We describe the pricing and hedging practices refraining from the use of probability. We encode volatility in an enhancement of the price trajectory and we give pathwise presentations of the fundamental equations of Mathematical Finance. In particular this allows us to assess model misspecification, generalising the so-called fundamental theorem of derivative trading (see Ellersgaard et al. [EJP17]). Our pathwise integrals and equations exhibit the role of Greeks beyond the leading-order Delta, and makes explicit the role of Gamma sensitivities. * King's College London. † Imperial College London. 1 An informal and brief account of the discussion given here, with a historical perspective on how this work developed and on the related literature, is given in [Bri19]. 1The quadratic variation is not a distributional feature of the price, in the sense that changing to an equivalent probability measure does not affect it. Notwithstanding its non-probabilistic nature, the quadratic variation is usually associated with the diffusion coefficient in a model described using Itô's theory of stochastic differential equations. In this setting the coefficients of the Black-Scholes PDE are seen as being derived from the characteristics of a certain diffusion process. These coefficients determine the parameters for the variances of diffusion's marginal laws, which can give rise to confusion between the concept of volatility and of variance which we propose to disentangle.To the best of our knowledge, C. Bender, T. Sottinen and E. Valkeila were the first to discern this distinction, see [BSV08]. Their work was anticipated by two earlier articles. The first is by Terry J. Lyons, [Lyo95], which focuses on replication arguments and observes how probability is only used to justify lower bounds for option prices. The main theorem in this paper, [Lyo95, Theorem 1], is effectively a precursor to the fundamental theorem of derivative trading. The second work is by D. Brigo and F. Mercurio, [BM00], where the consequences of changing probability measures are addressed. The authors showed, by constructive examples, that the change of measure used in martingale pricing can massively disrupt the distributional features of physical dynamics on arbitrarily fine time grids: close physical evolutions for stock prices can be transformed into pricing counterparts that imply arbitrarily different option prices. We expand on this insight that martingale pricing, although probabilistic in nature, actually entails that only pathwise properties of physical stock evolutions are relevant for option pricing models.Relying on the result of pathwise stochastic analysis in the book by P. Friz and M. Hairer ([FH14]), we reformulate the Black-Scholes technical apparatus without using probability. Moreover, we use the notion of a rough bracket as an extension of quadratic variation. This is not merely a linguistic exercise: on the one hand, it reveals that any linear transformation of level two of the price path's signature 2 can take the role of the quadratic ...

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We present a measurement of price impact in order-driven markets that does not require averages across executions or scenarios. Given the order book data associated with one single execution of a sell metaorder, we measure its contribution to price decrease during the trade. We do so by modelling the limit order book using state-dependent Hawkes processes, and by defining the price impact profile of the execution as a function of the compensator of a stochastic process in our model. We apply our measurement to a data set from NASDAQ, and we conclude that the clustering of sell child orders has a bigger impact on price than their sizes.

We compare optimal static and dynamic solutions in trade execution. An optimal trade execution problem is considered where a trader is looking at a short-term price predictive signal while trading. When the trader creates an instantaneous market impact, it is shown that transaction costs of optimal adaptive strategies are substantially lower than the corresponding costs of the optimal static strategy. In the same spirit, in the case of transient impact, it is shown that strategies that observe the signal a finite number of times can dramatically reduce the transaction costs and improve the performance of the optimal static strategy.

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