3.1 Pricing functionals
Let \((\Omega ,\mathscr {F},{\mathbb {P}})\) be a probability space endowed with a filtration \((\mathscr {F}_t)_{t\in [0,T]}\), with \(T>0\) a fixed time horizon, and let \(S :\Omega \times [0,T] \rightarrow {\mathbb {R}}_+\) be the price process of an asset. We assume also that \(\beta :\Omega \times [0,T] \rightarrow ]0,\infty [\) is the price process of a further asset used as numéraire, normalized with \(\beta _0=1\) and uniformly bounded from below, and that the market where both assets are traded is free of arbitrage, so that the set \(\textsf{Q}\) of probability measures \({\mathbb {Q}}\) equivalent to \({\mathbb {P}}\) such that the discounted price process \(\beta ^{-1}S\) is a \({\mathbb {Q}}\)-local martingale is not empty. For any \(\mathscr {F}_T\)-measurable claim X such that \(\beta _T^{-1}X\) is bounded, the value \({{\mathbb {E}}}_{\mathbb {Q}}\beta _T^{-1}X\) is an arbitrage-free price at time zero of X for every \({\mathbb {Q}}\in \textsf{Q}\). From now on we shall fix a measure \({\mathbb {Q}}\in \textsf{Q}\). For any measurable bounded function \(g :{\mathbb {R}}_+ \rightarrow {\mathbb {R}}\), the bounded \(\mathscr {F}_T\)-measurable random variable \(g(S_T)\) is the payoff of a European option on S with payoff profile g, the price of which at time zero is \({{\mathbb {E}}}_{\mathbb {Q}}\beta _T^{-1} g(S_T)\).
We shall call
pricing functional the map
$$\begin{aligned} \pi :g \longmapsto {{\mathbb {E}}}_{\mathbb {Q}}\beta _T^{-1} g(S_T) = {{\mathbb {E}}}\frac{d{\mathbb {Q}}}{d{\mathbb {P}}}\beta _T^{-1} g(S_T), \end{aligned}$$
defined first on the set of measurable bounded functions
\(g :{\mathbb {R}}_+ \rightarrow {\mathbb {R}}\). Let
\(\mu \) be the measure on
\({\mathscr {F}}_T\) defined by
$$\begin{aligned} \mu (A) := {{\mathbb {E}}}_{\mathbb {Q}}\beta _T^{-1} 1_A, \end{aligned}$$
(3.1)
that is,
\(\mu \) is the measure on
\({\mathscr {F}}_T\) the Radon-Nikodym derivative of which with respect to
\({\mathbb {P}}\) is
$$\begin{aligned} \frac{d\mu }{d{\mathbb {P}}} = \frac{d{\mathbb {Q}}}{d{\mathbb {P}}} \beta _T^{-1}. \end{aligned}$$
Note that
\(\mu \) is (in general) not a probability measure: in fact,
\(\mu (\Omega ) = {{\mathbb {E}}}_{\mathbb {Q}}\beta _T^{-1}\) need not be one, and could be interpreted as the price at time zero of a zero-coupon bond maturing at time
T with face value equal to one. In this case,
\(\mu \) is a sub-probability measure, i.e.
\(\mu (\Omega ) \leqslant 1\). The pricing functional can then be written as
$$\begin{aligned} \pi :g \longmapsto \int _\Omega g(S_T)\,d\mu . \end{aligned}$$
Denoting the pushforward of
\(\mu \) through
\(S_T\) by
\(S_*\mu \), i.e. the measure on the Borel
\(\sigma \)-algebra of
\({\mathbb {R}}\) defined by
$$\begin{aligned} S_*\mu :B \longmapsto \mu (S_T^{-1}(B)), \end{aligned}$$
one has
$$\begin{aligned} {{\mathbb {E}}}_{\mathbb {Q}}\beta _T^{-1} g(S_T) = \int _\Omega g(S_T)\,d\mu = \int _{\mathbb {R}}g\,d(S_*\mu ). \end{aligned}$$
Therefore, denoting the distribution function of the measure
\(S_*\mu \) by
F, i.e.
$$\begin{aligned} F(x):= \mu (S_T \leqslant x) = {{\mathbb {E}}}_{\mathbb {Q}}\beta _T^{-1} 1_{\{S_T \leqslant x\}}, \end{aligned}$$
the pricing functional can be written as
$$\begin{aligned} \pi :g \longmapsto \int _{{\mathbb {R}}_+} g\,dF. \end{aligned}$$
In other words, the pricing functional can be identified with
F, or with
\(S_*\mu \). Note also that the pricing functional can naturally be extended to every
\(g \in L^1(dF)\). If
\(g \geqslant 0\), as is mostly the case for payoff functions, then
\(\pi (g)\) is simply the norm of
g in
\(L^1(dF)\).
3.2 Measurements and representations
Depending on the problem at hand, the pricing functional \(\pi :g \mapsto dF(g)\) may or may not be known. If dF is assumed a priori to be known, as, for instance, in the Black-Scholes model with given volatility, then \(\pi \) is trivially known. An analogous situation arises in the case where dF is assumed to belong to a family of finite measures \((dF_\theta )_{\theta \in \Theta }\) parametrized by a finite-dimensional parameter \(\theta \): once an estimate \(\hat{\theta }\) is obtained, so that \(dF_{\hat{\theta }}\) is used in the definition of \(\pi \), one falls back into the previous (quite tautological) case. On the other hand, in many other situations, for instance when no parametric assumptions on dF are made, the pricing functional \(\pi \) is only known through its action on a set of “test functions” \((g_j)_{j \in J}\), e.g. with \(g_j\) the payoff profile of a call or put option with a strike price indexed by \(j \in J\). The next definition is hence natural.
A typical situation of practical relevance is given by \((g_j)\) being a collection of payoff profiles of (European) options. For instance, for any \(j \geqslant 0\), let \(g_j\) be the payoff function of a put option with strike price j, that is, \(g_j :x \mapsto (j-x)^+\). If the price of the put option with strike j is known for every \(j>0\), then we have a measurement \(M=(g_j,\pi _j)_{j \in J}\) setting \(J=]0,+\infty [\), \(g_j :x \mapsto (j-x)^+\), and \(\pi _j=dF(g_j)\).
Measurement sets can be ordered by inclusion, hence they can be compared. If M is a measurement set, the vector space generated by M, itself a measurement set, will be denoted by \(\hat{M}\).
Apart from the natural inverse problem of recovering the measure
dF from a sufficiently rich collection of option prices, possibly providing an algorithm to do so, it is interesting also to describe relations between measurement sets. For instance, if one needs
F only to price a certain set of options, instead of reconstructing
F it could suffice to identify a measurement set that already allows to accomplish the task. In the simplest case, if
g is the payoff profile of the option to price and
g belongs to the vector space generated by an available measurement set
M, there is clearly no need to recover
F. In spite of its simplicity, this is precisely how one can proceed to price options with continuous piecewise linear payoff profile. In fact, as is well known, these options can be priced in terms of linear combinations (independent of
F!) of prices of put options with strikes at the “juncture” points of the piecewise linear profile. A more sophisticated fact is that call option prices for every positive strike price allow to price option with arbitrary convex payoff. In this case, however, if
g is an arbitrary convex function and
\(\textrm{pr}_1 M\), the projection on
\(L^1(dF)\) of the measurement set
M, is the vector space generated by
\((x \mapsto (x-k)^+)_{k \in {\mathbb {R}}_+}\), it is not true in general that
\(g \in \textrm{pr}_1 M\). It is true, however, that
g is an accumulation point of
\(\textrm{pr}_1 M\), as discussed in Sect.
5 below.
It was mentioned above that it would not be meaningful to extend a measurement set
M taking its closure in
\(L^1(dF) \times {\mathbb {R}}\), as
F is considered unknown. However, one can indeed add some cluster points, if they are defined by procedures that do not involve
F. In particular, at least two possibilities exist:
(a)
let \((g_n) \subseteq \textrm{pr}_1 M\) be a sequence that converges pointwise to g and for which there exists \(h \in L^1(dF)\) such that \(\left|{g_n(x)}\right| \leqslant h(x)\) for all \(x \in {\mathbb {R}}_+\). The dominated convergence theorem then implies that \(g \in L^1(dF)\) and that \(\pi (g)=\lim _{n \rightarrow \infty } \pi (g_n)\);
(b)
let
\((g_n) \subseteq \textrm{pr}_1 M\) be such that
\(g_n \uparrow g\), i.e.
\((g_n)\) is an increasing sequence that converges pointwise to
g, and such that
\((\pi (g_n))\) is bounded from above, i.e.
\(\sup _n \pi (g_n)<\infty \). Then
\(0 \leqslant g_n-g_0 \uparrow g-g_0\) and, by the monotone convergence theorem,
$$\begin{aligned} \pi (g-g_0) = \lim _n \pi (g_n-g_0) = \sup _n \pi (g_n) - \pi (g_0) <\infty , \end{aligned}$$
hence
\(g-g_0 \in L^1(dF)\), i.e.
\(g \in L^1(dF)\) with
\(\pi (g) = \sup _n \pi (g_n)\).
The cluster points constructed in (a) and (b) do not depend on knowing
F, hence they could reasonably be added to the measurement set
M. The measurement sets obtained by adding to
\(\hat{M}\) the cluster points described in (a) and (b) will be denoted by
\(M^d\) and
\(M^m\), respectively. We shall see that if
\(M_1\) is the measurement set of all call options, and
\(M_2\) the measurement set of all convex options, then
\(M_1^m\) is finer than
\(M_2\). Since
\(M_2\) is finer than
\(M^m_1\) (the pointwise supremum of a family of convex functions is convex),
\(M_1^m\) and
\(M_2\) are equivalent measurement sets. In other words, one cannot replicate a convex payoff with just call payoffs, but one can approximate a convex payoff by a combination of call payoffs with any pricing accuracy.
Taking suitable limits of sequences of measurements is not the only possible way to enrich a measurement set. In fact, one can also perform several operations on \((\pi _j)_{j\in J} \subseteq {\mathbb {R}}\), using the structure of \({\mathbb {R}}\): they can for instance be added, multiplied, and functions \(\phi :{\mathbb {R}}^n \rightarrow X\) can be applied to n of them, with X suitable sets, and so on. Note that M could also be seen as a linear map from the space of finite measures \(\mathscr {M}^1({\mathbb {R}}_+)\) to \({\mathbb {R}}^J\), mapping dF to \((dF(g_j))_{j \in J}\). Viewing elements of \({\mathbb {R}}^J\) as functions from J to \({\mathbb {R}}\), the problem at hand may imply that these functions in the codomain have additional properties, for instance they may be monotone, or convex, or continuous, or differentiable, depending on the inputs \((g_j)\). Depending on the range of M in the codomain \({\mathbb {R}}^J\), different operations may be applied. For instance, taking derivatives on \({\mathbb {R}}^J\) or on C(J) would not make sense, but it would make sense on \(C^1(J)\), or in the a.e. sense if we knew that the range is made of Lipschitz continuous functions. We shall see that this point of view is also fruitful, showing that the right derivative of put prices, seen as a function P of the strike price, is equal to F. We shall also see that the price of an option with arbitrary convex payoff can be written in terms of an integral of C, where C(k) is the price of the call option with strike k.
In some cases one does not observe a measurement directly, but a function of a measurement. This is the case, for instance, of implied volatility. If \(g_k :x \mapsto (k-x)^+\) is the payoff function of a put option with strike k, there is a one-to-one correspondence between \(\pi _k:=\pi (g_k)\) and the Black-Scholes implied volatility, given by a function \(v :{\mathbb {R}}_+ \rightarrow {\mathbb {R}}_+\) such that \(\pi _k = \textsf{BS}(S_0,k,T,v(\pi _k))\). Here \(\textsf{BS}(S_0,k,T,\sigma )\) denotes the Black-Scholes price at time zero of a put option on an underlying with price at time zero equal to \(S_0\), strike k, time to maturity T, volatility \(\sigma \), and interest rate as well as dividend rate equal to zero. In particular, if the implied volatility is known for every strike \(k>0\), inverting the function v we obtain the measurement set of put prices \(M=(g_k)_{k \geqslant 0}\), which is a representation. In other words, implied volatility for all strikes uniquely determines the pricing functional or, equivalently, the measure dF. We may then say, with a slight abuse of terminology, that implied volatility is a representation.
Let
X be a locally compact space and
\(\phi :{\mathbb {R}}\rightarrow X\) be a measurable isomorphism, i.e. a bijection such that both
\(\phi \) and
\(\phi ^{-1}\) are measurable. This is the case, for instance, if
\(\phi \) is a homeomorphism. Then, for any
\(g \in L^1(dF)\), one has
$$\begin{aligned} dF(g) = \left\langle {g},{dF}\right\rangle = \big \langle {\phi ^*(\phi ^{-1})^*g},{dF}\big \rangle = \big \langle {(\phi ^{-1})^*g},{\phi _*dF}\big \rangle . \end{aligned}$$
This change of parametrization can also be interpreted in terms of measurement sets, saying that the measurement set
\(M=(g_j,dF(g_j))\) of
dF is in bijective correspondence with the measurement set
$$\begin{aligned} M' = \bigl ( (\phi ^{-1})^*g_j,\phi _*dF((\phi ^{-1})^*g_j) \bigr ) = \bigl ( (\phi ^{-1})^*g_j,dF(g_j) \bigr ) \end{aligned}$$
of
\(\phi _*dF\). Even though the two measurements are isomorphic (as sets), they may have quite different properties. Let us consider, for instance, the reparametrization from price to logarithmic return: setting
\(S_T=S_0\exp (\sigma X_T + m)\), where
\(\sigma >0\) and
m are constants, the pricing functional can be written as
$$\begin{aligned} \pi :g \longmapsto \int _{\mathbb {R}}g(S_0e^{\sigma x + m})dF_X(x), \end{aligned}$$
where
\(F_X\) is the distribution function of the measure
\((X_T)_*\mu \), the support of which is
\({\mathbb {R}}\). If
g is the payoff function of a put option with strike
k, then
\(x \mapsto g(S_0e^{\sigma x + m}) = {(k - S_0e^{\sigma x + m})}^+\) does
not have compact support. This is clearly in stark contrast to the expression of
\(\pi (g)\) in terms of
dF, where the intersection of the supports of
g and
dF is compact. As will be seen, several analytic arguments strongly depend on this property, that hence cannot be used with the new parametrization, even though the values of the corresponding integrals are the same.
Finally, we remark that it is sometimes useful to extend the definition of measurement set allowing for an extra collection
\((dF_j)\) of (possibly signed) measures for which a relation to
dF is known. For instance, let
\(g_k\), for any
\(k \geqslant 0\), be the payoff function of a put option with strike price
k, that is,
\(g_k :x \mapsto (k-x)^+\). Moreover, let
\((dF_n)\) be a sequence of Radon measures converging weakly to
dF as
\(n \rightarrow \infty \). Each measure
\(dF_n\) can be thought of as an approximation to the law
dF, and
\(dF_n(g_k)\) as the price of a put option with strike
k under the approximating law
\(dF_n\). If all such prices can be observed, then we have an “extended” measurement set
\(\widetilde{M}:= (g_j,\pi _j,dF_j)_{j \in J}\), where
\(J={\mathbb {R}}_+ \times {\mathbb {N}}\),
\(F_j=F_{kn}\),
\(F_{kn}=F_n\) for every
k,
\(g_j=g_{kn}=g_k\) for every
n, and
\(\pi _j=\pi _{kn}=F_n(g_k)\). Note that
\(g_k \in C_b({\mathbb {R}})\) for every
k, hence
\(F_n(g_k) \rightarrow F(g_k)\) as
\(n \rightarrow \infty \). In particular, if we define the (standard) measurement set
\(M=(g_j,\pi _j)_{j \in {\mathbb {R}}_+}\) as
\(g_j:x \mapsto (j-x)^+\) and
\(\pi _j=dF(g_j)\), then we could say that
\(\widetilde{M}\) “implies”
M. That is, for every
\(g \in \textrm{pr}_1 M\) there exists a sequence
\((dF_n) \subset \textrm{pr}_3 M\) such that
\(\pi (g)=dF(g)\) is the limit of
\(dF_n(g) \subset \textrm{pr}_2 M\). An example of this type, motivated by empirical nonparametric option pricing, is discussed in Sect.
7 below.