# Meet our Financial Math Alumni- Brandon Blevins

Meet Brandon Blevins, Product Controller at Credit Suisse in New York City. Brandon graduated from the Financial Math program in 2009. We were glad to catch up with him in Manhattan and learn about his job and life in the city.

Part I: Education & Job Background

1) How did the program prepare you for your job?

Brandon: The Financial Mathematics program gave me the background about quantitative finance. It provided me with the basics on how financial assets work, how models applied to assets, and how interest rate curves work.

Brandon: Currently, my job is Product Controller at Credit Suisse. The main point of Product Control is to ensure that the Profit and Loss (PL) generated by portfolios reviewed gets to the general ledger of the bank, which then is reported to shareholders and board members who make financial decisions based on Credit Suisse earnings.

My day involves reviewing risks on books, making sure risks are within tolerances, and P/L are in line with the risks. For example, suppose you have net Vega on a single position of \$100K. The volatility moved on the position by 100 basis points and you did not make or lose any money on that position. Did that make sense? It is the Product Controller’s job to make sure it does makes sense. If something is wrong, we flag it. We make comments on any big moves, big losses, and provide reasons why money is lost.

Part II: Analytic techniques

3) Does your company use stochastic models? If it does, what kind of models are used? Is there any reason for choosing these models?

Brandon: We use Black-Scholes formula to build up implied volatility curve. Options with the same underlyings and maturity but different strike prices have different implied volatility. The same options with different maturities may also have different implied volatility. Thus, implied volatility is a function of strike price and maturity, and we can define a volatility surface. We also use jump-diffusion models to model certain protocols. For example, is there a big court case coming up in the next few years for a specific company, or does this specific company have any big products coming out in a few years? That is where jump-diffusion comes into play.

Part III: Risk management

4) How does the crisis and the regulation policies enacted afterwards affect the behavior of your company?

Brandon: Radically. Since then, many parts of businesses have been shut down. Interest rate products have been drastically cut by 90%, because the Feds have kept rates low. There is a huge push to move everything onto exchange and standardize all products. Any flow business has been hit hard such as the credit default swap (CDS) market. Junk bond market has been on fire lately. But it did excite the mortgage back portion.

5) The goal of risk management is to achieve a balance between returns and risks. Thus, with lots of capitals and human resource spent, risk management may, to some extent, reduce a company’s profits. Now suppose you are a leader of a financial institution. Driven by the motivation of maximizing the profits, will you pay enough attention for risk management?

Brandon: Lead traders will listen to risk management and work in conjunction to set risk limits and VAR measures. If the limits get breached, everyone will look at it.

6) You used to be an interest rate derivative analyst, but now you are focusing on equity derivatives. So in your opinion, what is the difference between the interest rate derivative market and the equity derivative market?

Brandon: Interest rate market and equity derivative market start to look alike with low volatility. The bond market did very well in the past, but now it hover sideways because of the flat yield curve environment. Equity market is experiencing the same problem. Rates are not moving and are low.

7) What skills set are important to succeed in your field? And what kind of courses will you recommend for current students to take.

Brandon: Networking! Make sure people like you, so they will recommend you. I landed all of my jobs because I knew people who worked for companies that I wanted to join as well. I got the interviews because people recommended me. Therefore, students should get out there, meet people, talk to them, learn from them, make relationship with them, and then they will recommend you for jobs. Just get connected! People can put you in positions to succeed, and give you opportunities to help you succeed. If they like you, they want to you succeed.

Thank you Brandon!

# 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:

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

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

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

Using this, the distribution of ST may be written as

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

which can then be written as

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

where

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

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

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:

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

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:

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

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.

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%.

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%.

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%.

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.

# Meet our Financial Math Alumni- Coffee Chat with Albert Hopping- Part 1

Meet Albert Hopping, ERP- Manager of Risk Consulting at SAS Institute in Cary, North Carolina. Albert graduated from the Financial Math program in 2007. He is a Board Member and an active alumnus of our program. We were happy to meet with Albert for coffee at his office at SAS.

Part I: Education & Job Background:

1) Why did you decide to get a Master’s degree in Financial Mathematics from North Carolina State University?

Albert: That I might receive more wages. That is the short answer. I finished my undergraduate degree before this program existed. When I joined the program, I was working full time. I had been out of my undergraduate program for a few years and I ended up in a risk analytics team at a diversified energy company. At that point, I didn’t have a risk background and didn’t know much about the field. I was on the risk team looking around and I saw these quant guys programming in Matlab. It looked like fun to me and I thought that it was a cool job. I wanted to be a part of what they were doing, so I started helping them with their work as much as I could.

Eventually, it got to the point where I was doing this risk work a majority of my time. I told my manager that I should be moved to the quant job family. I was told that a master’s degree in Financial Mathematics was a prerequisite for the job. Not having the degree was a roadblock for me. I applied to the Financial Mathematics program that week and I am glad I did.

2) How did the program prepare you for your job?

Albert: I was already doing risk work; I was self-taught to a certain extent. This put me in a different position compared to most students, and I got different things out of the program than other people might have, as a result. Most students learn theory first and practice second. I started the program with the perspective of a practitioner. While in the program, I learned about models I used on the job. I used these tools at work, but I didn’t really know about stochastic partial differential equations. I used Black-Scholes, but I could not derive it.

What I received from the classes is a much deeper conceptual understanding and a firmer foundation from which to see my work. What I really took from this program is a fundamental, basic understanding of financial mathematics. I also learned new models and conceived great ideas to use in practice.

Albert: At SAS Institute, I am a Manager on the Risk Solution team of Professional Services & Delivery (PSD). Let me explain from the top down. PSD is the consulting, customization, and delivery arm of SAS. Many customers of SAS software want services, consulting, or even staff augmentation. PSD provides these services. We are the ones who go to the customer site and work with the customer to help them get the most benefit from our products. Our team within PSD specializes in the risk management domain. We work with all the risk solutions and provide consulting for all risk topics. Our team has about 20 members and is growing.

Within risk, I specialize in three industries Energy, Financial Services, and Healthcare. I am responsible for leading customer projects, providing industry and risk domain expertise to the sales teams, mentoring fellow team members, and most importantly providing value to the customer. Note that the views and opinions I express today are my own.

Part II: Analytic techniques

4) “Big Data” is a hot specialization in the field. Do you see this as a long term trend or something that might pass as a fad?

Albert: Big Data is definitely a long term trend. In fact, I would go beyond that; I would say it is going to be the new norm. It will progress to the point where big data is simply the paradigm. I will even extend that to unstructured data. Companies, who are not using big data and unstructured data to their advantage, are starting to fall behind. They are tomorrow’s luddites.

5) The trend of “Big Data” implies that people believe historical data can shed light on future prediction. However, this contradicts with “efficient market hypothesis” to some degree. What are your thoughts about this?

Albert: One of the things I would like to point out in terms of the “efficient market hypothesis,” is the irrationality in the market. A simple example comes to mind: technical traders discuss how a stock index will meet resistance or break through a barrier. But what are those points where the index meets resistance or breaks through? They are numbers with lots of zeroes on the end, round numbers. Why are those numbers important? They are only important because we tend to be emotional and we have ten fingers. I propose that if we had a different number of fingers we would use a different base for our number system. The round numbers where the stock index meets resistance would be different numbers.

Clearly, these barriers are irrational, as they are based on how many fingers we have. This means I cannot be a full believer in the “efficient market hypothesis.” The question remains, is all this historical data priced into the market already? To the extent that people are doing analytics on big data, perhaps yes. Was it priced in before? No. Was the data available? Mostly, but people could not convert the data into knowledge. It was impossible - until analytics on this big data was possible.

Now, we are in the place where something can be done because of the advancements in software and the physical hardware. Data can be restructured and put into use in the market. The fact that the data is available is clearly important, but prior to these advancements one could not glean actual insights. The data must be converted into information that helps those insights that yield a better price or a better model. Acting upon those insights is what makes the market more “efficient”.

-------

# 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

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

The variance is the same as the mean

We define the compensated Poisson process as

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

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

If we define the compensated compound Poisson process as

then 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

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

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

This is equivalent to the equation

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

For the next step, we need some notation. Define

where

and

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

With a little work, the price can be rewritten as

where

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.

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.

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.

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 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)