April 2014
Beginner to intermediate
830 pages
15h 21m
Chinese
Content preview from Mathematica 錦囊妙計
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582
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第十四章
編譯完成後，僅需 18.5 毫秒便可對 10 個不同履約價格進行定價。[Page-584]
討論說明
americanPutCompiled
函數回傳 2 個緊實陣列（packed array），第一個為股價格點，第
二個則是各股價所對應的美式選擇權價格，使用 Mathematica 提供的
Interpolation
指
令建立內插函數便可取得非格點的中間股價之選擇權價格。
藉由檢查
americanPutCompiled
的第 4 個元素，可得知此函數是經過充分編譯的，因為
第 4 個元素內含的所有元素均是數值型態。
本例所用的美式選擇權定價演算法屬於自由邊界值問題（free boundary value problem）
的線性互補形式（linear complementarity），參數
a
、
nn
、
T0
與
mm
（及其對應的
s
與
h
）
用來設定格點的分布，
a
與
nn
決定空間（即股價）格點，
T0
與
mm
則決定時間格點。對
顯式數值問題而言，兩方向的格點分布須滿足特定條件，否則數值運算將無法收斂至正
確解，完全顯式的數值方法須滿足
alpha=s/h^2<=1/2
不等式，這意味著如果空間的步長
減為十分之一，則時間步長便須減為百分之一，這並非精確度問題，而是穩定性問題。
舉例而言，如果
mm
值降低為 15，則
alpha
不等式便無法成立，數值解的不穩定性便十
分明顯，若降低至 10 甚至是 5，求解過程更是一場災難。傳統的美式選擇權計算定價時
使用的是二項樹法（binominal tree），但卻存
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