Abstract
摘要:衍生性金融商品之價值決定於標的物如股票,而標的物價格則為一連續時間形隨機程序。衍生性金融商品之定價問題等同於在適當機率測度空間(稱為風險中立機率測度空間)下,計算其收益的數學期望值。強烈路徑相關型衍生性金商品的收益,則取決於整個標的物之歷史價格,其計價問題在效率、精確度、及收歛性方面皆十分具挑戰性。
路徑相關型衍生性金融商品如亞式選擇權、障礙選擇權、與亞式障礙選擇權等,不但在資本市場上十分受到歡迎,而且在學術界亦受到廣泛地研究。雖然已有不少逼近演算法存在,但它們具有收歛震盪或執行時間非最佳之缺點。本計畫將對這些衍生性金融商品提出最有效率演算法,所使用的數學工具包括組合數學、生成函數、Lagrange乘數、內差、及外差,並特別探討所提出演算法的收歛行為。
除了上述演算法外,本計畫將研究亞式選擇權、障礙選擇權、及亞式障礙選擇權的次指數 精確演算法。由於目前此類演算法複雜度皆為 ,因此這項結果將會是理論界一大突破。
彩虹型選擇權含多個而非單一標的物,因此其定價問題本質上為多維度問題。根據所謂維度之詛咒,其定價問題預料極為困難,所以彩虹型選擇權之定價(除退化情形外)在實務上是經由蒙地卡羅
Abstract: Derivatives are financial instruments whose payoff depends on some underlying asset such as stock. The underlying asset is described by a continuous-time stochastic process. It is known that derivative pricing is equivalent to calculating the expected value of its payoff function under a suitable probability measure called the risk-neutral probability measure. Among the derivative securities, strongly path-dependent derivatives have payoff functions that depend nontrivially on the history of the underlying asset’s prices. Their pricing problems pose a challenge in terms of efficiency, accuracy, and convergence.
Path-dependent derivatives such as Asian options, barrier-type options, and Asian barrier options have been very popular in the capital markets and intensively studied in the literature. Although many approximation pricing algorithms have been proposed, they usually have oscillatory convergence behavior and/or suboptimal running time. This proposal intends to obtain the most efficient pricing algorithms for these options. The necessary mathematical techniques will include combinatorics, generating functions, Lagrange multipliers, interpolation, and extrapolation. Special attention should also be given to the convergence behavior of the algorithms.
Besides the approximation algorithms mentioned above, this proposal shall investigate exact algorithms that converge to the correct Asian option value or Asian barrier option value with a subexponential running time of . This will be theoretical breakthrough as existing exact algorithms all run in exponential time .
Rainbow options are options with multiple underlying assets instead of only one. Their pricing is therefore an inherently multidimensional problem and hard to compute because of the well-known curse of dimensionality. In practice, their prices are mostly obtained by Monte Carlo simulation except in degenerate cases. The partial derivatives (called the Greeks in finance) of the option price are essential to risk management and hedging. With simulation for pricing, calculating the Greeks requires resimulation to yield numbers for finite differences. Unfortunately, the Greeks are always biased because higher-order terms have been ignored in the Taylor expansion by finite differences. Furthermore, the right amount that the parameter should be varied in order for finite differences to yield good results is usually not known. It is not a simple matter of choosing the smallest possible as that will result in unstable Greeks.
We plan to develop a mathematical theory based on generalized functions that produce unbiased estimates for the Greeks of the rainbow options. These methods should not require resimulation. Furthermore, they should not employ finite differences. Hence they will completely do away with the problem of choosing the right in finite-difference schemes.
The GARCH model has emerged as one of the most influential stochastic-volatility models for stock prices. Applyi
Keyword(s)
障礙選擇權
亞式選擇權
衍生性金融商品
GARCH
希臘字母
演算法
路徑相關型衍生性金商品
組合數學
蒙地卡羅模擬
泛函
彩虹型選擇權
barrier option
Asian option
derivative
GARCH
Greek
algorithm
path-dependent derivative
combinatorics
Monte Carlo simulation
generalized functions
rainbow option