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.

 

These students made it through the rigorous Financial Math program at NC State

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(Left to Right- Xue Miao, Xinyuan Huang, Director- Jeff Scroggs, Director of Career Services- Leslie Bowman, Zhe Wang, Zhexing Zhang, Meenakshi Ramchurn, Sohaila Shaukat). Not pictured- Rana Kashif, Haozhi Wei, Zhengran Zhou, Samuel Busch, Kathy Varga, Xiangju Wang, Cheng Yu, Shihao Zuo, Meng Yang, Wen Zhong, Ying Xu)

The students (pictured above) proudly graduated on May 9, 2014 and received their Masters in Financial Mathematics (MFM) degree. They are excited and happy to achieve this meaningful accomplishment. Their hard work and long hours of studying paid off!

Sohaila Shaukat shares more details:

What was the most rewarding assignment or project of the program?
"The most rewarding project of the program was the one we did on asset pricing in 'Computational Methods in Economics and Finance' course. The project solidified my interest in derivative pricing and helped me build on what I had previously learned in Monte Carlo Methods for Financial Mathematics and Asset Pricing. Also, this added a lot to my resume, as employers are constantly seeking people who can build financial models and have a little bit of experience in it. It was also the reason I landed with an internship over the summer with Tata Consultancy Services."
What was the most interesting or favorite course and why?
"Most of the courses were rewarding. But my two most favorite courses are 'Computational Methods in Economics and Finance' and 'Time Series Analysis'. Both are difficult courses with brilliant professors, and helped me enhance my skills in data modeling, derivatives pricing and financial modeling. These courses also introduced me to R-programming and enhanced my skills in Matlab."
How many hours a week did you spend studying (on average)?
"I studied 20 to 30 hours per week on average. 20, when we didn't have to submit assignments in every course, and 30 or more usually when exam week/ mid terms are near."
Anything you would have done differently throughout your time here?
"I would have worked harder on the courses that involved a lot of Statistics and Stochastic Calculus. Since I had a non-mathematics background, I should have spent a lot more time on them. Also, I would have started applying for full time positions in July, 2013, instead of delaying it till January, 2014. This is because most major banking/wealth management firms hire their graduate trainees between July to December."
Sohaila's hard work paid off since she had several interviews and received a job offer. Look forward to a future post about her story.

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(Xinyaun Huang, Xue Miao, Zhe Wang, Zhexing Zhang)

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(Sohaila Shaukat, Director, Jeff Scroggs, Meenaski Ramchurn)

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(Time to celebrate!)

We congratulate them and wish many, many years of success!

To learn more about the program: Financial Math program details

 

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.