April 2014
Beginner to intermediate
830 pages
15h 21m
Chinese
Content preview from Mathematica 錦囊妙計
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14.9 使用有限差分法（Finite Difference Method）計算 Black-Scholes 公式
|
577
以下計算所使用的是顯式
有限差分法
（
finite difference method
，FDM）。此方法中，衍
生性金融商品
V
j,i-1
的價值將從
V
j+1,i
、
V
j,i
與
V
j-1,i
計算而得，原理將於「討論說明」
部分闡述。[Page-579]
本例中，股價的離散格點數目
n
可任意指定，增加格點的數目可提升準確性，但代價是
提高計算時間。以下計算中的格點數目設為 20。
股價計算區間應該要包含目前股價，但格點不宜過密，本例所使用的計算區間為 0 至履
約價格的兩倍，由此可算出格點間距
δS
。使用
NestList
指令可建立格點串列，程式碼
中的
#1 + δS &
是一個用來進行逐步疊加的純函數。
接著對股價的格點串列中的各點進行履約動作，以此串列作為初始條件。
X
X
顯式有限差分法的收斂條件是要有足夠大的時間步數，取決於時間步長、股價、波動
性及履約價格。時間步數可以下式獲得（詳細推導請見「延伸資訊」中 Wilmott 所著專
書）
時間步長則是：
在下 列
pricingFunc
定價 函數中，Γ 與 Δ 兩項將 直接 影響計 算的速 度，因為 這兩 項
均 位 於
Do
迴 圈 中，Mathematica 內 建 的
ListConvolve
指 令 可 以 很 有 效 率 的 執 行 有 限
差分 的計算。
Do
迴圈 結束後， 串列
V
內含 選擇權 的價 值，分別 對應 至各股 價格 點，
Interpolation ...
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