统计代写|金融统计代写financial statistics代考| Black–Scholes Option Pricing Model

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

统计代写|金融统计代写financial statistics代考|Black–Scholes Differential Equation

Simple, generally accepted economic assumptions are insufficient to develop a rational option pricing theory. Assuming a perfect financial market (Sect. 2.1) leads to elementary arbitrage relations which the options have to fulfil. While these relations can be used as a verification tool for sophisticated mathematical models, they do not provide an explicit option pricing function depending on parameters such as time, stock price and the options underlying parameters $K, T$. To obtain such a pricing function the value of the underlying financial instrument (stock, currency, … ) has to be modelled. In general, the underlying instrument is assumed to follow a stochastic process either in discrete or in continuous time. While the latter is analytically easier to handle, the former, which we will consider as an approximation of a continuous time process for the time being, is particularly useful for numerical computations. In the second part of this text, the discrete time version will be discussed as a financial time series model.

A model for stock prices which is frequently used and is also the basis of the classical Black-Scholes approach, is the so-called geometric Brownian motion. In this model the stock price $S_{t}$ is a solution of the stochastic differential equation
$$d S_{t}=\mu S_{l} d t+\sigma S_{t} d W_{t}$$
Equivalently, the process of stock price returns can be assumed to follow a standard Brownian motion, i.e.
$$\frac{d S_{l}}{S_{t}}=\mu d t+\sigma d W_{I}$$
The drift $\mu$ is the expected return on the stock in the time interval $d t$. The volatility $\sigma$ is a measure of the return variability around its expectation $\mu$. Both parameters $\mu$ and $\sigma$ are dependent on each other and are important factors in the investors’ risk preferences involved in the investment decision: The higher the expected return $\mu$, the higher, in general, the risk quantified by $\sigma$.

统计代写|金融统计代写financial statistics代考|Black–Scholes Formula for European Options

In this section we are going to use the Black-Scholes equation to compute the price of European options. We keep the notation introduced in the previous chapter. That is, we denote
$$C(S, t)=C_{K, T}(S, t), \quad P(S, t)=P_{K, T}(S, t)$$

统计代写|金融统计代写financial statistics代考|Numerical Approximation

(a) 正态分布可以用以下方式近似：

有限元方法代写

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MATLAB代写

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