On certain representations of pricing functionals

Let S, \(\beta :\Omega \times [0,T] \rightarrow {\mathbb {R}}_+\) denote the price processes of a risky asset and of a numéraire (that we shall assume to be the money-market account, for simplicity), respectively, in an arbitrage-free market, modeled on a filtered probability space \((\Omega ,{\mathscr {F}},({\mathscr {F}}_t)_{t\in [0,T]},{\mathbb {P}})\), where \(T>0\) is a fixed time horizon and \({\mathscr {F}}_0\) is the trivial \(\sigma \)-algebra. Assuming that pricing takes place with respect to a risk-neutral probability measure \({\mathbb {Q}}\), the price at time zero of a European option with maturity T and payoff profile \(g:{\mathbb {R}}_+ \rightarrow {\mathbb {R}}\) on the asset with price process S is given byWe shall call the map \(g \mapsto \pi (g)\), defined on the set of all measurable functions g such that the right-hand side is finite, the pricing functional.

A rather general and natural question, of clear relevance also for practical purposes, is the following: suppose that the action of \(\pi \) is known on a set of functions G, i.e. that \(\pi (g)\) is known for every \(g \in G\). Is it possible to enlarge the set of functions G where \(\pi \) is determined, i.e. to compute \(\pi (f)\) for some functions f that do not belong to G? We are going to discuss some questions of this type (although not in this generality), through the representation of the pricing functional as a Lebesgue–Stieltjes measure, that iswhere F is the right-continuous version of the distribution function of \(S_T\) with respect to the measure \((d{\mathbb {Q}}/d{\mathbb {P}})\beta _T^{-1} \cdot {\mathbb {P}}\), i.e. the measure with density with respect to \({\mathbb {P}}\) equal to the stochastic discount factor.

If a collection G of payoff profiles g is such that the prices \(\pi (g)\) of the corresponding options are known, the set \(M:=(g,\pi (g))_{g \in G}\) will be called a measurement set, and a measurement set that determines F will be called a representation. That is to say, if knowing M allows one to reconstruct F, then knowing M is equivalent to knowing the pricing functional itself, which is why we say that it is a representation (of dF, or of \(\pi \)). An interesting and important example of a representation is given by prices of put options: if G is composed of all functions \(g_k:x \mapsto (k-x)^+\), \(k \in {\mathbb {R}}_+\), then M as defined above is a representation. More precisely, it is shown in Sect. 4 (see also Sect. 6 for an alternative derivation) that if \(P(k)=\pi (g_k)\) denotes the price at time zero of the put option with maturity T and strike \(k \geqslant 0\), then \(D^+P(k)=F(k)\) for every \(k \in {\mathbb {R}}_+\), where \(D^+P(k)\) denotes the right derivative of P at k. The problem of reconstructing the law of the underlying from option prices was probably considered first in Breeden and Litzenberger (1978), where the authors showed that, denoting the price of a call option with maturity T and strike k by C(k), the second derivative of C is the density of \(S_T\) with respect to the measure \((d{\mathbb {Q}}/d{\mathbb {P}})\beta _T^{-1} \cdot {\mathbb {P}}\), i.e. the first derivative of F (the function C is implicitly assumed to be of class \(C^2\) in Breeden and Litzenberger 1978). This result, known as the Breeden–Litzenberger formula, has found many applications, e.g. in static hedging, nonparametric density estimation, and local volatility models (see, e.g., Ait-Sahalia and Lo 1998; Bossu et al. 2021; Itkin 2020; Marinelli and d’Addona 2017 and references therein, as well as Talponen and Viitasaari 2014, where an interesting extension to the multidimensional setting is presented and further references regarding static hedging are given). In general, F is not of class \(C^1\), hence C is not of class \(C^2\), but, if they are, then the Breeden–Litzenberger formula follows immediately from \(D^+P=F\) and the put-call parity relation. Note, however, that for pricing purposes it suffices to determine F rather than its derivative, and the relation \(D^+P=F\) is obtained here without any a priori assumptions on F.

The reconstruction of F from a set of option prices is interesting in its own right, but sometimes less information is enough for the problem at hand (roughly speaking, this is just the idea behind static hedging). Using the above terminology, if one needs a measurement set M, it may be possible to determine a measurement set \(M'\) that contains M, without necessarily recovering F first. The simplest example is the pricing of options with continuous piecewise linear payoff profile (such as straddles, strips, and strangles—see, e.g., Hull 2015, §12.4). Another one is the pricing of options with payoff function equal to the difference of convex functions in terms of call options. Even though, in the latter case, the measurement set of all call options is already a representation, there is an alternative pricing formula that avoids the differentiation of C, which might be preferable for numerical purposes. Such pricing formula for options with convex payoff profiles is not new, but we give nonetheless several proofs: a very concise one, a longer one that highlights the role of convexity, and a third one that is extremely simple if sufficient regularity is present. We also show that similar ideas can be used to “localize” the pricing formula, i.e. to price options with payoff profiles that are piecewise the difference of convex functions.

The main content is organized as follows: we collect in Sect. 2 some auxiliary facts from measure theory, convexity, and the theory of distribution. Definitions, motivations, basic properties, and examples pertaining to pricing functionals, measurement sets, and representations are given in Sect. 3. Qualitative properties of the functions P and C, as defined above, are discussed in Sect. 4, without any assumption on F. Moreover, we show that F is the right derivative of P by two methods, that is, using the integration-by-parts formula for càdlàg functions of finite variation and by a denseness argument, respectively. In Sect. 5 we revisit the fact that prices of options with convex profile are determined by prices of call options for all positive strikes. This is proved in two ways: by an integration argument, that uses essentially only the Fubini theorem, and via the above-mentioned integration-by-parts formula. The results of the previous two sections are derived by yet another approach in Sects. 4–5, that is, using the theory of distributions. An interesting aspect of this method is that it provides a particularly handy way to make computations, also in cases that do not directly follow from the setups of the previous two sections. We conclude in Sect. 7 where, motivated by empirical issues in nonparametric pricing of European options treated in Marinelli and d’Addona (2017), we consider a kind of representation where a sequence of measures converging towards dF intervenes. In particular, assume that the distribution of logarithmic returns admits a density f and that there exists a sequence of functions \((f_n)\) converging to f in \(L^2({\mathbb {R}})\). Considering \(f_n\) as an approximation of f and computing approximating put prices for all strikes accordingly, we show that the knowledge of such approximating prices suffices to uniquely determine the true put option prices for all strikes.

We shall use some elementary facts from measure theory and convexity, that we recall for convenience. Let \((X,\mathscr {A})\) and \((Y,\mathscr {B})\) be measurable spaces, and \(\mu \) a measure on the former. If \(\phi :X \rightarrow Y\) is a measurable function, then the image measure or pushforward of \(\mu \) through \(\phi \) is the measure on \((Y,\mathscr {B})\) defined by \(\phi _*\mu :B \mapsto \mu (\phi ^{-1}(B))\). If \(g :Y \rightarrow \overline{{\mathbb {R}}}\) is a measurable function, thenin the sense that g is \(\phi _*\mu \)-integrable if and only if \(g \circ \phi \) is \(\mu \)-integrable, and in this case the integrals coincide (see, e.g., Cohn 2013, §2.6.8). Interpreting precomposition as pullback, hence writing \(\phi ^*g:= g \circ \phi \), and using the notation \(m(f):= \left\langle {m},{f}\right\rangle := \left\langle {f},{m}\right\rangle := \int f\,dm\) for any function f integrable with respect to a measure m, the identity (2.1) can be written in the simple and suggestive formWe shall extensively use an integration-by-parts formula for Lebesgue–Stieltjes integrals (see, e.g., Dellacherie and Meyer 1980, p. 343). Let the functions F, \(G:{\mathbb {R}}\rightarrow {\mathbb {R}}\) be càdlàg (i.e. right-continuous with left limits) and with finite variation (i.e. having bounded variation on every bounded interval). Then, for any two real numbers \(a<b\),If G is continuous, one can obviously replace \(G(x-)\) by G(x). Whenever dG is an atomless measure we shall just write \(\int _a^b F\,dG\) instead of denoting the interval of integration as a subscript. Moreover, we set \({\mathbb {R}}_+:= [0,+\infty [\).

Let us recall a few facts about convex functions (see, e.g., Simon 2011, Chapter 1). Let \(I \subset {\mathbb {R}}\) be an open interval and \(f :I \rightarrow {\mathbb {R}}\) be a convex function. Then f is everywhere left- and right-differentiable, that is, for any \(x \in I\) the left and right derivativesexists and are finite, and \(D^-f(x) \leqslant D^+f(x)\). Moreover, both \(D^-f\) and \(D^+f\) are increasing functions, \(D^-f\) is left-continuous, and \(D^+f\) is right-continuous. It follows that f is differentiable except at the countable set of points where \(D^-f\) and \(D^+f\) do not coincide. The subdifferential of f at x is defined asIt can be shown that \(\partial f(x)=[D^-f(x),D^+f(x)]\) and that, for any \(x_1,x_2 \in I\), \(x_1 < x_2\), it holds \(D^+f(x_1) \leqslant D^-f(x_2)\), hence \(\partial f(x_1) \cap \partial f(x_2)\) is either empty, if the last inequality is strict, or equal to \(\{D^-f(x_2)\}\), if the last inequality is an equality. The right derivative \(D^+f\), being increasing, hence of bounded variation, and right-continuous, defines a (Lebesgue–Stieltjes) measure m via the prescriptionIn this sense, the positive measure m can be interpreted as the second derivative of f.

We shall also use elementary properties of distributions, for which we refer, e.g., to Schwartz (1966) or Duistermaat and Kolk (2010). If E is an open subset of the real line, the space of infinitely differentiable functions from E to \({\mathbb {R}}\) with compact support will be denoted by \({\mathscr {D}}(E)\). Elements of the dual \({\mathscr {D}}'(E)\) of \({\mathscr {D}}(E)\), i.e. linear continuous functionals on \({\mathscr {D}}(E)\), are called distributions. The indication of the set E will be omitted if it coincides with \({\mathbb {R}}\). The pairing between distributions and infinitely differentiable functions with compact support will be denoted by \(\left\langle {\cdot },{\cdot }\right\rangle \). We recall that, given a distribution \(F \in {\mathscr {D}}'(E)\), its distributional derivative \(F'\) is the distribution defined byA function \(f \in L^1_{\textrm{loc}}({\mathbb {R}})\) induces the distribution, denoted by the same symbol, defined byAssume that \(f :{\mathbb {R}}\rightarrow {\mathbb {R}}\) is piecewise of class \(C^1\), with discontinuity points \((x_n)\). Denoting the Dirac measure at \(a \in {\mathbb {R}}\) by \(\delta _a\), and using the standard notations \(\Delta f(x):= f(x+)-f(x-)\), one has (see Schwartz 1966, p. 37)where \(f'\) stands for the derivative of f in the sense of distributions, and \([f']\) for the derivative in the classical sense over the open intervals \(]x_n,x_{n+1}[\) where f is continuously differentiable (a corresponding result for higher-order derivatives can be obtained by induction). If f is a function with finite variation, then \(f'\) coincides, in the sense of distributions, with the Lebesgue–Stieltjes measure df. We shall need to consider functions f that are piecewise differences of convex functions, i.e. of the formwhere the sum is (at most) countable, and for every n there exist convex functions \(h^1_n\), \(h^2_n\) on \({\mathbb {R}}\) such that \(f_n = h^1_n-h^2_n\) on \([a_n,a_{n+1}[\). Then f is càdlàg and has finite variation. We are going to compute the first and second distributional derivatives of f. It is clear that it is enough to consider, without loss of generality, f with support equal to [a, b] and \(f=h^1-h^2\) on [a, b[, for \(h^1\) and \(h^2\) convex functions on \({\mathbb {R}}\).

Let us define the functions \(P,\,C :{\mathbb {R}}_+ \rightarrow [0,+\infty ]\) byNote that P(k) is finite for all k as \({\mathbb {R}}_+ \ni x \mapsto (k-x)^+\) is bounded (in fact, it is continuous with compact support), but C(k) is finite if and onlyhence C is everywhere finite if and only dF has a finite meanThe financial interpretation of \(\overline{dF}<\infty \) is that \({{\mathbb {E}}}_{\mathbb {Q}}\beta _T^{-1}S_T\) is finite. This is clearly not a limitation. Moreover, we recall that \(F(\infty ):=\lim _{x\rightarrow \infty } F(x)\) can be identified with the price at time zero of a zero-coupon bond expiring at time T and with unit face value, therefore \(F(\infty ) \leqslant 1\). However, we shall explicitly mention hypotheses on F because some of the considerations to follow may be interesting for general F, independently of the underlying financial interpretation.

The functions P and C will play a central role, so we discuss some of their properties. They are all rather basic, but they are fully justified for completeness.

By a direct computation one can obtain estimates for local and global Lipschitz constants. In fact, the 1-Lipschitz continuity of \(x \mapsto x^+\), hence also of \(k \mapsto (k-x)^+\), yields, for any \(k_1,k_2 \geqslant 0\),where \(F_-\) stands for the left-continuous version of F, defined by \(F_-(x):=F(x-):=\lim _{h\rightarrow 0+} F(x-h)\). The same estimate holds for P replaced by C. One can actually show, using subdifferentials, that the Lipschitz continuity estimates thus obtained are sharp. In fact, for any \(k_1, k_2 \geqslant 0\), convexity implieswhere¹\(\partial \) stands for the subdifferential in the sense of convex analysis. Hence, if \(k_1 \geqslant k_2\), \(P(k_1) \geqslant P(k_2)\) because P is increasing, which also implies that \(\partial P(x) \subset {\mathbb {R}}_+\) for every \(x>0\), henceSimilarly, if \(k_1 \leqslant k_2\),Recalling that \(\partial P(k) = [D^-P(k),D^+P(k)]\) for every \(k>0\), it easily follows thatAs \(D^+P=F\) and the left-continuous version of \(D^+P\) is \(D^-P\), it follows that \(D^-P=F_-\).

We are going to show that F is the right derivative of P, and that a similar relation holds between C and F. We give two proofs, one that relies on the integration-by-parts formula for càdlàg functions, and a slightly more indirect one based on a denseness result. Namely, we show that the set of put payoffs are total in \(L^1(dF)\), i.e. that for any \(g \in L^1(dF)\) there exist a sequence of finite linear combinations of put payoffs that converges to g in \(L^1(dF)\). Then we show that the two approaches are in fact equivalent. A third approach, using distributions, will be given in Sect. 6 below.

We are going to show that prices of call options for all strikes determine the price of any option with arbitrary convex payoff function (the result is not new—see, e.g., Karatzas and Shreve 1998, pp. 51–52, with a different proof), thus also for options with payoff function that can be written as the difference of two convex functions.

Using the language of Sect. 3, let \(M_1=(g,dF(g))_{g \in G}\) be the measurement set with G the set of convex functions on \({\mathbb {R}}_+\) (satisfying the assumption below), and \(M_2=(g_k,dF(g_k))_{k\in {\mathbb {R}}_+}\), \(g_k :x \mapsto (x-k)^+\), the measurement set of call options (for all strikes). It is evident that \(M_1\) is finer than \(M_2\). We shall show that \(M_2^m\) is finer than \(M_1\), hence that \(M_1\) and \(M_2^m\) are equivalent (in particular, \(M_1\) is a representation). The proof will actually establish that, for any \(g \in G\), \(\pi (g)\) can be written in terms of an integral of the function \(C :k \mapsto \pi (g_k)\). This will then be shown to belong to \(M_2^m\).

Throughout this section we assume that \(g :{\mathbb {R}}_+ \rightarrow {\mathbb {R}}\) is the restriction to \({\mathbb {R}}_+\) of a convex function h on an open set \(I \supset {\mathbb {R}}_+\). In particular, \(D^+g(0)>-\infty \). In order to avoid trivialities, we also assume that \(g \in L^1(dF)\). We recall that g is continuous, differentiable almost everywhere, right-differentiable on \([0,\infty [\), and that \(D^+g\) is increasing and càdlàg. In particular, \(D^+g\) has finite variation, thus generates a Lebesgue–Stieltjes measure that we shall denote by m, or also by \(dg'\). The positive measure m can also be identified with the second derivative of g in the sense of distributions.

Note thati.e. (5.1) could be written in the more symmetric formAnalogously, since(5.1) could also be written as

Slightly more generally, one can also writehence, taking conditional expectation with respect to \({\mathscr {F}}_t\), for any \(t \in [0,T]\), and multiplying by \(\beta _t\),where \(C_t(k):= \beta _t {{\mathbb {E}}}_{\mathbb {Q}}\bigl [\beta _T^{-1} (S_T-k)^+ \big \vert {\mathscr {F}}_t\bigr ]\) is the price at time t of the call option with strike k.

It is actually possible to prove Proposition 5.1 using only the integration-by-parts formula (2.2). Even though the proof is longer than the previous one, some of its ingredients may be interesting in their own right. We begin with a useful reduction step.

Note that the “normalizing” assumptions \(g(0)=0\) and \(D^+g(0)=0\) imply thatand thatrespectively. The former is evident, and the latter follows from \(m(\{0\})=D^+g(0)=0\). Therefore

Let us show that \(\int _{]0,\infty [} C\,dm \in M_2^m\). By Tonelli’s theorem,For any \(n \in {\mathbb {N}}\), let \((k_i)_{i=0,\ldots ,2^n}\) be a dyadic partition of ]0, n]. Thenas \(n \rightarrow \infty \), hence, again by Tonelli’s theorem,Thendefines a sequence of elements in the vector space generated by \(M_2\) that monotonically converges pointwise to the function \(x \mapsto \int _{]0,\infty [} (x-k)^+\,dm(k)\), which belongs to \(L^1(dF)\) by assumption, therefore also to \(M_2^m\).

We are going to show that most properties of the functions F, P and C discussed in the previous sections can also be obtained using Schwartz’s distributions. The main advantage of this approach is that several results reduce, in their formal aspect, to simple calculus for distributions. Some work is needed, however, to remove the regularity assumptions on test functions typical of this approach.

Throughout this section, the functions F, P, and C (the last one if \(\overline{dF}\) is finite) are assumed to be extended to \({\mathbb {R}}\) setting them equal to zero on \(]-\infty ,0[\). All of them obviously belong to \(L^1_{\textrm{loc}}({\mathbb {R}})\), hence they can be considered as distributions in \({\mathscr {D}}'\).

Let us start with a simple but useful observation.

In the next subsection we derive the main results of Sect. 4 using distributions.

In the standard Black-Scholes (BS) model one assumes that \(S_T = \exp (\varsigma \sqrt{T} Z -\varsigma ^2T/2)\) in law, where the volatility \(\varsigma \) is constant and Z is a standard Gaussian random variable. This family of random variables (indexed by \(\varsigma \), with time to maturity T fixed as before) can be embedded in the larger class defined by \(S_T = \exp (\sigma X + m)\), where \(\sigma \) and m are constants, and X is a random variable with density \(f \in L^2:=L^2({\mathbb {R}})\). This rather general family of laws can be used as setup for empirical nonparametric option pricing, essentially by projecting the density f on radial basis functions (see Marinelli and d’Addona 2023). More precisely, one considers expansions of f in terms of Hermite functions, so that the lognormal distribution of returns corresponds exactly to the zeroth-order expansion of f. The approach can thus be thought of as a perturbation of the BS model at fixed time to maturity. The following problem then arises: let \((f_n) \subset L^1 \cap L^2\) be a sequence of functions converging to f in \(L^2\), and let \(P_n(k)\) be the “fictitious” price of a put option with strike k, obtained replacing the density f with its approximation \(f_n\). Suppose that the \(P_n(k)\) are known for all \(k \in {\mathbb {R}}_+\) and \(n \in {\mathbb {N}}\). Is this information enough to determine the function P, i.e. to obtain the put option prices in the “true” model?

Assuming, for simplicity, that \(S_0=1\), \(\sigma =1\) and \(m=0\), and denoting the distribution function of \(X=\log S_T\) with respect to the measure \(\mu \) (see (3.1)) by \(F_X\), it is immediately seen that \(F_X(x) = F(e^x)\) for every \(x \in {\mathbb {R}}\) and thathenceNote that \(F_X\) and f are supported on the whole real line and that F and \(F_X\) are in bijective correspondence, hence \(F_X\) is in bijective correspondence also with P.

The sequence \(P_n(k)\) does not necessarily converge to P(k) as \(n \rightarrow \infty \), because the function \(x \mapsto (k-e^x)^+\) belongs to \(L^\infty \) but not to \(L^2\), hence it induces a continuous linear form on \(L^1\), but not on \(L^2\). Moreover, convergence in \(L^2({\mathbb {R}})\) does not imply convergence in \(L^1({\mathbb {R}})\).

We are going to show that the function P can be reconstructed from approximation to option prices with payoff of the typeMore precisely, to identify the pricing functional P, it suffices to know, for any sequence \((f_n)\) converging to f in \(L^2\), the values \(\left\langle {\theta _{k_1,k_2}},{f_n}\right\rangle \) for all \(k_1,k_2>0\) and all \(n \geqslant 0\), where \(\left\langle {\cdot },{\cdot }\right\rangle \) stands for the scalar product of \(L^2\). In fact, for any \(k_1,k_2>0\), the function \(\theta _{k_1,k_2}\) is in \(L^2\), hence, for any sequence \((f_n) \subset L^1 \cap L^2\) converging to f in \(L^2\) (weak convergence in \(L^2\) would also suffice), one hasMoreover,where \((k_1 - e^x)^+ \in ]0,k_1]\) for all \(x \in {\mathbb {R}}\), hence \(\frac{k_2}{k_1} (k_1 - e^x)^+ \in ]0,k_2]\) for all \(x \in {\mathbb {R}}\), andso thatTherefore the function \(x \mapsto \frac{k_2}{k_1} (k_1 - e^x)^+\) converges to zero as \(k_1 \rightarrow 0\) in \(L^p\) for every \(p \in [1,\infty [\) by the dominated convergence theorem. In particular, since \(f \in L^2\),We have thus shown thatthus also the following statement.

Completely analogously, if \(P(k_1)\) is known, then

Note that Proposition 7.1 can be interpreted as providing a sort of generalized representation of dF, but, as discussed at the end of Sect. 3, it cannot be formulated in the language introduced there. Even the extended measurements of the type \((g_j,\pi _j,dF_j)\), with \(dF_j\) a family of measures weakly converging to dF, is not enough. In fact, it is not difficult to check that the pushforward of \(f_n\,dx\) through \(x \mapsto e^x\), denoted by \(dF_n\), does not converge weakly to dF, in general. However, setting \(M_1 = (g_k,dF_n(g_k),dF_n)_{k>0,\,n \in {\mathbb {N}}}\), where \(g_k :x \mapsto (k-x)^+\), we have shown that \(M_1\) “implies” \(M_2=(\theta _{k_1,k_2},\pi _{k_1,k_2})_{k_1,k_2>0}\), where implication is meant as in the last paragraph of Sect. 3, and that \(M_2^m\) is finer than M, the measurement set composed of put prices, which is a representation.