財務筆記: Derivatives Valuation - Forwards & Futures (part 30) @ 經濟,財務,統計學,數理科學與政治評論

寫在前面:我們再過來將介紹衍生性金融商品的四個基本積木(four fundamental building blocks):Forward, Futures, Options and Swaps,與其相關的利率衍生性金融商品與其相對應財務風險管理的應用,然後我們會回頭談Alternative investing中關於Commodity與Hedge Funds的投資策略與績效評估,接著談Portfolio Management,談到海外投資的績效評估中我們會談到International CAPM,這個部分會與Economics&finance中常用的幾個parities(PPP,IRP, Fisher effect等等一起討論),最後我們會談Econometics,經濟成長理論,與Private Equity Valuation(這個部份算是Alternative Asset Valuation)與該重視的其他財務理論與實務!!

Hello there:

這裡先談Forward與Futures的評價與異同!!

I. Forward是交易雙方根據自身的需求,在契約內明定未來某一個特定時點,交易的數量,價格,交易品的品質等等內容,來進行交易的契約!!Forward在一開始到結束之前都沒有實際的現金流,直到交割那天才進行履約的動作,因此需要面對相當大交易對手的違約風險!!再進行Forward contract的訂價時,有幾點是需要注意的,因為Forward contract是交易雙方進行約定,在到期日時才進行交易,因此持有forward在day zero的價值為零!!另外我們總假定交易者能夠在一個完美市場中進行交易,稅負與其他因子都暫不考量,而所有的資訊成本都相當低或者已經充分反映在影響價格的因子當中!!而我們使用的計算原則一般稱之為無風險套利原則(Risk-free Non-arbitrage principle),簡單的說若是在期末時發現兩個投資組合會產生出完全相同的現金流量,則這兩個投資組合在期初時其價值將是相同的,因為兩者均採用相同的無風險利率進行折現!!

我們先談幾個常用的Forward contracts與其相關計算公式!

a. the equity forward contract

(1) at the beginning day:

F(0,T) = (So - Present Value of Dividends)*(1+Rf)**T

F(0,T) = (So - PVD)*exp(Rfc*T)

(2) at the specified day t:

long position: Vt = (St - PVD(t,T)) - F(0,T)/[(1+Rf)**(T-t)]

or Vt = (St - PVD(t,T)) - F(0,T)*exp(-Rfc*(T-t))

b. the stock index forward contract

(1) at the begining day:

F(0,T) = So*exp[(Rfc+SC-CY)*T]

其中 Rfc:Continuous-type interest rate,

SC:Storage cost,

CY:Convenience yield.....

(2) at the specified day t:

long position: Vt = St*exp[(SC-CY)*(T-t)] - F(0,T)*exp[-Rfc*(T-t)]

c. the fixed-income forward contract

(1) at the beginning day:

F(0,T) = (So - PVC)*(1+Rf)**T

= (So - PVC)*exp(Rfc*T)

其中 PVC = present Value of Coupon

Rf = discrete risk-free interest rate

Rfc = continuous risk-free interest rate

(2) at the specfied day t:

long position: Vt = [St - PVC(t,T)] - F(0,T)*exp[-Rfc*(T-t)]

c': the fixed-income forward contract

(1)' at the beginning day with coupons CI:

F(0,T) = [Bo(T+Y) - PV(CI,0,T)]*(1+Rf)**T

Forward price

= (Bond price - Present value of coupons over the life )*(1+Rf)**T

= (Bond price)*(1+Rf)**T - Future value of coupons over life of contract

(2)' at the specific dat t:

long position: Vt = Bt(T+Y) - PV(CI,t,T) - F(0,T)/(1+Rf)**(T-t)

d. FRA(Forward Rate Agreement)

(1) at the beginning day, forward rate (or price):

FRA(0,h,m)

= [([1 + Lo(h+m)*(h+m)/360]/[1 + Lo(h)*(h/360)]) - 1]*(360/m)

(2) at the day g, the value of FRA:

Vg(0,h,m)

= 1/[1 + Lg(h-g)*(h-g)/360]

- [1 + FRA(0,h,m)*(m/360)]/[1 + Lg(h+m-g)*(h+m-g)/360]

(3)at the expiration, the FRA payoff:

( Notional principal*

[(Underlying rate at expiration - Forward contract rate)*

(Days in the underlying rate/360)] )

= --------------------------------------------------------------------

[ 1 + Underlying rate at expiration*

(Days in the underlying rate/360) ]

V(0,h,m) = [Lh(m) - FRA(0,h,m)]/[1+Lh(m)*(m/360)]*(m/360)

where forward contract rate is the rate the two parties agree will be paid and days in the underlying rate is the number of days to maturity of the instriment on which the underlying rate is based...

e.Foreign currency forward contract

(A direct quote: units of the domestic currency per unit of foreign currency)

(1) at the beginning day:

F(O,T) = So*[(1+R(L))**T]/[(1+R(F))**T]

where (we use the direct quote)

R(L) is the local interest rate

R(F) is the foreign interest rate

(2) at the specific day t:

long position: Vt = St/[(1+R(F))**(T-t)] - F(0,T)/[(1+R(L))**(T-t)]

我們展現的是一般稱之為的Covered Interest Rate Parity: With a direct quote, the forward rate exceeds the spot rate as the domestic interest rate exceeds the foreign interest rate!!

(小結)另外值得注意的是,當我們在計算FRA時,我們採用一年360天當作計算,一個月30天當作計算基礎,而Zero Coupon Bond也是360天/一年;而到後面談Interest Rate Swaps時也是採用360天/一年!!其他的Futures比如Equity, Currency, Options,與傳統有附息的債券,都採用的是365天/一年為基礎!!

II. 再過來我們談Futures.....

先談Futures的交易機制,futures主要是在交易所交易,因此其交易對手的違約風險,可以透過交易所每天結算其所持有的期貨部位損益,來進行相對應的風險管理,而管理的基礎在於Mark-to-market,透過買賣雙方所繳交的起始保證金(initial matgin)與後面作為計算基礎的維持保證金(maintenance margin)來幫助結算的機能,避免信用風險的產生!!另外值得注意的是,在債券期貨上面還需要注意Cheapest-to-deliver的機能與最後結算的相關,這個部份就有賴您自行參閱了!!另外一些定義也是:Backwardation (Fo < So), Contango (Fo > So), Normal Contango (Fo > E(S(T) with expectation), Notmal backwardation (Fo < E(S(T) with expectation)!!一般在做期貨價格計算時,與上述的Forward contract都大同小異,但前提是利率水準的變動是已知或完全可以預期!!!

a.期貨與遠期契約在到期日之前的任一天價格差異:

一般說來,當利率水準在契約簽訂至到期日之間都是常數或為已知時,期貨與遠期契約在到期前任一天的交易價格差異不大!!(1)當利率變動和期貨價格呈現正相關(positively correlation)時,持有期貨者(the long position of futures)相對於遠期契約持有者來說是比較有利的,因為當利率上升時,期貨能夠因為有Mark-to-market的機能將資金從期貨部位取出,再投入其他利率高的金融工具中,而當利率走貶時,期貨多頭者能夠去金融市場上面借到比較便宜的資金來彌補其損失;此時,相對於遠期契約,期貨價格是較高的!!一般說來,股票期貨或股票遠期契約的關係是可能出現這類情況的!!(2)當利率變動與期貨價格呈現負相關時,交易者將偏好遠期契約而非期貨,理由就與前者相反,因為當期貨價格上漲而利率下跌時,我們將出現從期貨部位抽取出的資金只能用更低的利率去投資,因此相對上面我們反倒會希望資金部位沒有這類再投資的風險,因此遠期契約價格將高於期貨價格!!一般說來債券價格與利率呈反向關係,因此債券期貨自然比債券遠期契約來得價格低!!

b.T-Bill Futures (you may skip this one)

fo(h)

= Bo(h+m)/Bo(h)

in terms of spot T-bills

as Bo(h) = 1 - rd(h)*(h/360)

= B0(h+m)*[1+Ro(h)]**(h/365)

It gives the discount rate

r(h) = [1 - fo(h)]*(360/m)

= [1 - ([1 - Ro(h+m)*(h+m)/360]/[1 - Ro(h)*h/360])]*(360/m)

c.Eurodollar Futures (you may skip this one)

(1) The futures price at expiration

Fh(h) = 1 - Lh(m)*(m/360)