# FE720 The volatility surface: risk and models

Course Catalog Description

# Introduction

Campus | Fall | Spring | Summer |
---|---|---|---|

On Campus | X | ||

Web Campus |

Instructors

Professor | Office | |
---|---|---|

Zhenyu Cui |
zcui6@stevens.edu | Babbio 514 |

More Information

# Course Description

In this course students will understand the implied volatility, and the empirical static and dynamic behavior of the volatility surface formed using option prices for all strikes and expiration. The students will also examine the volatility risk, stochastic volatility and local volatility models, numerical methods for volatility surface calibration, Monte Carlo simulation of stochastic volatility models, and pricing options through fast Fourier transform. Topics include: the Black-Scholes implied volatility, empirical statics and dynamics of the volatility surface, volatility risk premium, stochastic volatility models (Heston, Hull-White, Stein-Stein, SABR, Bates, Scott, etc), Dupire’s local volatility model, Heston-Nandi GARCH model, arbitrage-free properties of the volatility surface, volatility surface parameterization and calibration, simulation of the Heston model, stochastic volatility model with jumps, option pricing based on fast Fourier transforms, and volatility derivatives (Variance swap, CBOE VIX futures and options, etc). Other advanced current research topics will be introduced as well. The students are required to have a solid working knowledge of stochastic calculus, and FE610 is a pre-requisite for this course. The course uses statistical software such as MATLAB or R throughout. A companion one credit of a relevant lab course is recommended if this knowledge is not acquired before.

# Course Outcomes

Upon successful completion of the course the students will be expected to have the following specific knowledge.

- A thorough knowledge of the statics and dynamics of the volatility surface, and basic volatility instruments.
- The ability to calibrate implied volatility surfaces from option surfaces and interpret the results.
- The ability to use software to visualize and interpret the volatility surface using for example the Bloomberg or Thomson Reuters terminal.
- Understand the CBOE white paper on construction of the VIX index.
- Understand Dupire’s local volatility model and its calibration.
- The ability to price equity and Foreign exchange (Forex) options using the Heston model.
- The ability to calculate option prices through the Fast Fourier Transform for stochastic volatility models with jump components.
- Understand and compare different discretization schemes in the Monte Carlo simulation of the Heston model.
- The ability to hedge volatility contracts using co-terminal European call and put options, and analyze the hedging error.
- The ability to price volatility derivatives, the CBOE VIX futures and VIX options.

Course Resources

# Textbook

- Jim Gatheral, The Volatility Surface, 1st Edition, Wiley Series in Finance, 2006
- Alan Lewis, Option Valuation under Stochastic Volatility, 1st Edition, Finance Press, 2000
- Gianluca Fusai, Andrea Roncoroni, Implementing Models in Quantitative Finance: Methods and Cases, 1st Edition, Springer Finance Series, 2008
- Sebastien Bossu, Advanced Equity Derivatives: Volatility and Correlation, 1st Edition, Wiley Series in Finance, 2014
- Umberto Cherubini, Giovanni Della Lunga, Sabrina Mulinacci and Pietro Rossi, Fourier Transform Methods in Finance, 1st Edition, Wiley Series in Finance, 2010

# Additional References

**Research papers (partial list):**

- Jim Gatheral and Antoine Jacquier, Arbitrage-free SVI Volatility Surfaces, Quantitative Finance, vol 14, issue 1, page 59-71, 2014.
- Peter Carr and Dilip Madan. Option Valuation Using the Fast Fourier Transform, Journal of Computational Finance, vol 2, page 61-73, 1998.
- Steven Heston and Siakat Nandi. A Closed-form GARCH Option Valuation Model, Review of Financial Studies, vol 13, number 3, page 585-625.
- Liuren Wu. Modeling Financial Security Returns using Levy Processes, chapter in the Handbook of Operation Research and Management Science-Volume 15 Financial Engineering, Editors: John.R.Birge and Vadim Linetsky, Elsevier North-Holland, 2008.

**Sample recent Master/Ph.D. thesis in the area of volatility modeling:**

- Pricing Variance Swaps and Corridor Variance Swaps under General Dividend Streams, Master Thesis available at: www.merlin.uzh.ch/contributionDocument/download/7504
- Pricing variance swaps by using two methods: replication strategy and a stochastic volatility model, Master Thesis available at: http://www.diva-portal.org/smash/get/diva2:239415/FULLTEXT01.pdf
- Hedging of Volatility, Master Thesis available at: http://uu.diva-portal.org/smash/get/diva2:718995/FULLTEXT01.pdf
- Pricing of Variance and Volatility Swaps in a stochastic volatility and jump framework, Master Thesis available at: http://pure.au.dk/portal-asb-student/files/39695930/thesis.pdf
- Volatility-of-Volatility Perspectives: Variance Derivatives and Other Equity Exotics, Ph.D. Thesis available at: http://www.math.ku.dk/noter/filer/phd11gd.pdf
- Managing Volatility Risk Innovation of Financial Derivatives, Stochastic Models and Their Analytical Implementation, Ph.D. Thesis available at: http://www.math.columbia.edu/~thaddeus/theses/2010/li.pdf
- Implied volatility asymptotics, Ph.D. Thesis available at: http://gradworks.umi.com/35/26/3526300.html

Grading

# Grading Policies

**Homework:**

There will be around 4 homework assignments throughout the semester. Collaboration is encouraged as it can be helpful to understand some of these concepts. Do not confuse collaboration for academic misconduct. Attempt each problem on your own before seeking help from another person. Make sure that you understand the entire assignment that you turn in, and could reproduce the work or solve a similar problem. Do not think that you can simply copy another person's assignment and expect to understand the material.

Late homework will be accepted under the following policy. If the homework is turned in within one week of the original due date, it will receive 2/3 of its score, going down by a third each week it is late. The homework will have a very firm deadline, of 11:55 PM on the due date. When I say this is a firm date, I mean that if the homework is submitted online at 11:56PM, its late, no exceptions. Plan ahead and submit your homework early to avoid problems due to internet or computer issues.

**Exams:**

There will be one final exam given in the class. If you miss an exam, you must provide a written explanation signed by proper authorities in order to be allowed the chance to take a replacement exam. The final exam is closed-book, but each student can bring two hand written pages of notes to the final. Calculators are permitted and encouraged, but cell phones and notebook computers are NOT allowed.

**Term Project:**

A list of term project topics will be assigned in Week 3 and students shall form a group of 2 to 4 students and decide on their topic by Week 5. Project proposal and plan is due by Week 7. The final term project presentation is scheduled in Week 14 and students will present their research findings in class. A final written term project report is due at the end of the term. Sample term project topics:

- Recovering future return densities: take options date on S&P500 index, and use closing prices as the spot, compute implied volatility for each option contract and construct smoother volatility surface (via smoothing and interpolation), then build risk-neutral return density for SPX across different maturities based on the implied volatility surface. Bonus work: try to recover the statistical (or physical) return distribution from options as done in the Ross recovery theorem in Ross (2015): “The Ross Theorem”, Journal of Finance, vol 70, issue 2, page 615-648, 2015.

- Efficient Fourier inversion for option pricing: compare different inversion methods in their computational efficient under different parameter environments for the Heston model: ie. High/low volatility, long/short maturity. Analyze how the control or dampening parameters in the Carr and Madan (1998) framework can be chosen optimally. Read the paper of Duffie, Pan and Singleton (2011): “Transform analysis and asset pricing for affine jump diffusions”, Econometrica, vol 68, issue 6, page 1343-1376, 2000, and apply the Fourier inversion method to option pricing under affine models. Read the papers of Fang and Oosterlee (2008) on the applications of Fourier Cosine series to the pricing of European options under affine jump diffusions. Apply their method to price Bermudan options, and discrete barrier options.

- Pricing volatility derivatives which are based on discrete sampling of log returns, read and implement the papers of Broadie and Jain (2008), Bernard and Cui (2014), and Zheng and Kwok (2014). Explore the pricing of a new volatility derivative called timer options introduced by the Societe Generale Investment Bank based on the Monte Carlo simulation methods in Bernard and Cui (2011) and the PDE perturbation method in Li and Mercurio (2014).

Lecture Outline

Topic | Reading | |
---|---|---|

Week 1 | Empirical features of stock returns time series and realized volatility; implied volatility smile and surface; Introduction to volatility market and trading. | Ch 1 in [1] and Slides |

Week 2 | Review of history and development of stochastic volatility and local volatility models | Ch 2 in [1] and Slides |

Week 3 | Theoretical analysis of stochastic and local volatility models | Slides |

Week 4 | Fast Fourier Transform for options pricing, affine Processes | Slides |

Week 5 | Monte Carlo simulation, numerical discretization schemes, Stochastic Alpha Beta Rho (SABR) model | Slides |

Week 6 | Special topic: Hermite series expansion technique, discussions on potential applications to stochastic volatility models. | Slides and research papers |

Week 7 | GARCH Model: basic topics | Slides |

Week 8 | GARCH Model: advanced topics | Slides and research papers. |

Week 9 | Heston-Nandi GARCH model and affine GARCH option pricing models. | Slides and research papers |

Week 10 | Valuing variance swaps. | Slides and research papers |

Week 11 | VIX and LETF derivatives | Slides and research papers |

Week 12 | Using Lévy process to model security returns | Slides and [9] |

Week 13 | Stochastic time change applied to Lévy processes, and corresponding stochastic volatility models, (if time allows) approximate stochastic volatility models using Markov Chains, or other topics to be determined. | Slides and research papers. |

Week 14 | Final Project Presentation |