1

30

在继续浏览本公司网站前，请您确认您或您所代表的机构是一名“合格投资者”。“合格投资者”指根据任何国家和地区的证券和投资法规所规定的有资格投资于私募证券投资基金的专业投资者。例如根据我国《私募投资基金监督管理暂行办法》的规定，合格投资者的标准如下：

一、具备相应风险识别能力和风险承担能力，投资于单只私募基金的金额不低于100万元且符合下列相关标准的单位和个人：

1、净资产不低于1000万元的单位；

2、金融资产不低于300万元或者最近三年个人年均收入不低于50万元的个人。(前款所称金融资产包括银行存款、股票、债券、基金份额、资产管理计划、银行理财产品、信托计划、保险产品、期货权益等。)

二、下列投资者视为合格投资者：

1、社会保障基金、企业年金等养老基金、慈善基金等社会公益基金；

2、依法设立并在基金业协会备案的投资计划；

3、投资于所管理私募基金的私募基金管理人及其从业人员；

4、中国证监会规定的其他投资者。

如果您继续访问或使用本网站及其所载资料，即表明您声明及保证您或您所代表的机构为“合格投资者”，并将遵守对您适用的司法区域的有关法律及法规，同意并接受以下条款及相关约束。如果您不符合“合格投资者”标准或不同意下列条款及相关约束，请勿继续访问或使用本网站及其所载信息及资料。

投资涉及风险，投资者应详细审阅产品的发售文件以获取进一步资料，了解有关投资所涉及的风险因素，并寻求适当的专业投资和咨询意见。产品净值及其收益存在涨跌可能，过往的产品业绩数据并不预示产品未来的业绩表现。本网站所提供的资料并非投资建议或咨询意见，投资者不应依赖本网站所提供的信息及资料作出投资决策。

与本网站所载信息及资料有关的所有版权、专利权、知识产权及其他产权均为本公司所有。本公司概不向浏览该资料人士发出、转让或以任何方式转移任何种类的权利。

本声明包含网络使用的有关条款。凡浏览本网站及相关网页的用户，均表示接受以下条款。

1、并非所有的客户都可以获得所有的产品和服务，您是否符合条件享受特别产品和服务，最终的解释权归我公司。我公司保留对该网页包含的信息和资料及其显示的条款、条件和说明变更的权利。

2、任何在本网站出现的信息包括但不限于评论、预测、图表、指标、理论、直接的或暗示的指示均只作为参考，您须对任何自主决定的行为负责。

3、本网站提供的有关投资分析报告、股市预测文章信息等仅供参考，股市有风险，入市须谨慎！本网站所提供之公司资料、个股资料等信息，力求但不保证数据的准确性，如有错漏，请以基金业协会公示信息报刊为准。本网站不对因本网资料全部或部分内容产生的或因依赖该资料而引致的任何损失承担任何责任。

4、互联网传输可能会受到干扰，中断、延迟或数据错误，本公司对于非本公司能控制的通讯设施故障可能引致的数据及交易之准确性或及时性不负任何责任。

5、凡通过本网站与其他网站的链结，而获得其所提供的网上资料及内容，您应该自己进行辨别及判断，我公司不承担任何责任。

6、本站某些部分或网页可能包括单独条款和条件，作为对本条款和条件的补充，如果有任何冲突，该等附加条款和条件将对相关部分或网页适用。

7、本人已阅读并同意《数字证书服务协议》。

1、并非所有的客户都可以获得所有的产品和服务，您是否符合条件享受特别产品和服务，最终的解释权归我公司。我公司保留对该网页包含的信息和资料及其显示的条款、条件和说明变更的权利。

2、任何在本网站出现的信息包括但不限于评论、预测、图表、指标、理论、直接的或暗示的指示均只作为参考，您须对任何自主决定的行为负责。

3、本网站提供的有关投资分析报告、股市预测文章信息等仅供参考，股市有风险，入市须谨慎！本网站所提供之公司资料、个股资料等信息，力求但不保证数据的准确性，如有错漏，请以基金业协会公示信息报刊为准。本网站不对因本网资料全部或部分内容产生的或因依赖该资料而引致的任何损失承担任何责任。

4、互联网传输可能会受到干扰，中断、延迟或数据错误，本公司对于非本公司能控制的通讯设施故障可能引致的数据及交易之准确性或及时性不负任何责任。

5、凡通过本网站与其他网站的链结，而获得其所提供的网上资料及内容，您应该自己进行辨别及判断，我公司不承担任何责任。

6、本站某些部分或网页可能包括单独条款和条件，作为对本条款和条件的补充，如果有任何冲突，该等附加条款和条件将对相关部分或网页适用。

7、本人已阅读并同意《数字证书服务协议》。

Chuan Shi 2017-03-09
本文章798阅读

**1**

**Introduction**

The Black-Scholes formula (also known as the Black-Scholes-Merton formula) for option pricing is very famous in quantitative finance. It is one great example that applies stochastic process to financial instrument pricing. The BS formula related questions are always asked, with no exception, in Wall Street quant job interviews. However, Nassim Nicholas Taleb, the author of *The Black Swan* despises the BS formula and he even co-authored an article entitled *Why we have never used the Black-Scholes-Merton option pricing formula* to denounce it.

Different people hold distinct views on how effective the BS formula is in investment practice. My point of view, however, is that the BS formula is just a result; it is a product from the rigorous derivation of stochastic calculus. **Looking at the nature of the phenomenon, there is a powerful mathematical system behind it that allows us to quantify the price of stocks, options, and other derivatives using stochastic processes.**** **A familiarity to this analytical approach is essential to those who want to achieve something in the field of quantitative finance.

It is by no means easy to master this mathematical system. If one searchs how to derive the BS formula on Google, the concepts such as the **Brownian motion**, **Ito's lemma**, and **stochastic differential equations** will pop up. They are all integral elements of this system, seamlessly linked with each together and perfectly fitted in the framework of **stochastic calculus**. People who are familiar with this can fully appreciate the beauty of mathematics behind the framework. For people who are not familiar with it, every concept may seem mumbo-jumbo; it is uneasy to figure out the logical relation between them even with higher mathematics knowledge.

To put it simple, the (standard) Brown motion is the simplest continuous stochastic process. It is the basic model for describing the stochastic nature of asset prices. For options and other derivatives, their prices are functions of the underlying asset price. **Since asset price is a stochastic process, the derivative price is a function of that stochastic process. The Ito's lemma provides a framework to differentiate the functions of stochastic process and this is of particular significance to derivative pricing** (before Ito's work, people did not know how to do it). **Ito's lemma allows us to derive the stochastic differential equation (SDE) for the price of derivatives. Solving such SDEs gives us the derivative pricing models. **The derivation of the BS formula is one simple example of this procedure.

Given the importance of stochastic calculus, we plan to talk about it in a series of two articles. As the first part, today's article will explain the Brownian motion and its properties, as well as present the basic form of the Ito's lemma. We tried hard to reveal the properties of the Brownian motion and its implications to stock price movement. In the second part of this series, we will start with a more general form of the Ito's lemma, and apply it to solve for the geometric Brownian motion, and finally derive the BS equation and the BS formula.

We hope that these two articles will provide you an intuitive understanding about the mathematical framework of stochastic calculus, and help you appreciate the beauty of mathematics where different elements seemlessly connect with each other to derive an elegant pricing formula.

**2**

**Brownian motion: development and mathematical definition**

In 1827, while looking through a microscope at particles trapped in cavities inside pollen grains in water, Scottish botanist Robert Brown noticed the random motion of the particles; but he was not able to determine the mechanisms that caused this motion. Albert Einstein published a paper in 1905 that explained in precise detail how the motion that Brown had observed was a result of the pollen being moved by individual water molecules. The development of Brownian motion in physics has been improving since then.

As a contrast, its development in mathematics is slower. The precise mathematical definition of the Brownian motion was not developed until 1918 by Norbert Winner. Therefore, **the Brownian motion is also referred to as the Wiener process.**

**The Brownian motion is ****a continuous-time stochastic process****, or a continuous-space-time stochastic process.** It is a stochastic process for which the index variable takes a continuous set of values, as contrasted with a discrete-time process for which the index variable takes only distinct values.

The figure above shows two examples of the Brownian motion. The one on the left is a two-dimentional Brownian motion where the two axes represent the space domain, while the one on the right is a one-dimension Brownian motion and the x-axis is the time domain. The Brownian motion on the right looks very similar to the movement of stock price, and this motivates people to use it to model stock price. The first person who considered this process is Bachelier, who used Brownian motion to evaluate stocks and options in his Ph.D thesis entitied *Théorie de la spéculation* in 1900.

The one-dimensional standard Brownian motion is defined as follows.

There exists a probability distribution over the set of continuous functions B: R -> R satisfying the following conditions:

1. B(0) = 0,

2. **(Stationary) **for all 0 ≤ s < t, the distribution of B(t) - B(s) is the normal distribution with mean 0 and variance t - s, and

3. **(Independent increment) **the random variables B(t_i) - B(s_i) are mutually independent if the intervals [s_i, t_i] are nonoverlapping.

We refer to such a process B(t) as the standard Brownian motion.

The definition indicates that the Brownian motion starts from some original point at t = 0. **In any given finite time interval ****Δt, B( Δt) satisfies a normal distribution with mean 0 and variance Δt, where the variance increases linearly with time. **Independent increment means that the movements of the Brownian motion in nonoverlapping intervals are independent. This is immediately followed by the fact that

**3**

**Properties of the Brownian motion**

Here are some facts about the Brownian motion, and they have important implications to modeling stock price movement using it:

1. The path crosses the x-axis (time axis) infinitely often.

2. B(t) has a very close relation with the curve x = y^2. At any time t, it does not deviate from this curve too much.

3. Let M(t) be max_{0 ≤ s ≤ t} B(t), it can show that Prob(M(t) ≥ a) = 2 × Prob(B(t) ≥ a)；

4. It is **nowhere** differentiable (this is very important).

To explain the first two properties, the following figure shows 15 sample paths of 15 standard Brownian motion in time interval 0 to t. Each path crosses y = 0 (the time axis) multiple times with the exception that only very few paths appear to be on the same side of y = 0 for the entire period. However, they will eventually cross the x-axis as t increases. The black parabola is the curve of t = y^2. We can see that although each sample path shows a distinct randomness, at any time t' ≤ t, they do not deviate too far from the parabola curve which is B(0) +/- the squart root of t. On the right of the figure is the probability density function of the normal distribution at t whose mean is 0 and variance is t. The range of the parabola corresponds to one standand deviation above and below the mean of the normal distribution.

Suppose we choose to use the Brownian motion to describe the high frequency intraday price movement (later in this article we will point out that a more accurate model is the **geometric Brownian motion with drift**, but let's now use the Brownian motion for a short while), then **the two properties above mean that the stock price will fluctuate around the open price and as time passes by, it will not deviate too much from the open price +/- the square root of t times the standard deviation of the price**. These properties are important to high frequency traders.

The third property shows how to derive the probability model for the extreme values of the Brownian motion. Since B(t) satisfies the normal distribution N(0, t), using Prob(M(t) ≥ a) = 2 × Prob(B(t) ≥ a), we can derive Prob(M(t) ≥ a) easily,

where Φ is the cumulative density function of the standard normal distribution. This can be proved by the Markovian property and the reflection principle. Likewise, let m(t) be the minimum value of B(t) in [0, t], i.e., m(t) = min_{0 ≤ s ≤ t} B(t). It can be shown that

These results can be used to quantify the **probability distribution about the extreme values of the stock price**, which can be of great importance in risk management.

**The last property is a crucial nature of the Brownian motion as a stochastic process. It says that although the Brownian motion is continuous, it is not differentiable everywhere **(this can be proved by contradiction with the usage of the mean value theorem and the third property). This is very intuitive to understand. Let's take a look at those 15 sample paths of the Brownian motion. Each of them has been fluctuating up and down, demonstrating its randomness. It is clear that **the trajectory of the Brownian motion is completely different from any continuous and smooth trajectory that we are familiar with.**

**The non-differentiability means that classical calculus is useless in analyzing the Brownian motion. **This was undoubtedly frustrating because people finally come up with a simple random process (to model stock price), but lacked the tools to further study it. **However, the pazzle was solved with the development of Ito calculus. It is no exaggeration to say that Ito calculus laid the foundation of modern financial mathematics.**

**4**

**Quadratic variation**

For a partition Π = {0 = t_0 < t_1 < t_2 < … < t_N = T} of an interval [0, T] and a continuous function f(t), its quadratic variation is defined as

If f is continuously differentiable in [0. T], then by using the mean value theorem of classical calculus, it can be shown that

**This means that as the partition becomes finer and finer, i.e., as max_i {t_{i+1} - t_i} approaches to 0, the quadratic variation of f(t) goes to zero.**

What if we change f(t) to the Brownian motion B(t) instead? Remember that B(t) is nowhere differentiable. Regarding the quadratic variation of B(t), the following theorem holds:

For a partition Π = {0 = t_0 < t_1 < t_2 < … < t_N = T} of an interval [0, T], let |Π| = max_i {t_{i+1} - t_i}. A Brownian motion B(t) satisfies the following equation with probability 1:

This can be proved by the law of large numbers. **It says that as a stochasitc process, the quadratic variation of the Brownian motion is T, rather than 0. **What does it mean?

Consider the following illustration. The blue curve is the path of a Brownian motion, and the red points show the location of B(t) at different partitioning points. Therefore, (B(t_{i+1} – B(t_i))^2 is the squared difference of the locations of two adjacent partition ing points. The quadratic variation is the cumulative sum of these squared differences.

For a regular function f(t) that is both continuous and differentiable, as the partition becomes finer and finer, its quadratic variation approaches to 0. However, this is not true for B(t) who is continuous but not differentiable. **This suggests that the randomness of B(t) makes it vary too much no matter how small a partitioning interval becomes. The cumulative sum of the flustration of B(t) from those tiny small intervals just won't go to 0. Instead, the limit goes to T, which is nothing but the lengh of the interval!** This is a key property of the Brownian motion.

The quadratic variation of the Brownian motion can also be written in the infinitesimal difference form:

As we will show in section 6 of this article, **this nonzero quadratic variation of the Brownian motion has significant implications in the derivation of Ito's lemma.**

**5**

**Use geometric Brownian motion to model stock price**

Previous section introduces the standard Brownian motion who follows normal distribution with mean 0 and variance t in the interval [0, t]. Now, we add a drift term μt as well as a scaling parameter σ. This leads to a Brownian motion with drift, denoted by X(t) = μt + σB(t). Note that μt is just a function of time and therefore it has no randomness. It is straightforward to see that X(t) follows a normal distribution with mean μt and variance (σ^2)t. In the infinitesimal form, it becomes

This is a **stochastic differential equation. It differs from a regular differential equation in that it has at least one stochastic term (in this case B(t)). **Note that this is not contradict to the fact that B(t) is not differentiable. Although B(t) is not differentiable everywhere, dB(t) still has a definite meaning. It represents the change of the Brownian motion within an infinitesimal time interval.

Even though we now have the Brownian motion with drift, it is still not the best stochastic process for modeling stock price movement. This is because X(t) or B(t) can take negative values as t passes by. However, the stock price cannot be negative. The return of the stock, on the other hand, can be both negative and positive. **Therefore, we can use X(t) to model stock returns.**

Let S(t) be the stock price, and dS(t) measures how S changes within an infinitesimal time interval. Therefore, dS(t)/S(t) is the stock return in this interval, and we have

This gives the SDE of S(t):

A stochastic process S(t) that satisfies the SDE is called a **geometric Brownian motion**. People like to use it to model the stock price because:

**1. Normal distribution: **Empirical evidence shows that the continuous compound return of stock approximately follows the normal distribution.

**2. Markovian property: **From the property of the Brownian motion, it is easy to see that S(t) is a Markov process, which means that the current stock price at t contains all the information needed to predict the future, and this complies with the weak form of the efficient-market hypothesis.

**3. The fact that B(t) and therefore S(t) are continuous but not differentiable agree withs with the actual movement of stock price.**

In order to study the stock price using S(t), we must be able to resolve the SDE above and find a close form of S(t). This can be done with the help of Ito calculus, and this is one of the topics in the second article of this series.

We close this section by talking about an interesting example of the Brownian motion with drift. Consider some positive real number μ and let X(t) = μt + B(t). Since the expectation of B(t) is 0, then the expectation of X(t) is E[X(t)] = μt. What we want to figure out is as time passes, which term will dominate X(t)? In fact, it can be shown that **X(t) is dominated by μt. For all fixed ε > 0, after long enough time, X(t) will always be between the lines by y = (μ – ε)t and y = (μ + ε)t.**

What does this example tell us? It implies that if we believe that the stock market will rise in the long run, i.e., **μ > 0, then we should** **gladly accept any short-term fluctuations it may have and hold the stocks (i.e., ignoring the randomness of B(T)). For a long time, stock price is determined by μt.** I guess Buffett must be a mathematician and he must understand this. With his value investment system, he earned a long term drift rate of μ that is higher than the US stock indexes. This allows him to earn stable excess return over the years.

**6**

**Ito's lemma**

The Brownian motion allows people to study stock prices. However, as for financial derivatives, their prices are functions of the underlying assets. **Let **

Let's see why classical calculus does not work first. To find df, where f is a continuous and smooth function of B_t, we apply the chain rule:

Since B_t is not differentiable, the differentiation dB_t/dt does not exit. **Hence, the formula above makes no sense. Our first try failed.**

One possible way to work around this problem is to try to describe the difference df in terms of the difference dB_t, rather than dB_t/dt. We have mentioned previously that dB_t has a clear meaning and it is the change of B_t in an infinitesimal time interval. We therefore have

This new formula at least makes sense since there is no need to refer to dB_t/dt which does not exist. In this expression, both f'(B_t) and dB_t can be computed. **Unfortunately, this does not quite work. Our second try failed again.**

To see why, consider the **Taylor expansion** of f(x) and it gives:

When Δx approaches to 0, the significant term is the first term f’(x)Δx and all other terms are of smaller order of magnitude, which can be ignored. Therefore, df = f’(x)dx is correct. However, is this true for x = B_t? **The answer is No.** For x = B_t we have

Again, the first term f’(B_t)ΔB_t is still significant. But can other terms be ignored comparing to it? **The answer is no due to quadratic variation, which says ****(dB)^2 = dt. **Since the quadratic variation of B_t is not 0, the second term is no longer negligable. The theory of Ito calculus essentially tels us that we can make the substitution (dB)^2 = dt, and the remaining terms are negligable. Hence, the equation above becomes

**It is the basic form of Ito's lemma.**

More generally, consider a smooth function f(t, x) which depends on two variables. In classical calculus, we will get

With x replaced by B_t and according to Ito calculus, we have

**Comparing the results above shows that, given the nonzero quadratic variation of the Brownian motion, to find df we must add an extra term to the results derived by classical calculus. **This extra term is the second order derivative of f to B_t (if f depends only on B_t) or the partial second order derivative of f to B_t (if f depends both on B_t and t). This conclusion changes everything and it permits the usage of calculus in the field of stochastic process.

In the next article of this series, we will apply Ito's lemma to solve the geometric Brownian motion and to derive the BS formula for option pricing.

**7**

**Summary**

The Brownian motion is effective in describing the movement of stock price. Its Markovian property agrees with the weak form of the efficient-market hypothesis. By using the reflection principle, it is easy to calculate the probability that the Brownian motion reachs some extreme value within a given period of time, and this is crucial to risk management. To take it further, a more precise model for stock price is the geometric Brownian motion with drift. In the long run, stock price is controlled by the drift rate, and it implies that we should adhere to long-term value investment while ignore the short-term volatility of the stock price called by randomness.

On the other hand, although it is continuous, the Brownian motion is nowhere differentiable. This meets with people's expectation that stock price flustrates a lot. In financial mathematics, it is important to analyze how the function of a stochastic process changes within a infinitesimal time interval. However, quadratic variation makes classical calculus useless. Ito Kiyoshi proposed a variant of classical calculus, the Ito calculus. It takes into account the quadratic variation of the Brownian motion, and provides a mean of using calculus framework to analyze stochastic process and its functions. This is the foundation of modern financial mathematics.

地址 : 北京市朝阳区领地 OFFICE 大厦 3 号楼 203

电话 : 8610 - 5601 3848 E-mail : info@liang-xin.com

Copyright © 2017-2020 北京量信投资管理有限公司版权所有 京ICP备17020789号