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.

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

Image

(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.

Image

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

Image

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

Image

(Time to celebrate!)

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

To learn more about the program: Financial Math program details