C++编程探索高级量化金融模型与方法

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《高级量化金融与C++》是一本由Alonso Peña博士编著的专业书籍，专注于将量化金融理论与C++编程实践相结合，帮助读者创建并实现数学模型。该书针对三个主要领域进行了深入探讨：金融（金融衍生品）、数学以及信息技术（C++编程）。在开篇的“引言”部分，作者首先介绍了量化金融的定义，强调了它如何通过结合金融、数学和计算机科学的原理来解决复杂的金融市场问题。书中提到，量化金融涉及的三个核心学科包括金融衍生品的定价和管理（如期权、期货等）、高级数学工具（如概率论、随机过程和微积分），以及C++这种高效的编程语言，用于编写处理大规模数据和复杂算法的代码。第二章“数学模型”详细讲解了各种金融市场的关键模型，包括股票市场（如股价模型）、外汇市场（涉及汇率预测）、利率和短期利率模型（如 Vasicek、Cox-Ingersoll-Ross模型）、市场模型（如Black-Scholes模型）、以及信用风险评估中的结构性模型（如Merton模型）和强度模型。每种模型都配有实例，帮助读者理解其背后的理论和实际应用。第三章“数值方法”是本书的核心内容之一，涵盖了两种重要的模拟技术：蒙特卡洛模拟法（Monte Carlo Simulation，用以估计难以解析的问题的解）和二叉树方法（Binomial Trees，常用于期权定价）。作者首先解释了蒙特卡洛方法的基本概念，包括算法步骤和一个具体案例，然后介绍了二叉树的构建和计算过程，同样配合了一个实例来加深理解。《高级量化金融与C++》旨在提供一个全面的框架，让读者掌握如何运用C++语言来解决量化金融领域的实际问题。无论是初学者还是经验丰富的专业人士，都能从这本书中找到提升金融工程技能的宝贵资源。同时，版权信息表明，未经出版商许可，不得复制或传播书中的任何内容，且书中的信息仅供参考，不承担任何法律上的责任。

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What is Quantitative Finance?

Quantitative Finance studies the application of quantitative techniques to the

solution of problems in nance. It spans diverse areas such as the management

of investment funds and insurance companies, the control of nancial risks for

manufacturing companies and banking industry, and the behavior of the nancial

markets. Quantitative Finance is eminently interdisciplinary building upon key

expertise from the disciplines of nance, mathematics, and informatics.

In this book, we will focus on one aspect of Quantitative Finance—the pricing

of nancial derivatives using the programming language C++. In the following

sections, we will describe the main features of the three key disciplines that

constitute Quantitative Finance:

• Finance

• Mathematics

• Informatics

Discipline 1 – nance

(nancial derivatives)

In general, a nancial derivative is a contract between two parties who agree to

exchange one or more cash ows in the future. The value of these cash ows depends

on some future event, for example, that the value of some stock index or interest rate

being above or below some predened level. The activation or triggering of this future

event thus depends on the behavior of a variable quantity known as the underlying.

Financial derivatives receive their name because they derive their value from the

behavior of another nancial instrument.

As such, nancial derivatives do not have an intrinsic value in themselves (in contrast

to bonds or stocks); their price depends entirely on the underlying.

What is Quantitative Finance?

[ 6 ]

A critical feature of derivative contracts is thus that their future cash ows are

probabilistic and not deterministic. The future cash ows in a derivative contract

are contingent on some future event. That is why derivatives are also known as

contingent claims. This feature makes these types of contracts difcult to price.

The following are the most common types of nancial derivatives:

• Futures

• Forwards

• Options

• Swaps

Futures and forwards are nancial contracts between two parties. One party agrees

to buy the underlying from the other party at some predetermined date (the maturity

date) for some predetermined price (the delivery price). An example could be a

one-month forward contract on one ounce of silver. The underlying is the price

of one ounce of silver. No exchange of cash ows occur at inception (today, t=0),

but it occurs only at maturity (t=T). Here t represents the variable time. Forwards

are contracts negotiated privately between two parties (in other words, Over The

Counter (OTC)), while futures are negotiated at an exchange.

Options are nancial contracts between two parties. One party (called the holder

of the option) pays a premium to the other party (called the writer of the option)

in order to have the right, but not the obligation, to buy some particular asset (the

underlying) for some particular price (the strike price) at some particular date in the

future (the maturity date). This type of contract is called a European Call contract.

Example 1

Consider a one-month call contract on the S&P 500 index. The underlying in this case

will be the value of the S&P 500 index. There are cash ows both at inception (today,

t=0) and at maturity (t=T). At inception, (t=0) the premium is paid, while at maturity

(t=T), the holder of the option will choose between the following two possible

scenarios, depending on the value of the underlying at maturity S(T):

• Scenario A: To exercise his/her right and buy the underlying asset for K

• Scenario B: To do nothing if the value of the underlying at maturity is below

the value of the strike, that is, S(T)<K

The option holder will choose Scenario A if the value of the underlying at maturity

is above the value of the strike, that is, S(T)>K. This will guarantee him/her a prot

of S(T)-K. The option holder will choose Scenario B if the value of the underlying at

maturity is below the value of the strike, that is, S(T)<K. This will guarantee him/her

to limit his/her losses to zero.

Chapter 1

[ 7 ]

Example 2

An Interest Rate Swap (IRS) is a nancial contract between two parties A and B

who agree to exchange cash ows at regular intervals during a given period of time

(the life of a contract). Typically, the cash ows from A to B are indexed to a xed

rate of interest, while the cash ows from B to A are indexed to a oating interest

rate. The set of xed cash ows is known as the xed leg, while the set of oating

cash ows is known as the oating leg. The cash ows occur at regular intervals

during the life of the contract between inception (t=0) and maturity (t=T). An

example could be a xed-for-oating IRS, who pays a rate of 5 percent on the agreed

notional N every three months and receives EURIBOR3M on the agreed notional N

every three months.

Example 3

A futures contract on a stock index also involves a single future cash ow (the

delivery price) to be paid at the maturity of the contract. However, the payoff in this

case is uncertain because how much prot I will get from this operation will depend

on the value of the underlying at maturity.

If the price of the underlying is above the delivery price, then the payoff I get

(denoted by function H) is positive (indicating a prot) and corresponds to the

difference between the value of the underlying at maturity S(T) and the delivery

price K. If the price of the underlying is below the delivery price, then the payoff

I get is negative (indicating a loss) and corresponds to the difference between the

delivery price K and the value of the underlying at maturity S(T). This characteristic

can be summarized in the following payoff formula:

( ( )) ( )H S T S T K= −

Equation 1

Here, H(S(T)) is the payoff at maturity, which is a function of S(T). Financial

derivatives are very important to the modern nancial markets. According to

the Bank of International Settlements (BIS) as of December 2012, the amounts

outstanding for OTC derivative contracts worldwide were Foreign exchange

derivatives with 67,358 billion USD, Interest Rate Derivatives with 489,703 billion

USD, Equity-linked derivatives with 6,251 billion USD, Commodity derivatives

with 2,587 billion USD, and Credit default swaps with 25,069 billion USD.

For more information, see http://www.bis.org/statistics/dt1920a.pdf.

What is Quantitative Finance?

[ 8 ]

Discipline 2 – mathematics

We need mathematical models to capture both the future evolution of the

underlying and the probabilistic nature of the contingent cash ows we encounter

in nancial derivatives.

Regarding the contingent cash ows, these can be represented in terms of the payoff

function H(S(T)) for the specic derivative we are considering. Because S(T) is a

stochastic variable, the value of H(S(T)) ought to be computed as an expectation

E[H(S(T))]. And in order to compute this expectation, we need techniques that allow

us to predict or simulate the behavior of the underlying S(T) into the future, so as to

be able to compute the value of ST and nally be able to compute the mean value of

the payoff E[H(S(T))].

Regarding the behavior of the underlying, typically, this is formalized using

Stochastic Differential Equations (SDEs), such as Geometric Brownian Motion

(GBM), as follows:

ds Sdt SdW

µ σ

= +

Equation 2

The previous equation fundamentally says that the change in a stock price (dS), can

be understood as the sum of two effects—a deterministic effect (rst term on the

right-hand side) and a stochastic term (second term on the right-hand side). The

parameter

is called the drift, and the parameter

is called the volatility. S is the

stock price, dt is a small time interval, and dW is an increment in the Wiener process.

This model is the most common model to describe the behavior of stocks,

commodities, and foreign exchange. Other models exist, such as jump, local

volatility, and stochastic volatility models that enhance the description of the

dynamics of the underlying.

Regarding the numerical methods, these correspond to ways in which the formal

expression described in the mathematical model (usually in continuous time) is

transformed into an approximate representation that can be used for calculation

(usually in discrete time). This means that the SDE that describes the evolution of

the price of some stock index into the future, such as the FTSE 100, is changed to

describe the evolution at discrete intervals. An approximate representation of an

SDE can be calculated using the Euler approximation as follows:

1t t t t

S S S t S d W

µ σ

− = ∆ + ∆

Equation 3

Chapter 1

[ 9 ]

The preceding equation needs to be solved in an iterative way for each time interval

between now and the maturity of the contract. If these time intervals are days and

the contract has a maturity of 30 days from now, then we compute tomorrow's

price in terms of todays. Then we compute the day after tomorrow as a function of

tomorrow's price and so on. In order to price the derivative, we require to compute

the expected payoff E[H(ST)] at maturity and then discount it to the present. In this

way, we would be able to compute what should be the fair premium

associated

with a European option contract with the help of the following equation:

exp( ) [ (S )] exp( ) [max(S ,0)]

T T

rT E H rT E K

= − × = − × −

Equation 4

Discipline 3 – informatics

(C++ programming)

What is the role of C++ in pricing derivatives? Its role is fundamental. It allows us

to implement the actual calculations that are required in order to solve the pricing

problem. Using the preceding techniques to describe the dynamics of the underlying,

we require to simulate many potential future scenarios describing its evolution. Say

we ought to price a futures contract on the EUR/USD exchange rate with one year

maturity. We have to simulate the future evolution of EUR/USD for each day for

the next year (using equation 3). We can then compute the payoff at maturity (using

equation 1). However, in order to compute the expected payoff (using equation 4),

we need to simulate thousands of such possible evolutions via a technique known

as Monte Carlo simulation. The set of steps required to complete this process is

known as an algorithm. To price a derivative, we ought to construct such algorithm

and then implement it in an advanced programming language such as C++. Of

course C++ is not the only possible choice, other languages include Java, VBA,

C#, Mathworks Matlab, and Wolfram Mathematica. However, C++ is an industry

standard because it's exible, fast, and portable. Also, through the years, several

numerical libraries have been created to conduct complex numerical calculations in

C++. Finally, C++ is a powerful modern object-oriented language.

It is always difcult to strike a balance between clarity and efciency. We have aimed

at making computer programs that are self-contained (not too object oriented) and

self-explanatory. More advanced implementations are certainly possible, particularly

in the context of larger nancial pricing libraries in a corporate context. In this book, all

the programs are implemented with the newest standard C++11 using Code::Blocks

(http://www.codeblocks.org) and MinGW (http://www.mingw.org).

www.allitebooks.com

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