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.

New Series- “Meet our Financial Math Alumni”- Coffee Chat with Jonathan Leonardelli

"Meet our Financial Math Alumni" is an up-close interview series with select Financial Math alumni to learn more about their career, experience and knowledge after receiving their Masters in Financial Math degree from NC State University. Alumni are important part of our program for many reasons. They provide support, vision and strategy to ensure the success of our program. They are role models and mentors for current students. They strengthen the reputation of the Financial Math program. They provide job leads and recruitment activity for students. Our alumni are intelligent and awesome! Thank you to those who participant in this series"- Leslie Bowman, Director of Career Services

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Meet Jonathan Leonardelli, FRM, Risk Consultant (Graduated 2004)

Interview conducted and summarized by Yi Chao and Xiaohong Chen, Financial Math Interns, May 2015 Graduates

Jonathan is currently a Risk Consultant at Financial Risk Group. He is also one of the first students to graduate from NCSU’s Financial Mathematics program. We were honored to have the opportunity to interview him.

Part I: Education & Job Background

1) Interviewer: Why did you decide to get a Masters in Financial Math at NC State?

Jonathan: I had just moved here and was interested in changing careers. I wanted to find a program that combined mathematics and finance. As it happened, NC State was in the process of creating the Financial Mathematics program. Although the start of the program was still a couple years off, that time allowed me to get the necessary mathematics background I needed. Entering the program produced the exact result that I wanted: it gave the mathematics I need to do interesting work in the banking industry.

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

Jonathan: The program really gave me the depth of math that I needed to work in the field of risk. It also taught me how to apply rigorous logic to a problem to help find a solution. Beyond this, though, I learned that life was filled with randomness. This randomness, as a result, causes the quantification of some metrics to be difficult.

Part II: Analytic techniques

3) Interviewer: "Big Data" is a hot specialization in the field. Do you see this as long term trend or something that might pass as a fad?

Jonathan: It is definitely not a fad. In this day and age many actions we take, especially when using a piece of technology, is probably captured and stored in some database. Now, think about all those action and all the data that comes along with it. What is a company going to do with this data? They are going to improve their sales, improve their risk practice…the list goes on. Having skills to work with big data, to be able to find the relevant information and then integrate it into a model, is very useful.

4) Interviewer: The trend of “Big Data” implies that historical data can shed some lights on future prediction. However, this contradicts with “efficient market theory” to some degree. What are your thoughts about this?

Jonathan: I think concerns should always be involved when using historical data because the tacit assumption is that the future is going to behave like the past. That being said, there are ways to mitigate these concerns. For example, when we calibrate models one of the first things we do is test it with a different period of data to see if the model is robust. Sometimes we might use the model on data representative of a stressed scenario (i.e., a scenario that is uncommon but still possible) and see how the model performs. If it performs badly, we try to assess why that is. Are the parameter estimates wrong? Are different variables needed?

5) Interviewer: Does your company use stochastic models to predict interest rate? What kind of models are used?

Jonathan: To be honest, in my current job the main stochastic (i.e., diffusion based) models I have used are CIR (Cox-Ingersoll-Ross) and GBM (Geometric Brownian Motion) with the occasional jump-diffusion model thrown into the mix. Most of the models I have worked with recently are linear regression, logistic regression, and Markov chains.

6) Interviewer: In your area of specialization, what is your favorite method or model and why? Do you believe it is perfect?

Jonathan: Markov chains and their resulting transition matrices. This comes from the years when I worked in the banking industry. At a glance, the transition matrix tells you the behavior of different segments of accounts. Depending on how the states of the Markov chain are defined, the transition matrix can tell you: 1) The probability of going to default, 2) the probability of paying off, 3) the probability of curing, 4) the probability of moving across multiple states over a given time, etc.

The transition matrix is a great summary tool. Of course, it is not perfect. Always keep that in mind when you build a model. Even though the model looks pretty and deals well with the data – now – it is not perfect. It is an easy move from complacency, when the model is performing well during good times, to anxiety when the model is performing poorly during a financial crisis.

Part III: Risk Management

7) Interviewer: How do the regulation policies enacted after crisis affect the behavior of your company?

Jonathan: The regulations have not impacted the behavior of our company. However as a risk consulting company, we have seen more requests from financial institutions asking us to help them comply with the regulations.

8) Interviewer: The goal of risk management is to achieve a balance between returns and risks. Thus, with a lot of capital and human resource spent, risk management may, to some extent, reduce a company’s profits. Driven by the motivation of maximizing the profits, will the companies pay enough attention for risk management?

Jonathan: I may be biased, given my chosen career path, but I think with the recent financial crisis still fresh in our memories as well as all the regulations that were created as a result of it, businesses will continue to pay enough attention to risk management. And, I think, it will be that way for a while.

9) Interviewer: Any tips for those interested in getting into the field?

Jonathan: First, be comfortable with the idea of randomness. Randomness is uncertainty and uncertainty is risk. Second, companies do not only seek candidates that are good at math and programming. They also seek those candidates who can clearly present their ideas in writing and presentations. Third, get perspective from areas outside of mathematics. I strongly suggest taking business classes because it gives you a different view of a business. A company is multifaceted and does not solely revolve around models.

10) Interviewer: What courses do you recommend?

Jonathan: The courses in the Financial Math program are very good. Among them, I definitely think the probability course is the most important one. That course takes you deep into the world of randomness and, hopefully, makes you comfortable with it. If you have an option of taking a course on risk or financial regulation that would be good. I took econometrics as an elective and, more often than not, draw upon the math tools I learned from that class more than others. Another very good course I took, and have used frequently, is time series analysis.

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"It is a great experience to interview Jonathan. He is humorous and smart. We learned a lot when we talked with him. Thanks Jonathan for participating in our interview! We are sure that your answers will shed some light for those interested in Financial Mathematics!"- Yi Chao and Xiaohong Chen