n 對於要以約定價格買進的那人，她既是權利，也是義務 A derivative is an instrument whose value depends on the values of other more basic underlying variables 衍生性金融商品可以歸類為：   1. 遠期、期貨、交換  n   對於要以約定價格買進的那人，她既是權利，也是義務 n   對於要以約定價格賣出的那人，她既是權利，也是義務 除了7-11，大部份的生意，多少帶有些遠期交易特性      

對於收受權利金，任『對方』行使以約定價格買進或是賣出『權』的那人，她擔負義務，並不享有權利   2. 選擇權、認股權證 n     對於付出權利金，『握有』約定價格買進或是賣出『權』的那人，她只享有權利，並不需要擔負義務 對於收受權利金，任『對方』行使以約定價格買進或是賣出『權』的那人，她擔負義務，並不享有權利 買房屋先付訂金，就是一種買「選擇權」的例子，您為了這選擇權利付出的代價，就是所付訂金；對方收受訂金的同時，等於是賣「選擇權」。   參加保險，就是一種買「選擇權」的例子；保險公司很類似賣「選擇權」。     n    

Derivatives Markets Exchange traded Over-the-counter (OTC) Traditionally exchanges have used the open-outcry system, but increasingly they are switching to electronic trading Contracts are standard there is virtually no credit risk (交易所居於其間) Over-the-counter (OTC) A computer- & telephone-linked network of dealers at financial institutions, corporations, & fund managers Contracts can be non-standard & there is some small amount of credit risk

Ways Derivatives are Used To hedge risks To speculate (take a view on the future direction of the market) To lock in an arbitrage profit To change the nature of a liability To change the nature of an investment without incurring the costs of selling one portfolio & buying another 4

Forward (遠期) Contracts A forward contract is an agreement to buy or sell an asset at a certain time in the future for a certain price (the delivery price) It can be contrasted with a spot contract which is an agreement to buy or sell immediately It is traded in the OTC market 10

Foreign Exchange Quotes on Aug 16, 2001 Bid Offer Spot 1.4452 1.4456 1-month forward 1.4435 1.4440 3-month forward 1.4402 1.4407 6-month forward 1.4353 1.4359 12-month forward 1.4262 1.4268

Forward Price The forward price for a contract is the delivery price that would be applicable to the contract if were negotiated today (i.e., it is the delivery price) The forward price may be different for contracts of different maturities 11

Terminology The party that has agreed to buy has what is termed a long position The party that has agreed to sell has what is termed a short position 10

Example On Aug. 16, 2001 the treasurer of a corporation enters into a long forward contract to buy £1 mil. in 6 months at an exchange rate of 1.4359 This obligates the corporation to pay $1,435,900 for £1 mil. on Feb. 16, 2002 What are the possible outcomes?

Profit from a Long Forward Position Price of Underlying at Maturity, ST K 14

Profit from a Short Forward Position Price of Underlying at Maturity, ST K 15

Futures (期貨) Contracts Agreement to buy or sell an asset for a certain price at a certain time Similar to forward contract Whereas a forward contract is traded OTC, a futures contract is traded on an exchange

Examples of Futures Contracts Agreement to: 1. buy 100 oz. of gold @ US$300/oz. in December (COMEX) 2. sell £62,500 @ 1.5000 US$/£ in March (CME) 3. sell 1,000 bbl. of oil @ US$20/bbl. in April (NYMEX) 9

1. Gold: An Arbitrage Opportunity? Suppose that: The spot price of gold is US$300 The 1-year forward price of gold is US$340 The 1-year US$ interest rate is 5% per annum Is there an arbitrage opportunity? (We ignore storage costs & gold lease rate)? 18

2. Gold: Another Arbitrage Opportunity? Suppose that: The spot price of gold is US$300 The 1-year forward price of gold is US$300 The 1-year US$ interest rate is 5% per annum Is there an arbitrage opportunity? 19

The Forward Price of Gold If the spot price of gold is S & the forward price for a contract deliverable in T years is F, then F = S (1+r )T where r is the 1-year (domestic currency) risk-free rate of interest. In our examples, S = 300, T = 1, & r =0.05 so that F = 300(1+0.05) = 315 20

1. Oil: An Arbitrage Opportunity? Suppose that: The spot price of oil is US$19 The quoted 1-year futures price of oil is US$25 The 1-year US$ interest rate is 5% per annum The storage costs of oil are 2% per annum Is there an arbitrage opportunity? 21

2. Oil: Another Arbitrage Opportunity? Suppose that: The spot price of oil is US$19 The quoted 1-year futures price of oil is US$16 The 1-year US$ interest rate is 5% per annum The storage costs of oil are 2% per annum Is there an arbitrage opportunity? 22

Exchanges Trading Options Chicago Board Options Exchange American Stock Exchange Philadelphia Stock Exchange Pacific Stock Exchange European Options Exchange Australian Options Market and many more (see list at end of book)

Options A put is an option to sell a certain asset by a certain date for a certain price (the strike price) A call option is an option to buy a certain asset by a certain date for a certain price (the strike price) 12

Long Call on Microsoft Profit from buying a European call option on Microsoft: option price = $5, strike price = $60 30 20 10 -5 40 50 60 70 80 90 Profit ($) Terminal stock price ($)

Short Call on Microsoft Profit from writing a European call option on Microsoft: option price = $5, strike price = $60 -30 -20 -10 5 30 40 50 60 70 80 90 Profit ($) Terminal stock price ($)

Long Put on IBM Profit from buying a European put option on IBM: option price = $7, strike price = $90 30 20 10 -7 90 80 70 60 100 110 120 Profit ($) Terminal stock price ($)

Short Put on IBM Profit from writing a European put option on IBM: option price = $7, strike price = $90 -30 -20 -10 7 90 80 70 60 100 110 120 Profit ($) Terminal stock price ($)

Payoffs from Options What is the Option Position in Each Case? K = Strike price, ST = Price of asset at maturity Payoff ST K

Types of Traders Hedgers Speculators Arbitrageurs Some of the large trading losses in derivatives occurred because individuals who had a mandate to hedge risks switched to being speculators 15

一. Hedging Examples A US company will pay £10 million for imports from Britain in 3 months & decides to hedge using a long position in a forward contract An investor owns 1,000 Microsoft shares currently worth $73 per share. A two-month put with a strike price of $65 costs $2.50. The investor decides to hedge by buying 10 contracts 16

二. Speculation Example An investor with $4,000 to invest feels that Cisco’s stock price will increase over the next 2 months. The current stock price is $20 & the price of a 2-month call option with a strike of 25 is $1 What are the alternative strategies? 17

三. Arbitrage Example A stock price is quoted as £100 in London & $172 in New York The current exchange rate is 1.7500 What is the arbitrage opportunity?

Futures Contracts Available on a wide range of underlying assets Exchange traded Specifications need to be defined: What can be delivered, Where it can be delivered, When it can be delivered Settled daily

Margins (保證金) A margin is cash or marketable securities deposited by an investor with broker The balance in the margin account is adjusted to reflect daily settlement Margins minimize the possibility of a loss through a default on a contract

期合約的交易制度   n   期貨合約的買、賣雙方均有交割之義務 n   多為沖抵(Offset)的方式結平，現貨交割(Delivery)（真的送豬肚到你家）只佔少數 n   某些金融商品無法實體交割 n   設有最小跳動金額(依期貨的種類而不同) n   部份有漲跌停板的限制 n   公開競價的方式交易 買賣雙方均要設置保證金帳戶

n  保證金成數(依期貨的種類而不同) ~ 如保證金是契約價值的百分之十，等於拿一塊錢便可以交易價值十塊錢的商品。 如果行情不利於交易人，致使保證金水位低於維持保證金，則交易人須補足保證金至原始保證金水準。

n Marked to Market [損益每日結平] ~   n   Marked to Market [損益每日結平] ~ 為控制風險，遂有每天計算保證金是否足夠規定（此為期貨交易之另一個特性，又稱為「每日結算」）。 如果發現保證金不足，交易人須於期貨商規定的時間內補足保證金 否則期貨商有權結清交易人的部位。

【參考】

國內期貨市場保證金類型，分為「結算保證金」、「原始保證金」、與「維持保證金」 「結算保證金」指期交所向結算會員收的保證金； 「原始保證金」為期貨經紀商向投資人收取的起始保證金；而當投資者保證金帳戶餘額低於「維持保證金」水準時，期貨經紀商會向投資人發出「補繳通知」（margin calls）。台灣期貨交易所會根據其風險控管機制，不定期公佈三種保證金的最低界線，

國內期貨市場保證金類型，分為「結算保證金」、「原始保證金」、與「維持保證金」 期交所不准許期貨經紀商低收「原始保證金」與低設「維持保證金」水準。根據期交所三項保證金數據分析，「維持保證金」為「結算保證金」的1.15倍，「原始保證金」則為「結算保證金」的1.5倍 「結算保證金」是計算其它兩種保證金的基礎。

客戶保證金 「臺灣期貨交易所股份有限公司結算保證金收取方式及標準」中第四條：「股價指數類期貨契約結算保證金金額為各契約之期貨指數乘以指數每點價值乘以風險價格係數。前項所稱風險價格係數，係參考一段期間內指數變動幅度，估算至少可涵蓋一日指數變動幅度百分之九十九點七信賴區間之值。」 台灣期交所的「結算保證金」制定標準，是「涉險值」（value-at-risk、VaR）觀念的應用

市場風險敏感性係數： 當市場投資組合的指數變動一單位時，投資組合價值變動百分比率，股票投資常常以貝他值 (Beta Coefficient)顯示其系統性風險，進而決定投資人要求的報酬率。如果投資人真的是充分分散投資標的，只有貝他值會影響他對於投資組合要求的報酬率。 E(rx) = rf + b[E(rm) - rf] 例如： E(rm) - rf = 0.08; rf = 0.03; - 全憑一個b係數在決定個股期望報酬 如 bx = 1.25 => E(rx) = .03 + 1.25(.08) = 13%

n Marked to Market [損益每日結平] ~   n   Marked to Market [損益每日結平] ~ 為控制風險，遂有每天計算保證金是否足夠規定（此為期貨交易之另一個特性，又稱為「每日結算」）。 如果發現保證金不足，交易人須於期貨商規定的時間內補足保證金 否則期貨商有權結清交易人的部位。

如果結算系統靈敏, 可以計算 價差保證金 避險保證金 整戶保證金 (每客戶或是每交易帳戶, 以淨額計算其保證金) 法人除期貨自營商, 只准許作避險, 目前主要是事後認定 期貨自營商可以作避險或是逐利交易

Example of a Futures Trade An investor takes a long position in 2 December gold futures contracts on June 5 contract size is 100 oz. futures price is US$400 margin requirement is US$2,000/contract (US$4,000 in total) maintenance margin is US$1,500/contract (US$3,000 in total)

Other Key Points About Futures They are settled daily Closing out a futures position involves entering into an offsetting trade Most contracts are closed out before maturity

Delivery [交割] If a contract is not closed out before maturity, it usually settled by delivering the assets underlying the contract. When there are alternatives about what is delivered, where it is delivered, & when it is delivered, the party with the short position chooses. A few contracts (for example, those on stock indices & Eurodollars) are settled in cash

Some Terminology Open interest (未平倉合約): the total number of contracts outstanding equal to number of long positions or number of short positions Settlement price (最後成交價格): the price just before the final bell each day used for the daily settlement process Volume of trading: the number of trades in 1 day

未平倉合約是期貨交易特有資料，它表達商品現有在倉數量，也是商品走勢動能資料。在買賣期指合約及期權合約時，買賣方合約都會被計入未平倉合約數量。 【參考】   未平倉合約(Open Interest) 未平倉合約是指期貨交易收盤後未平倉的期貨契約單邊買或賣的數量。期貨市場買方或賣方可等待合約到期，或利用與先前買賣方向相反合約結束責任，稱為平倉。 未平倉合約是期貨交易特有資料，它表達商品現有在倉數量，也是商品走勢動能資料。在買賣期指合約及期權合約時，買賣方合約都會被計入未平倉合約數量。 期交所中交易之買賣方要存入保證金，參與者持有未平倉數量愈大，需支付保證金愈高，成本亦高，所以若他相信後市與之前所預測多空方向相反，他會進行平倉。

期貨市場名詞： 正價差 (期貨價格高於現貨價格時) 與 逆價差 (現貨價格高於期貨價格時)

逆價差 與 正價差 　權值股漲勢稍歇，期貨市場空單回補壓力暫紓解，多單追價意願降低，成交量略縮，台期指、電子類指與摩台指均維持價平與小幅逆價差，觀望氣氛濃。分析師表示台股短線漲幅已高，加上美科技類股回檔壓力，除非權值股漲勢重啟，下周四台指拉高結算行情恐落空。 　四大期指今表現乏善可陳，台指期貨欠缺昨台積電急拉軋空力道，空單回補不積極，多單短線漲幅已高，也不貿然追價。中信期貨表示，昨台指未平倉量降低，多方自希將堅守最後防線的空單逼出場，倘使權值股持續盤整，原信心岌岌可危的空單，將因盤整拖長逐步消化，周四預期拉高結算行情恐難現。 　摩台指今走勢亦不若昨剽悍，雖與摩根現貨仍維持正價差，但僅維持強勢整理態勢，分析師研判外資回補台股趨勢未變，但買超幅應逐縮小，又近日費城半導體指數趨弱，恐降低外資買盤加碼意願。 - 摘錄自【期指動態】觀望 台期指狹幅震盪 中時晚報 920613 股市理財

Convergence of Futures to Spot Price Spot Price Futures Price Spot Price Time Time (a) (b)

Regulation of Futures Regulation is designed to protect the public interest Regulators try to prevent questionable trading practices by either individuals on the floor of the exchange or outside groups

Accounting & Tax If a contract is used for Hedging: (避險; 以負相關觀念認定) it is logical to recognize profits (losses) at the same time as on the item being hedged Speculation: (逐利) it is logical to recognize profits (losses) on a mark to market basis

Forward Contracts vs Futures Contracts FORWARDS FUTURES Private contract between 2 parties Exchange traded Non-standard contract Standard contract Usually 1 specified delivery date Range of delivery dates Settled at maturity Settled daily Delivery or final cash Contract usually closed out settlement usually occurs prior to maturity 16

Foreign Exchange Quotes [美國有外匯期貨] Futures exchange rates are quoted as the number of USD per unit of the foreign currency Forward exchange rates are quoted in the same way as spot exchange rates. This means that GBP, EUR, AUD, & NZD are USD per unit of foreign currency. Other currencies (e.g., CAD & JPY) are quoted as units of the foreign currency per USD. 17

Determination of Forward & Futures Prices Production, Consumption vs Investment Assets Investment assets are assets held by significant numbers of people purely for investment purposes (Examples: gold, silver) Production & Consumption assets are assets held primarily for production or consumption (Examples: copper, oil)

Short Selling Short selling involves selling securities you do not own Your broker borrows the securities from another client & 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 You must pay dividends & other benefits the owner of the securities receives

Measuring Interest Rates The compounding frequency used for an interest rate is the unit of measurement

Continuous Compounding In the limit as we compound more & more frequently we obtain continuously compounded interest rates $100 grows to $100eRT when invested at a continuously compounded rate R for time T $100 received at time T discounts to $100e-RT at time zero when the continuously compounded discount rate is R

Conversion Formulas Define Rc : continuously compounded rate Rm: same rate with compounding m times per year

Notation S0: Spot price today F0: Futures or forward price today T: Time until delivery date r: Risk-free interest rate for maturity T

Gold Example For gold F0 = S0(1 + r )T (assuming no storage costs) If r is compounded continuously instead of annually F0 = S0erT

Extension of the Gold Example For any investment asset that provides no income & has no storage costs F0 = S0erT

When an Investment Asset Provides a Known Dollar Income F0 = (S0 – I )erT where I is the present value of the income

When an Investment Asset Provides a Known Yield F0 = S0 e(r–q )T where q is the average yield during the life of the contract (expressed with continuous compounding)

Valuing a Forward Contract Suppose that K is delivery price in a forward contract 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

Forward vs Futures Prices Forward & 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 & the asset price implies the futures price is slightly higher than the forward price A strong negative correlation implies the reverse

Stock Index F0 = S0 e(r–q )T Can be viewed as an investment asset paying a dividend yield The futures price & spot price relationship is therefore F0 = S0 e(r–q )T where q is the dividend yield on the portfolio represented by the index

Stock Index (continued) For the formula to be true it is important that the index represent an investment asset (changes in the index must correspond to changes in the value of a tradable portfolio) The Nikkei index viewed as a dollar number does not represent an investment asset

Index Arbitrage When F0>S0e(r-q)T an arbitrageur buys the stocks underlying the index & sells futures When F0<S0e(r-q)T an arbitrageur buys futures & shorts or sells the stocks underlying the index

Index Arbitrage (continued) Index arbitrage involves simultaneous trades in futures & many different stocks Very often a computer is used to generate the trades Occasionally (e.g., on Black Monday) simultaneous trades are not possible & the theoretical no-arbitrage relationship between F0 & S0 does not hold

套利：股價指數期貨套利交易 Cash Flows from Carrying Stock 今日Borrow $100 for one year at 10%. +100 Buy one share of Widget, Inc. -100  半年後 Receive dividend of $2. +$2 Invest $2 for 6 months at 10%. -$2   一年後 Collect proceeds of $2.10 from dividend investment. +2.10 Sell Widget, inc. for P1. + P1 Repay debt. -110.00 Total Profit: P1 + $2.10 - $110.00

Information for Computing Fair Value Today’s date: July 6 Futures expiration: September 20 Days until expiration: 76 Index: Equally weighted index of 2 stocks Index divisor: 1.80 Interest rates: All interest rates are 10 percent 秀策企業股票 Today’s price: $115 Projected dividends: $1.50 on July 23. Days dividend will be invested: 59 rA: 0.10(59/360) =.0164   清源企業股票 Today’s price: $84 Projected dividends: $1.00 on August 12 Days dividend will be invested: 39 rB: 0.10(39/360) = .0108

Sell one SEP index futures contract for 115.00.   股票現貨買入持有套利交易 Cash–and–Carry Index Arbitrage   Date Cash Market Futures Market July 6 Borrow $199 for 76 days at 10%. Buy 秀策企業股票 & 清源企業股票 for a total outlay of $199. Sell one SEP index futures contract for 115.00. July 23 Receive dividend of $1.50 from 秀策企業股票 & invest for 59 days at 10%.   August 12 Receive dividend of $1.00 from 清源企業股票 & invest for 39 days at 10%.  

September 20 Assume any values for stock prices at expiration. We assume that stock price did not change. Therefore, the index value is still 110.56. Receive proceeds from invested dividends of $1.52 & $1.01. Sell 秀策企業股票 for $115 & 清源企業股票for $84. Total proceeds are $201.53. Repay debt of $203.20. At expiration, the futures price is set equal to the spot index value of 110.56. This gives a profit of 4.44 index units. In dollar terms, this is 4.44 index units times the index divisor of 1.8. Loss: $1.67 Profit: $7.99 Total Profit: $7.99 -$1.67 =$6.32

  股票現貨賣出套利交易 Reverse Cash-and–Carry Index Arbitrage Date Cash Market Futures Market July 6 Sell 秀策企業股票 & 清源企業股票 for a total of $199. Lend $199 for 76 days at 10%. Buy one SEP index futures contract for 105.00. July 23 Borrow $1.50 for 59 days at 10% & pay dividend of $1.50 on 秀策企業股票.   August 12 Borrow $1.00 for 39 days at 10% & pay dividend of $1.00 on 清源企業股票.

September 20 For Illustrative purposes, assume any values for stock prices at expiration. We assume that stock price did not change. Therefore, the index value is still 110.56. Receive proceeds from investment of $203.20. repay $1.52 & $1.01 on money borrowed to pay the dividends. Buy 秀策企業股票 for $115 & 清源企業股票 for $84. Return stocks to honor short sale. At expiration, the futures price is set equal to the spot index value of 110.56. This gives a profit of 5.56 index units. In dollar terms, this is 5.56 index units times the index divisor of 1.8. Profit: $1.67 Profit: $10.01 Total Profit: $1.67 + $10.01=$11.68

外匯市場的IRP: Futures & Forwards on Currencies 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

消費財 Futures on Consumption Assets F0  S0 e(r+u )T where u is the storage cost per unit time as a percent of the asset value. Alternatively, F0  (S0+U )erT where U is the present value of the storage costs.

The Cost of Carry The cost of carry, c, is the storage cost plus the interest costs less the income earned For an investment asset F0 = S0ecT For a consumption asset F0  S0ecT The convenience yield on the consumption asset, y, is defined so that F0 = S0 e(c–y )T

Futures Prices & Expected Future Spot Prices Suppose k is the expected return required by investors on an asset We can invest F0e–r T now to get ST back at maturity of the futures contract This shows that F0 = E (ST )e(r–k )T

Futures Prices & Future Spot Prices (continued) If the asset has no systematic risk, then k = r & F0 is an unbiased estimate of ST positive systematic risk, then k > r & F0 < E (ST ) negative systematic risk, then k < r & F0 > E (ST )

PROBLEM SET: MECHANICS OF FUTURES MARKETS 1.When an investor is obligated to buy the underlying asst in a futures position, it is a: A. Basis trade. B. Long-futures position. C. Short-futures position. D. Hedged-futures position.

128 2.Which of the following is NOT a characteristic specified by a futures contract? A. Asset quality. B. Asset quantity. C .Delivery month. D. Delivery method.

3.An investor enters into a short position in a gold- futures contract with the following characteristics: The initial margin is $3,000. The maintenance margin is $2,250. The position was entered at $293.60. Each contract controls 100 troy ounces. If the price drops to $291.00 at the end of the first day and $285.00 at the end of the second day, which of the following is closest to the variation margin at the end of the second day? A. $0. B. $260. c. $600. D. $860.

Hedging Strategies Using Futures Long & Short Hedges n   買入避險(The Long (Buy) Hedge) 預計未來要買入現貨部位者，為了避免現貨價格上漲的風險，而買入期貨合約 A long futures hedge is appropriate when you know you will purchase an asset in the future and want to lock in the price

避險   n   賣出避險(The Short (Sell) Hedge) 擁有現貨部位者，為了規避現貨價格下跌的風險，在市場上賣空期貨合約 A short futures hedge is appropriate when you know you will sell an asset in the future & want to lock in the price Question: 出口商要買進還是賣出美元遠期避險？玉米加工業者要買進還是賣出玉米期貨合約避險？

  風險完全規避 7/1 台灣玉米進口商向美中西部農戶買入5000 Bushel玉米@$2.1，預定8/1交貨，屆時此進口商將以時價賣給台灣食品製造商 n   What if the price of corn goes down in Aug? n   Strategy: 拋空期貨合約 Spot Future Basis 7/1 2.1 賣期貨 2.2 0.1 8/1 2.0 買期貨 2.1 0.1 (0.1) + 0.1 = 0

風險不完全規避 Spot Future Basis 7/1 2.1 賣期貨 2.2 0.1 8/1 2.0 買期貨 2.15 0.15   風險不完全規避 Spot Future Basis 7/1 2.1 賣期貨 2.2 0.1 8/1 2.0 買期貨 2.15 0.15 (0.1) + 0.05 = (0.05)

什麼情況下價格風險才能經由對沖完全免除？ 什麼情況下價格風險才能經由對沖完全免除？  n   either 期貨與現貨價格永遠相等•只有在到期日那一時點才出現(price convergence)   n   or 現貨與期貨價格變動的幅度完全相等時，價格風險才能完全由對沖免除 •基差(Basis) = 期貨價格 - 現貨價格 •基差隨時都會變動，此為基差風險 •買賣期貨是以價格風險交換基差風險 期貨契約規格固定也是風險難以完全消除之因

Arguments in Favor of Hedging Companies should focus on the main business they are in and take steps to minimize risks arising from interest rates, exchange rates, and other market variables

Arguments against Hedging Shareholders are usually well diversified and can make their own hedging decisions It may increase risk to hedge when competitors do not

Arguments against Hedging Explaining a situation where there is a loss on the hedge and a gain on the underlying can be difficult E.g., 電子廠商因為海外公司債券部位超過應收外幣貨款部位出現匯兌損失, 但是的確新台幣弱勢時東西好賣 E.g., 玉米商以空期貨規避玉米價格下跌風險, 但是當玉米價格上升, 他帳面上會有 . . .

Convergence of Futures to Spot Price Spot Price Futures Price Spot Price Time Time (a) (b)

Basis Risk Basis is the difference between spot & futures Basis risk arises because of the uncertainty about the basis when the hedge is closed out

期貨市場名詞： 正價差 (期貨價格高於現貨價格時) 與 逆價差 (現貨價格高於期貨價格時)

逆價差 與 正價差 　權值股漲勢稍歇，期貨市場空單回補壓力暫紓解，多單追價意願降低，成交量略縮，台期指、電子類指與摩台指均維持價平與小幅逆價差，觀望氣氛濃。分析師表示台股短線漲幅已高，加上美科技類股回檔壓力，除非權值股漲勢重啟，下周四台指拉高結算行情恐落空。 　四大期指今表現乏善可陳，台指期貨欠缺昨台積電急拉軋空力道，空單回補不積極，多單短線漲幅已高，也不貿然追價。中信期貨表示，昨台指未平倉量降低，多方自希將堅守最後防線的空單逼出場，倘使權值股持續盤整，原信心岌岌可危的空單，將因盤整拖長逐步消化，周四預期拉高結算行情恐難現。 　摩台指今走勢亦不若昨剽悍，雖與摩根現貨仍維持正價差，但僅維持強勢整理態勢，分析師研判外資回補台股趨勢未變，但買超幅應逐縮小，又近日費城半導體指數趨弱，恐降低外資買盤加碼意願。 - 摘錄自【期指動態】觀望 台期指狹幅震盪 中時晚報 920613 股市理財

Long Hedge Suppose that F1 : Initial Futures Price F2 : Final Futures Price S2 : Final Asset Price You hedge the future purchase of an asset by entering into a long futures contract Cost of Asset = 屆時街上的 Spot Price 買進價格 - 幫助避險期貨市場的盈虧 = S2 – (F2 – F1) = F1 + Basis

Short Hedge Suppose that F1 : Initial Futures Price F2 : Final Futures Price S2 : Final Asset Price You hedge the future sale of an asset by entering into a short futures contract Price Realized = 屆時街上的 Spot Price 賣出價格 + 幫助避險期貨市場的盈虧 = S2+ (F1 –F2) = F1 + Basis

Choice of Contract Choose a delivery month that is as close as possible to, but later than, the end of the life of the hedge [期貨交割月份晚於避險期間終止日] [Why?] 因為某些期貨到期時Long方須Take Delivery. When there is no futures contract on the asset being hedged, choose the contract whose futures price is most highly correlated with the asset price. There are then 2 components to basis

Two components to basis: 如果要Time Out鴻海股價波動,大股東只能夠空電子期貨 電子期貨到期日大股東總部位價值是 鴻海股價2 + (電子期貨契約價值 1 - 電子期貨契約價值2 ) = 電子期貨契約價值 1 + (電子指數現貨價值2 -電子期貨契約價值2 ) + (鴻海股價2 -電子指數現貨價值2 ) =電子期貨契約價值 1 + 電子指數期貨與現貨間基差 + 鴻海股與電子指數間現貨基差

Hedge Ratio 用以避險期貨契約規模 / 避險標的暴險大小

將總部位變異數降到最低: Optimal Hedge Ratio (避險率) Proportion of the exposure that should optimally be hedged is where sS is the standard deviation of dS, the change in the spot price during the hedging period, sF is the standard deviation of dF, the change in the futures price during the hedging period r is the coefficient of correlation between dS and dF. 如果 r = 1, sS = 3 X sF => Hedge Ratio = 3

如果是用台股指數期貨幫助規避馬蓋先建設股價隨大盤波動部份的風險: 如果是股價指數期貨避險, 風險變數是Beta To hedge the risk in a portfolio the number of contracts that should be shorted is [現貨市值除以期貨合約價值, 再乘以現貨投資組合的Beta值] where P is the value of the portfolio, b is its beta, and A is the value of the assets underlying one futures contract

Reasons for Hedging an Equity Portfolio [喊暫停 ] Desire to be out of the market for a short period of time. (Hedging may be cheaper than selling the portfolio and buying it back.) [有擇股能力者] Desire to hedge systematic risk (Appropriate when you feel that you have picked stocks that will out-perform the market.)

Example Value of S&P 500 is 1,000 Value of Portfolio is $5 million Beta of portfolio is 1.5 What position in futures contracts on the S&P 500 is necessary to hedge the portfolio?

position in futures contracts = 1.5 X (5,000,000 / 250,000) = 30

Changing Beta What position is necessary to reduce the beta of the portfolio to 0.75? What position is necessary to increase the beta of the portfolio to 2.0?

例子 ~ 電子期貨避險： 洪七可賣出持股以規避股價下跌損失，或賣空電子類股價指數期以規避股價下跌風險。   洪七九月持有之電子股價股價及股數如下： 股票 股價 股數 市值 台積電 65元 20,000股 1,300,000元 華碩 32元  640,000元 總價值   1,940,000元  洪七於九月六日以230點賣出二張十月份電子期貨契約：  230點電子期貨契約價值為： 4,000元ｘ230點 = 920,000元

應賣出避險部位為： 1,940,000/920,000元= 2張(取整數為2張契約)  避險結果：  十月三日股票收盤價格如下： 股票 股價 股數 市值 台積電 62元 20,000股 1,240,000元 華碩 30元  600,000元 總價值   1,840,000元  股票損失：  1,940,000元-1,840,000元=100,000元

十月三日電子類股價指數期貨：215點   電子期貨獲利：4,000元x(230-215點)x2 = 120,000元 避險結果：120,000元-100,000元 = 20,000元

金融期貨交易策略例子 洪七手中握有績優金融保險類股股票，市價約100萬元，預期金融保險類股可能走弱，遂於期貨市場600點價格賣出2張元月份金融期貨，元月十七日以550點價格買進，此時洪七手中金融保險類股票市值跌至92萬元。 其避險結果：  股票損失：1,000,000元- 920,000元 = 80,000元 賣出避險部位：1,000,000/600x1,000 =2 (取整數為2張)  期貨獲利：1,000元x(600點-550點)x2張=100,000元

例子 ~ 看多台積電股票表現的投資人，可以(1) 買進台積電股票、賣出台灣50指數期貨； (2) 買進台積電股票、賣出電子期貨   例子 ~ 看空台積電股票表現的投資人，可以(1) 融券台積電股票、買進台灣50指數期貨； (2) 融券台積電股票、買進電子期貨 例子 ~ 看多大型股票表現的投資人，可以買進 台灣50指數期貨、賣出台股期貨； 看空大型股票表現的投資人，可以賣出 台灣50指數期貨、買進台股期貨；

如果欲避險期間終止日比一切市場上期貨到交割日都晚很多, 只好玩接力賽了 (Rolling The Hedge Forward) We can use a series of futures contracts to increase the life of a hedge Time 1: 空電子期貨契約C1 Time 2: 結束電子期貨契約C1 空電子期貨契約C2 Time 3: 結束電子期貨契約C2 空電子期貨契約C3 Time 4: 結束電子期貨契約C3

n項基差風險 ~ Each time we switch from 1 futures contract to another we incur a type of basis risk 此一避險期間, 我們要面對3項基差風險, (1) C1期貨價格與C2期貨價格間的基差, (2) C2期貨價格與C3期貨價格間的基差, (3) C3期貨價格與避險期間終止日現貨價格間的基差,

PROBLEM SET: HEDGING STRATEGIES USING FUTURES 1.Which of the following is closest to the correct value for the basis associated with a spot position valued at $15per unit, and a futures contract with a value of $18 per unit? A. -$3.0. B. $5.0 C. $2.0 D. $3.0

Which of the following is the correct trade for our trader? 135 2.A.U.S. –based currency trader is trying to hedge an Irish pound position with the British pound contract. Assume the following: His Irish pound position has a value of GBP 312,500 million. Each British pound contract controls GBP 62,500. The chosen contract matures in December. The trader wishes to hedge the position for a few months through November. The British pound contract is currently trading at $1.40 per pound. The current exchange rate between the U.S dollar and the Irish pound is $1.15 per Irish pound. The correlation between the spot and futures is 0.90. The annual standard deviation of the Irish pound exchange rate is 0.067. The annual standard deviation of the British pound futures is 0.15. Which of the following is the correct trade for our trader? A. Short 5British pound contracts. B. Long 5British pound contracts. C. Short 2British pound contracts. D. Long 2British pound contracts.

P135 Which of the following is closest to the basis for the GBP contract described in the preceding question? A. ABB 250 million. B. $1.26. C. GBP 0.25. D. $0.25.

交易人須依期貨商所定保證金標準（期交所對此標準有基本要求）繳交保證金後才可從事臺股期貨交易。  需要準備多少資金才可從事臺股期貨交易？  交易人須依期貨商所定保證金標準（期交所對此標準有基本要求）繳交保證金後才可從事臺股期貨交易。 例如期貨商規定臺股期貨之原始保證金為新台幣12萬元，維持保證金為新台幣10萬元，則交易人需準備新台幣12萬元以上方可從事臺股期貨交易。 交易人如存入比原始保證金更充裕資金於期貨商保證金專戶，較能避免價格劇烈波動時因未能及時補繳保證金而遭斷頭出場。 臺灣期貨交易所「證交所股價指數期貨契約規格」 項目 內容 交易標的 ● 證交所發行量加權股價指數 中文簡稱 ● 臺股期貨

英文代碼 ● TX 交易時間 ● 證交所正常營業日上午8:45~下午1:45 契約價值 ● 臺股期貨指數乘上新臺幣200元 契約到期交割月份 ● 自交易當月起連續二個月份，另加上三月、六月、九月、十二月中三個接續的季月，總共有五個月份的契約在市場交易 每日結算價 ● 每日結算價原則上為當日收盤時段成交價，若收盤時段無成交價，則依期交所「證交所股價指數期貨契約交易規則」訂定之 每日漲跌幅 ● 最大漲跌幅限制為前一營業日結算價上下7%

升降單位 ● 指數1點（相當於新臺幣200元） 最後交易日 ● 各契約的最後交易日為各該契約交割月份第三個星期三，其次一營業日為新契約的開始交易日 最後結算日 ● 最後結算日為最後交易日之次一營業日 最後結算價 ● 以最後結算日證交所依本指數各成分股開盤十五分鐘為基礎，先算該段時間內各成分股成交量加權平均價，再訂定最後結算價 。 交割方式 ● 以現金交割，交易人於最後結算日依最後結算價差額，以淨額進行現金之交付或收受

● 交易人於任何時間持有各月份契約未平倉部位總和限制如下： 1. 自然人三百個契約 2. 法人機構一千個契約 部位限制 ● 交易人於任何時間持有各月份契約未平倉部位總和限制如下： 1. 自然人三百個契約 2. 法人機構一千個契約 3. 法人機構基於避險需求得向期交所申請豁免部位限制 4. 期貨自營商之持有部位不在此限 保證金 ● 期貨商向交易人收取之交易保證金及保證金追繳標準，不得低於期交所公告之原始保證金及維持保證金水準 ● 期交所公告原始保證金及維持保證金，以「期交所結算保證金收取方式及標準」結算保證金為基準，按期交所成數加成計算

臺灣期貨交易所「證交所電子類股價指數期貨契約規格」 項目 內容 交易標的 ● 證交所電子類股價指數 中文簡稱 ● 電子期貨 英文代碼 ● TE 交易時間 ● 證交所正常營業日上午8:45~下午1:45 契約價值 ● 電子期貨指數乘上新臺幣4,000元 契約到期交割月份 ● 自交易當月起連續二個月份，另加上三、六、九、十二月中三個接續季月，總共五個月份的契約在市場交易 每日結算價 ● 每日結算價原則為當日收盤時段成交價，若收盤時段無成交價，依期交所「證交所電子類股價指數期貨契約交易規則」訂定

● 最大漲跌幅限制為前一營業日結算價上下7％ 每日漲跌幅 ● 最大漲跌幅限制為前一營業日結算價上下7％ 升降單位 ● 指數0.05點（相當於新臺幣200元） 最後交易日 ● 各契約的最後交易日為各該契約交割月份第三個星期三，其次一營業日為新契約的開始交易日 最後結算日 ● 最後交易日之次一營業日 最後結算價 ● 以最後結算日證交所依本指數各成分股開盤十五分鐘為基礎，先算出該段時間內各成分股成交量加權均價，再訂定最後結算價 。

● 以現金交割，交易人於最後結算日依最後結算價差額，以淨額進行現金之交付或收受 交割方式 ● 以現金交割，交易人於最後結算日依最後結算價差額，以淨額進行現金之交付或收受 部位限制 ● 交易人於任何時間持有各月份契約未平倉部位總和限制如下： 1. 自然人三百個契約 2. 法人機構一千個契約 3. 法人機構基於避險需求得向期交所申請豁免部位限制 4. 期貨自營商之持有部位不在此限 保證金 ● 期貨商向交易人收取之交易保證金及保證金追繳標準，不得低於期交所公告之原始保證金及維持保證金水準 ● 期交所公告原始保證金及維持保證金，以「期交所結算保證金收取方式及標準」計算結算保證金為基準，按期交所成數加成計算

臺灣期貨交易所「證交所金融保險類股價指數期貨契約規格」 項目 內容 交易標的 ● 證交所金融保險類股價指數 中文簡稱 ● 金融期貨 英文代碼 ● TF 交易時間 ● 證交所正常營業日上午8:45~下午1:45 契約價值 ● 金融期貨指數乘上新臺幣1,000元 契約到期交割月份 ● 自交易當月起連續二個月份，另加上三、六、九、十二月中三個接續季月，總共五個月份的契約在市場交易

● 每日結算價原則為當日收盤時段成交價，若收盤時段無成交價，則依期交所「證交所金融保險類股價指數期貨契約交易規則」訂定 每日漲跌幅 ● 漲跌幅限制為前一營業日結算價上下7% 升降單位 ● 指數0.2點（相當於新臺幣200元） 最後交易日 ● 各契約的最後交易日為各該契約交割月份第三個星期三，其次一營業日為新契約的開始交易日 最後結算日 ● 最後交易日之次一營業日

● 以最後結算日證交所依本指數各成分股開盤十五分鐘為基礎，先計算該段時間各成分股成交量加權平均價，再訂定最後結算價 。 交割方式 ● 現金交割，交易人於最後結算日依最後結算價差，以淨額現金收付 部位限制 ● 交易人於任何時間持有各月份契約未平倉部位總和限制如下： 1. 自然人三百個契約 2. 法人機構一千個契約 3. 法人機構基於避險需求得向期交所申請豁免部位限制 4. 期貨自營商之持有部位不在此限 保證金 ● 期貨商向交易人收取之交易保證金及保證金追繳標準，不得低於期交所公告之原始保證金及維持保證金水準 ● 期交所公告原始保證金及維持保證金，以「期交所結算保證金收取方式及標準」計算結算保證金為基準，按期交所成數加成計算

臺灣期貨交易所「證交所股價指數小型期貨契約規格」 項目 內容 交易標的 ● 證交所發行量加權股價指數 中文簡稱 ● 小型臺指期貨 英文代碼 ● MTX 交易時間 ● 證交所正常營業日上午8:45~下午1:45 契約價值 ● 小型臺指期貨指數乘上新臺幣50元 契約到期交割月份 ● 自交易當月起連續二個月份，另加上三月、六月、九月、十二月中三個接續的季月，總共有五個月份的契約在市場交易  

● 每日結算價與「證交所股價指數期貨契約」之每日結算價相同 每日漲跌幅 ● 漲跌幅限制為前一營業日結算價上下7% 升降單位 ● 指數1點（相當於新臺幣50元） 最後交易日 ● 各契約的最後交易日為各該契約交割月份第三個星期三，其次一營業日為新契約的開始交易日 最後結算日 ● 最後交易日之次一營業日

● 以最後結算日證交所依本指數各成分股開盤十五分鐘為基礎，先計算出該段時間內各成分股之成交量加權平均價，再予以訂定最後結算價 。 交割方式 ● 現金交割，交易人於最後結算日依最後結算價差額，以淨額進行現金交付或收受 部位限制 ● 交易人於任何時間持有之各月份契約未平倉部位總和限制如下： 1. 自然人六百個契約 2. 法人機構二千個契約 3. 法人機構基於避險需求得向期交所申請豁免部位限制 4. 期貨自營商之持有部位不在此限

臺灣期貨交易所股份有限公司 「臺灣證券交易所臺灣50指數期貨契約」規格 項目 內容 交易標的 ● 臺灣證券交易所臺灣50指數 中文簡稱 ● 臺灣50期貨 英文代碼 ● T5F 交易時間 ● 臺灣證交所營業日上午8:45~下午1:45 契約價值 ● 臺灣50期貨指數乘上新臺幣500元

契約到期交割月份 ● 自交易當月起連續二個月，另加三月、六月、九月、十二月中三個接續的季月，總共有五個月份的契約在市場交易 每日結算價 ● 每日結算價原則上為當日收盤時段成交價，若收盤時段無成交價，則依「證交所臺灣50指數期貨契約交易規則」訂定 每日漲跌幅 ● 最大漲跌幅限制為前一營業日結算價上下7% 升降單位 ● 指數1點（相當於新臺幣500元）

最後交易日 ● 各契約的最後交易日為各該契約交割月份第三個星期三，其次一營業日為新契約的開始交易日 最後結算日 ● 最後交易日之次一營業日 最後結算價 ● 最後結算日證交所本指數各成分股當日交易開始後十五分鐘內均價計算指數。前述平均價係採每筆成交價成交量加權平均。但當日市場交易開始後十五分鐘內仍無成交價者，以當日市價升降幅基準價替代 交割方式 ● 以現金交割，交易人於最後結算日依最後結算價之差額，以淨額進行現金之交付或收受    

【參考】臺灣期貨交易所三十天期商業本票利率期貨契約規格草案 項目 內容 中文簡稱 n    三十天期利率期貨 英文代碼 n    CPF 交易標的 n    面額新台幣一億元之三十天期融資性商業本票 契約到期 交割月份 n    交易當月起連續之十二個月份 報價方式及 最小升降單位 n    本契約交易以百分比為報價單位，報價方式採一百減利率 n    最小升降單位為0.005，每一最小升降單位價值以411元計算

交易時間 n    銀行業營業日上午八時四十五分至十二時 每日結算價 n    每日結算價採收盤時段成交價 n    若當日收盤時段無成交價，則依「期交所三十天期商業本票利率期貨契約交易規則」訂定 每日漲跌幅 n    以前一交易日結算價上下各0.5為限 最後交易日 n    到期月份之第三個星期三

交割方式 n    現金交割 最後結算日 n    最後結算日同最後交易日 最後結算價 n    以一百減最後交易日上午十二時期交所選定機構所公布之一月期成交累計利率指標，向下取至最接近最小升降單位整數倍數值 部位限制 n    單一月份不超過500口；各月份合計不超過2,000口 保證金 n     期貨商向交易人收取交易保證金及保證金追繳標準，不得低於期交所原始保證金及維持保證金水準 n    原始保證金及維持保證金，以「期交所結算保證金收取方式及標準」結算保證金為基準，按訂定成數加成計算

【參考】政府債券期貨契約 項 目 內 容 中文簡稱 英文代碼 交易標的 可交割債券 n 十年期公債期貨 n GBF 項 目 內 容 中文簡稱 n         十年期公債期貨 英文代碼 n         GBF 交易標的 n         面額五百萬元，票面利率5%之十年期政府債券 可交割債券 n         到期日距交割日在七年以上十一年以下，一年付息一次，到期一次還本中央登錄公債

契約到期 交割月份 (三、六、九、十二季月循環) 報價方式 最小 升降單位 交易時間 每日結算價 每日漲跌幅 n 交易當月起接續之三個季月

項 目 內 容 最後交易日 n         交割月份第二個星期三 交割方式 n         實物交割 交割日 n         最後交易日後之第二個營業日 最後結算價 n         以最後交易日收盤前十五分鐘內所有交易之平均價訂之。但該時段內不足二十筆交易者，以當日最後二十筆交易剔除最高及最低各二筆後之平均價替代之 n         前項平均價係採每筆成交價之成交量加權平均得之

項 目 內 容 部位限制 n         單一月份不超過1,000口；各月份合計不超過2,000口 保證金 n         期貨商向交易人收取之交易保證金及保證金追繳標準，不得低於期交所原始保證金及維持保證金水準 n         原始保證金及維持保證金，以「期交所結算保證金收取方式及標準」結算保證金為基準，按訂定之成數加成計算

選擇權 執行期間~ 多在一年與七年間( 選擇權則多在一年以內)   –美式 American Style ~ 認購權證期滿前，如標的物市價高於執行價格隨時可執行合約 –歐式 European Style ~ 只能在權證滿期時才可執行合約

權利金係市場供需決定，當其他因素不變時，各單項因素上漲對買權與賣權之影響方向︰ 影響權利金的因素？ 權利金係市場供需決定，當其他因素不變時，各單項因素上漲對買權與賣權之影響方向︰   買權價值 賣權價值 現貨價格 + - 履約價格 存續期間 波動率 利率 現金股利  

選擇權權利金怎麼訂定？ 以外匯選擇權為例 Example: Formula for valuing a European call on spot FX:   Where r = the constant domestic (NT$) riskless interest rate rfor = the constant foreign riskless interest rate T = the time to expiration S = the spot price of one unit of FX K = the strike price S* = Se-rforT = the present value of the spot price of FX, discounted at the foreign riskless interest rate N(d1), N(d2) = the cumulative standard  

Normal distribution functions of d1 and d2, respectively   σ= the standard deviation of the percentage changes in the price of FX For example, find the value of a call on 31, 250 units of FX, given the following: S (現行匯率) = NT$1.50/FX; T (到期時長) = 0.5 year; rfor (美元利率) = 0.04/year(foreign interest rate) r (新台幣利率) = 0.06/year (NT$ interest rate); σ(匯率波動性(年化標準差) ) = 0.20/year; K = 1.50 (履約匯率)

權利金C = S*N(d1)-Ke-rTN(d2) =(1.470298)(0.55625)-(1.5e-(0.6)(0.5))(0.5) = 0.09   The value of a call on one unit of FX is NT$0.09. The value of a call on 31,250 units of FX is therefore NT$2812.50

權證名稱 金鼎12 履約價 87.4 發行數量－仟 24000 標的名稱 微星 發行時標的股價 76 發行金額－億 4.012 發行日期 2003/6/11 發行時Moneyness 115% 存續期間 0.75年 發行價格 16.72 權證型態 基本型 執行比例 1 發行波動率 74.30% 溢價率 22% 名目槓桿比率 4.55 最新標的股價：76.000　　權證理論價：16.596

認股權證的價格 時間價值 Warrant Price = Parity + Time Value 履約價值 Call:Stock Market P - Strike P Put: Strike P - Market P >0, 價內 =0, 價平 <0, 價外

PROBLEM SET: MECHASNICS OF OPTIONS MARKETS Use the following data to answer Questions 1 and 2. An investor owns a stock option that currently has a strike price of $100. P145 1.If the stock experiences a 4 to 1 split, the strike price becomes: A. $20. B. $25 C. $50. D. $100

P145 2.The number of shares now controlled by the option is: A. 100 B. 200 C. 300 D. 400 3.A June 45 call: A. Expires in June and has a strike price of $45. B. Expires in June and has a current stock price of $45. C. Was purchased in June and has a strike price of $45. D. Was purchased in June and has a current stock price of $45

P145 4.If an option is quoted at 2 3/4, the cost of one contract to the potential buyer is closest to: A. $0.275. B. $2.75 C. $275.00 D. $2,750.00.

PROBLEM SET: PROPERTIES OF STOCK OPTIONS 1.Which of the following has the same impact on all stock option prices? A. An increase in volatility. B. An increase in the stock price. C. An increase in the risk-free rate. D. A decrease in time to expiration.

P156 2.Consider a call option on a stock currently priced at $50 with a strike price of $55. Which of the following CANNOT be the price of the call option? A. $10 B. -$15 C. $50 D. $55

P156 3.Consider a European put option in the stock of the preceding question. The put option has an expiration of 6 months, a strike price of $40, and the risk-free rate is 5 percent. The price of the put option is closest to: A. $10.99 B. -$1.23 C. $0.01 D. $39.50

P156 4.Consider a 1-year European put option that is currently valued at $5 on a $25 stock and a strike of $27.50. The 1-year risk-free rate is 6 percent. Which of the following is closest to the value of the corresponding call option? A. $0.00 B. $3.89 C. $4.10 D. $5.00

P156 5.Consider an American call and put option in the same stock. Both options have the same 1-year expiration and a strike price of $45. The stock is currently priced at $50 and the annual interest rate is 10 percent. Which of the following could be the difference in the two option values? A. $4.95. B. $7.95 C. $9.35 D. 12.50