Signatures, path dependence, and deep hedging

In the world of derivatives, European option payoffs can be generated by combinations of hockey stick payoffs or monomials. Signatures can also be used to generate path-dependent options. In this talk, Bruno Dupire, Head of Quantitative Research in the Office of the CTO at Bloomberg, covered signatures, path dependency, and deep hedging, explaining that the concept of signatures lies at the intersection of several branches of mathematics, including calculus, functional analysis, geometry, and combinatorics. This research is conducted jointly with Valentin Tissot-Daguette, Guixin Liu, and Bryan Liang.

Signatures: The process and uses

Signatures can be used as features of paths for machine learning, including time series and ideograms, for Generative Adversarial Networks (GANS) to generate market-mimicking data, as a basis of path dependent options, and for option pricing and hedging. In the signature definition, for a d-dimensional path, a series of words is defined, where each word corresponds to a dimension. The signature for the path will then be a collection of iterated integrals, each one corresponding to a word. As an example, one asset and its price time series can be examined. The signatures will be iterated Stratonovich integrals with respect to the variables for time and price, and they will be described by a word with letters of {t, x} standing for time and price, and denoted as binary strings, where t corresponds to 0 and x corresponds to 1.

Do the shuffle

From the initial exercise, it is possible to take a collection of words denoted by alpha and beta, for example, that shuffles the words, while keeping them in the same order, respective of each other. This allows for the representation of a non-linear function of the signature as a linear combination of higher order signatures. Extending to properties of the words, by making use of Chen’s Identity, there will be a concatenation of paths, where the results comprise all possible partitions of alpha. This provides a systematic way to compute the signatures of a longer path by knowing each of the shorter segments of that path. The methods can also be applied to produce signatures of linear functions, monomials, and hockey sticks, all through manipulating the terms and coming up with segments and solutions. In terms of path reconstruction, computing signatures from a path is certainly possible, but can the path be reconstructed from the signatures; is it reversible? The answer is yes, and a subset of words will be adequate as they provide a complete set of weighted averages of the path.

Functional Taylor expansion

The Functional Itô Calculus is a framework for analyzing path dependence. Its main ingredients are paths and functionals, whose value depends on the whole path, as opposed to depending only on the current value. Within this framework, there is a Taylor expansion for functionals that allows us to approximate the value of a functional on a path by a sum over words of products of two terms. The first term is the functional derivative with the order of the differentiation given by the word; it depends only on the functional and not on the path. The second term is the signature of the path associated with the word; it depends only on the path, not on the functional. In summary, signatures are natural building blocks of path dependent options.

An alternative to deep hedging

A hedging strategy that only depends on the past price history is a functional and can be expressed as a linear combination of signatures. Given an objective function such as the variance of the P&L, looking for the optimal hedge boils down to finding the coefficients of the combination. The approach provides a fast alternative to deep hedging as it is an easier task to solve a linear problem than to optimize the weights of a neural net as deep hedging does. Examples given include European calls and forward start options.

The references for this talk include Dupire 2009 “Functional Itô Calculus”, which was published on SSRN.

Lightning talks

Following the short Q&A session, Bruno Dupire kicked off a series of “lightning talks,” 5-minute presentations where industry experts, researchers, and academics present a wide range of subjects to stimulate fresh thinking and interaction between various disciplines. Each talk examines a way that the industry is evolving and serves as an essential exploratory aspect of the Bloomberg Quant (BBQ) Seminar series.

In this session, Dhruv Madeka of Amazon explained his work with complex heavy tailed data and the use of causal inference in supervised learning, Janna Levin of Barnard College, Columbia University provided a brief history of cosmology and the nature of black holes, and Roman Paolucci of James Madison University presented work on variational autoencoders, which can solve challenges in generating realistic images by modeling latent distributions.

About the Bloomberg Quant seminar series

The Bloomberg Quant (BBQ) seminar series is held each month and covers a wide range of topics in quantitative finance. The BBQ seminar is offered in a virtual format and scheduled to accommodate participants from across EMEA and the Americas. Each session is chaired by Bruno Dupire, head of Quantitative Research at Bloomberg L.P., and features a keynote speaker presenting on his or her current research. This presentation is followed by several “lightning talks” of 5 minutes each in quick succession. This format gives the audience the opportunity to be exposed to a wider variety of topics.

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