# Asymptotics and dynamics of forward implied volatility

Lead Research Organisation:
Imperial College London

Department Name: Mathematics

### Abstract

Asymptotic methods represent a set of tools (from probability, PDE theory, geometry) allowing to study systems when some parameters become small or large. It is particularly useful when, say, an equation does not have an explicit solution, but the latter can be written as a series expansion when some parameter is small. This therefore yields approximate yet accurate understanding of the behaviour of the solution (up to some small error). In mathematical finance, many stochastic) models have been proposed and used in the past four decades in order to reflect the dynamics of asset prices and financial markets. Based on these processes, pricing equations can be written and solved numerically. This can be performed, either from a probabilistic point of view, where computing expectations boils down to (often complex) numerical integration, or from an analytic perspective, where the solution of the problem solves some partial (integro-) differential equation. Even though powerful numerical methods exist, they are often computer-intensive and do not provide easy (and intuitive) understanding of the behaviour of the solution.

The cornerstone of such models is the so-called Black-Scholes model, for which European call option prices have a trivial closed-form expression. However, in most models, option prices do not have closed-form representations, and have to be computed numerically. This is even more so for the corresponding implied volatility, which is just a standardised option price (now universally used in practice as a quoting mechanism). Over the past fifteen years, active research has been carried out to obtain explicit analytical approximations for this implied volatility, thus effectively replacing the highly demanding numerical computations by some simple approximate) solution. Lee was one of the pioneers of this stream, providing a precise link between the behaviour of the implied volatility and the tail distribution of the stock price. This result has since been extended and improved by several authors, including Benaim-Friz, Gulisashvili-Stein, De Marco-Hillairet-Jacquier. Other important results in this direction were obtained by Henry-Labordere (using differential geometry), Jacquier, Keller-Ressel and Mijatovic (using probabilistic tools) and Deuschel, Friz, Jacquier and Violante (using both geometric and probabilistic methods). All these results however do not give any information on the dynamic behaviour of the implied volatility, which is essential in order to accurately model the time-evolving nature of financial markets.

The goal of this project is to understand this dynamic behaviour of the implied volatility for a large class of models, and to propose a tractable formula describing it. This has been partially achieved in the static case, but the question remains wide open in the dynamic case. In order to do so, the PI intends to follow two main directions:

- determine the asymptotic behaviour of the dynamic implied volatility for a large class of stochastic models;

- extend to the dynamic case the existing arbitrage-free implied volatility parameterisation.

Progress in either of these directions would immediately yields a better understanding of the models currently used in practice: are they accurate enough? Do they possess realistic properties to model the behaviour of financial markets? It would also provide deeper insight on so-called model risk, namely the risk associated to the use of a statically tested model for dynamic purposes. Ultimately this could yield a classification of models according to their actual usefulness.

The cornerstone of such models is the so-called Black-Scholes model, for which European call option prices have a trivial closed-form expression. However, in most models, option prices do not have closed-form representations, and have to be computed numerically. This is even more so for the corresponding implied volatility, which is just a standardised option price (now universally used in practice as a quoting mechanism). Over the past fifteen years, active research has been carried out to obtain explicit analytical approximations for this implied volatility, thus effectively replacing the highly demanding numerical computations by some simple approximate) solution. Lee was one of the pioneers of this stream, providing a precise link between the behaviour of the implied volatility and the tail distribution of the stock price. This result has since been extended and improved by several authors, including Benaim-Friz, Gulisashvili-Stein, De Marco-Hillairet-Jacquier. Other important results in this direction were obtained by Henry-Labordere (using differential geometry), Jacquier, Keller-Ressel and Mijatovic (using probabilistic tools) and Deuschel, Friz, Jacquier and Violante (using both geometric and probabilistic methods). All these results however do not give any information on the dynamic behaviour of the implied volatility, which is essential in order to accurately model the time-evolving nature of financial markets.

The goal of this project is to understand this dynamic behaviour of the implied volatility for a large class of models, and to propose a tractable formula describing it. This has been partially achieved in the static case, but the question remains wide open in the dynamic case. In order to do so, the PI intends to follow two main directions:

- determine the asymptotic behaviour of the dynamic implied volatility for a large class of stochastic models;

- extend to the dynamic case the existing arbitrage-free implied volatility parameterisation.

Progress in either of these directions would immediately yields a better understanding of the models currently used in practice: are they accurate enough? Do they possess realistic properties to model the behaviour of financial markets? It would also provide deeper insight on so-called model risk, namely the risk associated to the use of a statically tested model for dynamic purposes. Ultimately this could yield a classification of models according to their actual usefulness.

### Planned Impact

This research will benefit both academics in the relevant areas (mathematical finance, stochastic analysis, applied probability) and practitioners. Dissemination in academia will be done through publications in leading scientific journals, such as Finance and Stochastics, Mathematical Finance and Annals of Applied Probability, and by speaking at major worldwide conferences. The PI also aims at circulating his results to a wider audience via publications in practitioners' oriented journals (Risk Magazine and Quantitative Finance). A major issue currently faced by many practitioners is the lack of consensus as to which models are relevant. There exists a plethora of models, each used for different purposes, and it is believed that the proposed research project could provide an answer to this. Different types of financial agents could thus benefit from it: it will provide investment banks with simple arbitrage-free tools to be implemented, and regulators (such as the Prudential Regulation Authority) could monitor more accurately Profit and Losses, and set precise maximal loss targets in a model-free way, thereby making these targets easily applicable to everyone. This will of course not be possible without actual implementation and calibration of the results, which the PI plans to do via the Python / IPython notebook framework. Python is a powerful programming language, with recent advances on its mathematical libraries, and its recent IPython notebook interface makes it ideal for collaboration and sharing. By doing so, the PI will thus increase the reproducibility of his results. It is not at all unreasonable to think that this output and implementation could be packaged as a software to be commercialised.

### Publications

Alòs E
(2018)

*The implied volatility of Forward-Start options: ATM short-time level, skew and curvature*in Stochastics
Badikov S
(2017)

*No-arbitrage bounds for the forward smile given marginals*in Quantitative Finance
Badikov S
(2016)

*No-arbitrage bounds for the forward smile given marginals*
Chassagneux J
(2016)

*An Explicit Euler Scheme with Strong Rate of Convergence for Financial SDEs with Non-Lipschitz Coefficients*in SIAM Journal on Financial Mathematics
De Marco S
(2017)

*Shapes of Implied Volatility with Positive Mass at Zero*in SIAM Journal on Financial Mathematics
De Marco S
(2016)

*Two examples of non strictly convex large deviations*in Electronic Communications in Probability
Guennoun H
(2014)

*Asymptotic Behaviour of the Fractional Heston Model*in SSRN Electronic Journal
Guennoun H
(2018)

*Asymptotic Behavior of the Fractional Heston Model*in SIAM Journal on Financial MathematicsDescription | We have improved our understanding of the forward implied volatility, and introduced a new class of models based on these findings. This new class of models seems to be able to fit market data more accurately. |

Exploitation Route | We have implemented some of our results and discussed part of them with academics and practitioners. The class of models we introduced is a first step, and we believe we could extend them further. In particular, we derived small- and large-time behaviour of option prices and implied volatilities in our model, and these asymptotics can be made more precise, which is something we are working on at the moment. |

Sectors | Financial Services and Management Consultancy Other |

Description | This is informal, but several quantitative groups in banks have mentioned they had been using some of the results developed through the grant for their daily activities. |

First Year Of Impact | 2015 |

Sector | Financial Services, and Management Consultancy |

Impact Types | Economic |

Description | Imperial College Platform Grant |

Amount | £1,400 (GBP) |

Organisation | Engineering and Physical Sciences Research Council (EPSRC) |

Sector | Public |

Country | United Kingdom |

Start | 05/2015 |

End | 06/2015 |

Description | Imperial College via EPSRC: - Imperial College Platform Grant (£ 1500; 2016 - 2016) |

Amount | £1,500 (GBP) |

Organisation | Engineering and Physical Sciences Research Council (EPSRC) |

Sector | Public |

Country | United Kingdom |

Start | 11/2016 |

End | 11/2016 |

Description | LMS Workshop Grant |

Amount | £4,800 (GBP) |

Organisation | London Mathematical Society |

Sector | Academic/University |

Country | United Kingdom |

Start | 08/2015 |

End | 09/2015 |

Description | Workshop Support Grant |

Amount | £10,000 (GBP) |

Organisation | Imperial College London |

Sector | Academic/University |

Country | United Kingdom |

Start | 11/2014 |

End | 11/2014 |

Description | Arbitrage-free implied volatility surfaces |

Organisation | Zeliade Systems |

Country | France |

Sector | Private |

PI Contribution | With Claude Martini (Zeliade Systems), we extended the results of Gatheral-Jacquier in order to provide more general arbitrage-free implied volatility surfaces, which can then be used as interpolators and extrapolators for pricing exotic options. |

Collaborator Contribution | With Claude Martini (Zeliade Systems), we extended the results of Gatheral-Jacquier in order to provide more general arbitrage-free implied volatility surfaces, which can then be used as interpolators and extrapolators for pricing exotic options. |

Impact | http://epubs.siam.org/doi/abs/10.1137/120900320 |

Start Year | 2015 |

Description | Asymptotics of the forward skew in rough volatility models |

Organisation | Ecole Polytechnique |

Country | France |

Sector | Academic/University |

PI Contribution | With the recent developments in volatility modelling, in particular, the "rough volatilty" approach pioneered by Gatheral-Jaisson-Rosenbaum, we have started looking at the behaviour of the forward implied volatility in those models. Due to the re-starting phenomenon, this yields SDEs (driven by fractional Brownian motion) with random initial datum. We are in the process of finishing a paper on this topic, with numerical implementations. |

Collaborator Contribution | Professor Rosenbaum and his student, Omar El Euch, have given mathematical foundations to rough volatility models, and we are trying to adapt these grounds to the case of forward-start options. |

Impact | There is no output yet. |

Start Year | 2016 |

Description | Asymptotics of the forward skew in rough volatility models |

Organisation | Zeliade Systems |

Country | France |

Sector | Private |

PI Contribution | With the recent developments in volatility modelling, in particular, the "rough volatilty" approach pioneered by Gatheral-Jaisson-Rosenbaum, we have started looking at the behaviour of the forward implied volatility in those models. Due to the re-starting phenomenon, this yields SDEs (driven by fractional Brownian motion) with random initial datum. We are in the process of finishing a paper on this topic, with numerical implementations. |

Collaborator Contribution | Professor Rosenbaum and his student, Omar El Euch, have given mathematical foundations to rough volatility models, and we are trying to adapt these grounds to the case of forward-start options. |

Impact | There is no output yet. |

Start Year | 2016 |

Description | Asymptotics of the forward smile using Malliavin calculus |

Organisation | University of Barcelona |

Department | Faculty of Mathematics and Computer Science |

Country | Spain |

Sector | Academic/University |

PI Contribution | With Professor Elisa Alos, we have been using tools from Malliavin calculus to determine the rate of explosion of the forward implied volatility in general classes of stochastic volatility models. I have further implemented the results numerically, and I am currently trying to develop more precise algorithms to increase the computation speed. |

Collaborator Contribution | Professor Elisa Alos is a world-expert on Malliavin calculus, and we have been extending her results on option price decomposition to the case of forward-start options and forwar implied volatility. |

Impact | So far, we are working on a paper, with implementations, which should be finished by Summer 2017. We hope to use this result to provide a classification of stochastic volatility models in terms of their usability to reliably calibrate forward-start based options. |

Start Year | 2016 |

Description | 9th Bachelier World Congress, NYC, July 2016 |

Form Of Engagement Activity | Participation in an activity, workshop or similar |

Part Of Official Scheme? | No |

Geographic Reach | International |

Primary Audience | Professional Practitioners |

Results and Impact | This is the largest academic conference (every two years) in Mathematical Finance. |

Year(s) Of Engagement Activity | 2016 |

URL | http://www.cvent.com/events/9th-world-congress-of-the-bachelier-finance-society/archived-e03d79a2ef3... |

Description | CSASC Meeting |

Form Of Engagement Activity | Participation in an activity, workshop or similar |

Part Of Official Scheme? | No |

Geographic Reach | International |

Primary Audience | Other audiences |

Results and Impact | This CSASC Meeting was attended by around 40 participants, from Europe, Asia, and the USA. |

Year(s) Of Engagement Activity | 2016 |

URL | http://csasc2016.espais.iec.cat/ |

Description | Global Derivatives, Budapest, May 2016 |

Form Of Engagement Activity | Participation in an activity, workshop or similar |

Part Of Official Scheme? | No |

Geographic Reach | International |

Primary Audience | Professional Practitioners |

Results and Impact | This is the largest conference for practitioners in Quantitative Finance. I was invited to give a talk there, and to discuss with practitioners. |

Year(s) Of Engagement Activity | 2016 |

URL | http://www.icbi-derivatives.com |

Description | Global Derivatives, Amsterdam, May 2015. |

Form Of Engagement Activity | Participation in an activity, workshop or similar |

Part Of Official Scheme? | No |

Geographic Reach | International |

Primary Audience | Professional Practitioners |

Results and Impact | Global Derivatives, Amsterdam, May 2015. This is the largest conference for practitioners in Quantitative Finance. I was invited to give a talk there, and to discuss with practitioners. |

Year(s) Of Engagement Activity | 2015 |

URL | http://www.icbi-derivatives.com/ |

Description | International Conference on Stochastic Analysis and Applications |

Form Of Engagement Activity | Participation in an activity, workshop or similar |

Part Of Official Scheme? | No |

Geographic Reach | International |

Primary Audience | Other audiences |

Results and Impact | International Conference on Stochastic Analysis and Applications, Hammamet, Tunisia, October This was an international workshop on Mathematical Finance. |

Year(s) Of Engagement Activity | 2015 |

URL | http://pinguim.uma.pt/Investigacao/Ccm/icsaa15/page8/page8.html |