Pathwise no-arbitrage in a class of Delta hedging strategies

where x·y denotes the euclidean inner product of two vectors x and y.

These two aspects imply that it is reasonable to require that price trajectories S of a risky asset possess all covariations 〈S _i ,S _j 〉 in the sense that the limit in (2.5) exists for all t∈ [ 0,T]. It was shown by Föllmer (1981) that, if in addition the covariations are continuous functions of t, Itô’s formula holds in a strictly pathwise sense (see also (Sondermann 2006) for additional background and an English translation of (Föllmer 1981)). Let us thus denote by Q V ^d the class of all continuous functions \(\mathbf {S}:[\!0,T]\to {\mathbb {R}}^{d}\) on [ 0,T] for which all covariations 〈S _i ,S _j 〉(t) exist along \(({\mathbb {T}}_{n})\) and are continuous functions of t. We point out that the existence and the value of the covariation 〈S _i ,S _j 〉(t), and hence the space Q V ^d , depend in an essential manner on the choice of the refining sequence of partitions, \((\mathbb {T}_{n})\); see, e.g., (Freedman 1983, p. 47). Moreover, Q V ^d is not a vector space (Schied 2016). It follows easily from Föllmer’s pathwise Itô formula that for the following class of “basic admissible integrands” ξ, the Itô integral \({\int _{r}^{t}}\boldsymbol {\xi }(s){\,\mathrm {d}}\mathbf {S}(s)\) exists for all t∈ [ r,u]⊂ [ 0,T] as the finite limit of Riemann sums in (2.4); see (Schied 2014, p. 86). This integral is sometimes also called the Föllmer integral.

Following Bick and Willinger (1994), we will from now on consider not just one particular price trajectory S, but admit an entire class of such trajectories so as to account for the uncertainty of the actual realization of the price trajectory. Specifically, we will consider the classes

and

where a(t,x)=(a _ij (t,x)) _i,j=1,…,d is a continuous function of \((t,\mathbf {x})\in \,[\!0,T]\times {\mathbb {R}}^{d}\) (respectively of \((t,\mathbf {x})\in [0,T]\times {\mathbb {R}}_{+}^{d}\) in case of ) into the set of positive definite symmetric d×d-matrices. Additional assumptions on a(t,x) will be formulated later on. Here, \({\mathbb {R}}_{+}:=(0,\infty)\), and we will write \({\mathbb {R}}^{d}_{(+)}\) to denote the two possibilities, \({\mathbb {R}}^{d}\) and \({\mathbb {R}}^{d}_{+}\), according to whether we are considering or . Similarly, we will write etc. Price trajectories in can arise as sample paths of multi-dimensional local volatility models. At least for d=1, the local volatility function \(\sigma (\cdot):=\sqrt {a(\cdot)}\) is often chosen by calibrating to the market prices of liquid plain vanilla options (Dupire 1997). Since in practice there are only finitely many given options prices, σ(·) is typically only determined on a finite grid (Bühler 2015), and so regularity assumptions on σ(·) can be made without loss of generality.

Our next goal is to introduce and characterize a class of self-financing trading strategies that may depend on the current value of the particular realization and includes candidates for hedging strategies of European derivatives. Before that, let us introduce some notation. By C(D) we will denote the class of real-valued continuous functions on a set \(D\subset {\mathbb {R}}^{n}\). For an interval I⊂[0,T] with nonempty interior, , we denote by \(C^{1,2}\left (I\times {\mathbb {R}}^{d}_{(+)}\right)\) the class of all functions in \(C\left (I\times {\mathbb {R}}^{d}_{(+)}\right)\) that are continuously differentiable in , twice continuously differentiable in x for all , and whose derivatives admit continuous extensions to \(I\times {\mathbb {R}}^{d}_{(+)}\). Let us also introduce the following second-order differential operators,

Now suppose that \(f:{\mathbb {R}}^{d}_{(+)}\to {\mathbb {R}}\) is a continuous function for which there exists a solution v to the following terminal-value problem,

For and t∈ [ 0,T), we can define

$$ \boldsymbol{\xi}^{\mathbf{S}}(t):=\nabla_{\mathbf{x}}v(t,\mathbf{S}(t)) \qquad \text{ and } \qquad \eta^{\mathbf{S}}(t):=v(t,\mathbf{S}(t))-\boldsymbol{\xi}^{\mathbf{S}}(t)\cdot\mathbf{S}(t). $$

(2.10)

Thus, (ξ ^S ,η ^S ) is a self-financing trading strategy with portfolio value V ^S (t)=v(t,S(t)). Since the function v is continuous on \([0,T]\times {\mathbb {R}}^{d}_{(+)}\), the limit \(V^{\mathbf {S}}(T):={\lim }_{t\uparrow T}V^{\mathbf {S}}(t)\) exists and satisfies

In this sense, (ξ ^S ,η ^S ) is a strictly pathwise hedging strategy for the derivative with payoff f(S(T)).

The preceding argument was first made by Bick and Willinger (1994, Proposition 3) in a one-dimensional setting. It is remarkable in several respects. For instance, consider the one-dimensional case with a(t,x)=σ ² x ² for some σ>0 so that (TVP⁺) becomes the standard Black–Scholes equation, which can be solved for a large class of payoff functions f. The preceding argument then shows that the Black–Scholes formula—which is nothing other than an explicit formula for v(0,S ₀)—can be derived without any probabilistic assumptions whatsoever. It follows, in particular, that the fundamental assumption underlying the Black–Scholes formula is not the log-normal distribution of asset price returns, but the fact that the quadratic variation of the asset prices is of the form \({\langle S,S \rangle }(t) = \sigma ^{2}{\int _{0}^{t}}S(s)^{2}{\,\mathrm {d}} s\). Let us now state general existence results for solutions of (TVP) and (TVP⁺), which in the case of (TVP) is taken from Janson and Tysk (2004). Recall that we assume that a(t,x) is positive definite for all t and x.

where 0=t ₀<t ₁<⋯<t _N =T denote the fixing times of daily closing prices and h is a certain function.

The condition on the local Lipschitz continuity of h in the preceding result can often be relaxed in more specific situations. Examples are the pathwise versions of the (d-dimensional) Bachelier and Black–Scholes models, which both correspond to the choice \(a_{ij}(t,\mathbf {x})=\widetilde {a}_{ij}\) for a constant positive definite matrix \((\widetilde {a}_{ij})\). In these cases the recursive scheme in Theorem 2.5 can be solved for large classes of payoff functions h without requiring local Lipschitz continuity. As a matter of fact, even the continuity of h can be relaxed so as to account for discontinuous payoffs as, e.g., in barrier options. This also applies to the strictly pathwise hedging argument that we are going to formulate next. However, these relaxations need case-by-case arguments. We therefore do not spell them out explicitly here and leave the details to the interested reader.

where ℓ is the largest k such that t _k <t. With these conventions, we obtain the following Delta hedging result.

Corollary 2.6.

Let H be an exotic option as in (2.12) and suppose that the recursive scheme in Theorem 2.5 holds for functions v _k , k=0,…,N. Then, for each , the strategy (2.13) is self-financing in the above sense and satisfies

$${\lim}_{t\uparrow T} \boldsymbol{\xi}^{\mathbf{S}} (t)\cdot\mathbf{S}(t) + \eta^{\mathbf{S}} (t)=v_{0}(0,\mathbf{S}(t_{0})) + {\int_{0}^{T}}\boldsymbol{\xi}^{\mathbf{S}}(s){\,\mathrm{d}} \mathbf{S}(s)=h(\mathbf{S}({t_{0}}),\dots, \mathbf{S}({t_{N}})). $$

In this sense, (ξ ^S ,η ^S ) is a strictly pathwise Delta hedging strategy for H.

The preceding corollary establishes a general, strictly pathwise hedging result for a large class of exotic options arising in practice. It also identifies v ₀(0,S(t ₀)) as the amount of cash needed at t=0 so as to perfectly replicate the payoff H for all price trajectories in . In continuous-time finance, this amount is usually equated with an arbitrage-free price for H. In our situation, however, the interpretation of v ₀(0,S(t ₀)) as an arbitrage-free price lacks an essential ingredient: We do not know whether our class of trading strategies is indeed arbitrage-free with respect to all possible price trajectories in . This question will now be explored in the subsequent section. Our corresponding result, Theorem 3.3, gives sufficient conditions under which trading strategies, as those in Corollary 2.6, do indeed not generate arbitrage in our pathwise framework. Theorem 3.3 will be the main result of this paper.

We are now going to study the absence of pathwise arbitrage within a class of strategies that is suggested by the pathwise Delta hedging strategies constructed in Theorem 2.5 and Corollary 2.6. We refer to the paragraph preceding Remark 2.7 for a motivation of this problem. Let us first introduce the class of strategies we will consider.

Definition 3.1.

Suppose that \(N\in \mathbb {N}\), 0=t ₀<t ₁<⋯<t _N =t _N+1=T, and v _k (k=0,…,N) are real-valued continuous functions on \([\!t_{k},t_{k+1}]\times \left ({\mathbb {R}}^{d}_{(+)}\right)^{k+1}\times {\mathbb {R}}^{d}_{(+)}\) such that, for k=0,…,N−1, the function (t,x)↦v _k (t,x ₀,…,x _k ,x) is the solution of (TVP⁽⁺⁾) with terminal condition f _k+1(x):=v _k+1(t _k+1,x ₀,…,x _k ,x,x) at time t _k+1. For we then define ξ ^S as in (2.13) and

$$ V^{\mathbf{S}}_{\boldsymbol{\xi}}(t):=v_{0}(0,\mathbf{S}(0))+{\int_{0}^{t}}\boldsymbol{\xi}^{\mathbf{S}}(s)\,\mathrm{d} \mathbf{S}(s), $$

(3.1)

where the pathwise Itô integral is understood as in (2.14). By we denote the collection of all pairs (v ₀(0,·),ξ ^·) that arise in this way.

Theorem 2.5 gives sufficient conditions for the existence of a family of functions (v _k ) as in the preceding definition, but these conditions are not necessary. In particular, as mentioned above, the local Lipschitz continuity of the terminal function v _N can be relaxed in many situations. We can now define our strictly pathwise notion of an arbitrage strategy.

Definition 3.2.

((Admissible) arbitrage opportunity) A pair is called an arbitrage opportunity for if the following conditions hold.

1. (a)

   \(V^{\mathbf {S}}_{\boldsymbol {\xi }}(T)\ge 0\) for all .

    
2. (b)

   There exists at least one for which \(V^{\mathbf {S}}_{\boldsymbol {\xi }}(0)=v_{0}(0,\mathbf {S}(0))\le 0\) and \(V^{\mathbf {S}}_{\boldsymbol {\xi }}(T_{0})>0\) for some T ₀∈(0,T].

    

An arbitrage opportunity (v ₀(0,·),ξ ^·) will be called admissible if the following condition is also satisfied.

1. (c)

   There exists a constant c≥0 such that \(V^{\mathbf {S}}_{\boldsymbol {\xi }}(t)\ge -c\) for all and t∈ [ 0,T].

    

Let us comment on the preceding definition. Condition (a) states that one can follow the strategy (v ₀(0,·),ξ ^·) up to time T without running the risk of ending up with negative wealth at the terminal time. Now let S be as in condition (b). The initial spot value, S ₀:=S(0), will then be such that v ₀(0,S ₀)=V ξ S(0)≤0. Hence, for any price trajectory with \(\widetilde {\mathbf {S}}(0)=\mathbf {S}_{0}\), only a nonpositive initial investment \(v_{0}(0,\mathbf {S}_{0})=V^{\widetilde {\mathbf {S}},\boldsymbol {\xi }}(0)\) is required so as to end up with the nonnegative terminal wealth \(V^{\widetilde {\mathbf {S}},\boldsymbol {\xi }}(T)\ge 0\). Moreover, for the particular price trajectory S, there exists a time T ₀ at which one can make the strictly positive profit \(V^{\mathbf {S}}_{\boldsymbol {\xi }}(T_{0})>0\). This profit can be locked in, e.g., by halting all trading from time T ₀ onward. In this sense, the strategy (v ₀(0,·),ξ ^·) is indeed an arbitrage opportunity. Condition (c) is a constraint on the strategy (v ₀(0,·),ξ ^·) that is analogous to the admissibility constraint that is usually imposed in continuous-time probabilistic models so as to exclude doubling-type strategies. Indeed, it follows, e.g., from Dudley’s (1977) result that standard diffusion models typically admit arbitrage opportunities in the class of strategies whose value process is not bounded from below (see also the discussion in (Jeanblanc et al. 2009, Section 1.6.3)). In our pathwise setting, an example of an arbitrage opportunity that does not satisfy condition (c) will be provided in Example 3.4 below. First, however, let us state the main result of our paper.

Recall from (2.12) our representation H=h(S(t ₀),…,S(t _N )) of the payoff of an exotic option, based on asset prices sampled at the N+1 dates 0=t ₀<t ₁<⋯<t _N =T. If N is large, it may be convenient to use a continuous-time approximation of the payoff H. For instance, the payoff \(H=\left (\frac {1}{N} \sum _{n=1}^{N}S^{1}_{t_{n}}-K\right)^{+}\) of an average-price Asian call option on the first asset, S ¹, can be approximated by a call option based on a continuous-time average of asset prices, \(H\approx \left (\frac {1}{T}{\int _{0}^{T}}{S^{1}_{t}}\,dt-K\right)^{+}\). Approximations of this type may be easier to treat analytically and are standard in the textbook literature. In this section, we extend our preceding results to a situation that covers such continuous-time approximations of (2.12). That is, we will consider payoffs of the form H(S), where S describes the entire path of the underlying price trajectory up to time T, and H is a suitable mapping from the Skorohod space \(D([\!0,T],{\mathbb {R}}^{d})\) to \({\mathbb {R}}.\) This will involve functional Itô calculus as introduced by (Dupire 2009) and further developed by Cont and Fournié (2010). In the sequel, we will use the same notation as in (Cont and Fournié 2010).

For a d-dimensional càdlàg path X in the Skorohod space \(D([\!0,T], \mathbb {R}^{d})\) we write X(t) for the value of X at time t and X _t =(X(u))_0≤u≤t for the restriction of X to the interval [0,t]. Hence, \(\mathbf {X}_{t}\in D([\!0,t], \mathbb {R}^{d})\). We will work with non-anticipative functionals as defined in (Cont and Fournié 2010, Definition 1), i.e., with a family F=(F _t ) _t∈[0,T] of maps \( F_{t}:D([0,t], \mathbb {R}^{d})\mapsto \mathbb {R}.\) For all further notation and relevant definitions, we refer to (Cont and Fournié 2010, Section 1).

The functional Itô formula (in the form of (Cont and Fournié 2010, Theorem 3)) yields that we can define general admissible integrands ξ in the following way, so as to ensure that the pathwise Itô integral \({\int _{r}^{t}}\boldsymbol {\xi }(s){\,\mathrm {d}}\mathbf {S}(s)\) exists for all t∈ [ r,u]⊂ [ 0,T] as a finite limit of Riemann sums; see (Cont and Fournié 2010, p. 1051).

In analogy to Proposition 2.3, we will now characterize self-financing trading strategies that may depend on the entire past evolution of the particular realization . For an interval I⊂ [ 0,T] with nonempty interior, , we denote by \({\mathbb {C}}^{1,2}(I)\) the class of all non-anticipative functionals on \(\bigcup _{t\in [a,b]} D([a,t], {\mathbb {R}}^{d}_{(+)})\) that are horizontally differentiable and twice vertically differentiable on and whose derivatives are continuous at fixed times and admit continuous extensions to I.

Thus, lifting the second-order differential operators and yields the following operators on path space,

The following proposition is a functional version of Proposition 2.3.

Now suppose that for suitably given \(H:D([\!0,T],\mathbb {R}^{d}_{(+)})\to \mathbb {R}\) there exists a solution F to the following path-dependent terminal-value problem,

Note that the terminal condition H has to be defined on the Skorohod space \(D([0,T],{\mathbb {R}}^{d}_{(+)})\) as opposed to \(C([0,T],{\mathbb {R}}^{d}_{(+)})\), because we need to take its vertical derivatives, which requires applying discontinuous shocks.

Then, for and t∈ [ 0,T), we can define

$$ \boldsymbol{\xi}^{\mathbf{S}}(t):=\nabla_{\mathbf{x}}F_{t}(\mathbf{S}_{t})\qquad\text{and}\qquad \eta^{\mathbf{S}}(t):=F_{t}(\mathbf{S}_{t})-\boldsymbol{\xi}^{\mathbf{S}}(t)\cdot\mathbf{S}(t). $$

(4.3)

whence we infer that (ξ ^S ,η ^S ) is a self-financing trading strategy with portfolio value V ^S (t)=F _t (S _t ). Since the functional F is left-continuous on [ 0,T] and S is continuous, the limit \(V^{\mathbf {S}}(T):={\lim }_{t\uparrow T}V^{\mathbf {S}}(t)\) exists and satisfies

Thus, (ξ ^S ,η ^S ) is a strictly pathwise hedging strategy for the derivative with payoff H=H(S).

In the next step, we will explore conditions yielding the existence and uniqueness of solutions to (FTVP) and (FTVP⁺). Path-dependent PDEs such as (4.1) are closely related to backward stochastic differential equations (BSDEs) generalizing the (functional) Feynman-Kac formula (Dupire 2009). In (Peng and Wang 2016), a one-to-one correspondence between a functional BSDE and a path-dependent PDE is established for the Brownian case. This was then generalized in (Ji and Yang 2013) to the case of solutions to stochastic differential equations with functionally dependent drift and diffusion coefficients.

We will now use (Ji and Yang 2013, Theorem 20) to formulate conditions such that (FTVP) and (FTVP⁺) admit unique solutions. To this end, we will need the following regularity conditions from (Peng and Wang 2016, Definition 3.1).

As above, the quantity F ₀(S ₀) can be identified as the amount of cash needed at t=0 so as to perfectly replicate the payoff H. But in order to interpret F ₀(S ₀) as an arbitrage-free price, we have to know whether our class of trading strategies is indeed arbitrage-free. Below we will formulate Theorem 4.7, which is a functional analogue of Theorem 3.3.

Definition 4.6.

Suppose that the non-anticipative functional F satisfying the conditions from Definition 4.1 is the solution of the path-dependent heat Eq. (4.1)

For , we then define ξ ^S as in (4.2) (on [ 0,T)) and

$$ V^{\mathbf{S}}_{\boldsymbol{\xi}}(t):=F_{0}(\mathbf{S}_{0})+{\int_{0}^{t}}\boldsymbol{\xi}^{\mathbf{S}}(s)\,\mathrm{d} \mathbf{S}(s). $$

(4.5)

By we denote the collection of all pairs (F ₀(·),ξ ^·) that arise in this way.

The notion of an (admissible) arbitrage opportunity for in the functional setting is defined in analogy to Definition 3.2; we only have to replace by .