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.

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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.

2) Describe your job.

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.

Part IV: Suggestion & Advice

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.

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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:

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%.

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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%.

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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%.

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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.

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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.

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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.

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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.