Join our new workshop- “Introduction to Financial Risk”

NC State's Financial Math program has partnered with 2004 Alumnus, Jonathan Leonardelli, to create a new workshop series "Introduction to Financial Risk" for all NC State students and faculty. The workshop is also opened to the public.

Students and faculty in Mathematics, Statistics, Economics, Finance, Operations Research, MBA and other related programs are welcome to join!

Here is what you will learn:

Risk Workshop Overview

Presenter information:

Jonathan Leonardelli, FRM, MFM

Jonathan Leonardelli, Risk Consultant at the Financial Risk Group, specializes in credit and market risk management. Over the course of his career he developed a diverse knowledge of retail banking risk as well as the technical skills needed to integrate risk assessment processes into a company’s business and technology infrastructure.

Jonathan’s career started with positions in the credit risk groups at Wells Fargo (Wachovia) and BB&T. In these positions, he developed expertise in acquisitions and portfolio risk management.  In his current position, Jonathan develops and implements processes that provide quantitative risk assessment and reporting capabilities for clients that include banks, hedge funds, and asset management companies.

Jonathan is an experienced presenter and author.  He is a certified SAS® Risk Dimensions Instructor. His papers in financial risk management covered topics such as the Dodd-Frank Act and its implications for risk professionals, as well as techniques for handling missing data. He has also authored a Webinar for the Insurance & Finance SAS® Users Group (IFSUG) regarding loss estimation using roll rate matrices.

Jonathan holds an Masters of Financial Mathematics from North Carolina State University and is a member of the Global Association of Risk Professionals (GARP).

FRM designation since 2010

SAS Certified Advanced Programmer for SAS 9

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Those interested- please contact Leslie Bowman, Director of Career Services- leslie_bowman@ncsu.edu 

Workshop begins Friday, September 5th 2014- registration deadline, August 28th

For new Financial Math students starting Fall 2014

The Financial Math program at NC State has students from diverse backgrounds and cultures; Yizhou Chen from Shanghai, China shares her story and tips for those moving to Raleigh, North Carolina.

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

"A new semester is coming! Since we have been living in Raleigh, North Carolina for one year, we have something to share with new classmates moving here as they start the Financial Math program at NC State.

First of all, I want to say that Raleigh is a great place to live. I am from Shanghai, China. Compare with Shanghai, the weather here is not too hot in summer and not too cold in winter. A lot of trees are around us and it is very quiet here compared to living in a big city. There are many ways to have fun with a several entertainment options.

The most important thing I think for new students, especially international students, is where you plan to live. Around NC State campus there are many different places to live from renting apartments, townhouses or sharing a house. Most students rent but there is options to buy as well. NC State has a great bus transportation system and is convenient for traveling back and forth to campus. However, a car is useful to run errands and see the rest of the city. If you can room with someone who has a car, that would be ideal. Many students end up having their own car after a year and move further from campus where rent is cheaper.

There are many restaurants near campus. Most places to eat are cheap and the food is decent. If you have a car, you can find some great restaurants in downtown Raleigh, North Raleigh and nearby cities of Cary, Morrisville, Chapel Hill and Durham. We enjoy venturing out to different nearby towns to eat at their restaurants. There are all types of cuisines here including Chinese, Indian, Japanese, Italian, Mexican, Southwest, American, European, French, etc.

Regarding coursework- It is good for students to register for four courses at the beginning in first semester. This way you can gage how much you can handle during the first few weeks. You can get the syllabus for each course and figure out the professor's style and expectations. Then evaluate your time and see if you can still handle four courses. You may find you need to drop one course.

Because every student's background is different, it is hard to know which courses will be hard and which ones will be easy. You will find after a few weeks that you can make friends who can help you study in courses that are challenging.

My advice is to not overload yourself in the first semester. Hard courses require a lot of time and energy. For example, I think MA 546 (Probability Theory and Stochastic Processes) is a very hard course, because I need to understand the different theories and figure out the proofs. However, MBA (Masters in Business Administration) courses are not very hard. Even though I need to read cases and prepare for presentations, these courses are not heavy brain work and time consuming.

For fun, we always go to mall for shopping. It is not very far and the shopping malls have a lot of interesting stores to check out. North Carolina has beaches and mountains which are not too far from Raleigh. We have taken day trips to the beach, hiked in the Smoky Mountains, and visited lakes to swim in close by. There are many forest parks and lake parks here. During the spring, summer and autumn, we can pick fruits in some farms. Sometimes we go to concerts, visit museums, and go to the theatre here.

People here are very polite and kindness, I think all of you will love living here. And, I think you will also enjoy the Financial Math program at NC State."- Yizhou Chenblogpostpic blogpostpic3 blogpostpic2

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.