Mathematical finance

Stochastic volatility

In statistics, stochastic volatility models are those in which the variance of a stochastic process is itself randomly distributed. They are used in the field of mathematical finance to evaluate derivative securities, such as options. The name derives from the models' treatment of the underlying security's volatility as a random process, governed by state variables such as the price level of the underlying security, the tendency of volatility to revert to some long-run mean value, and the variance of the volatility process itself, among others. Stochastic volatility models are one approach to resolve a shortcoming of the Black–Scholes model. In particular, models based on Black-Scholes assume that the underlying volatility is constant over the life of the derivative, and unaffected by the changes in the price level of the underlying security. However, these models cannot explain long-observed features of the implied volatility surface such as volatility smile and skew, which indicate that implied volatility does tend to vary with respect to strike price and expiry. By assuming that the volatility of the underlying price is a stochastic process rather than a constant, it becomes possible to model derivatives more accurately. The early history of stochastic volatility has multiple roots (i.e. stochastic process, option pricing and econometrics), it is reviewed in Chapter 1 of Neil Shephard (2005) "Stochastic Volatility," Oxford University Press. (Wikipedia).

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FRM: Volatility approaches

Lots of ways to estimate volatility. In this map, I parse out implied volatility (forward looking) and deterministic (constant) and focus on stochastic volatility: volatility that changes over time, either via (conditional) recent volatility and/or random shocks. For more financial risk vi

From playlist Volatility

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Risk Management Lesson 4A: Volatility

First part of Lesson 4. Topics: - Definitions of volatility - Basic assumptions (do they hold?) - Arch and G-arch models (brief overview)

From playlist Risk Management

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IDTIMWYTIM: Stochasticity - THAT'S Random

Hank helps us understand the difference between the colloquial meaning of randomness, and the scientific meaning, which is also known as stochasticity. We will learn how, in fact, randomness is surprisingly predictable. Like SciShow: http://www.facebook.com/scishow Follow SciShow: http://

From playlist Uploads

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Time Varying Volatility and GARCH in Risk Management

These classes are all based on the book Trading and Pricing Financial Derivatives, available on Amazon at this link. https://amzn.to/2WIoAL0 Check out our website http://www.onfinance.org/ Follow Patrick on twitter here: https://twitter.com/PatrickEBoyle In Todays video let's learn abo

From playlist Risk Management

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What are Volatility Swaps? Financial Derivatives - Trading Volatility

In todays class we learn about what a volatility swap is. These classes are all based on the book Trading and Pricing Financial Derivatives, available on Amazon at this link. https://amzn.to/2WIoAL0 Check out our website http://www.onfinance.org/ Follow Patrick on twitter here: https:/

From playlist The Term Structure of Volatility

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Volatility Trading - Call and Put Options - Trading Tutorial

These classes are all based on the book Derivatives For The Trading Floor, available on Amazon at this link. https://amzn.to/3GdLi2s Check out our website http://www.onfinance.org/ Follow Patrick on twitter here: https://twitter.com/PatrickEBoyle What is volatility trading? Volatility

From playlist Class 4 The Greeks & Dynamic Hedging

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FRM: Volatility: Moving Average Approaches

Within stochastic volatility, moving average is the simplest approach. It simply calculates volatility as the unweighted standard deviation of a window of X trading days. Here I show the three "flavors:" population variance (volatility = SQRT[variance]), sample, and simple. For more financ

From playlist Volatility

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What is Implied Volatility? Options Trading Tutorial.

These classes are all based on the book Trading and Pricing Financial Derivatives, available on Amazon at this link. https://amzn.to/2WIoAL0 Check out our website http://www.onfinance.org/ Follow Patrick on twitter here: https://twitter.com/PatrickEBoyle

From playlist The Term Structure of Volatility

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Introduction to the paper https://arxiv.org/abs/2002.06707

From playlist Research

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8 2 Stochastic Volatility Part 2

BEM1105x Course Playlist - https://www.youtube.com/playlist?list=PL8_xPU5epJdfCxbRzxuchTfgOH1I2Ibht Produced in association with Caltech Academic Media Technologies. ©2020 California Institute of Technology

From playlist BEM1105x Course - Prof. Jakša Cvitanić

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Martin Larsson: Affine Volterra processes and models for rough volatility

Abstract: Motivated by recent advances in rough volatility modeling, we introduce affine Volterra processes, defined as solutions of certain stochastic convolution equations with affine coefficients. Classical affine diffusions constitute a special case, but affine Volterra processes are n

From playlist Probability and Statistics

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Grégoire Loeper: Reconstruction by optimal transport: applications in cosmology and finance

Abstract: Following the seminal work by Benamou and Brenier on the time continuous formulation of the optimal transport problem, we show how optimal transport techniques can be used in various areas, ranging from "the reconstruction problem" cosmology to a problem of volatility calibration

From playlist Mathematics in Science & Technology

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Dylan Possamaï: Principal Agent Modelling - lecture 3

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From playlist Probability and Statistics

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Risk Management of Option Books with Arbitrage-Free Neural-SDE Market Models (SIAM FME)

SIAM Activity Group on FME Virtual Talk Series Join us for a series of online talks on topics related to mathematical finance and engineering and running every two weeks until further notice. The series is organized by the SIAM Activity Group on Financial Mathematics and Engineering. Spe

From playlist SIAM Activity Group on FME Virtual Talk Series

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6 6 Black Scholes Merton pricing Part 3

BEM1105x Course Playlist - https://www.youtube.com/playlist?list=PL8_xPU5epJdfCxbRzxuchTfgOH1I2Ibht Produced in association with Caltech Academic Media Technologies. ©2020 California Institute of Technology

From playlist BEM1105x Course - Prof. Jakša Cvitanić

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8 1 Stochastic Volatility Part 1

BEM1105x Course Playlist - https://www.youtube.com/playlist?list=PL8_xPU5epJdfCxbRzxuchTfgOH1I2Ibht Produced in association with Caltech Academic Media Technologies. ©2020 California Institute of Technology

From playlist BEM1105x Course - Prof. Jakša Cvitanić

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Christa Cuchiero: Rough volatility from an affine point of view​

Abstract: We represent Hawkes process and their Volterra long term limits, which have recently been used as rough variance processes, as functionals of infinite dimensional affine Markov processes. The representations lead to several new views on affine Volterra processes considered by Abi

From playlist Probability and Statistics

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The Volatility Smile - Options Trading Lessons

The volatility smile is a real-life pattern that is observed when different strikes of option, with the same underlying and same expiration date are plotted on a graph. These classes are all based on the book Trading and Pricing Financial Derivatives, available on Amazon at this link. htt

From playlist The Term Structure of Volatility

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10 9 Forward rates models Part 3

BEM1105x Course Playlist - https://www.youtube.com/playlist?list=PL8_xPU5epJdfCxbRzxuchTfgOH1I2Ibht Produced in association with Caltech Academic Media Technologies. ©2020 California Institute of Technology

From playlist BEM1105x Course - Prof. Jakša Cvitanić

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Wiener process | Black–Scholes model | Geometric Brownian motion | Volatility smile | Mean | Markov switching multifractal | Stochastic differential equation | Stochastic volatility jump | Stochastic process | SABR volatility model | Risk-neutral measure | Markov chain Monte Carlo | State variable | Directed set | Variance | Volatility (finance) | R (programming language) | Heston model | Mathematical finance | Expected value | Correlation | Subordinator (mathematics)