2010年4月1日 星期四

[衍生商品] Futures and Forward Pricing - No-Arbitrage Pricing Examples

這次要介紹 期貨與遠期契約的定價:

首先定義需要用到的符號:

$S_0:=$ 當前股價 (at time $0$)
$S_T:=$ 到期股價 (at time $T$)
$F_0:=$ 當前期貨價格(at time $0$)
$F_T:=$ 到期期貨價格 (at time $T$)
$T :=$ 到期時間 (以年計算)
$r:=$ 無風險年利率 (連續複利)
$D:=$ 配發股息
$q:=$ 配法股息利率 (連續複利)

無套利機會 (No-Arbitrage Opportunity ) 的遠期契約價格
\[
F_0^* = S_0 e^{(r-q)T}
\]上式等價
\[
F_0^* e^{rT}= S_0 -PV(D) + PV(Cost)
\]

現在考慮下面例子:

Example 1
考慮一個六個月股票遠期契約,且配發年股息 $3.96 \%$,當前股價為 $25$,無風險年利率為 $10 \%$,且股息與利率皆為連續複利。

(a) 試求 無套利價格 $F_0^*=$?
(b) 假設 $F_0=27$,是否存在套利機會?

Solution (a)
首先改寫已知資訊
\[
T=6/12, q=0.0396, S_0=25, r=0.1
\]計算無套利價格
\[
F_0^* = S_0 e^{(r-q)T} = 25 e^{(0.1-0.0396) \times \frac{6}{12}} = 25.766
\]

Solution (b)
由於 $F_0 = 27 > F_0^* = 25.766$,故存在套利機會 (存在買低賣高的機會):
也就是說 現階段的 遠期契約價格高於合理價格,我們可以賣出 (Short) 此遠期契約,並透過借款買入當前股票達成套利
\[\begin{array}{l}
\begin{array}{*{20}{c}}
{}&{Today}&{At\begin{array}{*{20}{c}}
{}
\end{array}Expiration}\\
\hline
{Short\begin{array}{*{20}{c}}
{}
\end{array}Forward}&0&{ + {F_0} - {S_T}}\\
{Long\begin{array}{*{20}{c}}
{}
\end{array}Stock}&{ - {S_0}{e^{ - qT}}}&{{S_T}}\\
{Borrow\begin{array}{*{20}{c}}
{}
\end{array}\$ }&{ + {S_0}{e^{ - qT}}}&{ - \left( {{S_0}{e^{ - qT}}} \right){e^{rT}}}\\
{Total\begin{array}{*{20}{c}}
{}
\end{array}Gain}&0&{{F_0} - {S_0}{e^{\left( {r - q} \right)T}}}
\end{array}\\
\\
 \Rightarrow \begin{array}{*{20}{c}}
{}&{Today}&{At\begin{array}{*{20}{c}}
{}
\end{array}Expiration}\\
\hline
{Short\begin{array}{*{20}{c}}
{}
\end{array}Forward}&0&{ + 27 - {S_T}}\\
{Long\begin{array}{*{20}{c}}
{}
\end{array}Stock}&{ - 24.51}&{{S_T}}\\
{Borrow\begin{array}{*{20}{c}}
{}
\end{array}\$ }&{ + 24.51}&{ - 25.766}\\
{Total\begin{array}{*{20}{c}}
{}
\end{array}Gain}&0&{ 1.2340}
\end{array}
\end{array}\]



Example 2
考慮一個 10 個月股票遠期契約,當前股價為 $50$,且已知會在第6個月配發股息 $5$ 元。且6個月無風險年利率為 $4 \%$,10個月無風險年利率為 $5 \%$ 。皆為連續複利。

(a) 試求 無套利價格 $F_0^*=$?
(b) 假設 $F_0=52$,是否存在套利機會?
(c) 假設 $F_0=45$,是否存在套利機會?

Solution (a)
首先改寫已知資訊
\[
T=10/12, D=5, S_0=50, r_{10}=0.05, r_{6}=0.04
\]計算無套利價格
\[\begin{array}{l}
F_0^*{e^{rT}} = {S_0} - PV(D) + PV(Cost)\\
 \Rightarrow F_0^*{e^{0.05 \times \frac{{10}}{{12}}}} = 50 - 5{e^{ - 0.04 \times \frac{6}{{12}}}} + 0\\
 \Rightarrow F_0^* = 47.02
\end{array}\]

Solution (b)
由於 $F_0 = 52 > F_0^* =47.02$,故存在套利機會 (存在買低賣高的機會):
現階段的 遠期契約價格高於合理價格,我們可以賣出 (Short) 此遠期契約,並透過借款買入當前股票達成套利
\[\small{\begin{array}{l}
\begin{array}{*{20}{c}}
{}&{Today}&{6\begin{array}{*{20}{c}}
{}
\end{array}month}&{At\begin{array}{*{20}{c}}
{}
\end{array}Expiration\left( {10\begin{array}{*{20}{c}}
{}
\end{array}month} \right)}\\
\hline
{Short\begin{array}{*{20}{c}}
{}
\end{array}Forward}&0&0&{ + {F_0} - {S_T}}\\
{Long\begin{array}{*{20}{c}}
{}
\end{array}Stock}&{ - {S_0}}&{ + D}&{{S_T}}\\
{Borrow\begin{array}{*{20}{c}}
{}
\end{array}\$ }&{ + {S_0} - D{e^{ - {r_4}T}}}&0&{ - \left( {{S_0} - D{e^{ - {r_4}T}}} \right){e^{{r_{10}}T}}}\\
{borrow\begin{array}{*{20}{c}}
{}
\end{array}\$ D}&{ + D{e^{ - {r_4}T}}}&{ - D}&0\\
{Total\begin{array}{*{20}{c}}
{}
\end{array}Gain}&0&0&{ + {F_0} - \left( {{S_0} - D{e^{ - {r_4}T}}} \right){e^{{r_{10}}T}}}
\end{array}\\
\\
 \Rightarrow \begin{array}{*{20}{c}}
{}&{Today}&{6\begin{array}{*{20}{c}}
{}
\end{array}month}&{At\begin{array}{*{20}{c}}
{}
\end{array}Expiration\left( {10\begin{array}{*{20}{c}}
{}
\end{array}month} \right)}\\
\hline
{Short\begin{array}{*{20}{c}}
{}
\end{array}Forward}&0&0&{ + 52 - {S_T}}\\
{Long\begin{array}{*{20}{c}}
{}
\end{array}Stock}&{ - 50}&{ + 5}&{{S_T}}\\
{Borrow\begin{array}{*{20}{c}}
{}
\end{array}\$ }&{50 - 4.9}&0&{ - \left( {45.1} \right){e^{{r_{10}}T}}}\\
{borrow\begin{array}{*{20}{c}}
{}
\end{array}\$ D}&{4.90}&{ - 5}&0\\
{Total\begin{array}{*{20}{c}}
{}
\end{array}Gain}&0&0&{4.9811}
\end{array}
\end{array}}
\]
Solution (c)
由於 $F_0 = 45 < F_0^* =47.02$,故存在套利機會 (存在買低賣高的機會):
現階段的 遠期契約價格低於合理價格,我們可以買入 (Long) 此遠期契約,並透過賣出當前股票達成套利
\[\small{\begin{array}{l}
\begin{array}{*{20}{c}}
{}&{Today}&{6\begin{array}{*{20}{c}}
{}
\end{array}month}&{At\begin{array}{*{20}{c}}
{}
\end{array}Expiration\left( {10\begin{array}{*{20}{c}}
{}
\end{array}month} \right)}\\
\hline
{Long\begin{array}{*{20}{c}}
{}
\end{array}Forward}&0&0&{ + {S_T} - {F_0}}\\
{Short\begin{array}{*{20}{c}}
{}
\end{array}Stock}&{ + {S_0}}&{ - D}&{ - {S_T}}\\
{invest\begin{array}{*{20}{c}}
{}
\end{array}\$ }&{ - \left( {{S_0} - D{e^{ - {r_4}T}}} \right)}&0&{ + \left( {{S_0} - D{e^{ - {r_4}T}}} \right){e^{{r_{10}}T}}}\\
{invest\begin{array}{*{20}{c}}
{}
\end{array}\$ D}&{ - D{e^{ - {r_4}T}}}&{ + D}&0\\
{Total\begin{array}{*{20}{c}}
{}
\end{array}Gain}&0&0&{ + \left( {{S_0} - D{e^{ - {r_4}T}}} \right){e^{{r_{10}}T}} - {F_0}}
\end{array}\\
\\
 \Rightarrow \begin{array}{*{20}{c}}
{}&{Today}&{6\begin{array}{*{20}{c}}
{}
\end{array}month}&{At\begin{array}{*{20}{c}}
{}
\end{array}Expiration\left( {10\begin{array}{*{20}{c}}
{}
\end{array}month} \right)}\\
\hline
{Long\begin{array}{*{20}{c}}
{}
\end{array}Forward}&0&0&{ + {S_T} - 45}\\
{Short\begin{array}{*{20}{c}}
{}
\end{array}Stock}&{50}&{ - 5}&{ - {S_T}}\\
{invest\begin{array}{*{20}{c}}
{}
\end{array}\$ }&{ - \left( {45.1} \right)}&0&{ + \left( {45.1} \right){e^{0.05 \times \frac{{10}}{{12}}}}}\\
{invest\begin{array}{*{20}{c}}
{}
\end{array}\$ D}&{ - 4.90}&{ + 5}&0\\
{Total\begin{array}{*{20}{c}}
{}
\end{array}Gain}&0&0&{2.0189}
\end{array}
\end{array}}\]