Futures,Forwards, and dswaps Financial Engineering gand Computations Dai, Tian-Shyr
此書內容 Financial Engineering & Computation 教課書的第 12 章 Forwards, Futures, Futures Options, Swaps Options, Futures, and Other Derivatives John C. Hull 的第七章 Swaps
Outline Brief Introduction of Derivatives Forwards Relationships Between Forward Price and Spot Price Futures Relationships between Forward and Futures Prices Cost of fcarry Hedging and Futures Minimum Variance Hedge Ratio Brief Introduction of Swaps
Derivatives Payoffs depend on other more fundamental assets: Underlying assets Commodity Index Interest est rate Other derivatives Four types of derivatives Forwards( 遠期契約 ) Futures( 期貨 ) Swaps( 交換合約 ) Options( 選擇權 )
遠期契約 (Forwards) 定義 : 遠期契約雙方同意在未來某一天以某一價 格 ( 履約價格 ) 交易某一特定資產或商品 Terms: Maturity date: 到期日 Strike price: 履約價格 Underlying asset: 標的物
Why we Need Forwards? A Simple Story 考慮 XYZ 公司欲投資 x 元生產棉花, 生產期為三個月 三個月後 : 棉花價格上升 : 公司大賺一筆 棉花價格狂洩 : 公司虧本甚至倒閉 如何避免棉花的價格風險?
A Simple Story on Forwards 假定有另一紡布公司 ABC, 三個月後需要棉花紡布 棉花不能今日進貨 : 倉儲問題 三個月後進貨 棉花價格上升 : 公司有虧本甚至破產之虞 棉花價格狂洩 : 公司大賺一筆 如何規避棉花的價格風險?
A Simple Story on Forwards XYZ 和 ABC 之間可簽訂合約 使用預定好的金額 K 於三個月後交易棉花 價格風險消失, 但也無法得額外利潤 XYZ (Forward 的報酬為 K-P) 不簽合約 遠期合約 簽合約 價格上升至 P1 P1( 額外利潤 ) K-P1 P1 價格下降至 P2 P2( 虧損 ) K-P2 P2 市場價格 總報酬鎖定為 K
A Simple Story on Forwards 上述為一遠期合約 標的物 : 棉花 到期日 : 三個月後 ( 假定棉花價格為 P) 履約價格 :K 買方 :ABC ( 報酬為 P-K) 賣方 :XYZ ( 報酬為 K-P) 交易方式 : 實物交易 現金清算
從 Buyer (ABC) 來看 Forward 的報酬 ΔV K V 1. 報酬構面 (payoff profile) 是直線的 2. 遠期合約期初價值為零, 期中無任何支付, 僅有期末會有支付產生
Forward 的缺點 當棉花價格上升,XYZ 失去大賺一筆的機會 當棉花價格下降,ABC 失去減少成本的機會 -Options Df Default risk( ik( 違約風險 ) -XYZ 可能失火, 導致無法履約 - 當棉花價格上升,XYZ 有違約的可能 - 當棉花價格下降,ABC 有違約的可能 採用期貨合約每日清算的制度, 減少違約風險
合約未標準化 Forward 的缺點 - 採用標準化的期貨合約 - 以台指期貨為例 交易標的物 : 加權指數 規格 : 近三個月和兩個季月到期的合約 Ex: Now: Oct., (Introduced later) 交易合約到期日 : Oct., Nov., Dec., Mar., Jun. 市場撮合和流通 - 期貨合約可在交易所流通
Reasonable Strike Price 在理想的假定下 ( no tax, transaction cost, etc.) 遠期的履約價格和標的物的現貨價格有一定的關係 當該關係不滿足時, 存在套利機會 凱因斯 (Keynes) 使用 Interest parity 指出該狀況 考慮遠期匯率合約 匯率和本國利率, 外國利率有連動的關係
Terms: Interest Rate Parity S: Spot exchange rate (Domestic/Foreign) F: Forward exchange rate (Maturity: T year) R f: Foreign interest rate R l : Local interest rate F R R = e S Then we have ( l f )T Otherwise, there exists arbitrage opportunity.
Assume Now: Arbitrage Opportunity F R e l R > S ( )T f 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: 1 R f T You get e foreign dollars. S Convert them at the rate F (Forward contract) F e S Since R f T R T get free lunch > e l
In Class Exercise F R R < e S ( l f )T Show that arbitrage opportunity exists as
Forward Price 遠期合約在期初時不會有現金流 遠期合約初始價值為 0( 買方不需付權利金給賣方 ) Forward price (F) 代表一合理的價格, 使得當履約價格 (K) 等於 F 時, 遠期合約的初始價值為 0 Forward price 會隨時間而改變 令 F t 為時間 t 的 Forward price 假定合約起始日為 0, 到期日為 T S T 為標的物在時間 T 的價格 可類推到期貨 (future price)
Contract Value 遠期合約在期初的價值為 0, rt 遠期合約在時間 t 的價值為 f = S Ke t t Proof: -At time t Long a forward contract Save Ke r ( T t) Short underlying asset -At time T ( S K ) + K S = 0 T + T Riskless, Benefit=0 ( t)
Forward Price 和 Spot Price 關係 ( 標的物於 T 前無支付現金 ) 在時間 0 時, 現貨價格為 S 0 假定標的物在 T 前不會產生 cash flow (like dividend) de d) 則可得 F = S e 0 0 如上式不滿足, 則存在套利空間 令 f S Ke rt ( 0) 0 = rt 0 0 F = K = S e 0 0 rt Enter a forward contract at time 0 with strike price K with strike price K 0 t T S 0 S t S T
Arbitrage Opportunity 假定 F > S e 0 0 rt 時間 0 時簽合約,K K= F 0 Cash flow 時間 0 借 S0 Buy the stocks Short Forward 0 時間 T 還 rt S e 得 S T 還 S T -K F 0 -S 0 e rt 0 (K=F 0 ) Se + S ( S F) > 0, 期初不用支付任何成本, 期末可得正的報酬! rt 0 T T 0
In Class Exercise 考慮 F < Se 0 0 rt 的套利機制
A Real Example r is the annualized 3-month riskless interest rate. S is the spot price of the 6-month zero-coupon bond. A new 3-month forward contract on a 6-month zerocoupon bond should command a delivery price of Se r/4. So if r = 6% and S = 970.87, then the delivery price is 970.87 e 0.06/4 = 985.54.
Forwards on Bond and Interest Rates Maturity of forward F is face value of the 6- month zero- coupon bond S is the spot price of the 6- month zero- coupon bond at time t t Maturity of zero bond At time A : At time B : 1 -r 1 6 S = F e 2 = 970.87 -rf A S 4 B = F e 1 0.06 r 985.54 F S e reward of buyer = S 4 4 970.87 e B = = = 985.5454 A A
Forward Price 和 Spot Price 關係 ( 標的物於 T 前支付固定報酬 ) 假定該報酬 ( 例如股息 ) 的現值為 I rt 可得 F 0 = ( S 0 I ) e rt 假定 F 0 > ( S 0 I) e -At time 0 short forward (K=F 0 ), long underlying asset, borrow S 0 Initial cash flow=0 -At time T rt rt S + F + S + Ie S e = F S I ) e 0 T 0 0 ( 0 > rt 0 T
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. 0.08 3 /12 0.08 6 /12 0.08 9 /12 I = 0.75e + 0.75e + 0.75e = 2. 162 where ( 50 2.162) 0.08 10 /12 F = e = $51. 14 I is the PV of the dividends
Homework 7 1.Show that arbitrage opportunity exist if F < ( S I ) e 0 ( 0 rt 2.Assume that the underlying stock pays a continuous dividend yield at rate q. Show that the forward price ( r q) T is F = S e 0 0
期貨合約 (Futures) 期貨合約和遠期合約很類似, 主要差別在於 : 1. 每日現金結算損益 (marked-to-market) 2. 買賣雙方都必須提供保證金 (margin) - 交易所藉由 marked-to-market 和 margin account 來降低違約風險 A B 保證金 結算所 保證金 雙方交易的共同保證人
期貨合約 (Futures) 3. 合約標準化 ( 期貨均在集中交易所交易 ) 4. 透過集中市場競價來決定交易價格 5. 可以反向契約來抵銷原有契約 Black: Futures 可視為一系列的遠期合約 在每日將前一天的 Forward 清算掉, 並重新訂約 有關期貨相關資訊可至台灣期交所 http://www.taifex.com.tw/
台灣的期貨市場 標的物 : 股票指數 Check: http://tw.stock.yahoo.com/future/
價格 走勢圖 ( 台指期 10) 6028 6020 6000 10:07:51 時, 成交價 6000 5991 昨收盤價 5985 08:45 10:00 時間
報價表 ( 名詞解釋 ) 名稱 : 台指期 10, 指交割月份為 10 月份的台指期貨 ( 期貨報價通常會揭露近三個連續近月與二個接續季月共五個資訊 ) 時間 :10:07:51, 指系統揭露成交的時間 漲跌 : 9, 指成交價減去昨天收盤價的價差 (6000-5991=9) 總量 :10806, 指 09:00~10:07 0:07 期間市場成交的總數量 基差 :-33.72, 指現貨價格減去期貨的價格, 由此報價表可得知台股指數約為 5966 點 ( 基差反應的是持有現貨的成本 收益與市場對 未來走勢的預期心理 ) 最高 :6028, 指 09:00~10:07 期間內最高價 平倉 : 以等量但相反買賣方向沖銷原有的部位 未平倉量 (open interest): 買 ( 賣 ) 的期貨契約在未平倉以前稱為未平倉量, 代表等量的多頭部位與空頭部位 ( 報價表中常見的名詞 )
保證金清算 替代原則 : 結算公司介入每筆交易, 成為買方的賣方, 也成為賣方的買方, 承擔雙方之信用風險 Initial margin Maintenance margin 保證金低至觸及維持保證金時, 投資人會收到保證金催繳通知, 必須將保證金補足至原始保證金水準, 否則期貨商可逕自予以平倉
Example: 期貨保證金清算 假設某投資人於 2006 年 11 月 23 日買進一口台股指數期貨契約, 成交價格是 5600 點, 原始保證金為 105,000 元, 維持保證金為 81,000 元, 此契約的總價值為 $1,120,000 元 ( 大台指契約乘數每點 200 元 ) 在 11 月 24 25 26 日時, 台指期收盤價為分別為 5500 5450 5700 點, 期貨保證金清算過程如下 :
Example: 期貨保證金清算 日期台指期收盤價當日損益保證金餘額 11/23 5600-105,000 11/24 5500-20,000000 85,000 11/25 5450-10,000 75,000 11/26 5700 50,000 155,000 保證金餘額低於維持保證金 $81,000, 該投資人會接到經紀商通知 (margin call), 並將保證金補足至原始保證金, 故 26 日當天一開始的餘額變為 $105,000 元
Daily cash flows 假定第 i 天的 future price 為 f i The contract cash flow at day i should be f i -ff i-1 Net cash flow: ( f f ) + ( f f ) + ( f f ) +... + ( f f ) = = S = F 1 0 2 1 3 2 T T 1 f T f 0 S T f 0 S T F T It may differ because of the reinvestment and the margin system. A futures contract has the similar A futures contract has the similar accumulated payoff to a forward contract.
Relationships between Forward and Futures Prices Forward price = futures prices, if the interest rate is not stochastic. Let Forward price is F Futures 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 r i. The interest rate for day i is r i One dollar at the beginning of day i grows to by day send end. Ri e r i
Proof Let fi be the futures price at the end of day i. So $1 invested in the n-day discount bond at the end of day n will be worth $R 1 2 r n 1 r r e... e n = R R Rn = R = 1 2... t t=1 e Consider the follow strategy: i R t Long futures positions at the end of day i 1 and t = 1 invest the cash flow at the end of day i in riskless bonds maturing on delivery day n. R
0 i Rt Proof 買入單位期貨 ( 假設無保證金 t = 1 ) 在第 i 日的損益為 ( ) i f f R i i 1 t t = 1 i-1 i i+1 n 到期時得 $ ( ) f f R i i 1 i n n ( ) i i 1 t t= 1 ( f f ) = ( f f ) = ( f f ) f f R R f f R f f R i i 1 t t i i 1 t i i 1 t= 1 t= i+ 1 t= 1 i i+1 n 拿 $ f f R 再投資無風險債券
i-1 i 0 n Long a futures Forwards 可視為 n 個 futures 所組成! The value at the end of day n is n i= 1 ( ) = ( ) = ( ) f f R f f R S f R i i 1 n 0 T 0 Recall the value of R units of forwards is ( F )R S T 0 Abi Arbitrage opportunity exists if f0 F0.
Relationships between Forward and Future Prices 當利率隨機波動時, future price 和 forward price 的理論價格不等 一般而言, 兩者差距不大 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 F = 0 S0e In general, U stands for the PV of storage cost incurred during the life of a futures contract, and the rt futures price on commodity is F = S U ) e F = ( S 0 + U ) e rt 0 ( 0 + rt F 0 = ( S 0 I ) e I is the present value of income U is the present vaule of storage cost, U = I o rt
Futures on Commodities (convenience yield) Because user of the commodity may feel that ownership of the physical commodity yprovides benefit that not obtained by holders of futures contract. The benefit are the convenience yield (y) provided by yt rt the product. F e = ( S + U ) e 0 ( 0 + Ex: Oil refiner would like to take crude oil The convenience yield for investment asset=0 to prevent arbitrage.
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 F o = Se o CT
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=s 0 -F 0 Fe = Se F = Se yt CT ( C y) T o o o o In case of consumption assets, the futures price is greater than the spot price (basis<0) reflecting that the cost of carry.
Convergence of Futures Price to Spot Price (a) () 在正常市場下, 基差 (basis) 為負, 反應的就是標的資產的持有成本, 隨著到期日接近基差就會越來越小 (b) Spot Price Futures Price Futures Price Spot Price Time Time 思考 : 為何現貨價格會大於期貨價格 ( 逆向市場 )?
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, t when the hedger owns an asset and expects to sell it at some time in the future. (see example)
Example-Short Hedge 假設今天 (1 月 25 日 ) 銅的現貨價格為每磅 120 美分, 某一國外銅開採公司預計其將在 5 月 15 日賣出 100,000 磅銅, 且在 COMEX 交易的五月份到期 ( 到期日假設為 5 月 15 日 ) 之銅期貨價格為每磅 120 美元 ( 契約乘數 ), 其中每一口銅期貨合約大小為 25,000 磅 ( 假設不用繳保證金 ) 如果 5 月 15 日銅的現貨價格為每磅 105 美元, 該國外銅開採公司如何利用 COMEX( 紐約商品期貨交易所 ) 銅期貨合約來規避未來銅的價格風險? 其淨收益為何?
該國外銅開採公司擔心未來銅價格下跌, 導致生產收入減少, 故它可以賣出四口 COMEX 銅期貨合約來避險 Spot Market Futures Market 1/25: 現貨 100000 磅 1/25: 賣出四口 5 月份到期之銅期貨 價值為 $120 100000=12,000,000 5/15: 賣出現貨 100000 磅 5/15: 結算四口銅期貨 現貨價值減少為 $105 100000=10,500,000 結算的利得為 $(120-105) 4 25000=$1,500,000 Opportunity Loss Gain = $1,500,000 = $12,000,000-10,500,000,,, =$1,500,000 價格鎖定 120 美分
空頭避險損益圖 value Spot position i value 0 gain loss 120 Spot price 現貨價格雖然在 5/15 下跌, 然而銅公司透過放空期貨可規避此價格的損失 Short futures position
In class Exercise 如果到期時銅的現貨價格為每磅 125 美分, 請分析其現金流 思考 : 假如市場上沒有想要避險現貨的標的資產之期貨? 或是現貨的交易日與期貨的到期日不相同? 該怎麼辦?
完全避險的條件 從事避險之期貨標的物要和現貨商品一模一樣 到期日要等於避險沖銷日 對應於要避險之現貨部位, 所買賣的期貨口數為某一整數
期貨合約避險之合約選擇 交叉避險 (Cross Hedge): 實務上要找到與現貨完全相同的標的資產之期貨合約非常困難, 故乃以與現貨相關性高標的資產替代, 此種策略稱為交叉避險 避險者使用期貨合約避險, 其效果好壞決定於期貨價格和現貨價格相關性高低 因此避險者選擇期貨合約需考慮的因素有二 : (1) 期貨合約標的資產的選擇 (2) 期貨合約交割月份的選擇
最適期貨避險數量 決定最適的期貨合約數量有二種方法 : (1) 單純避險法 (Naive Hedge Method) (2) 最小變異數避險比率法 (Minimum Variance Hedge Ratio Method)
單純避險法 又稱完全避險法 (Perfect Hedge Method), 指避險者買進或賣出和欲避險之現貨部位金額相同, 但部位相反的期貨合約 假設基差風險不存在, 亦即現貨價格和期貨價格之變化是完全一致 公式 : 期貨合約口數 = 欲避險之現貨部位金額 每口期貨合約價值 實際上現貨價格的變化和期貨價格的變化未必會完全一致, 因此單純避險法並不一定能將現貨價格風險完全規避
Example 假設某一基金經理人持有價值新台幣 20 億的股票, 而台股指數期貨目前價格為 5000 點, 其每一點值新台幣 200 元, 那麼該基金經理人為了防止其現股價格下跌的風險, 他應該出售多少口台股指數期貨來避險? 每口台股指數期貨價值 = 5000 200 = 1,000,000 應出售的期貨合約口數 = 2,000,000,000 1,000,000= 2,000 ( 口 )
最小變異數避險比率法 定義 : 找出使避險投資組合風險 ( 變異數 ) 最小的避險比率的方法 又稱為迴歸分析法 假設 : 公式 : S: 避險期間內現貨價格的變動 F: 避險期間內期貨價格的變動 σ s : 現貨價格變動的標準差 σ F : 期貨價格變動的標準差 ρ: S 和 F 間的相關係數 h* ( 最小變異數避險比率 ) = σ ρ s σ F
最小變異數避險比率 如果 ρ=1 且 σ F = σ s, 則 h * =1 ( 此時最小變異數避險比率 = 單純避險法之避險比率, 因為期貨合約價格的變動和現貨價格的變動完全一致 ) 假設其他情況不變, 若 σ s 愈大或 ρ 愈大, 則 h 愈大 假設其他情況不變, 若 σ F 愈大, 則 h 愈小
最小變異數避險比率的推導 Suppose we expect to sell N A units of an assets at time t 2 and to hedge at time t 1 by shorting futures contracts on N F units of a similar asset. N -The hedge ratio is F h = S 1 and S 2 are the asset prices at time t 1 and t 2, and F 1 and F 2 are the futures prices at time t 1 and t 2. N A
最小變異數避險比率的推導最小變異數避險比率的推導 Total amount realized for the asset denoted by Y A N F F F N S Y ) ( 1 2 2 = When the variance of S h F is minimized the F A ) ( 1 2 2 ) ( ) ( ) ( 1 1 2 1 2 1 F h S N N S N F F N S S N S Y A A F A A Δ Δ + = + = When the variance of S-h F is minimized, the variance of Y is minimized. S S S F F S F S h F h S MinVar h h h F h S Var ρσ σ σ ρ σ ρσ σ σ ρσ σ σ = Δ Δ + = + = Δ Δ 0 ) ( ) ( ) ( 2 ) ( 2 2 2 2 2 2 2 2 F S S F h h σ σ ρ ρσ σ = = 0 ) (
避險者的部位之變異數和避險比率之關係 部位之變異數 h* 避險比率
Example 假設台股指數價格變動百分比和大台指期貨價格變動百分比的相關係數為 0.8, 而台股指數價格變動百分比之標準差為 0.6, 而期貨合約價格變動百分比之標準差為 0.4, 某一基金經理人持有價值新台幣 20 億的股票, 而台股指數期貨目前價格為 5000 點, 其每一點值新台幣 200 元, 那麼該基金經理人為了防止其現股價格下跌的風險, 他應該出售多少口台股指數期貨來避險? Ans: 最小避險變異數比率為 08 (0 0.8 (0.6 0.4)=1.2 4)=12 應出售的口數為 2000 1.2=2400( 口 )
Swaps Swaps are agreements by two parties to exchange cash flows in the future according to a prearranged formula. There are two basic types of swaps: interest rate swaps Currency swaps Swaps on commodities are also available.
Interest rate Swaps An example for a Plain Vanilla Interest rate Swaps An agreement by Microsoft to receive 6- month LIBOR & pay a fixed rate of 5% per annum every 6 months for 3 years on a notional principal of $100 million
Cash Flows to Microsoft ---------Millions of Dollars--------- LIBOR FLOATING FIXED Net Date Rate Cash Flow Cash Flow Cash Flow Mar.5, 2004 4.2% Sept. 5, 2004 4.8% +2.10 2.50 0.40 Mar.5, 2005 5.3% +2.40 2.50 0.10 Sept. 5, 2005 5.5% +2.65 2.50 +0.15 Mar.5, 2006 5.6% +2.75 2.50 +0.25 Sept. 5, 2006 5.9% +2.80 2.50 +0.30 Mar.5, 2007 6.4% +2.95 2.50 +0.45
Typical Uses of an Interest Rate Swap Converting a liability or an investment from fixed rate to floating rate floating rate to fixed rate
Intel and Microsoft (MS) Transform a Liability 5% 5.2% LIBOR+0.1% Intel MS LIBOR Suppose that Microsoft has arranged to borrow $100 million at LIBOR plus 10 basis points. Suppose that Intel has a 3-year $100 million loan outstanding on which it pays 5.2%.
Using the Swap to Transform a Liability For Microsoft, the swap could be used to transform a floating-rate loan into a fixed rate loan. After Microsoft has entered into the swap, it has the following three sets of cash flows: 1.It pays LIBOR plus 0.1% to its outside lenders. 2.It receives LIBOR under the terms of the swap. 3.It pays 5% under the terms of the swap.
Using the Swap to Transform a Liability For Intel, the swap could have the effect of transforming a fixed-rate loan into a floating rate loan. After Intel has entered into the swap, it has the following three sets of cash flows: 1.It pays 5.2% to its outside lenders. 2.It pays LIBOR under the terms of the swap. 3.It receives 5% under the terms of the swap.
Intel and Microsoft (MS) LIBOR-0.2% Transform an Asset Intel 5% MS 4.7% LIBOR Suppose that Microsoft owns $100 million in bonds that will provide interest at 4.7% per annum over the next 3 years. Suppose that Intel has an investment of $100 million that yields LIBOR minus 20 basis points.
Using the Swap to Transform an Asset For Microsoft, the swap could have the effect of transforming an asset earning a fixed rate of interest into an asset earning a floating rate of interest. After Microsoft has entered into the swap, it has the following tree sets of cash flows: 1It 1.It receives 47% 4.7% on the bonds. 2.It receives LIBOR under the terms of the swap. 3.It pays 5% under the terms of the swap.
Using the Swap to Transform an Asset For Intel, the swap could have the effect of transforming an asset earning a floating rate of interest into an asset earning a fixed rate of interest. After Microsoft has entered into the swap, it has the following tree sets of cash flows: 1.It receives LIBOR minus 20 basis points on its investment. 2. IT pays LIBOR under the terms of the swap. 3. It receives 5% under the terms of the swap.
Quotes By a Swap Market Maker Maturity Bid (%) Offer (%) Swap Rate (%) 2 years 6.03 6.06 6.045 3 years 6.21 6.24 6.225 4 years 6.35 6.39 6.370 5 years 6.47 6.51 6.490 7 years 6.65 6.68 6.665 10 years 6.83 6.87 6.850
Using Swap Rates to Bootstrap the LIBOR/Swap Zero Curve Consider a new swap where the fixed rate is the swap rate When principals are added to both sides on the final payment date the swap is the exchange of a fixed rate bond for a floating rate bond The floating-rate rate bond is worth par. The swap is worth zero. The fixed-rate bond must therefore also be worth par This shows that swap rates define par yield bonds that can be used to bootstrap the LIBOR (or LIBOR/swap) zero curve
假設 F= swap rate, P=par, si=i-th period (year) spot rate, N=# of swap periods, 1*1 2 *2 * * * s N * * s P F e + P F e +... + P(1 + F) e s N = P
Valuation of an Interest Rate Swap that is not New Interest rate swaps can be valued as the difference between the value of a fixed-rate bond and the value of a floating-rate bond Alternatively, they can be valued as a portfolio of forward rate agreements (FRAs)
#Homework 8 請分析如何使用拔靴法 (bootstrap method) 和市場新台幣利率交換 (Interest Rate Swap,IRS) 報價資料來計算零息債券的利率 (zero coupon rate), 並撰寫計算零息利率的程式 假定一年交換一次, 第 4,6 年資料用內插估計 到期日 ( 年 ) Bid (%) Offer (%) Swap rate (%) 1 225 2.25 227 2.27 226 2.26 2 2.26 2.29 2.275 3 2.27 2.30 2.285 5 2.33 2.38 2.355 資料來源 ( 此為參考資料 ) 2008 年 9 月 15 日 Swap rate 為買賣價之平均值 KGI 凱基證券網站 http://www.kgieworld.com.tw/bond/bond_01 _02.htm 7 2.41 2.47 2.44
Currency Swaps A currency swap involves two parties to exchange cash flows in different currencies. Consider the following fixed rates available to party A and party B in U.S. dollars and Japanese yen: Dollars Yen A DA% YA% B DB% YB% Suppose A wants to take out a fixed-rate loan in yen, and B wants to take out a fixed-rate loan in dollars.
Currency Swaps A straightforward scenario is for A to borrow yen atya% and B to borrow dollars at DB%. But suppose A is relatively more competitive ii in the dollar market than the yen market, and vice versa for B. USD JPY YB + DA < DB +YA. BA(A) 3% 1% Compare spread Alternative arrangement: JAL(B) 5% 2% A borrows dollars. B borrows yen. They enter into a currency swap with a bank as the intermediary.
Currency Swaps (concluded) The counterparties ti exchange principal i at the beginning and the end of the life of the swap. This act transforms A s dollar loan into a yen loan and B s yen loan into a dollar loan. The total gain is ((DB DA) (YB YA))%: The total interest rate is originally (YA + DB)%. ) The new arrangement has a smaller total rate of (DA + YB)%. Transactions will happen only if the gain is distributed so that the cost to each party is less than the original.
Swaps & Forwards A swap can be regarded as a convenient way of packaging forward contracts The value of the swap is the sum of the values of the forward contracts underlying the swap Swaps are normally at the money initially This means that it costs nothing to enter into a swap It does not mean that each forward contract underlying a swap is at the money initially