Option pricing models without probability: a rough paths approach

We describe the pricing and hedging of financial options without the use of probability using rough paths. By encoding the volatility of assets in an enhancement of the price trajectory, we give a pathwise presentation of the replication of European options. The continuity properties of rough‐paths allow us to generalize the so‐called fundamental theorem of derivative trading, showing that a small misspecification of the model will yield only a small excess profit or loss of the replication strategy. Our hedging strategy is an enhanced version of classical delta hedging where we use volatility swaps to hedge the second‐order terms arising in rough‐path integrals, resulting in improved robustness.

The authors would like to dedicate this paper to their late colleague Mark Davis . His acumen, brilliance, and determination in facing fundamental questions, his disarming laughter and good-natured common sense will be missed. Each of the authors benefited greatly from discussions with Mark over the years and did so, in particular, during the preparation of an early version of this manuscript. One aspect of the presentation below is a perspective on the so-called Fundamental Theorem of Derivative Trading. Mark often stressed the importance of this result to the understanding and effectiveness of real-world derivatives trading; indeed he included a version of it in his entry "Black-Scholes Formula" in the Encyclopedia of Quantitative Finance.

INTRODUCTION
The theory of rough-paths provides a framework for understanding differen tial systems driven by irregular input signals. An asset-price process arising from a diffusion model may be associated a rough-path. Conversely, we will find a necessary condition for a rough-path to arise from a given diffusion model, and we will call a rough-path satisfying this condition a diffusive rough-path. An investment strategy gives rise to a rough differential equation (RDE) describing the evolution of the profit and loss (P&L) of the strategy under a given asset price signal. Given an option with a smooth payoff function, we will show that the P&L of a modified version of the classical delta hedging strategy replicates the option payoff for any diffusive rough-path. The modification we make to achieve this replication is to augment the delta-hedging strategy with additional trades determined by a particular type of volatility swap. By assuming that the price of these swaps is well controlled we see that, in the continuous time limit, purchasing these swaps will not influence the P&L. A core property of RDE solutions is their continuity with respect to the input rough-path. A first consequence therefore of our rough-path approach is robustness of our proposed hedging strategy: if the true asset price signal is close to a diffusive signal, our hedging strategy will still approximately replicate the option payoff. This relates to the classical Fundamental Theorem of Derivative Trading (Cont, 2010;Ellersgaard et al., 2017;El Karoui et al., 1998), which shows that if one hedges according to a given diffusion model but the actual asset price process is determined by a nearby diffusion model, the error of the delta hedging strategy will be small. Our approach goes beyond this in that it allows for asset price signals that do not arise from any diffusion model at all. Due to phenomena such as market-impact and front-running, any differential equation describing the dynamics of the P&L of an investment strategy in terms of the asset price dynamics is likely to contain some error. A perturbation of the second-order term of the asset price dynamics allows us to model such an error, and hence explain the robustness of hedging strategies in more realistic markets than those given by diffusion models.
A second consequence of our rough-path approach is that it demonstrates that a theory of hedging is possible without the need for probability theory, despite the central role of probability in the classical treatment of hedging (Harrison & Kreps, 1979;Harrison & Pliska, 1981). Our work clarifies the use of probability theory in justifying prices by identifying two steps: (i) showing that the asset price paths of a diffusion model satisfy our diffusivity condition; and (ii) deducing the uniqueness of the price of an option from the existence of a replicating strategy via a no-arbitrage argument. In a market with an arbitrage any price is possible, so there is no hope of obtaining uniqueness without invoking a no-arbitrage condition, and hence involving probability theory. We see, therefore, that the correct probability-free analogue of classical pricing is demonstrating the existence of a replicating strategy for a given initial endowment. In this way, we may interpret our theory as giving a probability-free approach to pricing.
In diffusion models, the quadratic variation is a well-defined pathwise notion which determines the price. Our definition of a diffusive rough-path identifies the exact property needed for the delta hedging strategy to work in a rough-path context. A continuous pricing signal is enhanced with a specification for its rough bracket to obtain a reduced rough path (see Friz & Hairer, 2014, Chap-ter 5) which we will term an enhanced price path. Our financial model will take the form of a specification for the properties of the rough bracket. Thus, our model specification is tantamount to a choice of enhancer, and it is this rough bracket which provides the appropriate analogue of quadratic variation for our asset pricing model. In our version of the Fundamental Theorem of Derivative Pricing, we will study the effect of a misspecification of the financial model by examining the sensitivity of our strategy to the choice of enhancer.
The purely pathwise nature of the enhancer, the price and hence the implied volatility is in marked contrast to the statistical (and therefore probabilistic) notion of historical volatility. This dichotomy between pathwise and probabilistic properties has been noted before. For example, it is exploited in (Brigo & Mercurio, 2000), which partly inspired the present work (see also Brigo, 2019), to give examples of diffusion models which are statistically indistinguishable using samples on a fixed time grid yet which have arbitrarily different option prices.
The usage of Foellmer calculus has the caveat that the continuous-time integrals depend upon the discrete-time approximating sequence, which more or less precludes obtaining robustness in their approach. To circumvent this dependence, our proposed strategy is an augmented version of delta-hedging where one also invests in volatility swaps in order to hedge the second-order part of the pricing signal. This yields a robust trading strategy, however at the expense of introducing assumptions on the price of volatility swaps to ensure our strategy is self-financing.
Moreover, the above-mentioned pathwise formulations required paths with semimartingale roughness, that is, of finite -variation for all > 2. Using Rough Path Theory we are able to accommodate paths of finite -variation for 2 < < 3, hence showing how the delta hedging can be extended to a wider class of price signals.
One additional assumption that we must make in our approach is that the option payoff is differentiable. We will show that for a European call option with strike , one can find diffusive rough-paths for which our strategy fails to replicate the option payoff. However, these rough-paths must have a stock price exactly equal to the strike at maturity. In a probabilistic theory, such paths occur with probability zero, so may be neglected. However, our interpretation is that the existence of such paths demonstrates a genuine lack of robustness of the classical delta hedging strategy. The need for a robust strategy becomes more important towards maturity as classical diffusion models break down and new phenomena occur such as the "pinning" of stock prices around exchange traded strikes (see e.g., Avellaneda & Lipkin, 2003;Avellaneda et al., 2012;Golez & Jackwerth, 2012;Jeannin et al., 2008). The failure of our strategy for certain stock paths indicates that one should switch strategy near maturity to a genuinely probabilistic strategy, such as a buy-and-hold strategy. This reflects actual trading practice, where delta hedging strategies are abandoned and quite different strategies adopted near maturity.
The article is organized as follows. In Section 2, we recall the classical theory of hedging and establish our notation. In Section 3, we describe the machinery on reduced rough path integration we will use. In Section 4, we define what is meant by a diffusive rough-path. In Section 5, we demonstrate formally how to obtain a pathwise formulation of the classical formulas of Mathematical Finance in continuous time. Section 6 shows how our continuous time trading strategy can be interpreted as a limit of discrete time trading strategies in volatility swaps. Section 7 demonstrates that our proposed strategy fails in the Black-Scholes model for call options when the stock price terminates at the strike. Section 8 presents our conclusions.

NOTATION AND PRELIMINARIES
We will develop a rough-path version of a classical diffusion model, and will begin by describing the classical model. Let 0 denote the value of a riskless asset at time , and assume that Let ∈ ℝ denote the price vector of non-dividend paying stocks, representing the risky asset. We let̃be the discounted price of the risky asset at time , namelỹ= ∕ 0 = − . We suppose that each component of the price vector̃displays the following dynamics in the pricing measurẽ= (̃) , = 1, … , ,̃0 = 0 ∈ ℝ , on a stochastic base (Ω, , , ( ) , ( ) ) carrying a standard -dimensional Brownian motion ( ) . We assume in −Ḧl loc (ℝ , ℝ × ), 0 < < 1. Einstein's summation convention on double indices is employed and will be throughout all the paper.
Let ( ) be the payoff of a Vanilla option on the underlying . We assume that is a continuous and bounded function on ℝ . Let andh ∶= − ℎ. The payoff is therefore equivalently written as ℎ(̃), and its discounted value is ℎ(̃).
The classical theory of (Harrison & Kreps, 1979;Harrison & Pliska, 1981) tells us that the option payoff can be replicated at time for a price,̃satisfying where ( ) is the semigroup on (ℝ ) generated by the infinitesimal operator of the dynamics of̃. We call the volatility operator. To ensure the absence of arbitrage, we must make some assumptions to ensure that the solutions to the Black-Scholes PDE are unique. In this paper will typically assume that the volatility operator is uniformly elliptic. The stochastic process̃is then a deterministic function = ( , ) of time and space applied after ( ,̃) which solves Equation (2) is the discounted version of the Black-Scholes partial differential equation. We will write ( , ) = ( , − ) for the undiscounted value function and will use following notation for the Greeks: In our setup, the pricing PDE is justified by the existence of a replicating strategy for the payoff. An investment strategy may be viewed as a pair ( 0 , ) indicating the quantities to purchase at each time of the riskless and risky asset. By Itô's formula, It follows that the delta hedging strategy = ( 0 , ) given by is such that the undiscounted portfolio process is self-financing, that is, it satisfies and replicates the option payoff, that is, ( ) = ( ).

PATHWISE INTEGRALS
In this section we review the elements of rough path theory that we will need. The results are standard, or minor variations of standard results and so proofs have been omitted, but may be found in the Arxiv version of this paper.
If is defined on the simplex but is additive, then it can be extended to an additive function on [0, ] × [0, ] by setting , ∶= − , .
Additivity characterizes those functions on [0, ] × [0, ] that descend from increments of paths, in the following sense.
Moreover, if is another path whose increments coincide with , then − is constant.
Given a partition of [0, ] and a time instant in [0, ], we adopt the following notational convention: If is additive, this notation is the usual -variation norm of the underlying path. For ≤ ≤ we introduce the symbol , , ∶= , − , − , .
Given a partition of [ , ] ⊂ [0, ] we may use a control function to measure the mesh-size.
Definition 3.3. The modulus of continuity of on a scale smaller or equal than the mesh-size | | is given by (10) By Proposition 3.5, we can regard the integral as the map By Proposition 3.1, we can unambiguously replace the range { additive functionals } with the space of continuous paths on  starting at 0 ∈ . Let -var ([0, ]; ) be the family of approximately additive functions Ξ ∶ {( , ) ∈ ℝ 2 ∶ 0 ≤ ≤ ≤ } →  that are of finite -variation, ≥ 1. Then, we have: Corollary 3.6. The restriction of the integral map ∫ to -var ([0, ]; ) takes value in the space -var 0 ([0, ]; ) of continuous paths on  that start at the origin 0 ∈  and are of finite -variation. Moreover,

Proposition 3.7 ("Young integral"). Let and be Young complementary and set
Then, Ξ is approximately additive and of finite -variation. As a consequence, the integral defines a continuous path in of finite -variation. The integral in (11) does not depend on whether Ξ is defined according to Ξ , = , or to Ξ , = , .
The continuity of is only used to show that the choice to evaluate at the beginning or at the end of the partition subintervals does not affect the integral. The two choices are respectively referred to as adapted evaluation and terminal evaluation. If is not continuous but of bounded variation, the Young integral is defined (because = 1), but depends on the evaluation choice. If is a partition of [0, ], we set which denotes the piecewise constant caglad approximation of on the grid . We let . be the Young integral of against with adapted evaluation, namely In this way, for continuous and of finite -variation, 1∕ + 1∕ > 1, we can write

Compensated integrals
When the complementary regularities of integrand and integrator are not sufficient for Young integration, we introduce the enhancement of a path and we will define the integral using compensated Riemann sums. In particular this is the case if and have the same -variation regularity for some greater than 2. As above, let be a continuous path of finite -variation with trajectory in the Banach space . Recall that denotes Hom(; ). We use the identification Hom(, ) ≅ Hom( ⊗ ; ), and we write Hom sym ( ⊗ ; ) for the subset of those in Hom( ⊗ ; ) such that ( ⊗ ) = ( ⊗ ) for all , ∈ . Also, the symbol  ⊙  will denote the symmetric tensor product of the Banach space , so that we can identify Hom sym ( ⊗ ; ) ≅ Hom( ⊙ ; ). We say that a continuous path ∶ [0, ] → admits a symmetric Gubinelli derivative ′ with respect to if there exists a continuous path ′ ∶ [0, ] → Hom sym ( ⊗ ; ) of finite -variation such that 1. and ∕2 are Young complementary; 2. , ∶= , − ′ , is of finite ∕( + )-variation.
In this case we say that the pair ( , ′ ) is -controlled of ( , )-variation regularity. Notice that the regularities of and of imply that is of finite -variation. The symbol [ ] will be referred to as volatility enhancer when the financial meaning of it is to be stressed. We say that Notice that does not depend on the enhancer because [ ] is additive; moreover, for all ≤ ≤ the following reduced Chen identity holds , , = , ⊙ , . Lemma 3.10. Let = ( , ) be an enhanced path and let ( , ′ ) be -controlled of ( , )variation regularity, with ′ being symmetric. Then, is approximately additive.
As a consequence of Lemma 3.10, the integral given by the compensated Riemann sum is well-defined. Analogously to (12), we write If is a time interval, and are non-negative integers and , are in [0,1), consider the space of ℝ -valued functions that are times continuously differentiable in time with the -th time derivative of local -Hölder regularity, and times continuously differentiable in space with all the -th order space derivatives of local -Hölder regularity. Notice that nothing is assumed about the cross derivatives in time and space of functions in + , + loc . Let Let  be the space Definition 3.11 (" -Moderation"). Let be in  and let be a continuous path on ℝ of finite -variation, with − 2 < < 1. We say that the pair ( , ) is -moderate if 1. the paths can be continuously extended up to [0, ], and ′ is of finite -variation for some 1 − 2∕ < 1∕ < ∕ ; 2. there exists a control function such that for all in the trace [0, ] and all 0 ≤ ≤ ≤ where Conv [0, ] is the convex hull of the trace of .

ENHANCED PATHS OF DIFFUSION TYPE
We now isolate the pathwise features of price trajectories that affect hedging practice. Until further notice, we adopt the perspective of discounted prices, so that only the secondorder part of is considered, with coefficients thought of as functions of the discounted stock price. Given an -Hölder volatility operator = ( ∇ 2 )∕2 and a continuous path ∶ [0, ] → ℝ of finite -variation, we can consider the ∫ ( ) -enhancement = ( , ) of given be For brevity we will henceforth call this the -enhancement of . Notice that such construction yields a bounded variation enhancement. The converse construction, which starts from a bounded variation enhancement and defines a differential operator, is formalized in the following Definition 4.2 ("Enhanced path of -diffusion type"). Let = ( , ) be an enhanced path of -variation regularity. We say that is of -diffusion type, − 2 < < 1, if by setting Remark 4.3. The ellipticity condition in Equation (15) allows us to apply the theory from (Lorenzi & Bertoldi, 2007, Chapter 2) to the existence and uniqueness of semigroups on (ℝ ) associated with the volatility operator . If the solution to the PDE associated with is known to posses a unique solution, the assumed ellipticity can be removed. This is the case for example of the classical Black-Schoels partial differential equation with volatility operator 2 2 2 ∕2.
Remark 4.4. Definition 4.2 is reminiscent of the class of price trajectories considered in (Schied & Voloshchenko, 2016). In both cases, the idea is to define the minimal pathwise requirements that link the dynamics of the underlying to a parabolic PDE. This link hinges on a differential operator. In the theory of Markov diffusions, this differential operator is the generator of the Markov semigroup and characterizes the probability law of the diffusion. However, the class of trajectories of Markov diffusions is strictly contained in the class of trajectories considered in (Schied & Voloshchenko, 2016), which in turn is strictly contained in the class of enhanced paths of diffusion type. Indeed, in (Schied & Voloshchenko, 2016) trajectories are only required to posses a quadratic variation: for example the sum = + of a standard one-dimensional Brownian motion and a fractional Brownian motion with Hurst exponent > 1∕2 is not a Markov diffusion but it is encompassed by (Schied & Voloshchenko, 2016) and by our framework. The case of a fractional Brownian motion with > 1∕3 is encompassed by our framework, but not by (Schied & Voloshchenko, 2016).
An enhanced path of -diffusion type is the minimal information that the PDE pricing approach requires from a probabilistic model. Indeed, assume that we wish to use the PDE approach to price a contingent claim ℎ( ), where ℎ is in (ℝ ) and is the terminal value of a continuous price path of finite -variation. Let = ( , ) be an enhancement of of -diffusion type and consider the equation Then, the Cauchy problem (16) for all 0 ≤ ≤ ≤ .
See Appendix 4 for the proof. We may use higher order sensitivities and pathwise integration to estimate errors arising from time discretization of integral quantities. Consider the cost of financing of a hedging strategy, defined as where ( 0 , 1 ) ∈ ℝ × ℝ is the strategy and 0 , are respectively the riskless asset and the risky asset. The symbols ( 0 . 0 ) and ( 1 . ) denote the time-marginals of the integral processes of 0 and 1 respectively against 0 and . Thus, the cost of financing in Equation (18) is the difference between the value of the portfolio at time and the cost of rebalancing the portfolio during the time window [0, ] in order to follow the hedging strategy. If continuous hedging were possible and one were able to take ( 0 , 1 ) = ( 0 , ) as defined in (4), then this cost 1 would match 0 = (0, 0 ), the price at time = 0 of the option, on a -full set. We remark that the probability is the measure of the stochastic base on which in the continuous-time case the Itô integral ( 1 . ) would be defined. In practice, the cost of financing has two components: the theoretical price 0 and the cost arising from time discretization, which is ( ) − 0 . For the latter, with replaced by the discretization ( 0 , ) of (4), we now provide a pathwise estimate that relies on integration bounds. Recall that in Proposition 4.5 plays the role of the discounted trajectorỹ = − .
Corollary 4.6. Assume the setting of Proposition 4.5. Let be the control function whose (2∕ + 1∕ )-th power asserts the approximate additivity of , + ′ , . Along any partition of [0, ], the discretized strategy ( 0 , ) stemming from (4) with̃= has a cost of financing ( 0 , ) that is bounded as follows: where osc( , | |) is the modulus of continuity of on a scale smaller or equal than the mesh-size of the partition, and −, is the difference between ( , ) =h( ) and the discounted value ( −, − ) of the option at the second last node of the partition. The path-dependent constant appearing in the bound is not greater than where is the ∕( + )-variation control of , − ′ , .
Proof. Let be the path ↦ ( , ). Fix a partition of [0, ] and recall the notation in (8). We preliminarily observe that where in the second line we have used summation by parts. Then, By adding and subtracting the compensation, we can apply the Sewing Lemma (Proposition 3.5) to complete the proof. □ So far, we have worked with the identification =̃, i.e. the enhanced path at hand has represented the actual enhanced path of the discounted stock price. In other words, the market models have been [̃]-compatible. This amounts to considering the square = of co-volatilities a true parameter. In Corollary 4.7 below, we no longer do so and we distinguish the modeled enhancer of from the actual enhancer of̃. The only assumption oñis that it is an enhanced path, that is, its tracẽis a continuous path of finite -variation, 2 < < 3, and its second-order process = (̃⊗̃− [̃])∕2 is a continuous two-parameter function of finite ∕2-variation with values in ℝ ⊙ ℝ ; the enhancer [̃] is not required to be of bounded variation and the integrals against it will be interpreted as Young integrals.
where the second summand on the right hand side is a well-defined Young integral. As a consequence, if denotes the strategy obtained by discretizing along the -delta hedging, then its cost of financing ( 0 , ) is bounded by where , and | −, | are as in Corollary 4.6 and See Appendix A for the proof.

PATHWISE FORMULATION OF FUNDAMENTAL EQUATIONS OF HEDGING
By adopting the perspective of undiscounted price paths, we recover the classical formulas of Mathematical Finance within our pathwise setting. Given a price path , we say that a model for has been specified when a choice for the enhancement = ( , ) is made. This means choosing the enhancer [ ], see Section 3. We speak of an -diffusive model specification if the enhancer is given by where , , 1 ≤ , ≤ are the coefficients of an -Hölder volatility operator and is the constant interest rate. In other words, an -diffusive model specification is the undiscounted counterpart to an -enhancement of some discounted price path, where is an -Hölder volatility operator as defined in Definition 4.1.
Theorem 5.1. Let ( ) be a contingent claim, where is in (ℝ ) and is the terminal value of a continuous -dimensional price path of finite -variation. Let = ( , ) be an -diffusive model specification, with > − 2, and let = , 2 , ∕2 be the corresponding volatility operator. Then, the Black-Scholes partial differential equation admits a solution in  and this solution is unique. Moreover, there exist a probability space (Ω, , , ( ) ) and a Markov diffusion process̃defined on it, such that for all 0 ≤ ≤ it holds where ℎ( ) ∶= ( ).
Proof of Theorem 5.1. The change of variable ∶= − allows us to rewrite Equation (23) as where ( , ) = − ( , ). Therefore, existence, uniqueness and regularity of the solution follow from those of Equation (16).
By applying the operator lim | |→0 ∑ ∈ to both sides of this expansion, we obtain (25) since solves the Black-Scholes partial differential Equation (23). □ The pathwise differential equation in (25) coincides with the classical SDE for the portfolio process in the delta hedging. In addition, the definition of the pathwise integral ( , ) .( , ) explicitly expresses the dependence on the gamma sensitivity, which is not captured by the classical stochastic integral.

Fundamental theorem of derivative trading
The formulas for pricing and hedging heavily depend on the diffusive model specification. In classical terms of Mathematical Finance, such specification amounts to specifying the diffusion coefficient (volatility) in Itô's price dynamics. Volatility is not directly observable and consequently a trader is liable to misspecify volatility and to use coefficients that do not faithfully represent the true price dynamics. The Fundamental Theorem of Derivative Trading addresses such misspecification. It provides a formula that computes the profit&loss that a trader incurs into when hedging with the wrong volatility -see (Cont, 2010;Ellersgaard et al., 2017;Karoui et al., 1998). Proposition 5.3 contributes to the assessment of model misspecification in two ways: on the one hand, it shows the pathwise nature of the P&L formula (this aligns with the unifying theme of the section); on the other hand, it provides a generalization of the classical P&L formula. The generalization consists in removing the assumption that the "true" price evolution is governed by an Itô SDE: we capture the misspecification that arises not just between two diffusive enhancements but between a diffusive enhancement (used by the trader) and a general enhanced path (the "true" dynamics).
Remark 5.4. If true arises from a diffusion model then as compensation terms in our integrals vanish in probability, our definition of the value of the portfolio,̂, can be justified as a selffinancing condition. We will justify this definition for general pricing signals in Section 6 below. In order to recognize the extension of the classical Fundamental Theorem of Derivative Trading, we rewrite the Young integral in Equation (26) as ) .
In the case where true is a diffusive enhancement, we have that [ true ] = ∫ 0 2 , true ( − ) , so that the integral is turned in the familiar form Proof of Proposition 5.3. We manipulate the Taylor expansion in the proof of Proposition 5.2 and, for 0 ≤ ≤ ≤ , we write where is the solution to the -dimensional Black-Scholes partial differential Equation (23), is a control function and > 1. We sum over the nodes of a partition and then we let the mesh-size shrink to zero, obtaining (26). The good definition of the Young integral of against [ true ] and [ ] holds as in Corollary 4.7. □

ENLARGED HEDGING STRATEGIES
Given an enhanced price path = ( , ), we interpreted the pathwise integral ( , ′ ) .( , ) as the portfolio trajectory arising from the position on the risky asset .
In this section, we explore the possibility to modify the interpretation of ( , ′ ).( , ). We will not only consider it as representing the values of the position on , but we will give a financial interpretation to the compensation ′ as well. This requires to analyze the mechanics of rebalancing portfolios during hedging periods.
Given a (continuous) path in ℝ and a partition we write piecewise constant caglad approximation Classically, given the partition and the discretized strategy ( 0 , ), the cost of rebalancing the portfolio from ( −, ] to ( , ′ ] is where, for 0 ≤ < ≤ and 1 ≤ ≤ ≤ , the amount , ( , ) = , ( , ) denotes the (exogenously-given) price at time of the swap , , with maturity . Notice that, since swap contracts are not primitive financial instruments, in the equation above the payoff −, at time is disentangled from the price ( , ′ ) required at time to take a unit position on the next swap , ′ . We assume that the price ( , ) of the swap contracts , defines a ℝ ⊙ ℝ -valued function on {( , ) ∈ ℝ 2 ∶ 0 ≤ ≤ ≤ }, null and right-continuous on the diagonal, 2 and such that ( , ) is of finite ∕2-variation. Let 2 be a continuous path of finite -variation on Hom(ℝ ⊙ ℝ ; ℝ), where and ∕2 are Young complementary. Then, the integral path ∶= ( 2 . ) 0, exists and represents the accumulated cost in the time interval [0, ] consumed by a continuously rebalanced enlarged strategy in order to adopt the positions 2 on the swap contracts.
Definition 6.1. Let ( ) be a contingent claim, where is in (ℝ ) and is the terminal value of a continuous -dimensional price path of finite -variation. Let = ( , ) be an -diffusive model specification, > − 2, and let , , and be as in Proposition 5.2. Let be a continuous real valued function on [0, ]. Then, the -enlarged delta hedging is the enlarged strategy defined as where ∶= ( 2 . ) 0, .
A desirable property of a hedging strategy is the self-financing condition, i.e. the fact that the strategy does nor require money to readjust its positions during the hedging period. The following Proposition 6.2 gives the explicit formula for in (28) that guarantees a null rebalancing cost of the -enlarged delta hedging.

Proposition 6.2. The continuous real valued function
where ∶= ( . ) 0, , is such that the -enlarged delta hedging has zero cost of continuous rebalancing.
Proof. We adopt the notation in Definition 6.1. Furthermore, we set We can write The cost of rebalancing along a partition is Hence, summing over ∈ , > 0, we have  (25) and (30). □ The classical delta hedging is such that the initial endowment 0 = (0, 0 ) is precisely what the replicating strategy requires in order to yield the amount ( ) at maturity . Therefore, the writer of an option invests 0 in the delta hedging strategy, and such strategy will cover the contingent claim at maturity. Since delta hedging has no additional costs of financing (i.e. rebalancing the portfolio does not consume money) the writer's profit&loss is null. For the -enlarged delta hedging in Proposition 6.2, the self-financing condition holds. Therefore, the option writer's P&L is exclusively given by the cost of replication, namely by the difference between the due payment ( ) and the final value 0 0 + 1 of the portfolio. Notice that the latter does not comprise the payoff of the swaps, because such endowments are consumed in the rebalancing process. Proposition 6.3. The profit&loss of the -enlarged delta hedging with given as in (29) is Proof. The profit&loss is given by the difference & = ( , ) − 0 0 + 1 . Hence, the statement follows immediately from the definitions in Equation (28) with given as in Equation (29). □

NON-SMOOTH OPTION PAYOUTS
We now consider the case of call options in the Black-Scholes model. We will see that as a result of the non-smooth payoff function one must employ a different, and truly probabilistic, trading strategy towards maturity.
Our setting is the one presented in Section 2, and we take the dimension equal to 1. The volatility operator is where > 0 is the volatility coefficient. Pricing a European option with payoff ( ) requires solving the partial differential Equation (2) where the terminal constrainth = − ℎ appearing in this PDE stands in relation to the payoff function as expressed in Equation (1).
The volatility operator in Equation (31) is not locally uniformly elliptic, i.e. it does not satisfy the requirement in Equation (15). However as pointed out in Remark 4.3, we do not require ellipticity itself only the existence and uniqueness of solutions to the equation in (16). Existence and uniqueness of solutions to the Black-Scholes PDE is well-known.
In our framework, the classical Black-Scholes model is specified by the following enhancer Under this specification, we now discuss the application of our pathwise framework to the case of European call options, where the payoff is for some fixed strike > 0. This payoff is not bounded, so in principle it is not included in the general discussion above. However despite the fact that the semigroup associated with the PDE pricing equation was defined on the set (ℝ), this semigroup extends to a wider class than (ℝ), hence allowing to treat the European call option. Even if the model specification did not allow for such an extension, pricing European call options could always be reduced to pricing European put options due to put-call parity.
In order to be able to apply Proposition 5.2, it remains to discuss the assumption on themoderation of the pair ( , ). Unfortunately, here we see that the non-smoothness of the payoff of the call option (or equivalently of the put option) prevents us from applying directly the results established above. We will discuss this in details now.
Recall the three conditions in Definition 3.11. Let and ′ be the delta and the gamma sensitivities namely = = ( , ) = ( 1 ( , )), where and denotes the distribution function of the standard normal distribution. The fulfilment of the three conditions in Definition 3.11 depends on the terminal value ot the price path. Depending on this terminal value we have the following asymptotics as ↑ : Instead, if = , then neither 1 nor 2 have a limit as ↑ . To see this we use the law of iterated logarithm, which gives a precise statement on the small time asymptotics of the Brownian path. We have that the terminal value is Hence, as ↑ we have The first factor on the right hand side is such that the limsup as ↑ is equal to 1, and the liminf is equal to −1. Therefore, if = , then lim sup ↑ 1 ( , ) = lim sup ↑ 2 ( , ) = +∞, lim inf ↑ 1 ( , ) = lim inf ↑ 2 ( , ) = −∞.
Because of Equation (38), conditions 1 and 2 in Definition 3.11 will not always be satisfied. Moreover, the singularity at will also impact condition 3.
One could circumvent this issue by a smooth approximation of the option payoff that could eliminate the point of non-differentiability. Here instead, we comment on what this says about option trading in practice, and on how these singularities, exposed by our pathwise framework, could be regarded as an underpinning of the practicality of option hedging.
The unstable behavior of the sensitivities when time is close to maturity is known in practice, in particular in the case of options that are at-the-money (i.e., the underlying has a price equal or very close to the strike). Because of this, it is common to stop the delta hedging before the actual option maturity, and to continue with a simpler strategy as buy-and-hold. This is described by introducing a time horizon̂smaller than the option maturity ; then the Black-Scholes price at is smooth and so our framework can be applied up to timêsubject to assuming that the option can be sold at this time at the Black-Scholes price.
After̂and in the limit as time approaches , the sensitivity in Equation (35) no longer controls of Equation (34) in the sense of Gubinelli. In the case of at-the-money options, the gamma sensitivity diverges to infinity as time approaches . This has an impact on the profit&loss formula of Proposition 5.3, as described in the following proposition.
Then, there always exists an arbitrary fine trading grid such that the profit&loss of the delta hedging on this trading grid diverges to −∞ as time approaches the option maturity.
Remark 7.2. Proposition 7.1 says that, in the case of at-the-money options, if the misspecification of the Black-Scholes model is such that the volatility is underestimated, then there exist trading times when following the delta hedging will make the trader incur in unbounded losses. Instead, in the cases of in-the-money and out-the-money options ( > and < respectively), the gamma sensitivity has a limit as time approaches maturity and this limit is zero. Therefore, in these two cases, the Young integral describing profit&loss can be bounded relying on the integration bounds of Section 3.
Proof of Proposition 7.1. Let be a trading grid up to the option maturity. Consider the approximation of the Young integral in Equation (26) The condition in Equation (39) We see that as ↓ 0 the quantity − 1− Γ − goes to −∞. □

CONCLUSIONS
In this work, we have shown that European options may be replicated in a framework that does not use probability. We instead study enhanced price paths defined in the spirit of Rough Path Theory. On the one hand, their enhancements are essential for pathwise integration, as discussed in Section 3. On the other hand, they encapsulate the specification of a model for the valuation of derivatives, carrying the information needed for the hedging (Section 4). Moreover, these enhancements allow to assess model misspecification: a P&L formula for the hedging under "wrong" volatility was proved, generalizing the so-called fundamental theorem of derivative trading (Section 5).
We stated the precise assumptions that allow for the application of Gubinelli integrals in the description of hedging strategies. These assumptions are satisfied in the standard Black-Scholes case of European call and put options only up to a timêthat strictly precedes the option maturity . On the one hand, this opens the question about suitable approximations for the limiting case aŝconverges to (without using probability); on the other hand, it provides a mathematical underpinning to some hedging practises linked to unstable option sensitivities, in particular in the at-the-money case.
The fact that our enhanced paths extend to trajectories other than semimartingales would make the no-arbitrage arguments suitable for models with transaction costs and other market imperfections. Indeed, in these cases price trajectories are usually less regular than semimartingales. Moreover, we would like to point out that the classical arguments for no-arbitrage under transaction costs is based on consistent price systems, see (Guasoni, 2006;Guasoni et al., 2008). This means that the absence of arbitrage is ultimately based on support theorems, hence presenting the opportunity to apply Rough Path Theory, whose application in support-type arguments has proved to be fruitful (see (Friz & Victoir, 2010, Chapter 19)). In this direction, a recent MSc Thesis at Imperial College London moved the first step (Pei, 2019).

A C K N O W L E D G M E N T S
Damiano Brigo is grateful to the participants of the conference in (Brigo, 2019) for helpful feedback. The work of Thomas Cass is supported by EPSRC Programme Grant EP/S026347/1

D ATA AVA I L A B I L I T Y S TAT E M E N T
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.