Option pricing models without probability: a rough paths approach

Abstract

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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“…In a pricing and hedging context there is an even more compelling reason to use signatures as a feature map for the time series: Not only does a signature-based generation yield a faster training and more stable convergence, but in a pricing and hedging context it eliminates pricing ambiguities while pure returns-based generation does not as Brigo explains in [9] in a framework consistent with statistical analysis of historical volatility that can lead to arbitrarily different options prices. Previously, [2,9] had shown that implied volatility is linked with a purely pathwise lift of the stock dynamics, confirming the idea that while historical volatility is a statistical quantity, implied volatility is a pathwise one. See also the end of Section 2.2.1 for a hedging perspective.…”

“…In fact, this test is based on a notion that is reminiscent of the role of moment generating functions on path space and hence characterises the distribution of the stochastic process uniquely, see Appendix A for details. Furthermore, it follows from [2,46], that in addition to the advantages above, signatures also provide the right framework to match hedging objectives and hence to bypass matter (4).…”

“…If however the paths are sampled on the level of signatures, this ambiguity of option prices does not occur. The findings of [2,9] therefore indicate that not only is the signature a more efficient way of encoding sample paths but one that that removes the ambiguity of the corresponding option prices and provides a meaningful control over hedging performance. See the end of Section 2.2.1 above for the last statement.…”

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“…It is worth mentioning recent lines of mathematical finance such as Armstrong, Bellani, Brigo, & Cass (2018) and Brigo (2019), which also place focus on probability-free frameworks, but these articles address more general questions than here, relating to pricing and hedging in the absence of such a measure, and depend upon rough path theory. Although we all target probability-free interpretations of existing results (for example Ellersgaard, Jönsson, & Poulsen (2017) in their case) and derivations of new ones, the underlying mathematics is ultimately very different.…”

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“…This appears to be a significant advantage compared to the classical notions of pathwise stochastic integration in [Bic81, WT89, Kar95, Nut12], which do not come with such stability estimates. In particular, the pathwise stability results of rough path theory allow one to prove a model-free version of the so-called fundamental theorem of derivative trading-see [ABBC18]-and may be of interest when investigating discretization errors of continuous-time trading in model-free finance; see [Rig16]. Furthermore, in contrast to Föllmer integration, rough integration allows one to consider general functionally generated integrands g(S t ), where g is a general (sufficiently smooth) function g: R d → R d , and not necessarily the gradient of another vector field f : R d → R. For instance, model-free portfolio theory constitutes a research direction in which it is beneficial to consider non-gradient trading strategies; see [ACLP21].…”