Finite Difference Method for derivative pricing from a student’s point of view

"As we know, the value of a certain derivative can be expressed as a stochastic differential equation (SDE). Since stochastic differential equation can be transformed into a corresponding partial differential equation (PDE), it is worth to learn some numerical methods to solve PDEs. And Finite Difference Method (FDM) is the one widely used in this area. In this article, I will give a brief introduction to FDM and how it could be applied in option pricing. We focus on the famous Black-Scholes partial differential equation in this article. More details about FDM, Matrix Solver and stability analysis will be given in the later articles.

BS Model

In the BS Model, the price of a European call option satisfies the following PDE:

where f is the price of European stock options, S is the price of the underlying stock, sigma is the volatility of the stock price per year and r is the riskless interest rate.

Initial and Boundary Conditions: In order to apply FDM, we also need to provide initial and boundary conditions. In this problem, the terminal condition is given, which is the payoff of the option at expiration time:

The terminal condition can be converted to initial condition by simply changing the sign of the first derivative with respect to time t in the original equation.

Then the PDE becomes:

Normally, we use Dirichlet boundary conditions to approximate this Cauchy problem, which can be expressed as:

Discretized Schemes

(1) Explicit-Euler scheme

(2) Implicit-Euler scheme

(3) Crank-Nicolson scheme

Conclusion:

Explicit-Euler scheme is an explicit method, which means the discretized system of equations can be solved explicitly. Therefore, this method runs fast on the computer. However, this method is only first-order accurate in time and has some stability issue. This means there is a restriction on the size of the time step for this method to be stable.

Implicit-Euler scheme is an implicit method, which means we need to solve a linear algebra system of equations. Fortunately, the matrix formed in this problem is tridiagonal, which, to some extent, reduces the storage and calculation cost. This method is also first-order accurate in time. But compared with Explicit-Euler, Implicit-Euler guarantees the stability.

The most popular scheme may be Crank-Nicolson, which is always stable and has second-order accuracy in time. Of course, this method is implicit and thus needs to deal with Matrix solving problem. This scheme seems perfect, but in fact it is not. We should notice that when the advection term is dominated, the so-called spurious oscillation may occur, which can cause great error. This issue will be addressed in detail in another following article." - Xiaohong Chen (May 2015 Graduate)

Bloomberg visit- women have more influence on the economy than men!

He is one of Bloomberg’s top economists.

He predicted the 2008 financial crisis in 2007.

He was hailed as “Nostradamus of the Financial Industry” by Bank Advisor in 2008.

His name is Richard Yamarone. Mr. Yamarone recently visited North Carolina State University and talked with Financial Math students about career paths in Wall Street, and well as outlook on the economy.

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Mr. Richard Yamarone started his speech by talking about how to prepare for jobs in Wall Street. He stressed the importance of networking, pointing out that even your classmates and alumni are great resources to socialize with. He also stated that grades may not have much impact on your career developments. Instead, your transferable skills and personal ability matters more and should be highlighted in your resume and interview.

In terms of preparing for interviews, Mr. Richard Yamarone suggested that we should read the news everyday to keep up to date on the latest trends and news in the industry. He also discussed the tough job market and getting used to rejection. He shared his personal experience as an example, and how he landed a job in Wall Street.

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Later, Mr. Richard Yamarone changed his topic to indicators for economic forecasting. He believes that there is no holy grail of economic indicators but the ‘Fab Five’ indicators he summarized may shed some lights on predicting. The ‘Fab Five’ refers to:

(1) Dining out

(2) Casino Gambling

(3) Jewelry & Watch

(4) Cosmetics & Perfumes

(5) Women’s Dresses

Yes, these are all related with women’s consumption. Mr. Richard Yamarone concluded that women have high influence on the economy not men!

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Thank you, Mr. Richard Yamarone for your wonderful and enlightening presentation!

Share your thoughts- Do you think women have more influence than men on the economy? Do you think grades matter or not?

-Xiaohong Chen, May 2015 Graduate, Career Ambassador

Meet Yi Chao- Career Ambassador of the Week

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"My name is Yi Chao (May 2015 Graduate), currently a first year graduate student in Financial Mathematics at NC State. I grow up in Baoji, a small city in the middle of China. During 4 years’ college life in Beijing, the capital of China, I found myself really enjoy modern city life. For example, I enjoyed spending all the day in museums and art galleries. After arriving at Raleigh, North Carolina, I quickly adapt life here such as enjoying nature by hiking in the parks around the city. I have also fallen in love with the warm sunshine and fresh air here.

Since I major in math during undergraduate, it may be a natural choice for me to pursue a Masters in Financial Math. I have enjoyed the feeling of applying the mathematical knowledge I learned in classes into reality by receiving several prizes in mathematical modeling contest. As my graduate studies have progressed, I realized Financial Mathematics is much more than I thought. Now I focus more on the risk management field by taking the extra steps with the Financial Risk Analysis and FRM certification this semester.

My interest in risk management came to be through an internship in the “Enterprise Risk Service” department at Deloitte in 2012. During that internship, our team worked hard building a risk appetite system for a large reinsurance company. We researched and modified the well-known Qis5 model according to the company’s real situation. Although this was challenging and we faced many technical obstacles, we finally made it out.

Comparing to myself at that time, I have much more knowledge both in math and finance now through the Masters in Financial Math program. Together with this, I engaged in leadership positions as a Career Ambassador and Financial Math Intern this semester. Our Director of Career Services, Mrs. Bowman, provided opportunities to understand more about networking with employers, and interview and presentation skills. I am looking forward to future opportunities to apply these new skills with my past work experience and enhanced knowledge in the risk management field."- Yi Chao

Student’s view on Financial Math core courses at NC State

"Having taken many courses so far, Masters of Financial Math (MFM) students have discovered interesting and useful courses. Below are examples of a few core courses I have found important and useful."- Yizhou Chen, May 2015 Graduate

Statistical Theory:

Statistical Theory I & II is important in providing fundamental theory and formulas. In Statistical Theory II, we developed the probabilistic tools and language of mathematical statistics. The course describes basic probability theory, probabilistic models for a properties of random variables, common probability distributions for univariate and multivariate random variables, and sampling distributions and convergence theory. We learn description of discrete and absolutely continuous distributions, expected values, moments, moment generating functions, transformation of random variables, marginal and conditional distributions, independence, order-statistics, multivariate distributions, and concept of random sample.

The Statistical Theory classes are designed to provide the basic tools of statistical inference and prepare us to understand the foundations behind statistical inference. Thus, the knowledge enables us to formulate appropriate statistical procedures. Additionally we learn sufficient, ancillary, and complete statistics; Methods of finding estimators, including maximum likelihood; Mean squared error and unbiasedness; Hypothesis testing, including maximum likelihood; Mean squared error and unbiasedness; Hypothesis testing, including likelihood ratio; Power functions; Neyman-Pearson Lemma; Uniformly most powerful tests; Confidence intervals; Asymptotic properties of estimators and tests.

Asset Pricing:

Asset Pricing is a core course in the first semester of the Financial Math program. We gained a lot knowledge about finance from this course, especially for the students who have little knowledge about finance.  This course is an introduction to the pricing of assets. The emphasis is on the mathematical methods used to derive pricing formulas, and there is additional time devoted to explaining the major types of paper assets that can be priced with those methods. Real assets, such as factories and machines, also can be priced with the same methods. The goal of this course is to introduce us to the major types of asset prices and give us an understanding at an intuitive level of the relation between asset prices and the mathematics that governs their evolution.

The content in this course: Introduction to major fundamental assets (stocks and bonds), interest rates, and derivative assets, such as put and call options. Arbitrage theorem, present value, risk aversion, hedging, duration, properties of derivative assets, binomial trees, elementary stochastic calculus, Black-Scholes option pricing formula, implied volatility, capital asset pricing model. Emphasis on mathematical methods used to price derivative assets.

Probability and Stochastic Processes:

Probability and Stochastic Processes I: This course is set as an alternative course to Statistical Theory II. This course is more theoretical and the key point of this course is different from Statistical Theory I. In Statistical Theory I, we developed the probabilistic tools and language of mathematical statistics. Probability and Stochastic Processes describes basic probability theory, probabilistic models for and properties of random variables, common probability distributions for univariate and multivariate random variables, and sampling distributions and convergence theory. It is a modern introduction to Probability Theory and Stochastic Processes. The choice of material is motivated by applications to problems such as queueing networks, filtering and financial mathematics. Topics include: review of discrete probability and continuous random variables, random walks, markov chains, martingales, stopping times, erodicity, conditional expectations, continuous-time Markov chains, laws of large numbers, central limit theorem and large deviations.

Financial Mathematics:

Financial Mathematics- This is a core course in second semester, and challenging; some say difficult! Probability and Stochastic Processes and Asset Pricing courses are necessary to prepare us for Financial Mathematics class. Understanding the history of mathematics evolving over time as they are subjected to random shocks and knowledge of the mathematics of asset pricing are essential tools for this course.

Financial Mathematics course focuses on the basic mathematical tools for finance. In particular, we cover time value of the money, simple interest rate, bank discount rates, compound interest, ordinary annuities, extending ordinary annuities, amortization, sinking funds, perpetuities and capitalized costs.

Content of this course: Stochastic models of financial markets, No-arbitrage derivative pricing, discrete to continuous time models, Brownian motion, stochastic calculus, Feynman-Kac formula and tools for European options and equivalent martingale measures. We also learn about Black-Scholes formula, Hedging strategies and management of risk, Optimal stopping and American options, Term structure models and interest rate derivatives, and Stochastic volatility.

Monte Carlo Methods:

Monte Carlo Methods with Application to Financial Mathematics- This course requires some knowledge of programming. We use Matlab to write functions, apply appropriate control structures, and import and export data. We implement the methods mentioned in the other learning outcomes in Matlab. Matlab is utilized to visualize the results. The homework of this course may not be so difficult, but it takes a lot of time. Because of plenty use Matlab, we need take some pre-courses to prepare for it.

In this course we learn Monte Carlo (MC) methods for accurate option pricing, hedging and risk management. Modeling using stochastic asset models (e.g. geometric Brownian motion) and parameter estimation. Stochastic models, including use of random number generators, random paths and discretization methods (e.g. Euler-Maruyama method), and variance reduction."

By- Yizhou Chen, May 2015 Graduate, Career Ambassador & Financial Math Intern

Learn more about NC State Masters of Financial Math.