### 金融代写|金融工程作业代写Financial Engineering代考|Black-Scholes Model

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• Foundations of Data Science 数据科学基础

## 金融代写|金融工程作业代写Financial Engineering代考|The Black-Scholes Model

In Black and Scholes [1973], the authors introduced their famous model for the price of a stock, together with a partial differential equation for the value of a European option. To do so, they assumed the following “ideal conditions” on the market:

• The short-term interest rate, also called risk-free rate, is known and constant until the maturity of the option.
• The log-returns of the stock follows a Brownian motion with drift.
• The option is European, i.e., it can only be exercised at maturity.
• There are no transaction costs, nor penalties for short selling.
• It is possible to buy or borrow any fraction of the security.
To make things simpler, we assume that the market is liquid and frictionless, that trading can be done in continuous time, and that the risk-free rate is constant during the life of the option. The latter hypothesis will be weakened in Chapter $3 .$

## 金融代写|金融工程作业代写Financial Engineering代考|Stock Exchange Data

What do we do when there are dividends or stock splits? In each case, there is a predictable but abnormal jump in the prices. In order to make historical data comparable, it is necessary to take these jumps into account.

For example, in the case of Apple, there were two $2: 1$ stock splits (June $21^{s t}, 2000$, and February $\left.28^{t h}, 2005\right)$, while the last dividend was reported on November $21^{s t}, 1995 .$

In many textbooks on Financial Engineering, e.g., Hull [2006] and Wilmott $[2006]$, it is suggested to replace the stock price $S_{i}$ at the ex-dividend period $i$ by $S_{i}+D_{i}$, where $D_{i}$ is the dividend. In fact, it is often more convenient to do just the opposite ${ }^{1}$, by subtracting the dividend from the pre-dividend price, i.e., the adjusted price for period $i-1$ is $S_{i-1}-D_{i}=S_{i-1} f_{i}$, with $f_{i}=$ $1-D_{i} / S_{i-1}$. In fact, all pre-dividend values $S_{j}, j<i$, are then multiplied by the same factor $f_{i}$. For splits, one proceeds in a similar way. For example, in the case of a 2:1 stock split, all pre-split prices are multiplied by 0.5. The returns are then calculated from these adjusted closing prices. This is in accordance to the standards of the Center for Research in Security Prices (CRSP). The same method is applied to the adjusted prices available on YAHOO!FINANCE website.

## 金融代写|金融工程作业代写Financial Engineering代考|Continuous Time Models

Let $S(t)$ be the (adjusted) value of an asset at time $t$. Before defining the Black-Scholes model, one needs to define what is a Brownian motion.

Definition 1.2.1 A stochastic process $W$ is a Brownian motion ${ }^{2}$ if it is a

continuous Gaussian process starting at 0 , with zero expectation and covariance function
$$\operatorname{Cov}{W(s), W(t)}=\min (s, t), s, t \geq 0 .$$
In particular, if $0=t_{0} \leq t_{1} \leq \cdots \leq t_{n}$, then the increments $W\left(t_{i}\right)-$ $W\left(t_{i-1}\right)$ are independent, have a Gaussian distribution with mean 0 and variance $t_{i}-t_{i-1}, i \in{1, \ldots, n}$.

Definition 1.2.2 (Black-Scholes Model) In the Black-Scholes model, one assumes that the value $S$ of the underlying asset is modeled by a geometric Brownian motion, i.e.,
$$S(t)=s e^{\left(\mu-\frac{\alpha^{2}}{2}\right) t+\sigma W(t)}, \quad t \geq 0,$$
where $W$ is a Brownian motion.
Note that $S$ is often defined as the solution to the following stochastic differential equation:
$$d S(t)=\mu S(t) d t+\sigma S(t) d W(t), S(0)=s .$$
Remark 1.2.1 The Black-Scholes model depends on two unknown parameters: $\mu$ and $\sigma$. Therefore they must be estimated. In practice, data are collected at regular time intervals of given length $h$. For example, observations can be collected every 5 seconds, daily, weekly, monthly, etc.

It is also important to characterize the parameters. For example, is $\mu$ an expectation? If so, it is the expectation of which variable? One can ask the same question for the parameter $\sigma$.

In what follows, we will see that $\mu$ and $\sigma$ can be interpreted in terms of the log-returns
$$X_{i}=X_{i}(h)=\ln {S(i h)}-\ln [S{(i-1) h}], \quad i \in{1, \ldots, n} .$$
First, we need to find the joint law of these returns. In addition to being important for the estimation of the unknown parameters, the distribution of the returns is needed in order to find expressions for option prices or when using Monte Carlo methods for pricing complex options, measuring risk or performance, etc.

## 金融代写|金融工程作业代写Financial Engineering代考|The Black-Scholes Model

• 短期利率，也称为无风险利率，在期权到期之前是已知的并且是恒定的。
• 股票的对数收益遵循带有漂移的布朗运动。
• 期权是欧式的，即只能在到期时行使。
• 没有交易成本，也没有卖空的处罚。
• 可以购买或借入任何部分的证券。
为简单起见，我们假设市场是流动的且无摩擦的，交易可以在连续的时间内完成，并且在期权的生命周期内无风险利率是恒定的。后一种假设将在本章中被削弱3.

## 金融代写|金融工程作业代写Financial Engineering代考|Continuous Time Models

d小号(吨)=μ小号(吨)d吨+σ小号(吨)d在(吨),小号(0)=s.

X一世=X一世(H)=ln⁡小号(一世H)−ln⁡[小号(一世−1)H],一世∈1,…,n.

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