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The chapter delves into the functional Taylor expansion in path space, highlighting its role as a central tool in calculus. It explores the analogues of derivatives and monomials in the path space, introducing the functional Itô calculus and signature as key components. The text discusses the functional Taylor expansion (FTE) and its applications in finance, including risk analysis and the pricing of exotic options. It also covers the functional Stratonovich formula, signature functionals, and their properties. The chapter concludes with practical examples and exercises, illustrating the theoretical concepts in real-world scenarios. Readers will gain insights into the functional Taylor expansion's role in approximating path-dependent functionals and its implications for financial modeling and risk management.
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Abstract
This chapter links the path signature to the functional Itô calculus (Dupire, Functional Itô calculus. SSRN (2009). Republished in Quantitative Finance 19(5), 721–729, 2019). The junction is the functional Taylor expansion (FTE), discussed in Sect. 5, which is a powerful tool to approximate functionals after an observed path.
In particular, the FTE decomposes the functional from future scenarios. In risk analysis, the FTE leads to new Greeks, one of particular importance being the Lie bracket between the space and time functional derivatives, which we call Libra. Once paired with the Lévy area, the Libra can be used to speed up the computation of risk measures such as Value at Risk and Expected Shortfall as explained in Sect. 6.1. In Sect. 6.3, we explain how the FTE can quantify the hedging error of replicating portfolios in a robust manner. In the context of financial derivatives, we demonstrate in Sect. 6.4 how the FTE generates approximations of exotic payoffs. The relevance of the FTE for cubature methods is finally outlined in Sect. 6.5.
1 Introduction
The Taylor expansion is a central tool in calculus. It requires two essential ingredients:
(i)
The derivatives of the target function at a given point.
and (i), (ii), correspond respectively to the derivatives \(\{f'(x), \ f''(x), \ldots \}\) and monomials \(\{1,y,y^2,\ldots \}\). We may therefore ask ourselves: what are the analogues of (i), (ii) in the path space?
With regards to (i), the functional Itô calculus [8] provides convenient notions of derivatives in the path space, recalled in Sect. 3. Moreover, as mentioned several times in this book, the signature can be regarded as a generalization of the monomials in path space. In fact, the signature pairs well with the functional derivatives and is therefore a good candidate for (ii). This is supported by our main result, the functional Taylor expansion (FTE), confirming that signature functionals are the building blocks of path dependence. We shall see in Theorem 5.1 that the FTE admits the general expression
where I is a word in the alphabet \(\{0,1\}\), where 0 refers to the time and 1 to the space variable. Also, \(\triangle _{I}f\), \(S^{I}\) are respectively the functional derivative and signature associated to I. Note that the FTE echoes the classical Taylor expansion in (1).
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Taylor expansions are classically used to either provide a local approximation of functions using a small truncation order or express real analytic functions as a power series similar to (1) and (2). Herein, we will focus on the former, with applications given in Sect. 6, and refer the interested reader to [9, Section 3.1.5] for the latter.
Notations
We fix throughout a time window \(T>0\) and consider the bundle
where for each \(t\in [0,T]\), \(\Lambda _t\) is the space of càdlàg paths (right continuous with left limit) on \([0,t]\) and taking values in \(\mathbb {R}\). Also, write \(X_{\mathcal {T}} = (X_t)_{t\in \mathcal {T}}\), \(\mathcal {T} \subseteq [0,T]\), so that \(X_{[0,s]} \in \Lambda _s \) is the restriction of X to \([0,s]\), \(s\le t\).1
A functionalf is a map from the path space \(\Lambda \) to \(\mathbb {R}\). We say that f is \(\Lambda \)-continuous if it is continuous with respect to the following metric,
$$\displaystyle \begin{aligned} {} d_{\Lambda}( X, Y) = \sup_{0 \le u \le t} |X_{u\wedge s} - Y_u | + t-s, \quad X \in \Lambda_s, \ Y \in \Lambda_t, \quad s \le t . \end{aligned} $$
(3)
Let \(*: \Lambda ^2 \to \Lambda \) be the path concatenation operation that preserves continuity as seen in chapter “A Primer on the Signature Method in Machine Learning”. If \(\mathbb {Q}\) is a probability measure on \(\Lambda \) and \(f:\Lambda \to \mathbb {R}\), the conditioned expectation under \(\mathbb {Q}\) of \(f(Z)\), \(Z\in \Lambda _t\), given \(Z_{[0,s]} = X \in \Lambda _s\) is defined as
Functionals express path dependence which is ubiquitous in finance, through the definition of exotic options (even vanillas; see Sect. 6.2) or the dynamics of the underlying asset price. For completeness, we recall from the previous section that:
Definition 2.1
A functional is a map \(f:\Lambda \to \mathbb {R}\).
To better distinguish functionals on fixed and varying horizons (i.e., on \(\Lambda \)), we also introduce the following notion.
Definition 2.2
Given \(T>0\), a T-functional is a map \(g:\Lambda _T \to \mathbb {R}\).2
In finance, functionals and T-functionals play different roles. First, T-functionals typically represent the payoff of a path-dependent claim with maturity T. Depending on the term sheet of g, it is usually not possible to specify the reward value having only observed the price path up to some intermediate time. For instance, the payoff of a forward start call option with maturity T and strike fixing date \(\tau \in (0,T)\) reads
Then g is a T-functional whilst the price of the option before maturity (that depends on the model) is a functional. On the one hand, one can directly define a T-functional from a functional \(f:\Lambda \to \mathbb {R}\) by defining the restriction \(g = f|_{\Lambda _T}\) of f to \(\Lambda _T\). For the reverse direction, we introduce the notion of embeddings.
Definition 2.3
We say that \(f:\Lambda \to \mathbb {R}\)embeds\(g:\Lambda _T \to \mathbb {R}\) into the space of functionals if f coincides with g on \(\Lambda _T\), that is \(f(X) = g(X)\) for all \(X\in \Lambda _T\).
Note that there are infinitely many functionals that embed a given T-functional. A natural candidate is the intrinsic functional, defined as
with the flat extension\(X^{* \delta t}_s = X_{s \wedge t}, s\le t+\delta t\), which is central in Definition 3.1 when introducing the functional time derivative. In other words, the enlarged path \(X^{*(T-t)} \in \Lambda _T\) is obtained by flat extending X until T; see Fig. 1. One family of particular interest in finance consists of the Black-Scholes embeddings\(g \mapsto f_{\sigma }\), defined as
where the concatenated process \(Y\in \Lambda _{T-t}\) is, under \(\mathbb {Q}_{\sigma }\), a geometric Brownian motion with zero drift and volatility parameter \(\sigma \). Concretely, \(Y_t = Y_0e^{\sigma W_t-\frac {1}{2}\sigma ^2t}\), where W is an \(\mathbb {Q}_{\sigma }\)-Brownian motion. Note that the initial value \(Y_0\) is irrelevant since Y is concatenated to X in a continuous way. If \(X\in \Lambda _T\), then \(Y = \emptyset \) in (6) which implies that \(f_{\sigma }(X) = g(X)\). In other words, \(f_{\sigma }\) coincides with g on \(\Lambda _T\) in line with Definition 2.3. In financial terms and absent of interest rates, \(f_{\sigma }(X)\) gives the price of the claim g in the Black-Scholes model \(\mathbb {Q}_{\sigma }\) having observed X.
Fig. 1
Illustration of the Black-Scholes embedding (6) with \(\sigma = 25\%\). The intrinsic functional (\(\sigma = 0\%\)) corresponds to a flat extension of X (solid line at level \(X_t\))
Another possibility is to use Bachelier embeddings where \(\mathbb {Q}_\sigma \) in (6) is replaced by the Wiener measure scaled by the (Bachelier) volatility \(\sigma \). Put differently, \(Y = Y_0 + \sigma W\), where W is an \(\mathbb {Q}_{\sigma }\)-Brownian motion.
The usefulness of the Black-Scholes and Bachelier embeddings will be revealed in Sect. 6 when discussing financial applications.
Exercise 2.4
Let \(g(X) = \max \limits _{t\le T} X_t - X_T\) which delivers the drawdown of \(X \in \Lambda _T\).
(a)
Compute the Bachelier embedding \(f_{\sigma }(X)\) explicitly for arbitrary \(\sigma >0\).
(b)
What happens to \(f_{\sigma }(X)\) when \(\sigma \downarrow 0\)? In other words, what is the intrinsic functional of g?
3 Functional Itô Calculus
As recalled in the introduction, any Taylor expansion requires a notion of derivative in the corresponding ambient space. When dealing with paths, natural candidates are the derivatives arising from the functional Itô calculus [8], which we now outline.
Definition 3.1
The functional time derivative of \(f:\Lambda \to \mathbb {R}\) is given for \(X\in \Lambda _t\) by
The image displays a mathematical formula involving limits and functions. The main equation is: [ Delta_t f(X) = lim_{delta t downarrow 0} frac{f(X^{* delta t}) - f(X)}{delta t} ] where X^{* delta t}_s = X_{s wedge t} and s leq t + delta t . The formula involves Greek symbols such as delta (Delta) and uses mathematical operations like limits and function evaluation.
(7)
when the limit exists. Moreover, the functional space derivative of \(f:\Lambda \to \mathbb {R}\) is
The time and space functional derivatives are respectively illustrated in Figs. 2 and 3.
Fig. 2
The functional time derivative \(\triangle _t f(X)\), defined in (7), is obtained by introducing a flat extension of the path at the spot level \(X_t\)
Unlike classical calculus, \(\triangle _x\) and \(\triangle _t\) need not commute in general, i.e. \(\triangle _{tx}f \neq \triangle _{xt}f\) where \(\triangle _{tx}f = \triangle _{t}(\triangle _{x}f)\) and similarly for \(\triangle _{xt}f\). In particular, the Lie Bracket
We gather from (11) that the Lie bracket thus captures the instantaneous path-dependence, i.e., how the order in which a path is shifted in space and extended in time affects a functional.
What are \(\triangle _t f\) and \(\triangle _x f\)?
(b)
Compute the Lie bracket \(L = \triangle _{xt} - \triangle _{tx} \) of f and conclude that \(\triangle _t, \triangle _x\) do not commute in general.
(c)
Construct a genuinely path-dependent functional whose Lie bracket is zero.
3.1 Functional Stratonovich Formula
As shown in [8], the functional derivatives can be used to generalize Itô’s formula to the path space. In line with the path signature and the functional Taylor expansion in Sect. 5, we here present the analogue of the functional Itô formula in terms of Stratonovich integrals. We also adopt a pathwise framework as in [9, 14].
Let \(\Pi = (\Pi ^N)\) be a sequence of partitions of \([0,T]\) with vanishing mesh size, e.g., the regular partition \(\Pi ^{N} = \{T n /N : n=0,\ldots , N\}\). We say that \(X \in \Lambda _t\) is a \(\Pi \)-integrator if it is continuous and its quadratic variation process along \(\Pi \), namely
is well-defined, finite, and continuous. We write \(\Lambda ^{\Pi }_t\) for the set of \(\Pi \)-integrators in \(\Lambda _t\) and \(\Lambda ^{\Pi } := \bigcup _{t\in [0,T]} \Lambda ^{\Pi }_t\). Important examples of \(\Pi \)-integrators are sample paths of continuous semimartingales. Indeed, it is shown in [14] that for every sequence of partitions \(\Pi \), almost all trajectories of continuous semimartingales belong to \(\Lambda ^{\Pi }\). We also note that \(\Lambda ^{\Pi }\) contains all continuous functions of bounded variation, in particular smooth paths. Next, we define
where \(\mathcal {J}_t\) contains all piecewise constant paths in \(\Lambda _t\) taking finitely many values. The enlarged space \(\bar {\Lambda }^{\Pi }\) is needed to compute the functional space derivative (8) which entails a discontinuity at the final time. We now state a generalization of the Stratonovich formula to path functionals. For a proof, see [9, Theorem 3.2].
Theorem 3.3 (Functional Stratonovich Formula)
Let\(X\in \Lambda ^{\Pi }_t\)for some\(t \in [0,T]\). If\(\triangle _{t}f,\triangle _{x}f,\triangle _{xx}f\)exist and are\(\Lambda \)-continuous, then
The next example shows that (13) is consistent with the standard Stratonovich formula.
Example 3.4
Let \(X\in \Lambda ^{\Pi }_t\) and the path-independent functional \(f(X) = \phi (t,X_t)\), \(\phi \in \mathcal {C}^{1,2}([0,T]\times \mathbb {R})\). It is easily seen that \(\triangle _t f = \partial _t \phi \), \(\triangle _{x}f = \partial _x \phi \), \(\triangle _{xx}f= \partial _{xx} \phi \). We can thus invoke Theorem 3.3 to recover the classical Stratonovich formula, namely,
The above theorem allows, in passing, to construct pathwise Stratonovich integrals \(\int \varphi (X) \circ dX\) where \(\varphi \) is continuously differentiable (i.e., \(\varphi \) is \(\Lambda \)-continuous and \(\triangle _t\varphi , \triangle _x\varphi \) exist and are \(\Lambda \)-continuous). Indeed, assume that \(X_0 =0\) for simplicity and define \( \Phi (X) = \int _{0}^{X_t} \varphi (X^{-h}) d h \), \(X\in \bar {\Lambda }_t^{\Pi }.\) Then it follows from Definition 3.1 that \(\triangle _x \Phi =\varphi \). We can thus apply Theorem 3.3 to \(\Phi \) and obtain
Noting also that \(\triangle _t \Phi (X) = \int _{0}^{X_t} \triangle _t \varphi (X_t^{-h}) d h\), we can rearrange (14) to arrive at
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \int_0^t \varphi(X_{[0,s]}) \circ dX_s = \int_{0}^{X_t} \varphi(X^{-h}) d h - \int_0^t \int_{0}^{X_s} \triangle_t \varphi(X_{[0,s]}^{-h}) d h \ ds. \end{array} \end{aligned} $$
(15)
In particular, when \(\triangle _t \varphi = 0\), we simply have \( \int _0^t \varphi (X_s) \circ dX_s = \int _{X_0}^{X_t} \varphi (X_t^{(\varepsilon )}) \ d \varepsilon . \) The Stratonovich integral in (15) is therefore well-defined pathwise. Incidentally, this allows us to properly define signature functionals introduced in the next section.
It is counterintuitive that the existence of the Stratonovich integral in (15) requires some conditions on the functional derivatives of the integrand \(\varphi \) (namely that \(\triangle _t\varphi , \triangle _x\varphi \) exist and are \(\Lambda \)-continuous). Assuming that \(\varphi \) is merely \(\Lambda \)-continuous turns out not to be sufficient in general, as shown in the next example.
Example 3.5
Consider the functional \(\varphi (X) = \int _0^t \kappa (t-s)\circ dX_s\) with the fractional kernel \(\kappa (\delta ) = \delta ^{H-1/2}\), \(H> 0\). It is easily seen that \(\varphi \) is \(\Lambda \)-continuous, while \(\triangle _x\varphi (X) = \lim _{s\uparrow t} \kappa _{H}(t-s) = \infty \) when \(H<1/2\). On the other hand, it is shown in [3] that \(\int \varphi (X)\circ dX\) diverges when X is a typical Brownian path (which belongs to \(\Lambda ^{\Pi }\) almost surely). This confirms that the \(\Lambda \)-continuity of \(\varphi \) is typically not enough to define the pathwise Stratonovich integral in (15).
4 Signature Functionals
In this chapter, the signature is regarded as a collection of functionals as we now explain. Introduce the extended path \((t,X)\) and write \(X^0_t = t\), \(X^1_t = X_t\) for convenience. Recall also the set \(\mathcal {W}\) seen in chapter “A Primer on the Signature Method in Machine Learning” containing all the words \(I = (i_1,\ldots ,i_k)\), \(k\ge 0\), herein on the alphabet \(\{0,1\}\). Then the signature is given by the collection \(S = \{S^{I}:\bar {\Lambda }^{\Pi } \to \mathbb {R} : I\in \mathcal {W} \}\), where \(S^{\emptyset } \equiv 1\) and for \(|I| := \text{length}(I) \ge 1\),
The symbol \(\circ \) indicates that the integrals in (16) are in the Stratonovich sense. The signature \(S^I(X)\) is well-defined pathwise for all \(X \in \bar {\Lambda }^{\Pi }\), \(I\in \mathcal {W}\) as a consequence of Theorem 3.3; see [9, Proposition 3.5].
Remark 4.1
We stress that it is necessary to enlarge X, here one-dimensional, with the time itself to enrich the signature. In the alphabet \(\{1\}\), S boils down to the scaled monomials \(1,X_t-X_0,\frac {(X_t-X_0)^2}{2!}, \frac {(X_t-X_0)^3}{3!},\ldots \) so it would not carry any information about the path apart from the increment \(X_t-X_0\). This is in contrast with the multi-dimensional case as in chapter “A Primer on the Signature Method in Machine Learning” where a path is described as a curve parametrized by t, e.g., \(t\mapsto (X_{t}^1,X_t^2) \in \mathbb {R}^2\) as depicted in the right panel of Fig. 4, and the signature is typically rich enough in this formulation.
Fig. 4
Contrast between \((t,X)\) paths (left) and \((X^1,X^2) = (X^1_t,X^2_t)_{t\in [0,T]}\) (right)
Moreover, adding time makes the signature map \(X \mapsto S(X)\) injective. We may therefore wonder how to reconstruct the unique path (when it exists) associated to a given sequence of signature elements. We refer the interested reader to Appendix for further details.
It turns out that the signature is compatible with the functional derivatives in the following sense: for fixed \(I = (i_1,\ldots ,i_k)\), then
From the above recursions, we deduce a more general property in part (a) of the next Proposition. We also collect other properties of the signature, proved in [9].
Proposition 4.2
The following properties holds:
(a)
Let\(I = (i_1,\ldots ,i_k) \in \mathcal {W}\). Then\(\triangle _{J} S^I \ne 0\)if and only if\(J = (i_\ell ,\ldots , i_k)\), for some\(\ell \le k\). If so, then\(\triangle _{J} S^{I} = S^{(i_1,\ldots , i_{\ell -1})}\).
(b)
The signature functionals are linearly independent, i.e., for all finite subset\(\mathcal {W}' \subseteq \mathcal {W}\), and coefficients\((c_{I})_{I\in W'} \subseteq \mathbb {R}\), then
The collection of T-functionals\(\mathcal {A}^k = \{S^{I} :\bar {\Lambda }_T^{\Pi }\to \mathbb {R} \ : \ |I|\le k \}\)is linearly dependent.
(d)
For all\(k\in \mathbb {N}\), the family of T-functionals\(\mathcal {B}^k = \{ S^{I}:\bar {\Lambda }_T^{\Pi }\to \mathbb {R} \ : \ I = (i_1,\ldots ,i_k,1), k< k \}\)is linearly independent. Moreover,\(\mathrm {span}(\mathcal {B}^k)=\mathrm {span}(\mathcal {A}^k)\).
A proof of part (d) in the above Proposition can be found in [9, Appendix A.3]. The family \(\mathcal {B}^k\) yields a basis for T-functionals spanned by the signature functionals up to order k. Moreover, it is incremental in the sense that \(\mathcal {B}^k\) contains \(\mathcal {B}^{k-1}\) (and in turn \(\mathcal {B}^{k}\) for all \(k\le k-2\)). The nested structure is of practical convenience as increasing the truncation level k does not alter the elements already present in the basis.
Exercise 4.3
Prove parts (a), (b), (c) of Proposition 4.2. Hint: Use (a) to show (b).
5 Functional Taylor Expansion
We now iterate the functional Stratonovich formula established in Theorem 3.3 to derive the functional Taylor expansion (FTE). As we shall see, the signature functionals appear naturally. We first state the FTE in general form (Theorem 5.1) and later discuss some important consequences. Write \(\triangle _0 = \triangle _t\), \(\triangle _1 = \triangle _x\) (recall that \(0\sim t\), \(1 \sim x\)) and define the higher order functional derivatives as \(\triangle _{I} = \triangle _{I_1} \cdots \triangle _{I_k}\), e.g., \(\triangle _{110}f = \triangle _{x} (\triangle _x(\triangle _t f))\). Note also that \(\triangle _{\emptyset }f=f\). Next, let \(|I|_0\) (respectively \(|I|_1\)) denote the number of 0’s (resp. 1’s) in \(I \in \mathcal {W}\) and introduce for \(k,\ell \in \mathbb {N}\) and \(\Omega \subseteq \Lambda \),
A related result can be found in Litterer and Oberhauser [18, Theorem 14], yet there are two important differences. First, the expansion in [18] is established for Itô diffusions, while our expansion is made in the pathwise setting introduced by Föllmer [14]. Second, the partition of the words is different: to properly reflect the scale of diffusions, [18] use the weighted length \(\lVert I \rVert = 2|I|_0 + |I|_1\) instead of \(|I|\) to truncate the expansion. The literature on functional expansions will be further discussed in Sect. 8.
Interestingly, the functional Taylor expansion can be used to recover Chen’s identity seen in chapter “A Primer on the Signature Method in Machine Learning”.
Corollary 5.2 (Chen’s Identity)
Let\(I = (i_1,\ldots , i_{k})\). Then for all\(X \in \Lambda ^{\Pi }_s, \ Y \in \Lambda ^{\Pi }_t\),
where we write\((i_1,\ldots , i_{0}) = (i_{k+1},\ldots , i_k) =\varnothing .\)
Proof
Let us apply the functional Taylor expansion of order \(k+1\) to the signature functional \(f = S^{I}\) defined on \(\bar {\Lambda }^{\Pi }\). This gives the expression
provided that \(S^{I}\) satisfies the regularity condition in Theorem 5.1 which we now verify. Using Proposition 4.2 (a), then \(\triangle _{J}S^{I}(X)\) equals \(S^{(i_1,\ldots , i_{\ell })}\) if \(J = (i_{\ell +1},\ldots , i_k)\), \(\ell \le k\) (where we recall that \((i_1,\ldots , i_{0}) = (i_{k+1},\ldots , i_k) =\varnothing \)), and is zero otherwise. In particular, \(\triangle _{J}S^{I} = 0\) for all word J of length greater than k, so the remainder \(R_{k+1}(X,Y)\) vanishes. We gather that \(S^{I}\) is smooth and conclude from Theorem 5.1 that
Oftentimes, we simply expand a functional around the initial value of the path. Given \(X\in \bar {\Lambda }^{\Pi }_t\), this amounts to choosing \(X = X_0\), \(Y = X\) in Theorem 5.1, leading to the so-called Functional Maclaurin expansion.
Corollary 5.3 (Functional Maclaurin Expansion)
Let f be a functional in\(\mathbb {C}^{k,k+1}(\bar {\Lambda }^{\Pi })\). Then for all\(t\in [0,T]\), the following expansion holds,
Formula (21) suggests that the signature can be used to approximate sufficiently regular functionals. Indeed, applying a first order functional Maclaurin expansion (Corollary 5.3) to the remainder \(R_k\), namely \(R_k(X) \approx \sum _{|I| = k} \triangle _{I}f(X_0) S^I(X) \), yields the approximation,
In other words, \(f^{k}\) approaches f via a linear combination of signature functionals. Naturally, the approximation in (23) is valid only if the truncation error \(f(X) - f^k(X)\) is small. For explicit remainder estimates, see [9, Section 3.1.5] and [18, 19] in the context of diffusions.
In finance, \(f^{k}\) is referred to as signature payoff [1, 20] or simply sig-payoff [6, 7]. By applying the Stone–Weierstrass theorem, it can be shown that signature payoffs form a family of continuous functionals which is dense in compact subsets of \(\Lambda \) [1, 6, 7, 20]. While this guarantees the existence of a signature payoff arbitrarily close to a target functional, the former is not given explicitly. In contrast, the functional Taylor expansion provides a recipe to approximate functionals as hinted by (23) and further explained in Sect. 6.4.
Another avenue is to use a regression approach to compute the coefficients \(\lambda ^I\), as carried out in [1, 20] and illustrated in Sect. 6.2. In this case, the regression is performed using paths from the reference measure, or from a pool of models (e.g. Black-Scholes with varying volatility) to improve the robustness of the coefficients. On the other hand, the FTE approach is model-free as the Taylor expansion is done pathwise. In Sect. 6.4, we shall see that an embedding must be nevertheless introduced in (23) when f is a T-functional, yet the auxiliary model that defines the embedding can be much simpler than the reference model. Moreover, Taylor expansions are incremental in the sense that increasing the order does not change the previously computed coefficients. This is not the case for regressions, where adding features affects all coefficients.
Exercise 5.4
Let \(f(X) = \int _{0}^t X_u^2 du\), \(X\in \Lambda _t\).
(a)
Write down explicitly the FTE of f with \(k=1\) and arbitrary paths \(X ,Y\).
(b)
Compute the functional Taylor approximation \(f^k\) (given in (23)) with \(k=3\). What do you observe?
Note: If X represents the volatility process of some asset, say S, then \(f(X)\) is the total realized variance of S.
6 Applications
6.1 Risk Analysis: A New Greek
Let us see how the general formulation of the FTE (Theorem 5.1) can be used for local approximation of functionals in portfolio risk management. Concretely, fix \(X\in \Lambda _{t}\) and scenario \(Y \in \Lambda _{\delta t}\), and increment \(\delta y = Y_{\delta t}-Y_0\). Notice that \(\delta y = S^{(1)}(Y)\). Then, a second order functional Taylor expansion of f after X gives (writing \(\triangle _I f=\triangle _I f(X)\)),
$$\displaystyle \begin{aligned} \begin{array}{rcl} f(X * Y) & \approx&\displaystyle f + \triangle_t f \delta t + \triangle_x f \delta y + \triangle_{xx} f \ \frac{\delta y^2}{2} + \triangle_{tt}f \ \frac{\delta t^2}{2}{} \end{array} \end{aligned} $$
The terms on the right side of (24) are classical, while the cross terms in (25) are new. Indeed, the latter cannot be grouped together since the functional derivatives \(\triangle _t,\triangle _x\) do not commute in general; see again Exercise 3.2. Nevertheless, we can introduce the symmetric-antisymmetric decomposition,
with the mixed derivative \(\partial _{tx} := \frac {\triangle _{xt} + \triangle _{tx}}{2}\), and Lie bracket \(L = \triangle _{xt} - \triangle _{tx}\) seen in (9). As \( S^{(0,1)}(Y)+ S^{(1,0)}(Y) = \delta t \delta y\), we can thus rewrite (25) as
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \triangle_{tx}f \ S^{(0,1)}(Y) + \triangle_{xt}f \ S^{(1,0)}(Y) = \partial_{tx} f \ \delta t \delta y + L \! f \mathcal{A}(Y), \end{array} \end{aligned} $$
(26)
where \(\mathcal {A}(Y)\) is the Lévy area of the enlarged path \((t,Y)\) as seen in chapter “A Primer on the Signature Method in Machine Learning”, namely
We recall that it represents the signed area enclosed between the graph of \(Y-Y_0\) and the segment \(\{(u,Y_0 + u\delta y) : u\le \delta t\}\). In the present context, \(\mathcal {A}(Y)\) gauges the convexity of the path Y . Indeed, \(\mathcal {A}(Y) < 0\) if \(u\mapsto Y_u\) is convex and \(\mathcal {A}(Y) >0\) if \(u\mapsto Y_u\) is concave; see Fig. 6. Expression (26) may be visually explained as follows:
Fig. 6
Observed path \(X \in \Lambda _t\) and scenario \(Y \in \Lambda _{\delta t}\), \(\delta y = Y_{\delta t}-Y_0\) (several realizations). The Lévy Area \(\mathcal {A}(Y)\) corresponds to the signed area between the graph of \(Y-Y_0\) and the diagonal \(\{(u,Y_0 + u\delta y) : u\le \delta t\}\)
Moving to applications in risk management, let \(f(X)\) represent the time t value of a portfolio contingent upon \(X \in \Lambda _t\), we can break down the profit and loss (P&L) between t and \(t+\delta t\) in terms of the path-dependent sensitivities \(\Theta = \triangle _tf(X)\), \(\triangle = \triangle _xf(X)\), \(\Gamma = \triangle _{xx}f(X)\) and a new Greek stemming from the Lie Bracket,
which we call Libra. Indeed, we obtain from (24) and (26) the P&L decomposition,
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \text{P\& L} = f(X*Y) - f(X)\approx \Theta \delta t + \Delta \delta y + \Gamma\frac{\delta y ^2}{2} + \mathcal{L} \mathcal{A}(Y). \end{array} \end{aligned} $$
(29)
While \(\Theta , \Delta , \Gamma \) are paired with the increments \(\delta t, \delta y\), the last term in (29) is genuinely path-dependent as it entails the Lévy area \(\mathcal {A}(Y)\). If the future path is a bridge, that is \(\delta y =0\), then (29) simply reads \( \text{P\&L} = \Theta \delta t + \mathcal {L} \mathcal {A}(Y)\). Hence the convexity of the future path Y also contributes to the profit-and-loss on top of the usual time decay.
For a given observed path X and price functional f, the Greeks can be precomputed offline and used thereafter to approximate the P&L efficiently. Schematically,
The functional Greeks in the offline phase can be estimated using finite difference (forward in t, centered in x). In particular, the Libra \(\mathcal {L}\) is approximated by the ratio
for small \(\delta t', h>0\). Note that the online phase in (30) is nearly instantaneous as it only requires to compute two simple quantities for each scenario Y : the increment \(\delta y\) and the Lévy area \(\mathcal {A}(Y)\). The approximated P&L can then be translated into risk measures, such as Value at Risk (VaR) and Expected Shortfall (ES) as illustrated in the next example.
Example 6.1
Consider a (simplified) portfolio consisting of a long position in one at-the-money Asian call options as in Sect. 6.4.2, that is
Introduce the value functional \(f(X) = \mathbb {E}^{\mathbb {Q}}[g(Z)|X]\), \(X\in \Lambda _t\), where \(\mathbb {Q}\) is the measure associated to the Black-Scholes model with \(\sigma =20\%\) and zero interest rates. Suppose also that \(X_0=100\), \(T=0.5\), \(t=T/3\) (4 months to maturity) and \(\delta t = \frac {5}{252}\) (one-week horizon). We then approximate the one-week P&L of the portfolio as in (29) assuming daily observations of the underlying.3 More precisely, we simulate 1000 scenarios \(Y = (Y_{t_0},\ldots ,Y_{t_5})\) and compute the Lévy area \(\mathcal {A}(Y)\) by discretizing the integral in (27).
The portfolio value and Greeks are reported in Table 1 for a fixed path \(X \in \Lambda _{t}\) depicted in Fig. 7. Since the average value of the transformed path \((X^{h})^{*\delta t }\) is larger than the one of \((X^{*\delta t})^{h }\) when h is positive and g is nondecreasing in the average functional A, it is then expect from (31) that the Libra \(\mathcal {L}\) is positive. This is confirmed in Table 1, and we conclude without much surprise that the convexity of Y , captured by \(\mathcal {A}(Y)\), has a negative impact on the value of the option portfolio.
Fig. 7
Observed path X, intrinsic time average \(A_0(X) = A(X) + X_t(1-t/T)\), and scenarios Y where \(T = 6\) months, \(t = \) 2 months, \(\delta t = 1\) week
Table 2 shows the Value at Risk (VaR) as well as Expected Shortfall (ES) at the \(95\%\) and \(99\%\) level, respectively. The small approximation error is offset by a significant speedup. Indeed, the run time for the exact calculation of the P&L (to retrieve VaR and ES) is 52.50 seconds, while the FTE approximation in (30) (offline and online phase) takes 0.49 seconds.
Table 2
Comparison of value at risk and expect shortfall in Example 6.1 using Monte Carlo simulation and the FTE
Value at risk
Expect shortfall
Level
\(95\%\)
\(99\%\)
\(95\%\)
\(99\%\)
Exact
\(-0.237\)
\(-0.255\)
\(-0.249\)
\(-0.261\)
FTE
\(-0.248\)
\(-0.252\)
\(-0.251\)
\(-0.254\)
6.2 Vanilla Options Are Path-Dependent
In this section, we make a somewhat counterintuitive statement: the Libra \(\mathcal {L}\) of vanilla options may be nonzero. Put differently, vanilla options are path-dependent!
We start with a simple example. Let \(T>0\) and suppose that we are given a call option with strike \(K>0\) and maturity \(\tau \in (0,T)\). If \(\mathbb {Q}\) is a risk-neutral measure, then the value of the option is
where in the case \(t \ge \tau \), we assume that the proceeds from the payoff is reinvested until time t with zero interest rates. Suppose that the price process Z is Markovian under \(\mathbb {Q}\), so there exists a function \(c:[0,T]\times \mathbb {R}\to \mathbb {R}\) such that \(f(X) = c(t,X_{t\wedge \tau })\), with \(c(t,x) = (x-K)^+\) if \(t\ge \tau \). We also assume that the Libra can be expressed as a double limit as in (11), namely \( Lf(X) = \lim _{h,\delta t \to 0} \frac {1}{h\delta t}[f((X^{h})^{*\delta t}) - f((X^{*\delta t})^h)].\) When \(t\le \tau \), the Markov representation of f implies that for all \(\delta t < \tau - t\),
If \(t>\tau \), then \(f((X^{h})^{*\delta t}) = f((X^{*\delta t})^h) = (X_{\tau }-K)^+\) as neither a spatial bump nor a flat extension after the expiration \(\tau \) changes the payoff of the option. Hence \(Lf(X) = 0\) for all \(X\in \Lambda _t\) with \(t\ne \tau \). Finally when \(t=\tau \) and \(X_{\tau } > K\), observe that for \(h\neq 0\) small enough,
We conclude that \(\mathcal {L} = Lf(X)\ne 0 \) (in fact, \(\mathcal {L} = \infty \)) when \(X\in \Lambda _{\tau }\). Indeed, the value of the payoff is unchanged if we first time extend the path since the maturity has then passed. On the other hand, starting with a spatial bump does affect the payoff when the spot price \(X_{\tau }\) is in-the-money; See also Fig. 8.
Fig. 8
Vanilla call option expiring at \(\tau <T\). The Libra associated to the in-the-money path (\(X^1\)), as shown on the right panel (zoom of left panel)
As announced at the beginning of this section, the Libra of vanilla options is indeed nonzero in general. We also stress that even when \(\tau = T\), a vanilla call option can have nonzero Libra if the underlying process is non-Markovian under \(\mathbb {Q}\). This situation notably arises in path-dependent or rough volatility models.
We end this section with a more general example.
Example 6.2
Consider a portfolio of vanilla call options of the form
for some weights \((\pi _{K ,\tau })\). Given a risk-neutral measure \(\mathbb {Q}\), then the value of the portfolio having observed \(X\in \Lambda _t\) is
Again, the price process Z is Markovian under \(\mathbb {Q}\). Hence for all \((K,\tau ) \in \mathbb {R}_+\times \ [0,T]\), there exists \(c_{K,\tau }:[0,T]\times \mathbb {R}\to \mathbb {R}\) such that \(\mathbb {E}^{\mathbb {Q}}[(Z_{\tau }-K)^+ \ | \ X] = c_{K,\tau }(t,X_{t})\) with \(c_{K,\tau }(t,x) = (x-K)^+\) when \(t\ge \tau \). Assuming that the Libra can be expressed as a double limit as in (11), we then observe that
As can be seen, \(\mathcal {L}\) is nonzero and corresponds to the cumulative portfolio weight of in-the-money options (i.e., the strike is below the spot \(X_t\)). Furthermore, rewriting the portfolio as
where \(\phi (\tau ,x) = \int _{\mathbb {R}_+} \pi _{K,\tau } (x - K)^+ dK\) is the aggregate vanilla payoff at time \(\tau \), then the Libra becomes \(\mathcal {L} = \partial _x \phi (t,X_t)\) and represents the sensitivity of the t-section \(\phi (t,\cdot )\) with respect to the spot level x.
6.3 Static Hedging
The FTE possesses also direct applications to static hedging. Consider a contingent claim \(g:\Lambda _T\to \mathbb {R}\) with price functional \(f \in \mathbb {C}^{k,k+1}(\Lambda )\), that is \(f(X) = \mathbb {E}^{\mathbb {Q}}[g(Z)|X]\) for a risk-neutral measure \(\mathbb {Q}\). Suppose we are given tradable instruments \(\varphi _1,\ldots , \varphi _M\) and we associate to any weight vector \(\pi = (\pi _1,\ldots ,\pi _M) \in \mathbb {R}^M\) the portfolio
The weights \((\pi _m)\) are thus obtained by solving a linear system, where we assume that a solution exists. Using the FTE, we are able to quantify the hedging error \(g-\varphi \). Indeed, applying Corollary 5.3 to the difference \(f-\varphi \) gives
with remainder given in (22). If \(f, \varphi \) belong to the space spanned by the signature functionals up to order k, we would obtain a perfect hedge, i.e. \(R_{k} \equiv 0\). Since \(f=g\) at maturity, we obtain in particular that
When \(R_{k} \ne 0\), the hedging error \(|g-\varphi |\) can be estimated as we now explain. In line with financial markets, we assume that X is the piecewise interpolation of tick data and is in particular Lipschitz continuous. If furthermore, there exists a constant \(C< \infty \) such that \( \sup _{|I| = k}|\triangle _{I}(f-\varphi )(X)| \le C\) for all Lipschitz path X in \(\Lambda _t\), then Proposition 3.16 in [9] gives \(R_{k}(X) = \frac {\mathcal {O}(t^k)}{k!}\). Choosing \(t=T\), the hedging error for the claim g is therefore bounded by
We emphasize that the above upper bound holds pathwise, matching the needs of exotic option traders to be protected against future scenarios.
6.4 Pricing of Exotic Options
Let \(\mathbb {Q}\) be a risk-neutral measure and \(g:\Lambda _T \to \mathbb {R}\) a path-dependent payoff such that \(\mathbb {E}^{\mathbb {Q}}[|g(Z)|]\) is finite. Regarding the signature functionals as hedging instruments, we can derive from the FTE a nearly replicating portfolio using \((S^I)\). The initial price is then approximated by taking expectation with respect to \(\mathbb {Q}\). However, the FTE can only be applied to functionals, not T-functionals such as g. A remedy is to employ embeddings (Sect. 2) as we now explain. Fix a functional \(f:\Lambda \to \mathbb {R}\) such that \(f=g\) on \(\Lambda _T\), e.g. using Black-Scholes or Bachelier embeddings. Provided that \(f \in \mathbb {C}^{k,k+1}(\Lambda )\) for some \(k\ge 0\), then Corollary 5.3 gives as in (23),
Importantly, the “model” \(\mathbb {Q}\)only materializes through the expected signature, which can be precomputed once and for all. This becomes greatly advantageous over a straightforward Monte Carlo approach when the simulation of paths under \(\mathbb {Q}\) is costly. We next illustrate the procedure with examples.
6.4.1 Vanilla Call Option
Let \(X_0=100\) and consider a three-month at-the-money vanilla call option, that is \(g(X) = (X_T-X_0)^{+}\), \(T=\frac {1}{4}\), in the Black-Scholes model with zero interest rates and volatility \(\sigma _{\text{ BS}} = 20\%\). As the Black-Scholes price \(p^{\text{ BS}} := \mathbb {E}^{\mathbb {Q}}[(Z_T - X_0)^+]\) is readily available, this example permits to verify the pricing procedure in (36) and (37). The reader is nevertheless encouraged to explore more complex risk-neutral dynamics of the underlying. Given \(X \in \Lambda _t\), introduce the Bachelier embedding \(f_{\sigma }(X) = \mathbb {E}^{\mathbb {Q}_{\sigma }}[(Z_T - X_0)^{+} \ | \ X] = p^{\text{ B}}(t,X_t,\sigma )\), where \(p^{\text{ B}}(t,x,\sigma )\) is the price given \(X_t = x\) of a call with maturity T in the Bachelier model with volatility4\(\sigma \) and zero interest rates. For completeness, recall that
where \(\phi ,\Phi \) is respectively the PDF and CDF of \(\mathcal {N}(0,1)\). Writing \(p_{ \sigma }(t,x) = p^{\text{ B}}(t,x,\sigma )\), we obtain \(\triangle _t f_{\sigma } = \partial _t p_{ \sigma }, \ \triangle _x f_{\sigma } = \partial _x p_{ \sigma }\) and the same holds for higher derivatives.
Owing to the functional Feynman-Kac formula [8], the price functional \(f_{\sigma }\) in (6) solves the path-dependent partial differential equation (PPDE)5
Writing, with a slight abuse of notations, \(f_{\sigma } = f_{\sigma }(X_0) \), \(p_{ \sigma } = p_{ \sigma }(0,X_0)\), and \(S^I = S^I(X)\) for fixed \(X\in \Lambda _T\), then a Taylor expansion of order \(k = 2\) reads
where we used that \(\triangle _{tx}f_{\sigma } = \triangle _{xt}f_{\sigma } =\partial _{tx}p_{ \sigma }\) and \((S^{(1,0)}+ S^{(0,1)})(X) = T (X_T-X_0)\), in the last equality. Omitting the higher order Greeks \(\partial _{tt}p_{ \sigma }\), \(\partial _{tx}p_{ \sigma } \), this gives
The call payoff is therefore replicated using a parabola in \(X_T\) with coefficients depending on the Bachelier volatility \(\sigma \) as seen in the left panel of Fig. 9.6 As Taylor approximations are inherently local, the above replication deteriorates as \(X_T\) moves away from \(X_0\) or for longer maturity T. Note that the Greeks \(\partial _t p_{ \sigma }, \partial _x p_{ \sigma }, \ldots \) are available in closed form in the Bachelier model, hence there is no numerical error induced by the computation of the FTE coefficients themselves in this simple case. For more general payoff, one can use finite difference.
Fig. 9
Left: Approximation of \(g(X) = (X_T-X_0)^+\) using the FTE with Bachelier embedding for varying volatility \(\sigma \) compared to the regression approach. The risk-neutral density of \(X_T\) is also displayed (bottom). Right: Price (dashed line) as function of \(\sigma \) and truncation order k. The solid line is the exact Black-Scholes price, surrounded by a \(5\%\) error band
In Fig. 9, we also report the approximation obtained from a regression approach similar to [1, 6, 20]. That is, we simulate trajectories \((X^j)_{j=1}^J\) with \(J=1000\) from the reference model \(\mathbb {Q}\) and perform the regression
As can be observed in Fig. 9, the fitted parabola is close to the Taylor expansion with Bachelier volatility \(\sigma = 20\).
We next take expectation with respect to the martingale measure \(\mathbb {Q}\) and compare the estimated prices \( p^{k}_{ \sigma }\) given in (37) with \(k=0,1,2\), respectively. In view of (41), we have \(p^{0}_{ \sigma } = p_{ \sigma }\) which is precisely the call price in the Bachelier model. It is therefore intuitive to align the volatility of the Bachelier embedding with the Black-Scholes volatility, even though the higher order terms would eventually compensate. Next, taking \(k=1\) gives
where \(\mathbb {E}^{\mathbb {Q}}[Z_T-X_0] =0\) follows from the martingality of Z under the risk-neutral measure \(\mathbb {Q}\). Moving finally to the second order, we obtain from (42) that
with the risk-neutral variance \(\mathbb {V}^{\mathbb {Q}}[Z_T] = \mathbb {E}^{\mathbb {Q}}[(Z_T-X_0)^2]\). We note that the incremental nature of Taylor expansions is computationally convenient as terms of lower orders can be directly added. The effect of \(\sigma \) on the price is illustrated on the right panel of Fig. 9. We observe that it is indeed beneficial to choose \(\sigma \) close to the scaled Black-Scholes volatility \(X_0 \sigma _{\text{ BS}} = 20\).
6.4.2 Asian Call Option
For fixed maturity \(T>0\), consider an at-the-money Asian call option with payoff
with the Bachelier price \( p^{\text{ B}}\) given in (38). As in [8], the functional \(A_0\) is introduced to avoid a numerically undesirable drift term in the associated pricing PDE, otherwise present if \(f_{\sigma }(X)\) is expressed in terms of \(\int _0^t X_udu\). Next, we compute the FTE coefficients \(\lambda ^{I}_{\sigma } = \triangle _I f_{\sigma }(X_0) \). It is easily seen (\(\to \) exercise) that \(\triangle _t A_0(X) = 0\), while \(\triangle _x A_0(X) = 1-t/T\). Writing \(p^{\text{ B}} = p^{\text{ B}}(0,X_0,\Sigma (0,\sigma ))\) (note that \(A_0(X_0) = X_0\)), the first order FTE coefficients read
Since \(A(X)\) is the average value of the asset over the horizon \([0,T]\), then \(A(X) - X_0\) captures the trend of X compared to the terminal value \(X_0\). As in Sect. 6.2, the partial derivatives \(\partial _tp^{\text{ B}}, \partial _x p^{\text{ B}}, \ldots \) can be computed explicitly from (48). Again, \(\mathbb {Q}\) is the risk-neutral distribution in the Black-Scholes model with volatility \( \sigma _{\text{ BS}} = 20\%\) and \(T=\frac {1}{4}\) (three-month maturity). Choosing \(\sigma = X_0\sigma _{\text{ BS}}\) gives a value of \(2.321\) using the FTE, which is close to the exact Black-Scholes price (equal to \(2.303\)).
Exercise 6.3
(a)
Derive formula (48). Hint: Recall that \(\int _0^t W_u du \sim \mathcal {N}(0,t^3/3)\) if W is Brownian motion.
(b)
Verify that \(da_t = \sigma (1-t/T)dW_t\), where \(a_t = A_0(Z_{[0,t]})\) and \(dZ_t = \sigma dW_t\).
(c)
Derive using (b) the PDE solved by \(p_{ \sigma }\) and compare it with the path-dependent PDE (39).
Exercise 6.4
Consider a floating strike Asian put option where \(g(X) = (X_T - A(X))^+ \) and \(A:\Lambda _T\to \mathbb {R}\) given in (46).
1.
Show that \(f_{\sigma }(X) = p_{\sigma }(t,X_t,A_0(X))\) for some function \(p_{\sigma }\) to be determined.
Hint: Compute the joint distribution of \((W_s, \int _0^s W_u du)\).
2.
Compute the functional derivatives \(\triangle _{I}f_{\sigma }\), where \(I\in \{0,1,00,01,10,11\}\) and write down the Taylor-Maclaurin approximation \(f_{\sigma }^k\) for \(k=2\).
3.
Write a code that returns the replication price \(p_{ \sigma }^2 = \mathbb {E}^{\mathbb {Q}}[f_{\sigma }^2(Y)]\) as function of \(\sigma \), where \(\mathbb {Q}\) is an arbitrary model.
6.4.3 Discussion: Barrier and Lookback Options
Barrier and Lookback claims are contingent upon the range of the underlying. For instance, a down-and-out digital option pays a notional amount (say, \(\$1\)) if the stock price has stayed above a given barrier level throughout the life of the contract and zero otherwise. Another example is a floating strike lookback put option that delivers at expiration the realized drawdown \(g(X) = \max _{t\le T} X_t - X_T\) seen in Exercise 2.4.
Similar to the previous example, we may apply the FTE to an embedding \(f_\sigma \) of g and obtain an approximation of the initial price. However, \(f_{\sigma }\) is typically not regular enough when g depends on the maximum (or minimum) of the path. Indeed, taking the drawdown example and \(\mathbb {Q}_{\sigma }\) to be the measure associated to the Black-Scholes model with volatility \(\sigma \), it can be shown that \(f_{\sigma }(X) = p_{ \sigma }(t,X_t,\max _{0 \le s \le t}X_s)\), \(X\in \Lambda _t\), where \(p_{ \sigma }:[0,T]\times \mathbb {R}^2 \to \mathbb {R}\) is a smooth function. However, the functional \(f_{\sigma }(X)\) is not smooth when the spot value equals the maximum. One remedy is to regularize the maximum as in [8, Example 2] and perform an expansion of the regularized functional. We note that the application of the procedure (36) and (37) to range-based options is still subject to ongoing research.
6.5 A Glimpse into Cubature Methods
Cubature is the analogue of Gaussian quadrature for multiple integrals. In short, a cubature rule is described by a collection of nodes \(x_1,\ldots ,x_M \in \mathbb {R}^d\) and weights \(\lambda _1,\ldots ,\lambda _M\) such that
for a collection of polynomials \((p_I)\). Put differently, the polynomials have to be exactly integrated under the discrete measure \(\sum _{m=1}^M \lambda _m \delta _{x_m}\). Similarly, cubature on Wiener space [17, 19] requires that the signature elements, up to a prescribed order, are “priced” perfectly under a discrete measure. The use of cubature methods in option pricing is well-established for non-exotic payoffs [5, 10], but little is known in the path-dependent case (see also the discussion in [2]). In this section, we briefly address the latter by connecting cubature on the path space \(\Lambda \) with the FTE.
Let \(\mathbb {Q}\) denote the Wiener measure such that the canonical process Z on \(\Lambda _T\) is Brownian motion. Akin to (49), we say that \((X,\lambda ) \in (\Lambda _T \times \mathbb {R})^M\) is a cubature rule of order k if
Setting \(\hat {\mathbb {Q}} = \sum _{m=1}^M \lambda _m \delta _{X^m}\) where \(\delta _{X}\) is the Dirac measure at some path X, we can then estimate the expectation of some \(\mathbb {Q}\)-integrable T-functional \(G:\Lambda _T\to \mathbb {R}\) as
In option pricing, the functional G would correspond to the composition \(G = g\circ \Phi \), where \(g:\Lambda _T\to \mathbb {R}\) is the payoff of an exotic option and \(\Phi :\Lambda _T \to \Lambda _T\) is the so-called Itô map linking Brownian motion to the underlying price process. For instance, in the Black-Scholes model with zero interest rate, volatility \(\sigma \) and initial asset value equal to one, then \(\Phi (X) = e^{\sigma X_T - \frac {1}{2}\sigma ^2T}. \) The price of the claim would then be approximated by the linear combination in (51).
In view of (50), it is expected that the FTE plays a central role in cubature methods. Indeed, if \(F\in \mathbb {C}^{k,k+1}(\Lambda )\) is an embedding of G, then \(F^k(X) = \sum _{|I|\, \le \, k} \triangle _{I}F (X_0) S^{I}(X)\) seen in (23) satisfies \(\mathbb {E}^{\mathbb {Q}}[f^k(Z)] = \mathbb {E}^{\hat {\mathbb {Q}}}[f^k(Z)]\) by linearity and (50). This leads to the following upper bound,
The above pricing error is therefore controlled as soon as the remainder \(R^k\) is small in magnitude. See [9, Section 3.1.5] for precise remainder estimates and [5, 19] for a comprehensive treatment of cubature methods.
7 Extension to the Multi-Dimensional Setting
The functional Taylor has been introduced for paths with one space dimension. However, the multi-dimensional setting is by no means irrelevant. In short, the path space becomes
and functionals are maps from \(\Lambda ^d\) to \(\mathbb {R}\). The signature is defined as in (16), with words in the alphabet \(\{0,1,...,d\}\). This general setting allows to tackle multi-asset exotic options, where the payoff is a T-functional \(g:\Lambda _T^d \to \mathbb {R}\). Examples include Asian basket options, where g depends on a weighted average of the form \(\frac {1}{T} \int _0^T\sum _{i=1}^d w_i X_t^i dt\).
The functional space derivatives \(\triangle _{x_1},...,\triangle _{x_d}\) again do not commute with \(\triangle _t\), while they actually commute with each other. Consequently, some terms in the functional Taylor expansion can be grouped together (e.g. \(\triangle _{x_1x_2} \ f S^{(1,2)}\) and \(\triangle _{x_2x_1}f \ S^{(2,1)}\)), leading to simplified approximations of multi-asset exotic payoffs. Towards a fully general expansion, the functional space derivatives need to be modified so as to satisfy \( \triangle _{x_ix_j} \ f \ne \triangle _{x_jx_i} \ f \) when \(i\ne j\). This goes, nevertheless, beyond the scope of this chapter.
8 Bibliographical Notes
More details on the functional Itô calculus can be found in the original paper [8] and [16] focusing on the calculation of Greeks. In particular, the authors express the functional derivative of options in terms of Malliavin weights, giving an alternative approach to finite difference seen in Sect. 6.1. For Stone-Weierstrass type results in the path space, see [18] for continuous paths of bounded variation and in [6, 7] for paths emanating from general semimartingales. Applications to finance, including regression approaches, can be found in [1, 6, 20].
Related generalizations of Taylor’s theorem were introduced by Fliess [11‐13] for T-functionals and continuously differentiable paths and more recently by Litterer and Oberhauser [18], called Chen-Fliess approximation. The latter formulation also uses the functional derivatives in [8] and assumes that the paths under consideration solve a stochastic differential equation with smooth coefficients. The pathwise Itô calculus was introduced by Föllmer in his seminal work [14]. The pathwise Stratonovich formula in Theorem 3.3 is similar to the approach in [4] where a pathwise functional Itô formula is proved.
Acknowledgements
Research partially supported by the Bloomberg Quantitative Finance Ph.D. Fellowship (second author).
We would like to thank Harald Oberhauser, Josef Teichmann, Christa Cuchiero, Peter Friz, as well as our colleagues Bryan Liang and Guixin Liu at Bloomberg for fruitful discussions.
Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.
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An important property of the signature is that it uniquely characterizes a path, up to the so-called tree-like equivalence [15]. Here, the enlargement \((t,X)\) and monotonicity of \(t\mapsto t\) rules out the existence of tree-like paths, which are specific loops [15]. Hence, there is exactly one path associated to a given collection of signatures \(\{S^I(X) : I\in \mathcal {W}\}\) and we may wonder how to rebuild it. In fact, a single “diagonal” in the signature tree is enough, as we now outline. See [23, Section 2.3] for additional details.
Set \(T=1, \ X_0=0\) for simplicity and focus on paths on the full horizon \([0,1]\). Let \(\mathcal {H} = L^2([0,1])\) be the space of square integrable functions on \([0,1]\) with inner product \((h_1,h_2) = \int _0^1 h_1(t)h_2(t)dt\). If \(X,Y\in \Lambda _1 \cap \mathcal {H}\), we also write \((X,Y) = \int _0^1 X_tY_tdt\). We start with a simple but crucial observation.
Lemma 9.1
Consider the Legendre words\(I_{k} :=(1,0,\ldots ,0)\in \{0,1\}^{k+2}\), \(k \ge 0\). Then for every\(X \in \bar {\Lambda }^{\Pi }_1 \cap \mathcal {H}\),
This is a straightforward adaptation of [23, Lemma 2.7]. □
Since the time-reversed monomials \(r_k\) in (52) form a total basis of \(\mathcal {H}\), we conclude from the above lemma states that the sequence \((S^{I_k}(X))_{k\ge 0}\) characterizes the path X in its entirety. However, nothing is said (yet) about how the path is reconstructed. One possibility, which we follow, is to carry out a Gram-Schmidt procedure that linearly transforms the time-reversed monomials into an orthonormal basis (ONB) and use Hilbert projections.
Let \((q_k)\) be the (normalized) Legendre polynomials on \([0,1]\) given by (see [22]),
Note that (54) gives an explicit recipe to reconstruct the original path. What remains is to express the \(L^2\) products \((X,q_k)\) in terms of \((S^{I_k}(X))\). This is immediate once observing that
Thereafter, the \(L^2\) products \((X,q_k)\) can be finally retrieved from \((X,m_{\ell })_{{\ell }\le k }\) by linearity. The algorithm is summarized below7 and illustrated in Fig. 10 when \(K\in \{5,10\}\). This procedure can be easily generalized to the multidimensional case. Indeed, each component \(i=1,\ldots ,d\) can be reconstructed separately from the words \((i, 0,\ldots ,0) \in \{0,1\}^{k+2}\), \(k\ge 0\).
Fig. 10
Path reconstruction. Brownian path X and projection \(X^K\), \(K\in \{5,10\}\)
We recall that the Bachelier volatility is expressed in dollar amount. That is, as \(X_0 = 100\), a \(20\%\) Black-Scholes volatility would roughly correspond to a Bachelier volatility equal to \(\sigma = 20\).
To be more precise (39) is satisfied in the topological support of the measure \(\mathbb {Q}_{\sigma }\), which consists of all continuous paths starting at zero in light of the Stroock-Varadhan support theorem [21].
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when the limit exists. Moreover, the functional space derivative of \(f:\Lambda \to \mathbb {R}\) is 
