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定量研究是在金融领域使用数值或定量技术的工作。类似的工作在其他大多数现代工业中完成;这样的工作称为定量分析。在投资行业中,进行定量分析的人,常常被称为数量分析专家。
虽然最初的数量分析专家关注风险管理和金融衍生品定价,但是随着时间的推移这个词的含义扩大了,包括了那些个人参与的几乎所有的金融领域数学应用。其中一个例子就是统计套利。
1.1历史-History
定量计算金融于20世纪30年代在美国当一些精明的投资者开始用数学公式来给股票和债券定价时开始发明的。
罗伯特·c·默顿,定量分析的先驱,将随机微积分引入金融研究。
哈里·马克维茨在1952年发表的博士论文“投资组合选择”是第一个正式将数学概念改编进金融的论文。马克维茨将回归概念形式化并且协方差的普通股让他可以在市场中量化“多元化”的概念。他展示了如何计算一个给定组合的均值回归和方差,认为在所有投资组合与给定的平均回报中投资者应持有那些方差最小的投资组合。虽然现在金融语言涉及到微积分,以风险最小化的量化方式构成现代理论。
定量研究计算金融经济-A Quantitative study of Computational Finance in the Economy
A quantitative study is working in finance using numerical or quantitative techniques. Similar work is done in most other modern industries; the work is called quantitative analysis. In the investment industry, people who perform quantitative analysis are frequently called quants.
Although the original quant’s were concerned with risk management and derivatives pricing, the meaning of the term has expanded over time to include those individuals involved in almost any application of mathematics in finance. An example is statistical arbitrage.
1.1历史-1.1 History
Quantitative computational finance started in the U.S. in the 1930s as some astute investors began using mathematical formulae to price stocks and bonds.
Robert C. Merton, a pioneer of quantitative analysis, introduced stochastic calculus into the study of finance.
Harry Markowitz's 1952 PhD thesis "Portfolio Selection" was one of the first papers to formally adapt mathematical concepts to finance. Markowitz formalized a notion of mean return and covariance’s for common stocks which allowed him to quantify the concept of "diversification" in a market. He showed how to compute the mean return and variance for a given portfolio and argued that investors should hold only those portfolios whose variance is minimal among all portfolios with a given mean return. Although the language of finance now involves It calculus, minimization of risk in a quantifiable manner underlies much of the modern theory.
In 1969 Robert Merton introduced stochastic calculus into the study of finance. Merton was motivated by the desire to understand how prices are set in financial markets, which is the classical economics question of "equilibrium," and in later papers he used the machinery of stochastic calculus to begin investigation of this issue.
At the same time as Merton's work and with Merton's assistance, Fischer Black and Myron Schools were developing their option pricing formula, which led to winning the 1997 Nobel Prize in Economics. It provided a solution for a practical problem, that of finding a fair price for a European call option, i.e., the right to buy one share of a given stock at a specified price and time. Such options are frequently purchased by investors as a risk-hedging device. In 1981, Harrison and Pliska used the general theory of continuous-time stochastic processes to put the Black-Scholes option pricing formula on a solid theoretical basis, and as a result, showed how to price numerous other "derivative" securities.
1.2量化和计算融资-1.2Quantitative and Computational Finance
‘Quantitative Finance’ as a branch of modern finance is one of the fastest growing areas within the corporate world. Together with the sophistication and complexity of modern financial products, this exciting discipline continues to act as the motivating factor for new mathematical models and the subsequent development of associated computational schemes. Alternative names for this subject area are Mathematical Finance, Financial Mathematics or Financial Engineering.
This is a course in the applied aspects of mathematical finance, in particular derivative pricing. The necessary understanding of products and markets required will be covered during the course. The overall theme of the course is to develop the Partial Differential Equation (PDE) approach to the pricing of options. As well as a two hour examination during the summer term, students will undertake a short computing project where they will use numerical and computational techniques to perform derivative pricing.
Simulation Methods in Finance
Brief introduction to Stochastic Differential Equations (SDEs) – drift, diffusion, It’s Lemma. The statistics of random number generation in Excel. Simulating asset price SDEs in Excel.
Financial Products and Markets
Introduction to the financial markets and the products which are traded in them: Equities, indices, foreign exchange, fixed income world and commodities. Options contracts and strategies for speculation and hedging.
Black-Schools framework
Similarity reduction and fundamental solution for the heat equation. Black-Schools PDE: simple European calls and puts; put-call parity.The PDE for pricing commodity and currency options.Discontinuous payoffs – Binary and Digital options.The Greeks: theta, delta, gamma, Vega &rho and their role in hedging.
Computational Finance
Solving the pricing PDEs numerically using Explicit, Implicit and Crank-Nicholson Finite Difference Schemes. Stability criteria. Monte Carlo Technique for derivative pricing.
Fixed-Income Products
Introduction to the properties and features of fixed income products; yield, duration &convexity.Stochastic interest rate models: stochastic differential equation for the spot interest rate; bond pricing PDE; popular models for the spot rate (Vesicle, CIR and Hull &White); solutions of the bond pricing equation;
1.3固定收益-1.3Fixed Income
The fixed-income market demands a vast selection of investment options with a variety of credit quality, maturities, and yields to meet investors' objectives. To accomplish this, fixed income groups frequently create and modify mathematical models to calculate bond pricing, perform yield analysis, calculate cash flows, and develop hedging strategies.
Fixed-income research groups use the thousands of prewritten math and graphics functions in Math Works products to access bond data, perform statistical analysis, calculate spreads, determine bond and derivative pricing, perform sensitivity analyses, and run Monte Carlo simulations.
Advanced graphics and rendering capabilities in MATLAB make reviewing cash flows, visualizing decision trees, plotting spot and forward curves, and creating deployable interactive 2- and 3-D models easy.
1.4股权-1.4 Equity
Smart security investing requires in-depth research and analysis. Measuring all the influencing factors is an essential part of risk management. As a result, research groups continually create and modify mathematical models to calculate stock value, review forecasts, and develop innovative risk strategies.
Equity research groups use the thousands of math and graphics functions in Math Works products to access stock data, perform statistical analysis, determine derivatives pricing, perform sensitivity analyses, and run Monte Carlo simulations. The graphics capabilities in MATLAB offer a variety of ways to review time series data, visualize portfolio risks and returns, and create forecasting graphs.
1.5投资管理和交易-1.5 Investment Management and Trading
To meet the investment needs of individuals, institutions, and governments, investment firms need to deliver a wide range of investment opportunities with risk-adjusted performance and consistent returns over time. To accomplish this, financial professionals need to develop and use mathematical models to optimize portfolios and develop trading strategies and systems that can respond to market conditions.
Investment management and trading research groups use the thousands of math and graphics functions in Math Works products to easily access securities data, perform statistical analysis, determine pricing, conduct sensitivity and principal component analyses, and implement buy and sell criteria. The graphics capabilities in MATLAB offer a variety of ways to easily review time series data, visualize portfolio risks and returns, and create forecasting graphs. With Math Works deployment tools, you can easily compile and integrate your MATLAB algorithms into your system.
1.6数学和统计方法-1.6Mathematical and statistical approaches
According to Fund of Funds analyst Fred Gem, "There are two types of quantitative analysis and, therefore, two types of quant’s. One type works primarily with mathematical models and the other primarily with statistical models. While there is no logical reason why one person can't do both kinds of work, this doesn’t seem to happen, perhaps because these types demand different skill sets and, much more important, different psychologies."
A typical problem for a numerically oriented quantitative analyst would be to develop a model for pricing and managing a complex derivative product.
A typical problem for statistically oriented quantitative analyst would be to develop a model for deciding which stocks are relatively expensive and which stocks are relatively cheap. The model might include a company's book value to price ratio, its trailing earnings to price ratio and other accounting factors. An investment manager might implement this analysis by buying the underpriced stocks, selling the overpriced stocks or both.
One of the principal mathematical tools of quantitative finance is stochastic calculus.
According to a July 2008 Aite Group report, today quant’s often use alpha generation platforms to help them develop financial models. These software solutions enable quant’s to centralize and streamline the alpha generation process.
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