指数收盘价日期
保证金余额低於维持保证金$81,000,该投资人会接到经纪
商通知(margin call),并将保证金补足至原始保证金,故26
日当天一开始的余额变为$105,000元.
Daily cash flows
假定第i天的future price为fi
The contract cash flow at day i should be fi-fi-1
Net cash flow:
It may differ because of the reinvestment and the
margin system.
10 21 32 1
00
( ) ( ) ( ) ... ( )TT
TT
ff ff ff ff
ffSf
+ + ++
= =
A futures contract has the similar
accumulated payoff to a forward contract.
∵ST = FT
Relationships between Forward and
Futures Prices
Forward price = futures prices, if the interest rate is
not stochastic.
Let Forward price is FFutures price is f
Consider forward and futures contracts on the same
underlying asset with n days to maturity.
The interest rate for day i is ri.
One dollar at the beginning of day i grows to
by day's end.
ir
iRe≡
Proof
Let Fibe the futures price at the end of day i.
So $1 invested in the n-day discount bond at the
end of day zero will be worth $R
Consider the follow strategy:
Long futures positions at the end of day i 1 and
invest the cash flow at the end of day i in riskless bonds
maturing on delivery day n.
RRRRReee
n
t
tn
rrrn===×∏
=1
21.........121
∏
=
i
t
tR
1
Proof
()∏
=
i
t
tiiRFF
1
1
() ()()111
11 1
in n
ii ttii tii
tti t
ff R R ff R ffR
==+ =
= = ∏∏ ∏∵
i-1ii+1
0
n
()1
1
i
ii t
t
ff R
=
∏
买入 单位期货
(假设无保证金)
拿$ 再投资无风险债券
在第i年的损益为
()1iiffR 到期时得$
i+1 n
∏
=
i
t
tR
1
The value at the end of day n is
Recall the value of R units of forwards is
Arbitrage opportunity exists if .
()()()100
1
n
ii n T
i
ffRffRSfR
=
= = ∑
i-1 i0n
Long a futures
Forwards 可视为n个futures所组成!
()RFST0
00fF≠
Relationships between Forward and
Future Prices
当利率随机波动时, future price和forward price的理
论价格不等
一般而言,两者差距不大
Unless stated otherwise, assume forward and futures
prices are identical.
Futures on Commodities
For a commodity hold for investment purposes and
with zero storage cost, the futures price is
In general, Ustands for the PV of storage cost
incurred during the life of a futures contract, and the
futures price on commodity is
rTeSF00=
rT
o
rT
eUSF
IUtstorageofvaulepresenttheisU
incomeofvaluepresenttheisIeISF
)(
,cos
)(
0
00
+=∴
=
=
∵
∵
rTeUSF)(00+=
Homework
You are provided the following information.
Current price of wheat = $19,000 for 5000 bushels
Riskless rate = 10 % (annualized)
Cost of storage = $200 a year for 5000 bushels
One-year futures contract price = $20,400 (for a contract
for 5000 bushels)
(a) What is F* (the theoretical price)
(b) How would you arbitrage the difference between F
and F* (Specify what you do now and at expiration
and what your arbitrage profits will be.)
Futures on Commodities
(convenience yield)
Because user of the commodity may feel that
ownership of the physical commodity provides
benefit that not obtained by holders of futures
contract.
The benefit are the convenience yield (y) provided by
the product.
Ex: Oil refiner would like to take crude oil
The convenience yield for investment asset=0 to
prevent arbitrage.
rTyTeUSeF)(00+=
Cost of Carry
The relationship between futures prices and spot
prices can be summarized in term of the cost of carry.
This measures the dollar cost of carrying the asset
over a period and consists of interest rate r, storage
cost at the rate of U, minus cash flow at the rate of q
generated by the asset.
-The cost of carry is C≡r+U-q
The futures price is CT
ooFSe=
Example
A manufacturer needs to acquire gold in 3 months.
The following options are open to her:(1) Buy the
gold now. (2) Long one 3-month gold futures contract
and take delivery in 3 months. If the T-bill are
yielding an annually compound rate of 6%. What is
the cost of carry for owning 100 ounces of gold at
$350 per ounce for a year
Example (Solution)
The cost of ownership is 6%, because it represents
the interest one would have earned if one had
bought a T-bill instead of the gold.
The cost of carry owning 100 ounces for 1 year is
C=100×$350×0.06=$2100.
The relationship cost of carry,
convenience yield and basis
basis=S0-F0
In case of consumption assets, the futures price is
greater than the spot price (basis<0) reflecting that
the cost of carry.
()yT CT C y T
oo ooFe Se F Se =→=
Convergence of Futures Price to Spot Price
TimeTime
(b)
Futures
Price
Futures
Price
Spot Price
Spot Price
(a)
在正常市场下,基差(basis)为负,反应的就是标的
资产的持有成本,随著到期日接近基差就会越来越
小.
思考:为何现货价格会大於期货价格(逆向市场)
Hedging with Futures
A long hedge is that involve taking a long position in
a futures contract, when the hedger will have to
purchase a certain asset in the future and wants to
lock in a price now.
A short hedge is that involve taking a short position
in a futures contract, when the hedger owns an asset
and expects to sell it at some time in the future.
(see example)
Example-Short Hedge
假设今天(1月25日)铜的现货价格为每磅120美分,某一国
外铜开采公司预计其将在5月15日卖出100,000磅铜,且在
COMEX交易的五月份到期(到期日假设为5月15日)之铜期货
价格为每磅120美分(契约乘数),其中每一口铜期货合约
大小为25,000磅.(假设不用缴保证金)
如果5月15日铜的现货价格为每磅105美分,该国外铜开采
公司如何利用COMEX(纽约商品期货交易所)铜期货合约来
规避未来铜的价格风险 其净收益为何
该国外铜开采公司担心未来铜价格下跌,导致生产收入
减少,故它可以卖出四口COMEX铜期货合约来避险.
Spot Market Futures Market
_______________________________________________
1/25: 现货100000磅 1/25: 卖出四口5月份到期之铜期货
5/25:卖出现货100000磅 5/25: 结算四口铜期货
_______________________________________________
Opportunity Loss Gain = $150,000
= $1,200,000-1,050,000
=$150,000
价值为
$120×10000=1,200,000
保证金为$0
现货价值减少为
$105×10000=1,050,000
结算的利得为$(120-105)
×4×25000=$150,000
故净收益为$0
Spot price
Spot position value
Short futures position
gain
loss
value
0
空头避险损益图
现货价格虽然在5/15下
跌,然而铜公司透过放空
期货可规避此价格的损
失.
120
In class Exercise
如果到期时铜的现货价格为每磅125美分 其净收
益为何
思考:假如市场上没有想要避险现货的标的资产之
期货 或是现货的交易日与期货的到期日不相同 该
怎麼办
Homework
It is March 1. A U.S. company expects to receive 50
million Japanese yen at the end of July. The
September futures price for the yen is currently 0.78.
We suppose the spot and futures prices when the
contract is closed out are 0.72 and 0.725 at the end of
July, respectively. One contract is for the delivery of
12.5 million yen. How would the U.S. company
manage the exchange risk Calculate the basis risk
when closing out the futures on currency.
(Hint: Short hedge Or long hedge )
完全避险的条件
从事避险之期货标的物要和现货商品一模一样.
到期日要等於避险冲销日.
对应於要避险之现货部位,所买卖的期货口数为
某一整数.
期货合约避险之合约选择
交叉避险(Cross Hedge):实务上要找到与现货完全相同的
标的资产之期货合约非常困难,故乃以与现货相关性高标
的资产替代,此种策略称为交叉避险.
避险者使用期货合约避险,其效果好坏决定於期货价格和
现货价格相关性高低.因此避险者选择期货合约需考虑的
因素有二:
(1)期货合约标的资产的选择
(2)期货合约交割月份的选择
最适期货避险数量
决定最适的期货合约数量有二种方法:
(1)单纯避险法(Naive Hedge Method)
(2)最小变异数避险比率法(Minimum Variance Hedge
Ratio Method)
单纯避险法
又称完全避险法(Perfect Hedge Method),指避险者买进或
卖出和欲避险之现货部位金额相同,但部位相反的期货
合约.
假设基差风险不存在,亦即现货价格和期货价格之变化
是完全一致.
公式:期货合约口数=欲避险之现货部位金额÷每口期货
合约价值.
实际上现货价格的变化和期货价格的变化未必会完全一
致,因此单纯避险法并不一定能将现货价格风险完全规
避.
Example
假设某一基金经理人持有价值新台币20亿的股票,而台股指
数期货目前价格为5000点,其每一点值新台币200元,那麼
该基金经理人为了防止其现股价格下跌的风险,他应该出售
多少口台股指数期货来避险
每口台股指数期货价值= 5000 ×200 = 1,000,000
应出售的期货合约口数= 2,000,000,000 ÷1,000,000= 2,000 (口)
最小变异数避险比率法
定义:找出使避险投资组合风险(变异数)最小的避险比率
的方法.又称为回归分析法.
假设:
△S:避险期间内现货价格的变动
△F:避险期间内期货价格的变动
σs:现货价格变动的标准差
σF:期货价格变动的标准差
ρ:△S和△F间的相关系数
公式:
h*(最小变异数避险比率) =
F
s
σ
σ
ρ
最小变异数避险比率
如果ρ=1且σF= σs,则h*=1.(此时最小变异数避险比率=单
纯避险法之避险比率,因为期货合约价格的变动和现货价格
的变动完全一致.)
假设其他情况不变,若σs愈大或ρ愈大,则h愈大
假设其他情况不变,若σF愈大,则h愈小.
最小变异数避险比率的推导
Suppose we expect to sell NAunits of an assets at
time t2and to hedge at time t1by shorting futures
contracts on NFunits of a similar asset.
-The hedge ratio is
S1and S2are the asset prices at time t1and t2, and
F1and F2are the futures prices at time t1and t2.
A
F
N
N
h=
最小变异数避险比率的推导
Total amount realized for the asset denoted by Y
When the variance of △S-h△Fis minimized, the
variance of Y is minimized.
FANFFNSY)(122 =
)()()(112121FhSNNSNFFNSSNSYAAFAA += +=
F
S
SF
SSSFFSFS
h
h
FhSMinVar
hhhFhSVar
σ
σ
ρ
ρσσ
σρσρσσσρσσσ
=
=
+ = +=
0)(
)(
)(2)(
2
2222222
∵
∵
避险者的部位之变异数和避险比率之关系
避险比率
部位之变异数
h*
Example
假设台股指数价格变动百分比和大台指期货价格变动百
分比的相关系数为0.8,而台股指数价格变动百分比之标
准差为0.5,而期货合约价格变动百分比之标准差为0.4,
某一基金经理人持有价值新台币20亿的股票,而台股指
数期货目前价格为5000点,其每一点值新台币200元,那
麼该基金经理人为了防止其现股价格下跌的风险,他应
该出售多少口台股指数期货来避险
Ans: 最小避险变异数比率为0.8×(0.5÷0.4)=1.2
应出售的口数为2000×1.2=2400(口)
Financial Engineering and Computations
Dai, Tian-Shyr
Outline
Brief Introduction of Derivatives
Forwards
Relationships Between Forward Price and Spot Price
Futures
Relationships between Forward and Futures Prices
Cost of Carry
Hedging and Futures
Minimum Variance Hedge Ratio
Derivatives
Payoffs depend on other more fundamental assets:
Underlying assets
Commodity
Index
Interest rate
Other derivatives
Four types of derivatives
Forwards(远期契约)
Futures(期货)
Swaps(交换合约)
Options(选择权)
远期契约(Forwards)
定义:远期契约双方同意在未来某一天以某一价
格(履约价格)交易某一特定资产或商品.
Terms:
Maturity date:到期日
Strike price:履约价格
Underlying asset: 标的物
Why we Need Forwards
A Simple Story
考虑XYZ公司欲投资x元生产棉花,生产期为三个月
三个月后:
棉花价格上升:公司大赚一笔
棉花价格狂泄:公司亏本甚至倒闭
如何避免棉花的价格风险
A Simple Story on Forwards
假定有另一纺布公司ABC,三个月后需要棉花纺布
棉花不能今日进货:仓储问题
三个月后进货
棉花价格上升:公司有亏本甚至破产之虞
棉花价格狂泄:公司大赚一笔
如何规避棉花的价格风险
A Simple Story on Forwards
XYZ和ABC之间可签订合约
使用预定好的金额K於三个月后交易棉花
价格风险消失,但也无法得额外利润
XYZ (Forward的报酬为K-P)
P2
P1
市场价格远期合约
K-P2P2(亏损)价格下降至P2
K-P1P1(额外利润)价格上升至P1
签合约不签合约
总报酬锁定为K
A Simple Story on Forwards
上述为一远期合约
标的物:棉花
到期日:三个月后(假定棉花价格为P)
履约价格:K
买方:ABC (报酬为P-K)
卖方:XYZ (报酬为K-P)
交易方式:
实物交易
现金清算
从Buyer (ABC)来看Forward的报酬
1. 报酬构面(payoff profile)是直线的
2. 远期合约期初价值为零,期中无任何支付,
仅有期末会有支付产生.
V
K
V
Forward的缺点
当棉花价格上升,XYZ失去大赚一笔的机会
当棉花价格下降,ABC失去减少成本的机会
-Options
Default risk(违约风险)
-XYZ可能失火,导致无法履约
-当棉花价格上升,XYZ有违约的可能
-当棉花价格下降,ABC有违约的可能
采用期货合约每日清算的制度,减少违约风险
Forward的缺点
合约未标准化
-采用标准化的期货合约
-以台指期货为例
交易标的物:加权指数
规格:近三个月和两个季月到期的合约
Ex: Now: Oct.,
交易合约到期日: Oct., Nov., Dec., Mar., Jun.
市场撮合和流通
-期货合约可在交易所流通
(Introduced later)
Reasonable Strike Price
在理想的假定下( no tax, transaction cost, etc.)远期
的履约价格和标的物的现货价格有一定的关系
当该关系不满足时,存在套利机会
凯因斯(Keynes)使用Interest parity指出该状况
考虑远期汇率合约
汇率和本国利率,外国利率有连动的关系
Interest Rate Parity
Terms:
S: Spot exchange rate (Domestic/Foreign)
F: Forward exchange rate (Maturity: T year)
Rf: Foreign interest rate
Rl: Local interest rate
Then we have
Otherwise, there exists arbitrage opportunity.
()TlfRRF
e
S
=
Arbitrage Opportunity
Assume
Now:
Borrow 1 domestic dollar, change to 1/S foreign
dollar, save it at a interest rate of Rf
Sign a exchange rate forward with price F.
t year later:
You get foreign dollars.
Convert them at the rate F (Forward contract)
Since get free lunch
()TlfRRF
e
S
>
T1
fRe
S
TTfl
RRF
ee
S
>
In Class Exercise
Show that arbitrage opportunity exists as ()TlfRRF
e
S
还ST-K
(K=F0)
Short Forward
F0-S0erT得ST还时间T
0Buy the stocks借S0时间0
0
rTSe
00()0 !rT
TTSe S S F + >,期初不用支付任何成本,期末可得正的报酬
Cash flow
In Class Exercise
考虑 的套利机制00
rTFSe
0)(0000> = +++ rTrTrT
TTeISFeSIeSFS
Example
Consider a 10-month forward contract on a stock
with a price of $50. We assume that the risk-free rate
interest is 8% per annum for all maturity, and that
dividends of $0.75 per share are expected after three
months, six months and nine months. Please
calculate the forward price.
()
dividendstheofPVtheisIwhere
eF
eeeI
14.51$162.250
162.275.075.075.0
12/1008.0
12/908.012/608.012/308.0
= =
=++=
×
× × × ∵
Homework
Show that arbitrage opportunity exist if
Assume that the underlying stock pays a continuous
dividend yield at rate q. Show that the forward price
is
rTeISF)(00 <
TqreSF)(
00
=
期货合约和远期合约很类似,主要差别在於:
1.每日现金结算损益(marked-to-market)
2.买卖双方都必须提供保证金(margin)
-交易所藉由marked-to-market和margin account来降低
违约风险
期货合约(Futures)
保证金保证金
BA
双方交易的共同保证人
结算所
期货合约(Futures)
3. 合约标准化(期货均在集中交易所交易) .
4. 透过集中市场竞价来决定交易价格.
5.可以反向契约来抵销原有契约.
Black: Futures可视为一系列的远期合约.在每
日将前一天的Forward清算掉,并重新订约.
有关期货相关资讯可至台湾期交所http://www.taifex.com.tw/
台湾的期货市场
标的物: 股票指数
Check: http://tw.stock.yahoo.com/future/
走势图(台指期10)
5985
5991
6000
6020
6028
10:0009:00时间
价格
昨收盘价
10:07:51时,成交价6000
报价表(名词解释)
名称:台指期10,指交割月份为10月份的台指期货.
(期货报价通常会揭露近三个连续近月与二个接续季月共五个资讯)
时间:10:07:51,指系统揭露成交的时间.
涨跌:△9,指成交价减去昨天收盘价的价差(6000-5991=9).
总量:10806,指09:00~10:07期间市场成交的总数量.
基差:-33.72,指现货价格减去期货的价格,由此报价表可得知台股指
数约为5966点.(基差反应的是持有现货的成本,收益与市场对
未来走势的预期心理)
最高:6028,指09:00~10:07期间内最高价.
平仓:以等量但相反买卖方向冲销原有的部位.
未平仓量(open interest):买(卖)的期货契约在未平仓以前称为未平仓
量,代表等量的多头部位与空头部位.(报价表中常见的名词)
保证金清算
替代原则:结算公司介入每笔交易,成为买方的卖方,也
成为卖方的买方,承担双方之信用风险
保证金低至触及维持保证金时,投资人会收到保证金催
缴通知,必须将保证金补足至原始保证金水准,否则期
货商可迳自予以平仓
Initial margin
Maintenance margin
Example:期货保证金清算
假设某投资人於2006年11月23日买进一口台股指
数期货契约,成交价格是台股指数5600点,原始
保证金为105,000元,维持保证金为81,000元,此
契约的总价值为$1,120,000元(大台指契约乘数每
点200元) .在11月24,25,26日时,台股指数收
盘价为分别为5500,5450,5700点,期货保证金
清算过程如下:
Example:期货保证金清算
155,00050,000570011/26
75,000-10,000545011/25
85,000-20,000550011/24
105,000-560011/23
保证金余额当日损益台股指数收盘价日期
保证金余额低於维持保证金$81,000,该投资人会接到经纪
商通知(margin call),并将保证金补足至原始保证金,故26
日当天一开始的余额变为$105,000元.
Daily cash flows
假定第i天的future price为fi
The contract cash flow at day i should be fi-fi-1
Net cash flow:
It may differ because of the reinvestment and the
margin system.
10 21 32 1
00
( ) ( ) ( ) ... ( )TT
TT
ff ff ff ff
ffSf
+ + ++
= =
A futures contract has the similar
accumulated payoff to a forward contract.
∵ST = FT
Relationships between Forward and
Futures Prices
Forward price = futures prices, if the interest rate is
not stochastic.
Let Forward price is FFutures price is f
Consider forward and futures contracts on the same
underlying asset with n days to maturity.
The interest rate for day i is ri.
One dollar at the beginning of day i grows to
by day's end.
ir
iRe≡
Proof
Let Fibe the futures price at the end of day i.
So $1 invested in the n-day discount bond at the
end of day zero will be worth $R
Consider the follow strategy:
Long futures positions at the end of day i 1 and
invest the cash flow at the end of day i in riskless bonds
maturing on delivery day n.
RRRRReee
n
t
tn
rrrn===×∏
=1
21.........121
∏
=
i
t
tR
1
Proof
()∏
=
i
t
tiiRFF
1
1
() ()()111
11 1
in n
ii ttii tii
tti t
ff R R ff R ffR
==+ =
= = ∏∏ ∏∵
i-1ii+1
0
n
()1
1
i
ii t
t
ff R
=
∏
买入 单位期货
(假设无保证金)
拿$ 再投资无风险债券
在第i年的损益为
()1iiffR 到期时得$
i+1 n
∏
=
i
t
tR
1
The value at the end of day n is
Recall the value of R units of forwards is
Arbitrage opportunity exists if .
()()()100
1
n
ii n T
i
ffRffRSfR
=
= = ∑
i-1 i0n
Long a futures
Forwards 可视为n个futures所组成!
()RFST0
00fF≠
Relationships between Forward and
Future Prices
当利率随机波动时, future price和forward price的理
论价格不等
一般而言,两者差距不大
Unless stated otherwise, assume forward and futures
prices are identical.
Futures on Commodities
For a commodity hold for investment purposes and
with zero storage cost, the futures price is
In general, Ustands for the PV of storage cost
incurred during the life of a futures contract, and the
futures price on commodity is
rTeSF00=
rT
o
rT
eUSF
IUtstorageofvaulepresenttheisU
incomeofvaluepresenttheisIeISF
)(
,cos
)(
0
00
+=∴
=
=
∵
∵
rTeUSF)(00+=
Homework
You are provided the following information.
Current price of wheat = $19,000 for 5000 bushels
Riskless rate = 10 % (annualized)
Cost of storage = $200 a year for 5000 bushels
One-year futures contract price = $20,400 (for a contract
for 5000 bushels)
(a) What is F* (the theoretical price)
(b) How would you arbitrage the difference between F
and F* (Specify what you do now and at expiration
and what your arbitrage profits will be.)
Futures on Commodities
(convenience yield)
Because user of the commodity may feel that
ownership of the physical commodity provides
benefit that not obtained by holders of futures
contract.
The benefit are the convenience yield (y) provided by
the product.
Ex: Oil refiner would like to take crude oil
The convenience yield for investment asset=0 to
prevent arbitrage.
rTyTeUSeF)(00+=
Cost of Carry
The relationship between futures prices and spot
prices can be summarized in term of the cost of carry.
This measures the dollar cost of carrying the asset
over a period and consists of interest rate r, storage
cost at the rate of U, minus cash flow at the rate of q
generated by the asset.
-The cost of carry is C≡r+U-q
The futures price is CT
ooFSe=
Example
A manufacturer needs to acquire gold in 3 months.
The following options are open to her:(1) Buy the
gold now. (2) Long one 3-month gold futures contract
and take delivery in 3 months. If the T-bill are
yielding an annually compound rate of 6%. What is
the cost of carry for owning 100 ounces of gold at
$350 per ounce for a year
Example (Solution)
The cost of ownership is 6%, because it represents
the interest one would have earned if one had
bought a T-bill instead of the gold.
The cost of carry owning 100 ounces for 1 year is
C=100×$350×0.06=$2100.
The relationship cost of carry,
convenience yield and basis
basis=S0-F0
In case of consumption assets, the futures price is
greater than the spot price (basis<0) reflecting that
the cost of carry.
()yT CT C y T
oo ooFe Se F Se =→=
Convergence of Futures Price to Spot Price
TimeTime
(b)
Futures
Price
Futures
Price
Spot Price
Spot Price
(a)
在正常市场下,基差(basis)为负,反应的就是标的
资产的持有成本,随著到期日接近基差就会越来越
小.
思考:为何现货价格会大於期货价格(逆向市场)
Hedging with Futures
A long hedge is that involve taking a long position in
a futures contract, when the hedger will have to
purchase a certain asset in the future and wants to
lock in a price now.
A short hedge is that involve taking a short position
in a futures contract, when the hedger owns an asset
and expects to sell it at some time in the future.
(see example)
Example-Short Hedge
假设今天(1月25日)铜的现货价格为每磅120美分,某一国
外铜开采公司预计其将在5月15日卖出100,000磅铜,且在
COMEX交易的五月份到期(到期日假设为5月15日)之铜期货
价格为每磅120美分(契约乘数),其中每一口铜期货合约
大小为25,000磅.(假设不用缴保证金)
如果5月15日铜的现货价格为每磅105美分,该国外铜开采
公司如何利用COMEX(纽约商品期货交易所)铜期货合约来
规避未来铜的价格风险 其净收益为何
该国外铜开采公司担心未来铜价格下跌,导致生产收入
减少,故它可以卖出四口COMEX铜期货合约来避险.
Spot Market Futures Market
_______________________________________________
1/25: 现货100000磅 1/25: 卖出四口5月份到期之铜期货
5/25:卖出现货100000磅 5/25: 结算四口铜期货
_______________________________________________
Opportunity Loss Gain = $150,000
= $1,200,000-1,050,000
=$150,000
价值为
$120×10000=1,200,000
保证金为$0
现货价值减少为
$105×10000=1,050,000
结算的利得为$(120-105)
×4×25000=$150,000
故净收益为$0
Spot price
Spot position value
Short futures position
gain
loss
value
0
空头避险损益图
现货价格虽然在5/15下
跌,然而铜公司透过放空
期货可规避此价格的损
失.
120
In class Exercise
如果到期时铜的现货价格为每磅125美分 其净收
益为何
思考:假如市场上没有想要避险现货的标的资产之
期货 或是现货的交易日与期货的到期日不相同 该
怎麼办
Homework
It is March 1. A U.S. company expects to receive 50
million Japanese yen at the end of July. The
September futures price for the yen is currently 0.78.
We suppose the spot and futures prices when the
contract is closed out are 0.72 and 0.725 at the end of
July, respectively. One contract is for the delivery of
12.5 million yen. How would the U.S. company
manage the exchange risk Calculate the basis risk
when closing out the futures on currency.
(Hint: Short hedge Or long hedge )
完全避险的条件
从事避险之期货标的物要和现货商品一模一样.
到期日要等於避险冲销日.
对应於要避险之现货部位,所买卖的期货口数为
某一整数.
期货合约避险之合约选择
交叉避险(Cross Hedge):实务上要找到与现货完全相同的
标的资产之期货合约非常困难,故乃以与现货相关性高标
的资产替代,此种策略称为交叉避险.
避险者使用期货合约避险,其效果好坏决定於期货价格和
现货价格相关性高低.因此避险者选择期货合约需考虑的
因素有二:
(1)期货合约标的资产的选择
(2)期货合约交割月份的选择
最适期货避险数量
决定最适的期货合约数量有二种方法:
(1)单纯避险法(Naive Hedge Method)
(2)最小变异数避险比率法(Minimum Variance Hedge
Ratio Method)
单纯避险法
又称完全避险法(Perfect Hedge Method),指避险者买进或
卖出和欲避险之现货部位金额相同,但部位相反的期货
合约.
假设基差风险不存在,亦即现货价格和期货价格之变化
是完全一致.
公式:期货合约口数=欲避险之现货部位金额÷每口期货
合约价值.
实际上现货价格的变化和期货价格的变化未必会完全一
致,因此单纯避险法并不一定能将现货价格风险完全规
避.
Example
假设某一基金经理人持有价值新台币20亿的股票,而台股指
数期货目前价格为5000点,其每一点值新台币200元,那麼
该基金经理人为了防止其现股价格下跌的风险,他应该出售
多少口台股指数期货来避险
每口台股指数期货价值= 5000 ×200 = 1,000,000
应出售的期货合约口数= 2,000,000,000 ÷1,000,000= 2,000 (口)
最小变异数避险比率法
定义:找出使避险投资组合风险(变异数)最小的避险比率
的方法.又称为回归分析法.
假设:
△S:避险期间内现货价格的变动
△F:避险期间内期货价格的变动
σs:现货价格变动的标准差
σF:期货价格变动的标准差
ρ:△S和△F间的相关系数
公式:
h*(最小变异数避险比率) =
F
s
σ
σ
ρ
最小变异数避险比率
如果ρ=1且σF= σs,则h*=1.(此时最小变异数避险比率=单
纯避险法之避险比率,因为期货合约价格的变动和现货价格
的变动完全一致.)
假设其他情况不变,若σs愈大或ρ愈大,则h愈大
假设其他情况不变,若σF愈大,则h愈小.
最小变异数避险比率的推导
Suppose we expect to sell NAunits of an assets at
time t2and to hedge at time t1by shorting futures
contracts on NFunits of a similar asset.
-The hedge ratio is
S1and S2are the asset prices at time t1and t2, and
F1and F2are the futures prices at time t1and t2.
A
F
N
N
h=
最小变异数避险比率的推导
Total amount realized for the asset denoted by Y
When the variance of △S-h△Fis minimized, the
variance of Y is minimized.
FANFFNSY)(122 =
)()()(112121FhSNNSNFFNSSNSYAAFAA += +=
F
S
SF
SSSFFSFS
h
h
FhSMinVar
hhhFhSVar
σ
σ
ρ
ρσσ
σρσρσσσρσσσ
=
=
+ = +=
0)(
)(
)(2)(
2
2222222
∵
∵
避险者的部位之变异数和避险比率之关系
避险比率
部位之变异数
h*
Example
假设台股指数价格变动百分比和大台指期货价格变动百
分比的相关系数为0.8,而台股指数价格变动百分比之标
准差为0.5,而期货合约价格变动百分比之标准差为0.4,
某一基金经理人持有价值新台币20亿的股票,而台股指
数期货目前价格为5000点,其每一点值新台币200元,那
麼该基金经理人为了防止其现股价格下跌的风险,他应
该出售多少口台股指数期货来避险
Ans: 最小避险变异数比率为0.8×(0.5÷0.4)=1.2
应出售的口数为2000×1.2=2400(口)
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