論文：二叉樹應用于面臨突變可能市場中的或有要求權(quán)定價

論文：二叉樹應用于面臨突變可能市場中的或有要求權(quán)定價

摘要：按Merton的觀點，股價的變化可分為兩類，一類是常規(guī)的邊際擾動，另一類是非常規(guī)的突變。為了更好地分析研究金融市場的變化性態(tài)，我們在運用二叉樹研究股價及其或有要求權(quán)價格常規(guī)變動的基礎(chǔ)上，進一步利用此工具研究在面臨突變可能的市場中或有要求權(quán)的定價問題，并推導出關(guān)于歐式或有要求權(quán)（期權(quán)）定價、標的資產(chǎn)（股票）估價的明確表達式。
關(guān)鍵詞：突變（跳），突變，歐式或有要求權(quán)（期權(quán)），定價，二叉樹
研究領(lǐng)域：市場有效性與資本資產(chǎn)定價研究

Binomial trees used in valuation of contingent claims
in a market in face of sudden breaks

Abstract: According to Prof. Merton, the total change in the stock price is composed of two types of changes, the normal, marginal vibrations and the abnormal, sudden and rare breaks. In order to analyze these marketing behaviors more simply and efficiently, we use the binomial trees to introduce a new method on Valuation of European Contingent Claims in a market in face of sudden and rare breaks (with jumps), on the
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stock that has more than a marginal effect on price, for e*ample, due to the Sept. 11 attacks, which caused disasters in the world as well as in the financial markets. Usually, such information will be specific to the firm or possibly its industry. It is reasonable to e*pect that there will be active times in the stock when such information arrives and quiet times when it dose not arrive although the active and quiet times are random. According to its very nature, important information arrives only at discrete points of time. This component is modeled by a jump process reflecting the non-marginal impact of the information (This jump process can also be considered as a model reflecting the sudden breaks). And on this occasion of abnormal vibrations, the Black-Scholes formula is not valid, even in the continuous limit, because the stock price dynamics cannot be represented by a stochastic process with a continuous sample path. We must use stochastic differential equations with jumps to study this problem. In order to simplify the research process, we introduce binomial trees to study the valuation problems.
Binomial trees may be one of the most popular methods to deal with valuation of ECC and EO in stochastic financial markets without jump. Hull and Kolb’s works [3, 4, 6] both involve these contents respectively. But these books have not involved those contents of binomial trees used in stochastic financial markets with sudden breaks (These sudden breaks may cause more severe reactions in the market than usual), whereas we will study this in the present paper. We give some e*plicit formulae to demonstrate the value of ECC and EO and the prices of underlying assets (stocks).

2.Valuation of European options and contingent claims using binomial tree

Figure 1 A Binomial Tree of Stock Prices Used to Value a Stock Option
In this section we use a binomial tree approach to value European call options (Abbreviated, ECO) and European contingent claims (Abbreviated, ECC) under the risk-neutral assumption, and in a discrete time version in a stochastic financial market without or with sudden breaks. We start by dividing the life of the option into a large number of small time intervals of length (the length of each time step is defined as ). We assume that in each time interval the stock price can move up by a multiplier factor or down by a multiplier factor . At time , there are two possible stock prices, and at time , there are possible stock prices considered. Since we are working in a risk-neutral world, the e*pected return from a stock is the risk-free continuous paying interest rate, an ……（未完，全文共14323字，當前僅顯示3407字，請閱讀下面提示信息。收藏《論文：二叉樹應用于面臨突變可能市場中的或有要求權(quán)定價》）