## Final Report Summary - ROBUSTFINMATH (Robust Financial Mathematics: model-ambiguous framework for valuation and risk management)

The use of mathematical models to assist in decision making is ubiquitous in the modern society. In some situations deterministic models are sufficient but for complex systems often a random component is needed. This, in particular, is true in economics and finance where the dynamics are often shaped by many interacting agents. The classical modelling approach first prescribes a fixed probability measure which fully describes the system's (random) evolution and then optimises a given criterion to obtain model's outputs. The process is vulnerable to model mis-specification, also known as the Knightian uncertainty, and does little to understand or quantify such risks.

RobustFinMath Project proposed a new modelling paradigm for problems in mathematical finance. Our approach, referred to as the "robust approach", allows to quantify the impact and risks of making modelling assumptions. The framework we developed interpolates between the two ends of the modelling spectrum: the agnostic model-free end and the classical probability-specific end described above. Our overarching narrative is pathwise: we think of modelling assumptions as specifying the set of feasible paths for the future evolution of quantities of interest (e.g. price processes). The model free approach considers all paths to be possible and as such will not produce unique outputs but rather a set of feasible outputs (e.g. intervals of no-arbitrage prices). The model-specific approach narrows down the set of paths so that outputs become strongly constrained, or even unique as in the classical Black-Scholes model. The process is data-driven, since the market information endogenously specifies modelling restrictions, but also user driven since an agent can add on their personal beliefs.

The project evolved along three inter-connected yet complementary directions and significant progress was achieved in each one of them. First, in a string of theoretical results we provided analogues of the classical no-arbitrage pricing theory within the new paradigm. In particular, we obtained robust versions of the so-called Fundamental Theorem of Asset Pricing and the Pricing-Hedging Duality. Second, we proposed to bridge the traditional gap between econometrics and mathematical finance, or between the so-called physical and risk-neutral measures, by incorporating in a coherent way time series data and market option prices as modelling inputs. This was achieved as a two-step process within the pathwise modelling narrative, as described above, as well as in a direct manner by designing statistical estimators for superheding prices. Third, we developed numerical methods to compute some of the framework's outputs and worked out proof-of-concept applications of our framework.

The project spanned a wide range of interrelated questions and topics. Our work relied on many different branches of mathematics, often involving their novel interactions and leading to results of interest to a wider mathematical community. The core developments in robust pricing, hedging and risk assessment, as described above, often led to additional exciting scientific explorations. We made substantial contributions to other topics from behavioural finance, robust dynamically-consistent portfolio optimisation, through pathwise stochastic integration, enlargement of filtrations, to Skorokhod embeddings, optimal stopping theory and martingale optimal transport problems. Our papers have been published in leading international peer-reviewed journals in the respective fields and the team members have been invited to present our results at seminars and international conferences. We were also able to attract many leading researchers to visit us in Oxford.

RobustFinMath Project proposed a new modelling paradigm for problems in mathematical finance. Our approach, referred to as the "robust approach", allows to quantify the impact and risks of making modelling assumptions. The framework we developed interpolates between the two ends of the modelling spectrum: the agnostic model-free end and the classical probability-specific end described above. Our overarching narrative is pathwise: we think of modelling assumptions as specifying the set of feasible paths for the future evolution of quantities of interest (e.g. price processes). The model free approach considers all paths to be possible and as such will not produce unique outputs but rather a set of feasible outputs (e.g. intervals of no-arbitrage prices). The model-specific approach narrows down the set of paths so that outputs become strongly constrained, or even unique as in the classical Black-Scholes model. The process is data-driven, since the market information endogenously specifies modelling restrictions, but also user driven since an agent can add on their personal beliefs.

The project evolved along three inter-connected yet complementary directions and significant progress was achieved in each one of them. First, in a string of theoretical results we provided analogues of the classical no-arbitrage pricing theory within the new paradigm. In particular, we obtained robust versions of the so-called Fundamental Theorem of Asset Pricing and the Pricing-Hedging Duality. Second, we proposed to bridge the traditional gap between econometrics and mathematical finance, or between the so-called physical and risk-neutral measures, by incorporating in a coherent way time series data and market option prices as modelling inputs. This was achieved as a two-step process within the pathwise modelling narrative, as described above, as well as in a direct manner by designing statistical estimators for superheding prices. Third, we developed numerical methods to compute some of the framework's outputs and worked out proof-of-concept applications of our framework.

The project spanned a wide range of interrelated questions and topics. Our work relied on many different branches of mathematics, often involving their novel interactions and leading to results of interest to a wider mathematical community. The core developments in robust pricing, hedging and risk assessment, as described above, often led to additional exciting scientific explorations. We made substantial contributions to other topics from behavioural finance, robust dynamically-consistent portfolio optimisation, through pathwise stochastic integration, enlargement of filtrations, to Skorokhod embeddings, optimal stopping theory and martingale optimal transport problems. Our papers have been published in leading international peer-reviewed journals in the respective fields and the team members have been invited to present our results at seminars and international conferences. We were also able to attract many leading researchers to visit us in Oxford.