Abstract
摘要:衍生性金融商品(derivative)為一種價值決定於某基本資產(如股票)的金融商品。基本資產價格為隨機程序,而衍生性金融商品定價問題相當於計算其收益之數學期望值。路徑相關型(path-dependent)衍生性金商品的價值取決於基本資產價格之全部或部分歷史,這方面的研究為財務計算(computational finance)領域的核心。我們的三年計畫將探討學術上重要或市場上有交易的路徑相關型衍生性金商品之定價。我們將路徑相關型衍生性金融商品分成兩類。
第一類衍生性金融商品有多項式時間演算法,如幾何型亞洲選擇權、障礙選擇權、巴黎式選擇權、重設選擇權、幾何平均重設選擇權等等(細節請見計畫內容)。計畫第一個目標將利用組合數學與特殊演算法技巧,設計與實作最快的計價演算法。 資訊科學在這方面可以對財務計算領域產生很大貢獻。在初步文獻蒐集中我們發現財務文獻充斥著不完全正確的演算法,很多錯誤來自於計算者的電腦實驗不夠嚴謹,因此計畫第二個目標就是更正這些嚴重錯誤。大部分演算法用離散時間模型來逼近連續時間模型,離散化產生之誤差導致演算法收歛很不規則,此現象一旦出現會使任何訂價演算法的結果受到極大質疑
Abstract: Derivatives are financial instruments whose payoff depends on some underlying asset such as stock. The underlying asset’s price is a stochastic process. Pricing such instruments amounts to calculating the expected value of their payoff. Path-dependent derivatives are derivatives whose payoff depends nontrivially on the history of the underlying asset’s prices. Pricing such derivatives is the core of computational finance. This three-year proposal intends to investigate the pricing of important and popular path-dependent derivatives found in the market and/or literature. We divide the derivatives into two categories.
The first category contains those derivatives with a polynomial-time pricing algorithm. Examples include geometric Asian options, barrier options, Parisian options, reset options, geometric-average-trigger reset options, and many others (see the proposal for more details). Our first goal is to design and implement the fastest pricing algorithms possible using combinatorics and special algorithmic techniques. We have encountered many incorrect pricing algorithms in the finance literature during our preliminary literature survey. Many errors arise because the computational scientists did not conduct their experiments rigorously. The second goal is hence to correct published results in the literature that are erroneous. Most algorithms are discrete-time approximations to the continuous-time model. But because of discretization error, the algorithms can converge in erratic ways. This phenomenon undermines any algorithms whether they are efficient or not. Understanding the behavior has been a key research topic in computational finance. Almost all the derivatives we chose above display such behavior. Our third goal is to research intensively what makes their convergence so erratic and how to handle them. The fourth goal is to search for a very general closed-form formula as a multi-dimension integration that can price a large class of derivatives. The formula can serve as a unifying pricing framework and a benchmark in the study of numerical methods. A correct formula can also point to a way to price the derivatives using Monte Carlo simulation. Again, we have found several erroneous formulas in the literature in this respect, and we shall correct them.
The second category of path-dependent derivatives contains those for which efficient pricing algorithms are not available. Derivatives in the second category are hence hard to price exactly. The most prominent example is the arithmetic Asian option. The payoff of an Asian option depends on the arithmetic average of the asset prices. Pricing this option accurately is one of the most important research issues in computational finance.
This proposal will investigate the pricing of arithmetic Asian options via approximation algorithms. First, we shall design and implement efficient algorithms with a guaranteed upper bound on the pricing error better than those found in the literature. Second
Keyword(s)
財務計算
衍生性金融商品
演算法
幾何型亞洲選擇權
障礙選擇權
巴黎式選擇權
重設選擇權
幾何平均重設選擇權
路徑相關型衍生性金商品
組合數學
多重積分
蒙地卡羅模擬
算術型亞洲選擇權
逼近演算法
computational finance
financial derivative security
algorithm
geometric Asian option
barrier option
Parisian option
reset option
geometric-average-trigger reset option
path-dependent derivative
combinatorics
multi-dimension integral formula
Mon