Bloomberg terminal for Financial Math students

By Xiaohong Chen, May 2015 Graduate, Financial Math Intern & Career Ambassador

The Bloomberg Terminal is a computer system provided by Bloomberg that enables professionals in finance and other industries to access Bloomberg Professional services through which users can monitor and analyze real-time financial market data and place trades on the electronic trading platform. Most large financial firms have subscriptions to the Bloomberg Professional service.  I am please to tell everyone that last spring NC State's Financial Math program received a Bloomberg Terminal!

So you may ask- what could you do with this Bloomberg Terminal? I have listed three most important features students may have interest in:

1) Market data: The most well-known function of Bloomberg terminal is the real-time financial market data, namely security values. This includes the data for stocks, fixed-income securities, derivatives, and foreign exchange. This function is powerful that one can even view historical pricing, read a description of the business, and view analyst reports.

2) News: The Bloomberg terminal can offer you the real-time news. By typing “NEWS” in the search bar, one can get the latest financial or non-financial news headlines from all over the world. The Bloomberg terminal also has job search function built inside of it. And you can filter by position and location to help pinpoint the right career that suits your needs.

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3) Education: If you are a beginner, don’t worry because Bloomberg terminal offers a lots of ways to help you learn it. You can choose to take Bloomberg Essentials Online Training program, which is designed to provide users with an introduction to the Bloomberg Professional Service. You can also find other seminars and events in Bloomberg University. And when you have no idea about how to use a terminal properly, Bloomberg services offers free help via phone and IM chat 24 hours a day.

Current students- are you excited about it? Don’t wait. Just go and play around with this powerful tool, and you will find more surprising features.

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%.

19

 

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.

 

Re-cap of Hedge Fund Challenge 2014

By Dendi Suhudby, May 2016 graduate

A couple of weeks ago a few students in the Financial Math program decided to attend the local Hedge Fund Challenge held nearby at the Washington Duke Inn in Durham. Once we heard about this event from local alumni and current students, we immediately became interested. I researched the event to learn that it was a challenge to pitch an investment strategy for a new startup hedge fund. This really sparked my interest, because hedge fund structures allows us to find strategies in broad range of asset classes compared to, for instance, finding investment strategies for mutual funds or traditional investment vehicles.

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I attended the event with our Program Director, Dr. Scroggs and current student, Bingxi Du (both pictured above). The other participating teams were from Duke, UNC Chapel Hill, Eastern Carolina University, University of Richmond, and Elon University. There were 5 scheduled speakers to gave lectures about 5 main points of an investment strategy buildup:

1. Idea generation

2. Valuation

3. Macroeconomic Analysis

4. Risk Management

5. Trade Structuring

Here are the summaries of the speakers lecture.

1. Idea generation

Idea generation is all about the brainstorming and finding inefficiencies in several asset classes for an opportunity to take an advantage for convergence of mis-pricing. For an example, in an idea generation there may be due diligence from a researcher, portfolio manager and the investor relations on the company’s on going corporate strategies. There might be more indirect idea generation by looking at data from Bloomberg and making a quantitative analysis pattern on data.

2. Valuation

The part of the valuation is where you pull up a spreadsheet and try to predict the outcome of the investment strategy in todays time. In finance lingo, it is the process to project future cash-flows and discount back to todays present value. The highest valuation of the investment opportunities are then chosen.

3. Macroeconomic Analysis

Macroeconomic analysis is done to find the possible directions of the environment say for example of interest rates, or business policies that are favorable for specific industries.

4. Risk Management

The portfolio manager and the buy side researcher then finds possible deviations of outcomes through risk analysis. For example using scenario analysis, they would scenario the investment strategy if there is a deviation of parameters within their models that change the outcome of investments. Within this process, the portfolio managers also find ways to hedge the investment if something goes in an adverse way or how to completely liquidate the investment if it is on loss.

5. Trade Structuring

After doing all those above, then portfolio managers make the decision to structure their trade either 1) buying the underlying asset or 2) enter a derivative contract. For example, if a portfolio manager has the view that the interest rates would increase he/she might want to short bonds or has the possibility to enter in interest rate future contracts in the Chicago Mercantile Exchange or even enter an OTC Credit Default Swap. Even these three investment strategy are betting on an increase in interest rate; there are several things that needs to be decided such as liquidity of each asset (futures liquidity > CDS liquidity > bond liquidity) where the future asset class might be more favorable than shorting the bond.

audienceHFchallenge

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The main conclusion of the kickoff event and the hedge fund lectures are that to conduct a trade takes a lot of effort, brainstorming, quantitative analysis, and due diligence, differs far from the public opinion that a hedge fund bets on reckless investments. Hedge funds also are much more flexible than other fund structures because of its unregulated nature, and it can also invest in exotic investment structures that mutual funds cannot invest in.- Dendi Suhudby.