contracts underlying assets 1 forwards 2 futures 3

contracts underlying assets 穀物 畜肉 金屬 能源 1. forwards 2. futures 3. options commodity financial asset 4. swaps derivatives matrix 股票、股票組合 通貨 利率(國庫券、公債) 例︰us$ forward , T-bill future, 台股期貨 特性︰載舟（hedging、high return） trade-off 覆舟（high risk）

right and obligation Fd 買方 （long） Fd 賣方 （short） cash flow at T 交付Fd價格，取得資產 -F(0, T)+S(T) 交付資產，取得Fd 價 -S(T)+F(0, T)

value at T 45° long Fd 0 F(0 ) ‧ S(T) short Fd 45° zero-sum game between long and short traders. ∵ Fd買方或賣方在(立約時點, T)之間無任何 cash flow 0 - F(0, T) + S(T) 之現值 ＝ 0

Why standardized and daily settlement ? transaction cost liquidity credit risk secondary mkt. delivery large turnover

6. 基差（basis） basis = spot future = S(t) F(t, T) storage cost and positive cash flow → basis 持有現貨不利 持有現貨有利（如 ︰原料欠缺） 基差收斂原則 F(T, T) = S(T) i. e. basis(T) = 0 if no transaction cost. Why ? arbitrage !

4. agents pay off 45° 0 k ap 0 45° k k 0 k 45°

5. factors factor premium ap k call + - + put Interest volatility maturity rate + - + +

future value 如何改成現值？ 每年m次複利，則t年后一元之 PV 二、PV & FV of cash flow stream time 0 now 1 2 n

Ch 2 Simple Arbitrage Relationships for Fd and Ft Contracts 一、Introduction 二、Forward and Spot Prices (一) no cash flow on the underlying asset (二) cash flow embedded the underlying asset (三) forward Contracts on Commodities

三、forward and future Prices ISSUE : Spot and Forward Prices CORE : no - arbitrage – opportunities (N-A-O) ISSUE : Forward and Future Prices

一、Introduction (一) arbitrage A-O no commitment now but possible gain after type(A) gain now and no loss after tpye(B) N-A-O Pricing securities mkt equil & END No START A-O? yes exploit A-O ( buy low , sell high ) prices changes

(二) assumptions A 1: perfect ( frictionless) A 2: no credit risk of counterparty A 3: competitive price A 4: A-O changes N-A-O

二、Forward and Spot Prices (一) no cash flow on the underlying between [ 0 , T] zero-coupon , non-dividend stock S(t); spot price of underlying asset at t F(t, T); forward (delivery) price of Fd contract issued at t and matures at T B(t, T); T時點確定（risk-free）之1元, 以t點貨幣表 示之 價值。T貨幣折現至t時點的無險折現因子

1. S(t) v. s F( t, T ) 2. Strategy I(date 0) CF(0) 3. (a) long forward contract 0 S(T)- CF(T) F(0, T) 4. F(0, T) 5. Logic of duplicate cash flow stream of Strategy I ? 6. Strategy Ⅱ(date 0) 7. (b) buy underlying asset - S(0) S(T) 8. (c) borrowing S(0) at riskless 9. interest rate and mature at T to get S(0) now 10. （i. e short-sale riskless asset） 1(0, T) S(0) 0 - S(0)B-1(0, T) S(T)-S(0)B-

兩策略在T 時點前CF 均是 0，只有在T各發生一筆CF > if CF(T) of S-II < F(0, T) < > S(0)B-1(0, T) What will happen in the mkts ? When will this A-O vanish ?

Ans : F(0, T) = S(0)B-1(0, T) in N-A-O setting or S(0) = F(0, T)‧B(0, T) CF(T) (b) buying underlying at 0 what is (b)+(c) ? 45° 0 -S(0)B-1(0, T) S(T) (c) borrowing

generalization : S(t)= F(t, T)B(t, T) (1) <Throrem 1> 1. 無險借入資金買現貨 long forward 賣空無險資產+買現貨 long forward synthetic forward 買入無險資產+賣空現貨 short forward 2. 賣空現貨投資無險資產 short forward

<Theorem 2> F(t, T) > S(t) and 直覺 why ? robustness of Theorem 1 and 2 : no cash flow underlying e. g. S(0)=$45 市場上90天Fd 報價＄46. 54 B（0, 90)=(1+0. 0485× 90/365)-1 ← 簡單利息計價 90天期年化無險利率 Fd理論價格 F(0, 90) (1 )= S(0)B-1(0, 90) = 45(1+0. 0485× 90/365) = $45. 54

可行操作策略（at date 0） (1) 市場short Fd risk if S↑ CF(0) (2) (i) 合成遠貨 1(0, 90) CF(90) 0 S(90)-S(0)B- （無險借入S(0), 買現貨） (ii) 市場賣出Fd 0 46. 54 - S(90) 0 45. 54 S(0)B- 1(0, 90) = 46. 54 45. 54

2. value of a forward contract Vt(0, T) ; value of Fd（0, T; F(0, T)）at t time pt. issuing maturity forward (delivery) price F(t, T) ; forward price of Fd （t, T; F(0, T)）at t time pt. (delivery) time pt. Comparing to option , F(t, T) in Fd Vt(t, T) in Fd in option

<Theorem 3> V(T)= [ F(t, T) - F(0, T)]B(t, T) … (2) implication : 同到期日（期限不同），同標的資產之 Fds. > 后發行之Fd的forward price = 早發行之Fd < 的forward price > if. f. 前發行之Fd契約價值<＝ 0 直覺 why？ robustness of <Th. 3> : Equally applied to no cash flow with cash flow embedded underlying

application : Fd 1 3/1(t=0)發行 180天（T＝ 9/1）u. s$DF, F(0, 180) Fd 2 6/1(t=0)發行 90天 （T＝ 9/1）u. s$DF, F(90, 180) what is the intrinsic value of Fd 1 at 6/1 ?

(二) known cash flow to the underlying d(t 1) ; CF of underlying at t 1 known today time long underlying long 0 t 1 T - S(0) d(t 1) S(T) Fd 0 0 - S(0) d(t 1) S(T)-F(0, T) Strategy I (date zero) (a) long underlying (b) risk borrowing d(t 1)B(0, t 1) - d(t 1) S(T) 0 PV 0(d(t 1)) matures +） at t 1 CF of S-I d(ti)B(0, t 1)-S 0 0 S(T)

Strategy Ⅱ(date zero) (c) long Fd (d) riskless invest 0 0 -F(0, T)B(0, T) S(T)-F(0, T) 0 S(T) PV 0(F(0, T)) matures +）at T CF of S -Ⅱ -F(0, T)B(0, T) 比較兩策略之CF知 N-A-O → F(0, T)B(0, T) = S(0) d(ti)B(0, t 1)

To generalize equation (3) , we get <Theorem 4> F(t, T)B(t, T) = S(t) PVt（all CFs over (t, T)） Discussion : 1. (4) 式亦可適用於 no CFs 之(1) 2. 如果(t, T)間存在d(t 1), d(t 2), … d(tm) 多筆CFs (可能是正或負), 則

§. value of a Fd contract (4) 此一Fd契約評價公式與 no CF 時相同﹗why？

2. Fd on foreign exchange S(t); spot exchange rate at t B(t, T); PVt(certainty one NTD at T) in terms of NTD

S-I (date 0) 0 S(T)-F(0, T) -B$(0, T)S(0) S(T) F(0, T)B(0, T) -F(0, T) (a) long a $ Fd S-Ⅱ(date 0) CF of S-Ⅱ F(0, T)S(0, T)-B$(0, T)S(0) Comparing CF of S-I & S-Ⅱ N-A-O F(0, T)B(0, T) = B$(0, T)‧S(0) S(T)-F(0, T)

3. concise representation of CIRP i. T ; Tw (t, T) 期限之無險利率 iu. s ; u. s (t, T) 期限之無險利率 . .

4. application of CIRP short run determination foreign exchange rate. 5. contract value. 同前 Try to derive it !

三、Forward Price versus Future Price 持有Fd和持有Ft有何差異？和標的是否和cash flow 有關？ Short – Term Rate 0 1 2 Long – Term Rate

FORWARD CONTRACT Forward Price 0 DATE 1 DATE 2 Cash Flow FUTURES CONTRACT Futures Price Cash Flow 0 F (0, 2) 0 DATE 1 F (1, 2) – F (0, 2) DATE 2 S(2) S (2) – F (1, 2)

strategy I 0 1 (a) long Fd (date 0) 0 0 0 F (1, 2) strategy 2(T) S(T)-F(0, 2) Ⅱ (b) long Ft (date 0) F (0, 2) F (T) F (1, 2) (c) 將date 1之收入(損失) 投資riskless asset(無險 0 -（F (1, 2)-F (0, 2)）（F (1, 2)-F (0, 2 借入) 到期日 2(=T) 兩策略期末cash flow 比較有何差異？ Ans : S -Ⅱ以 0 觀點有再投資風險

<Theorem 5> F(t, T)=F (t, T) - B(t, T)-1 × PVt （all margin account interest gains of Ft over (t, T)） <Lemma> 契約存續期間，如果期貨與利率同(反)向變化。 則到期價大于(小于)遠期價，若Cox & Ingersoll & Ross(1981)JFE ; farrow & cidfield(1981)JFE <Theorem 6> If all future interest rates are certainly known at t → F(t, T) = F (t, T)

proof : F 0 F 1 F 2 F 3 Fk F k+1 0 1 2 3 k k+1 T define F(t, T) ≡F t Strategy A part 1 date 0 : 買B(1, T)單位Ft（i. e. 本期買進B(2, T) -B(1, T) 單位Ft） date 1: 部位調整至B(2, T)單位Ft date 2: 部位調整至B(3, T)單位Ft date k : 部位調整至B(k+1, T)單位Ft date T-1 : 部位調整至B(T, T) =1單位Ft

Strategy 2 : long Fd ( date 0 ): 此一策略只有到期日產生 S(T) F(0, T)的cash flow. 以上兩策略現值應相等而為 0. F（0, T）＝ F（0, T） price of Ft contract ? Q. E. D.

(三) Fd 的期中價格 此一結果和標的是否有cash flow 無關 二、Forward Price versus Future Price F F F