Bloomberg terminal for Financial Math students

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

The Bloomberg Terminal is a computer system provided by Bloomberg that enables professionals in finance and other industries to access Bloomberg Professional services through which users can monitor and analyze real-time financial market data and place trades on the electronic trading platform. Most large financial firms have subscriptions to the Bloomberg Professional service.  I am please to tell everyone that last spring NC State's Financial Math program received a Bloomberg Terminal!

So you may ask- what could you do with this Bloomberg Terminal? I have listed three most important features students may have interest in:

1) Market data: The most well-known function of Bloomberg terminal is the real-time financial market data, namely security values. This includes the data for stocks, fixed-income securities, derivatives, and foreign exchange. This function is powerful that one can even view historical pricing, read a description of the business, and view analyst reports.

2) News: The Bloomberg terminal can offer you the real-time news. By typing “NEWS” in the search bar, one can get the latest financial or non-financial news headlines from all over the world. The Bloomberg terminal also has job search function built inside of it. And you can filter by position and location to help pinpoint the right career that suits your needs.

.IMG_0435 IMG_0436

3) Education: If you are a beginner, don’t worry because Bloomberg terminal offers a lots of ways to help you learn it. You can choose to take Bloomberg Essentials Online Training program, which is designed to provide users with an introduction to the Bloomberg Professional Service. You can also find other seminars and events in Bloomberg University. And when you have no idea about how to use a terminal properly, Bloomberg services offers free help via phone and IM chat 24 hours a day.

Current students- are you excited about it? Don’t wait. Just go and play around with this powerful tool, and you will find more surprising features.

Stochastic models for option pricing- stochastic volatility model

By- Xiaohong Chen, May 2015 Graduate

Introduction:

With the Black-Scholes model, the volatility of stock price is assumed to be constant, but we have observed that the implied volatilities of tradable options vary from day to day. This can be caused by changing risk preference of market participants like in the jump-diffusion model. An alternative explanation is that the instantaneous volatility of a stock itself follows a stochastic process. One of the first research papers of stochastic volatility model was published by Hull and White [1]. In this blog post, we focus on discussing the content of that paper below:

Stochastic model and risk-neutral pricing:

Generally, volatility is chosen to follow a diffusive process. Let us consider a derivative f with the underlying S. Assume the instantaneous variance V2. Then, in Hull and White’s paper [1], we have the following stochastic processes:

1

The variable φ is a parameter that may depend on S, σ, and t. The variable μand ξ may depend on σ and t, but it is assumed, for the present, that they do not depend on S. The Brownian motions z and w have correlation ρ. Also, we assume the risk-free rate, which will be denoted by r, is constant or at least deterministic.

A quick application of Ito’s lemma shows that the drift of stock price must be the risk-free rate r in risk-neutral world. Invoking Girsanov’s theorem, we conclude that all risk-neutral measures are associated to the process of the form

2

Where w tilt and z tilt are Brownian motions with correlation coefficient ρunder risk-neutral measure. Notice that the drift of instantaneous variance ais arbitrary and could be any reasonable function of σ and t. This reflects the fact that volatility is not a tradable quantity. Hence our market has two sources of uncertainty but only one underlying and so is incomplete.

A closed form formula in the uncorrelated case:

Hull and White [1] deduce a closed-form formula for pricing European options when the correlation coefficient ρ is zero. By using risk-neutral pricing formula, the price of an option can be expressed as

3

where

T :  time to maturity;

St : security price at time t;

σt : instantaneous standard deviation at time t;

p~(ST|St, σ2t): conditional distribution of ST given the security price and variance at time t under the risk-neutral world;

f(ST, σt, T): max[0, S-K] and K is strike price.

Define V bar as the mean variance over the life of the derivative security defined by the stochastic integral

4

Using this, the distribution of ST may be written as

5

where g~(ST|V-bar) and h~(V-bar|σ2t) are conditional distributions of ST andV-bar under risk-neutral world respectively. Thus one can get

6

which can then be written as

7

Under the assumptions that ρ=0, μ and ξ are independent of S, the inner term is the Black-Scholes price for a call option on a security with a mean varianceV-bar, which will be denoted as C(V-bar) and expressed as

8

where

9

and N(x) is the cumulative density function of standard normal distribution. Thus, the option value is given by

10

Monte Carlo simulation procedure

In this part, we relax some assumptions made before. We now allow the correlation coefficient ρ to be nonzero and let V follow a mean-reverting process. One example is

11

where κ and θ are constants. Here, the instantaneous variance follows a CIR process [2]. It reverts to level θ at rate κ.

Now we introduce a Monte Carlo simulation procedure described in Hull & White [1], we divide the time interval T - t into n equal subintervals. Two independent normal variates ui and vi are sampled and used to generate the stock price Si and variance Vi at time i in a risk-neutral world using the formula:

12

where Δt = (T t)/n.

In order to speed up the simulation, we need to apply some variance reduction techniques. Hull & White proposed a procedure as following. The value of

13

is calculated to give one “sample value”, P1, of the option price. A second price, P2, is calculated by replacing ui with -ui (1 ≤ i n) and repeating the calculations; P3 is calculated by replacing vi with -vi (1 ≤ i n) and repeating the calculation; P4 is calculated by replacing ui with -ui and  vi with -vi (1 ≤ in) and repeating the calculations. Finally, two sample values of the B-S price q1 and q2 are calculated by simulating S using { ui } and { -ui }, respectively, with V kept constant at V0. This provides the following two estimates of the pricing bias:

14

These estimates of bias are averaged over a large number of simulations, and the final estimator of the option price is

15

where C(V0) is the B-S price with V kept constant at V0.

This procedure uses the antithetic variables technique twice and the control variate technique. More details about these techniques are described in Glasserman [3].

Stochastic volatility smile

Since the possibility of the stochastic volatility getting large increases the possibility of the large movement of the underlyings, the model, therefore, gives rise to a fatter tails distribution for the terminal log stock price. This leads to implied-volatility smiles which pick up out-of-money. See Figure 1.

One can introduce skewness by letting the underlying and the volatility correlated. Roughly speaking, the smile is downwards sloping when the correlation is negative while it becomes upwards sloping for large moneyness when the correlation is positive. See Figure 2 and 3.

The major difference between stochastic volatility model and jump-diffusion model is in their time decay. The amount of stochasticity in the volatility increases over time and this leads to long-maturity smiles not decaying. However, the time behavior could be controlled by the mean-reversion parameter to some degree. The faster the mean-reversion, the flatter long-time smiles will be. See Figure 1 and 4.

16

Figure 1 Stochastic volatility smiles (Heston model) for time horizons of one through five years. Spot is 110 and volatility is uncorrelated with spot. The reversion speed is 1 and the volatility of variance is 0.5. Initial volatility is 10%.

17

Figure 2 Stochastic volatility smiles (Heston model) for time horizons of one through five years. Spot is 110 and volatility is negatively correlated (-0.6) with spot. The reversion speed is 1 and the volatility of variance is 0.5. Initial volatility is 10%.

18

Figure 3 Stochastic volatility smiles (Heston model) for time horizons of one through five years. Spot is 110 and volatility is positively correlated (0.6) with spot. The reversion speed is 1 and the volatility of variance is 0.5. Initial volatility is 10%.

19

 

Figure 4 Stochastic volatility smiles (Heston model) for time horizons of one through five years. Spot is 110 and volatility is uncorrelated with spot. The reversion speed is 2 and the volatility of variance is 0.5. Initial volatility is 10%.

Reference

[1] J. Hull and A. White, “The Pricing of Options on Assets with Stochastic Volatilities,” Advances in Futures and Options Research, vol. 3, pp. 27-61, 1988.
[2] J. C. Cox, J. E. Ingersoll and S. A. Ross, “A Theory of the Term Structure of Interest Rates,” Econometrica, vol. 53, pp. 385-407, 1985.
[3] P. Glasserman, Monte Carlo Methods in Fiancial Engineering, New York: Springer Science+Business Media, 2003.
[4] M. Joshi, The Concepts and Practice of Mathematical Finance, Cambridge University Press, 2003.
[5] J. C. Hull, Options, Futures, And Other Derivatives, Pearson Education Limited, 2012.
[6] S. E. Shreve, Stochastic Calculus for Finance II, Springer Science+Business Media, LLC, 2004.

 

Stochastic models for option pricing- Jump-diffusion model

Background

The derivative pricing model developed by Black, Scholes and Merton is a huge success in financial engineering area. It says that there exists an arbitrage-free price for plain vanilla options and the investors can perfectly hedge them by constructing a self-finance portfolio. However, the empirical observation demonstrates that this model is not perfect. For one thing, two different options on the same underlying with the same expiry date but different strike prices can imply different volatility. Indeed, if one plots the implied volatility as a function of the strike price of an option, the curve is roughly smile-shaped. For another thing, the stock and foreign exchange prices are simply not log-normally distributed as the model assumes. And in fact, the actual distribution of the logs of asset price changes have fat tails. To cope with these problems, we need to introduce more sophisticated models.

Introduction

A big shortcoming of Black-Scholes model is that it assumes the asset price is a continuous function. But in reality, the stock market undergoes crash periodically. We, therefore, wish to permit the possibility of jumps in our model. In this post, we briefly discuss the jump-diffusion model presented by Merton. And in order to illustrate it, we first briefly discuss the properties of Poisson process.

Poisson process

The Poisson process with intensity lambda counts the number of jumps that occur at or before time t and its distribution is

1

Its increments are stationary and independent. The expectation of the increment is

2

The variance is the same as the mean

3

We define the compensated Poisson process as

4

Then M(t) is a martingale.

Now let Y1, Y2,… be a sequence of identically distributed random variables with mean Beta=EYi. We assume the random variables Y1, Y2,… are independent of one another and also independent of the Poisson process N(t). We define the compound Poisson process

5

Like the simple Poisson process, the increments of the compound Poisson process are stationary and independent, and the expectation is

6If we define the compensated compound Poisson process as

7then it is a martingale.

Asset driven by a Brownian motion and a compound Poisson process

In this section, the stock price will be modeled by the stochastic differential equation

8

where S is the stock price, W(t) is a Brownian motion, and Q(t) is a compound Poisson process. Lambda is the intensity of the jump and Beta is the expectation of the jump size Yi.

Under the original probability measure, the mean rate of return on the stock is a. We assume that the jump size yi > -1 for i = 1,…, M in order to guarantees that although the stock price can jump down, it cannot jump from a positive to a negative value or to zero. We begin with a positive initial stock price S(0), and the stock price is positive at all subsequent times.

By the property of Doleans-Dade exponential, one can find that the solution to the above SDE as

9

We now undertake to construct a risk-neutral measure. The probability measure is risk-neutral if and only if

10

This is equivalent to the equation

11

which is the market price of risk equation for this model. Here the letters with tilt represent the corresponding variables in risk-neutral world. Obviously, there is no unique risk-neutral measure in this situation because one can find infinitely many combinations satisfying the market price of risk equation. One can choose a risk-neutral measure by matching the market. Here, we assume a certain risk-neutral measure is chosen.

Closed form formula for European call option

A jump-diffusion model with a continuous jump distribution was first treated by Merton, who considered the case in which one plus the jump size has a log-normal distribution

12

For the next step, we need some notation. Define

13

where

14

and

15

is the cumulative standard normal distribution function. In other words,kappa(tau, x, r, sigma) is the standard Black-Scholes-Merton call price on a geometric Brownian motion with volatility sigma when the current stock price is x, the expiration date is tau time units in the future, the interest rate is r, and the strike price is K.

Now define tau = T t. We give the closed-form formula for the European Call option without proof

16

With a little work, the price can be rewritten as

17

where

18

These formulas were originally derived by Merton using PDE approach. Although the formula is an infinite series, it converges very fast and the first several terms can produce quite good approximation.

Jump-diffusion smile

In this part we discuss about the properties associated with the volatility smile generated by jump-diffusion model. It is straightforward that an option on a stock with a jump component is more valuable than an option on a stock without jump component. In fact, the effect of adding jumps can give rise to a heavy tail for the distribution of log stock price. Therefore, the out-of-money options become more valuable, a consequence leading to an implied-volatility smile.

The jump intensity lambda tilt controls the frequency of the happening of jumps. Increasing intensity makes the stock price more volatile, and thus the smile shape become steeper. On the other hand, the lower the jump intensity is, the flatter the smile would be. Also, the smile will be much sharper for short-term maturities. Over long time periods, the smile becomes more horizontal as the diffusive component of the model becomes dominant. See Figures 1 and 2.

19

Figure 1 Jump-diffusion smiles for time horizons of one through three years. Spot is 110, jump intensity is 0.1, and jump size are log-normal distributed with mean equal to -1.

The distribution of jump size Y determines the shape of volatility smile. Using symmetric distribution of jumps will lead to a symmetric smile shape. While if we let jump size follows a log-normal distribution, like what we use in this post, the smile becomes downwards sloping. The parameter mu can affect the skewness of the smile. Usually, we pick mu < 0, which means the stock price more likely goes down when jump occurs. This causes a downwards sloping smile. If we let mu > 0, the smile becomes upwards sloping for large moneyness. See Figure 2 and 3.

20

Figure 2 Jump-diffusion smiles for time horizons of one through three years. Spot is 110, jump intensity is 0.01, and jump size are log-normal distributed with mean equal to -1.

21

Figure 3 Jump-diffusion smiles for time horizons of one through three years. Spot is 110, jump intensity is 0.01, and jump size are log-normal distributed with mean equal to 1.

I have implemented the final closed-form formula for pricing European options in an Excel add-in. One can use the functions in this library to check the shape of "volatility smile" generated by the model as what I did in the last part of the article. One can download this Excel add-in from the following link below. Add it into your Excel (only for windows system) it's free to use!- (by- Xiaohong Chen, May 2015 Graduate, Financial Math Intern, Career Ambassador)

 

Reference

[1] F. Black and M. Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, vol. 81, pp. 637-659, May/June 1973.
[2] R. C. Merton, “Theory of Rational Option Pricing,” Bell Journal of Economics and Management Science, vol. 4, pp. 141-183, Spring 1973.
[3] R. C. Merton, “Option Pricing When Underlying Stock Returns Are Discontinuous,” Journal of Financial Economics, vol. 3, pp. 125-44, March 1976.
[4] M. Joshi, The Concepts and Practice of Mathematical Finance, Cambridge University Press, 2003.
[5] J. C. Hull, Options, Futures, And Other Derivatives, Pearson Education Limited, 2012.
[6] S. E. Shreve, Stochastic Calculus for Finance II, Springer Science+Business Media, LLC, 2004.

 

Finite Difference Method for Derivative Pricing- part two

By- Xiaohong Chen, May 2015 Graduate
This is part two from my first post about Finite Difference Method.  In the last post, I discussed about some traditional finite difference schemes used for pricing European options. Among them, the Crank-Nicolson scheme is the most popular because of its unconditional stability and high order accuracy. But as I pointed out before, this scheme is not perfect. It suffers from a so called “spurious oscillation phenomenon” in some situations. In this post, I try to present two different kinds of schemes, which are designed to overcome this shortcoming.Using the same notation from the last article, we deal with the following PDE:

1

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.

For call option, the initial and boundary conditions are:

2

Last time, we have discussed the Crank-Nicolson scheme. It is unconditionally stable and has second order accuracy in both time and space. However, when the convection term is dominated, the spurious oscillation phenomenon will occurs, and thus making the method ineffective. In the rest part, we discuss two other schemes, which can avoid spurious oscillation.

Implicit-Explicit method

Implicit-Explicit (IMEX) methods are fully coupled methods that are designed to handle some terms implicitly and others explicitly. A simple example is obtained by combining forward Euler with backward Euler:

3

In this scheme, backward Euler is used for diffusion term and zero term while forward Euler is used for convection term.

It is not difficult to find that the diffusion term and zero term are unconditionally stable. As for the convection term where forward Euler is used, it would be unstable if central difference is applied. Therefore, upwind scheme is used instead. It shows that the stability condition for this scheme is:

4

where

5

rearrange the inequality, we get:

6

Thus, there is a restriction on the time step size. The matrix of the linear algebra system formed by using this scheme is strictly diagonal dominant. Thus, this method avoid the oscillations phenomenon occurred when using Crank-Nicolson scheme. But it is only conditionally stable and first order accuracy in both time and space.

Exponentially fitted scheme

In general, for convection-diffusion equation:

7

We have scheme:

8

where the fitting factor is

9

As what has been discussed above, Implicit Euler method is unconditionally stable. However, when the convection term is dominated, spurious oscillation will occur. The modified exponentially fitted scheme is designed to overcome this weakness.

Here we take an extreme case as example. Suppose the coefficient of convection term is positive and that of diffusion term is zero. Then the equation degenerates to a hyperbolic equation. In this case, the fitting factor becomes:

10

Inserting this result into the degenerated hyperbolic equation gives us the first-order scheme:

11

This is the so-called implicit upwind scheme which is stable and convergent. Like the IMEX method, the exponentially fitted scheme successfully avoid oscillation phenomenon at the cost of lowering the accuracy.

To summarize, the Crank-Nicolson scheme is very popular due to its higher order accuracy and non-conditional stability. However, it is notorious that this scheme will give rise to “spurious oscillation” when the convection term is dominated. To overcome this problem, we introduce ‘IMEX’ scheme and exponential fitted scheme. Both of these two methods eliminate oscillation phenomenon by scarifying the accuracy and stability. There is no perfect scheme here. But one could choose a suitable one for the certain situation.

Test drive the Quantitative Finance BETA site from Stack Exchange

Quantitative Finance Stack Exchange is a question and answer site for finance professionals and academics. This website http://quant.stackexchange.com/ covers questions on real-life problems you face such as:

  • securities valuation
  • risk modeling
  • market microstructure
  • portfolio management
  • financial engineering
  • econometrics

And this is not a discussion forum...Quantitative Finance Stack Exchange is all about answers. Pretty straight forward- Ask a question, get an answer! Great resource to check out while it is in BETA.

Other resources for Quantitative Finance:

1. Wilmott.com

Wilmott.com is a quantitative financial portal created by Paul Wilmott. One can find job postings, technical articles, up-to-date news, and other useful resources.

http://www.wilmott.com/

2. QuantStart

QuantStart is a personal website discussing Algorithmic Trading strategy research, development, backtesting and implementation. The author behind this website once worked in a hedge fund as a quantitative trading developer in London. Therefore, his articles are very practical. There are several articles on career development, which are very useful for those unfamiliar with the industry.

http://www.quantstart.com/

3. QuantLib

The QuantLib project is aimed at providing a comprehensive software framework for quantitative finance. QuantLib is a free/open-source library for modeling, trading, and risk management in real-life. It is written in C++ with a clean object model, and is then exported to different languages such as C#, Objective Caml, Java, Perl, Python, GNU R, Ruby, and Scheme.

http://quantlib.org/

4. Yahoo Finance

You can get free stock quotes, option prices, up to date news, portfolio management resources, international market data, message boards, and mortgage rates from here.

http://finance.yahoo.com/

5. Data and Charts of U.S. Department of the Treasury

Great resource to find all kinds of interest rates related with the U.S. treasury.

http://www.treasury.gov/resource-center/data-chart-center/Pages/index.aspx

6. QuantNet

It is a leading resource on Financial Engineering education and news, but the main focus is education. Quantnet provides detail information about Financial Engineering programs in North America.

https://www.quantnet.com/

*****
By- Xiaohong Chen, Financial Math Intern, May 2015 Graduate

Database trends in financial services that quants should know

Image

Recent trip to New York City included a small alumni meet-up and Data Summit 2014. At Data Summit 2014 we learned about several database trends in financial services well beyond the popular RDBMS (relational databases) including Hadoop Big Data Platforms, NoSQL, NewSQL, and in-memory databases.

Quants know SQL, and it's important for them to be aware of the above database trends and what's driving them in financial services - such as risk analytics and reporting, market data feeds, high frequency trading, regulation, among other use cases driving demand for high volume and scalable, specialized databases.  While many quants are proficient in programming, it's not reasonable to expect them to learn each programming language driving these technologies to access data (Erlang, Javascript, C#, Java, etc).  This is not unique to quants as we're seeing SQL enable wider adoption of the Hadoop Big Data Ecosystems.

Sumit Sarkar of Progress Software (Gold sponsor of our program) talks about how professionals such as those in quantitative finance can easily work with data in the growing landscape of highly specialized database technologies, MongoDB for example, using standard based SQL interfaces such as ODBC and JDBC.

Image

(Alumni Left to Right- Emmanuel Sanchez with Allianz; Director of Career Services, Leslie Bowman; Yoshi Funabashi with Credit Suisse; Brandon Blevins with Credit Suisse)

Keep a lookout for their  "Meet our Financial Math Alumni" interviews.

We will be back again in October, 2014- so all NYC alumni, plan for another fun gathering!

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)

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.