A pseudo-analytic generalization of the memoryless property for continuous random variables and its use in pricing contingent claims

We explore an extension of the memoryless property for continuous random variables by using the concept of pseudo-sum. Subsequently, we demonstrate the practicality of this approach through two financial applications in which pseudo-sums characterize the values of arbitrage-free contingent claims. Moreover, we are able to establish new interesting connections between different probability distributions.


Introduction
Aczél characterizes all strictly increasing associative binary operations in abstract algebra, by relating them to the associativity functional equation [1,2].In probability theory, one can characterize all memoryless continuous random variables, by relating their probability laws to the Cauchy exponential functional equation [3,4].
This article connects these two functional equations to each other.We relate Aczél's characterization [1] of all associative and strictly increasing binary operations, as isomorphic to ordinary addition, to the well-known characterization of all continuous memoryless random variables as exponentially distributed.As a result, we obtain a generalization of the memoryless property in the continuous setting, which proves useful in financial applications and distribution theory.In particular, we discuss the case of two arbitrage-free contingent-claim values and the emerging relationships among some extreme value distributions.
Our work can be seen as a further application of the non-Newtonian approach developed by Grossman & Katz [5], and more recently extended by Pap [6] and co-authors.As we shall observe, our findings also share the rationale of some pioneering research by Kolmogorov [7] and de Finetti [8], among others.
This article is organized as follows.In §2, we define the generalized memoryless property (GMP).In §3, we present our view on how to characterize the GMP for a continuous random variable.In §4, we illustrate our findings with two examples, leading towards the financial applications of §5.In §6, exploiting the GMP, we provide new connections among notable distributions of the financial and actuarial literature, as well as of extreme value theory.Finally, §7 summarizes this article and suggests interesting future research.In the appendices, we collect the longer proofs and extra results, including some incidental findings that could be useful in the study of economic inequality.
It is well-known that the only strictly decreasing continuous solution to equation (2.2) with S(0) = 1 is a ≥ 0, for some positive constant λ [1].
In other words, the only continuous law satisfying the memoryless property of equation (2.1) is the exponential distribution, whose survival function (SF) is actually given in equation (2.3).
The generator of a monoid surely needs to be strictly monotonic, but it does not necessarily need to be exponential.By changing the generator S, one can, in fact, change the binary operation that is isomorphic to ordinary addition.Our GMP simply replaces the ordinary sum x 1 + x 2 on the left-hand side of equation (2.1) with the so-called pseudo-sum Following the pioneering work of Grossman & Katz [5], further elaborated by Pap [6], a pseudo-sum is the result of applying the binary operation ⊕ in a monoid (A: ⊕ , e), where A is now a connected subset [ℓ, u] of the extended real line [ − ∞, ∞].As a result, we have the following properties: (ii) existence of an identity element: there exists e ∈ A, such that x ⊕ e = e ⊕ x = x, for all x ∈ A; and (iii) associativity: for x 1 , x 2 , x 3 ∈ A, one has that ( In the general setting of [5] and [6], the monoid (A: ⊕ , e) is not required to be commutative, or cancellative, or to have an inverse element in the set.
In this article, the set A = [ℓ, ℎ] over which the monoid of interest is defined will be the support of the law of a continuous1 random variable X.
When the associativity requirement of a monoid is dropped, the resulting algebraic structure is called an unital magma [10].Let us thus consider an unital magma (S: ⊕ ^G , e) where A = [ℓ, ℎ] is a connected subset of the extended real line [ − ∞, ∞], possibly [ − ∞, ∞] itself, and whose binary operation ⊕ ^G is strictly increasing but not necessarily associative.It turns out that, in this context, additionally demanding the associativity of the binary operation ⊕ ^G puts a severe constraint on its form.Let ⊕ G denote the binary operation in the unital magma (A: ⊕ G , e), when it is required to be both strictly increasing and associative.Then, (A: ⊕ G , e) is also a monoid, and there must exist a strictly monotonic function G: ℝ ↦ A such that [2]: (2.5) where G −1 : A ↦ ℝ denotes the inverse of the generator G.
Therefore, 2 requiring that the strictly increasing binary operation ⊕ ^G in the magma (A: ⊕ ^G , e) also be associative forces the resulting binary operation ⊕ G to be isomorphic to ordinary addition + over the reals, where the isomorphism is given by equation (2.5).The strictly monotonic function G: ℝ ↦ A in equation (2.5) is called the generator of the strictly increasing binary operation ⊕ G in the monoid (A: ⊕ G , e).It turns out that this monoid must be commutative, cancellative, and that, for the identity element, e = G(0).Definition 2.1 (GMP).The law of a continuous random variable X, with support A = [ℓ, ℎ], is said to enjoy the GMP if, for all x 1 , x 2 ∈ A, the probability measure ℙ satisfies either where ⊕ G indicates a pseudo-sum, which is strictly increasing in both x 1 ∈ A and x 2 ∈ A.
From now on, we restrict the domain of the strictly monotonic generator G to the non-negative real line [0, ∞).Moreover, we demand that the strictly monotonic generator G maps all of [0, ∞) onto the support A = [ℓ, u] of the continuous random variable X .Since the generator G is a strictly monotonic map from all of [0, ∞) onto A, its inverse map G −1 must also be a strictly monotonic map from all of A onto [0, ∞).
As a consequence, we have that: (i) if G is strictly increasing, then so is G −1 , and the continuous random variable X with support A = [ℓ, u] enjoys the GMP as per equation (2.6); and (ii) if G is strictly decreasing, then so is G −1 , and the continuous random variable X with support A = [ℓ, u] enjoys the GMP as per equation (2.7).
All in all, the law of a continuous random variable X supported on A = [ℓ, ℎ] enjoys the GMP, with + replaced by the strictly increasing associative binary operation ⊕ G , if and only if the strictly monotonic generator This restriction on the type of monotonicity of G, the domain of G and the image of G is one of the contributions of this article.Together with the probabilistic and financial interpretations of the pseudo-sums involved in the GMP, it distinguishes our work from similar intuitions in references [3,11].
The reader might have noticed that the GMP naturally connects to other useful concepts of probability and statistics, like, for example, the generalized means of Kolmogorov, Nagumo, de Finetti & Chisini [7,8,12].As observed in [5], thanks to different appropriate choices of the generator G, the operation ⊕ G can define 'alternative universes', in which addition is, for instance, replaced by multiplication, the natural integral is the geometric integral, and the familiar arithmetic mean is somehow replaced by the geometric one.In such a multiplicative universe, the random variable showing the memoryless property would not be the exponential but rather the Pareto (see §4), and the random variable playing the role of the normal in many useful convergence results could be the lognormal (see §7 for an intuition).A universe like that would therefore be a natural place to study phenomena of growth, contagion, cascading failures and so on [5].Moving across all these 'universes' thus often provides ways to simplify or extend known facts, discovering interesting relations and applications, as we try to do in the next sections.
3. Bijections between [0, ∞) and A that allow for the generalized memoryless property Again, let X be a continuous random variable supported on a connected interval A = [ℓ, u], which is a subset of the extended real line [−∞, ∞].We allow ℓ = −∞ and/or u = ∞.The SF of X is a strictly decreasing map S X (x) from A to the unit interval, [0, 1].We require that S X (ℓ) = 1 and that S X (u) = 0. Let I be a strictly increasing function I : [ℓ, u] ↦ [0, ∞) with I(ℓ) = 0 and I(u) = ∞.We call the function where λ > 0 is a positive constant called the Laplace parameter [2].
The cumulative distribution function (CDF) of X, denoted by F X (x), is defined by Obviously, the CDF of X is a strictly increasing map from the support A of X to the unit interval [0, 1], with F X (ℓ) = 0 and F X (u) = 1.Let D(x) be a strictly decreasing function D: [ℓ, u] ↦ [0, ∞) with D(ℓ) = ∞ and D(u) = 0. We call the function D(x) the Laplace exponent in the Laplace representation of F X (x) if where again λ > 0 is a positive constant.
Since I( ⋅ ) is a strictly increasing map of all of A onto [0, ∞), its inverse I −1 ( ⋅ ) is a strictly increasing map of all of [0, ∞) onto A. Likewise, since D( ⋅ ) is a strictly decreasing map of all of A onto [0, ∞), its inverse D −1 ( ⋅ ) is a strictly decreasing map of all of [0, ∞) onto A.
Let a denote an arbitrary element in the additive monoid ([0, ∞): + , 0).Suppose that a monotonic generator G(a) produces an isomorphic monoid (A: ⊕ , e), where A is as usual a connected open subset of ℝ, ⊕ is the monoid's binary operation and e is the monoid's identity element.If the generator G(a) is strictly monotonic, then we denote the binary operation by ⊕ G and the identity element by G(0).From equation (2.5), the binary operation ⊕ G in the isomorphic monoid (A: ⊕ G , G(0)) is strictly increasing.
We henceforth focus our attention on the strict subset of the class of all strictly monotonic generators which enforce a bijective map between [0, ∞) and A, which is the support of the random variable X.We call such generators feasible for X.
If the X −feasible generator G is strictly increasing, then we set G(a) = I −1 (a), so that When we obtain the Laplace exponent I(x) (or D(x)) by inverting an X −feasible generator G, then the SF (or the CDF) defined by equation (3.1) (or equation (3.3)) is guaranteed to strictly decrease from 1 to 0 (or to strictly increase from 0 to 1) as required.
If the X −feasible generator G(a) is strictly increasing, then the SF generated by equation (3.1) satisfies the functional equation 4 royalsocietypublishing.org/journal/rsos R. Soc.Open Sci.11: 231690 which leads to the GMP in equation (2.6), that is, once again, If the X −feasible generator G(a) is instead strictly decreasing, then the CDF generated by equation (3.3) satisfies the functional equation which leads to the GMP in equation (2.7), restated here as where again x 1 , x 2 and x 1 ⊕ G x 2 are all in the support A = [ℓ, u] of X.
To illustrate these results, let us first consider the identity map for the generator G, that is, G(a) = a for a ≥ 0.
Since this generator is a strictly increasing map from all of [0, ∞) to [0, ∞), it follows that G(a) = a is feasible for a continuous random variable X supported on [0, ∞).The inverse of G is clearly an increasing map, so we set I(x) = x in equation (3.1) for the SF of X.The continuous random variable X is thereby identified as being exponentially distributed with parameter λ > 0.
Trivially, when G(a) = a, one gets that ⊕ G = +, and the GMP reduces to the standard memoryless property of the exponential random variable [9], as per equation (2.1).

Examples of bijective generators
Let us now take into consideration some examples of random variables manifesting the GMP.Of two of them, we will discuss the financial implications in §5, to show their applicability in financial engineering.

The generator G is an increasing power function
A first basic example of a strictly increasing generator is the power function Since this generator maps all of [0, ∞) onto [0, ∞), it is feasible for a continuous random variable X supported on [0, ∞).
Letting x ≡ a b ≥ 0, the inverse of the generator G defined in equation (4.1) is This inverse is also strictly increasing. Setting , the resulting SF is that of a Weibull [13], that is: ), the corresponding binary operation ⊕ b is an ℓ p norm of the vector [x 1 , x 2 ], with p = 1/b: , and b > 0, the following GMP holds for the Weibull-distributed random variable X ≥ 0, whose SF is in equation ( 4.3): royalsocietypublishing.org/journal/rsos R. Soc.Open Sci.11: 231690 It is worth noticing that equation (4.5) recalls the functional equation that Wang [14] provides to characterize Weibull-distributed random variables.

The generator G is a negated logarithm
Suppose now that the generator is the following negated (natural) logarithm: Note that the generator defined by equation (4.6) is strictly decreasing.Since this G maps all of [0, ∞) to (−∞, ∞], the generator G defined in equation (4.6) is feasible for a continuous random variable X supported on (−∞, ∞]. Letting x ≡ −blog a, for a ≥ 0, b > 0, we observe that x ∈ (−∞, ∞].As a result, the inverse of the generator G defined in equation (4.6) is This inverse is also strictly decreasing.

Other possibilities
Many examples of the GMP emerge if one chooses different generators.For example, when G(a) = e a , a ≥ 0, one has x 1 ⊕ G x 2 = x 1 × x 2 , and the random variable showing the GMP is a standard Pareto.Interestingly, as observed by Chisini [12] and de Finetti [8], when speaking about means, this is also the situation in which the arithmetic mean we are all familiar with is naturally replaced by the geometric mean as the reference mean.This comes from the fact that + is replaced by ×, thanks to G.
Simple generalizations can be obtained by adding, for instance, a constant c > 0, so that G(a) = exp(a + c).These generalizations all lead to different Paretian random variables, like type II and type III [13].
Alternatively, choosing G(a) = log e a − 1, with a > 0, generates the pseudo-sum , and the GMP belongs to a standard logistic random variable.Interestingly, in this case, G −1 (x) = log e x + 1, which is nothing more than the integrated CDF of the same logistic random variable [13], linked to the well-known softplus activation function in machine learning [15,16].
Many other examples can naturally be thought of, but our goal here is not to provide a complete taxonomy of all possible cases of GMP, as it would be unrealistic.We rather prefer to show the usefulness of this alternative way of approaching memorylessness via pseudo-sum, when considering some financial and probabilistic applications.For this reason, in the rest of this article, we focus our attention on the two generators of §4.1 and §4.2.

Financial interpretations of ⊕ b and ⊕ −b
The pseudo-sums x 1 ⊕ b x 2 and x 1 ⊕ −b x 2 , as defined in equations (4.4) and (4.9) of §4, can be both interpreted as the arbitrage-free values of particular contingent claims.This strengthens the idea of the usefulness of non-Newtonian calculus in finance, as already observed in [17,18].
As a matter of fact, pseudo-sums seem to emerge naturally in many relevant aspects of financial mathematics.For instance, one could already notice that, in the basic Black-Scholes-Merton setting (BSM) [19], when prices follow a geometric Brownian motion, the change in measure between ℙ (market or physical measure) and ℚ (risk-neutral measure), controlled by the so-called Wang transform  [20] is just example of a pseudo-sum.
Let S(t) ≥ 0 be the price of a given asset at time t ∈ [0, T] under the physical measure ℙ.Let K > 0 be a strike price.Fix a value S(0) ≥ 0 and assume that where μ ∈ ℝ, σ > 0 and B(t) is a standard Brownian motion under ℙ.
Hence, we have that, in T, under ℙ: where Φ( ⋅ ) is the CDF of a standard normal.
If we introduce ℚ as the risk-neutral measure equivalent to ℙ, we also know [19] that, underℚ, log S(T) ∼ Φ log S(0 where r ≥ 0 is the so-called risk-free rate. The connection between ℙ and ℚ in the basic BSM framework has been studied explicitly in [20], showing that (5.1) where is clearly the Sharpe ratio [19], and Φ −1 ( ⋅ ) is the quantile function associated with Φ( ⋅ ).
Equation (5.1) thus shows that ℚ is a distortion of ℙ, obtained via a quantile shift, which induces a re-weighting of the probabilities.Simple manipulations also bring to Equation (5.1) is an example of application of the function known as Wang transform [20], which represents a useful distortion function in the actuarial literature.Such a function is convex for c > 0 and concave for c < 0. When c = 0, no distortion clearly takes place.Now, it is not difficult to observe that the Wang transform is just a special case of pseudo-sum-or, if one prefers, of pseudo-difference-in fact This is just the first simple example.In fact, one could also play with models for which the Wang transform is substituted by the more general where F is the CDF of a given unimodal random variable linked to the log returns of an asset of interest.Incidentally, both equations (5.3) and (5.5) also represent two proper Lorenz functions of the socioeconomic inequality literature [13,21], whose application in risk management has been investigated in [22], among others.Further examples could be given, for example, by introducing new derivatives like Stoptions [17].
However, let us now come back to our pseudo-sums ⊕ b and ⊕ −b and to their financial interpretations, in line with the price generator views of [23].

The pseudo-sum ⊕ b and a zero-coupon default-free convertible bond
When the induced binary operation is ⊕ b , then for x 1 , x 2 ∈ [0, ∞), and b ∈ (0, 1), one can observe that (5.6) where D b is a mean-one conjugate power Dagum (CPD) distributed random variable (see appendix A for more details) with CDF: (5.7) The right-hand side of equation (5.6) is nothing but the arbitrage-free value of a zero-coupon defaultfree convertible bond with face value x 1 ≥ 0 and conversion value x 2 ≥ 0. The positive random variable D b > 0 is the gross return on the convertible bond's underlying stock.Incidentally, notice that D b also defines a Radon-Nikodym derivative [19], and this further justifies the use of CPD random variables, as, for instance, in [24].
To prove that equation (5.6) holds, notice that its right-hand side can be written as .This put value is the integral of the distribution function of the mean-one CPD random variable, that is: (5.9) . This proves that equation (5.6) holds.

The pseudo-sum ⊕ −b and a zero-coupon defaultable bond
When the induced binary operation is ⊕ −b , then for x 1 , x 2 ∈ (−∞, ∞], and b > 0, one can notice that (5.10) where Z is a standard logistic random variable with a distribution function [13]: (5.11) The right-hand side of equation (5.10) is now the arbitrage-free value of a zero-coupon defaultable bond with face value x 1 ∈ (−∞, ∞] and whose underlying security has initial value x 2 ∈ (−∞, ∞].The real-valued random variable bZ is the change in the value of the operations of the firm. To prove that equation (5.10) holds, notice that its right-hand side can be written as (5.12) The multiplier of b is the value of a put written on Z and struck at b .This put value is the integral of the standard logistic distribution function, that is: (5.13)This proves that equation (5.10) holds.

New connections between notable distributions
Exploiting the pseudo-sums of §4 and the relative GMP, we now offer new ways of connecting random variables largely used in the financial, actuarial and economic inequality literature [13].These new links may open to better characterizations, as well as to more reliable modelling of important economic quantities.Interestingly, the random variables under scrutiny are all recurrent objects in extreme value theory [25], suggesting the existence of some non-trivial research paths for the future, as we sketch in §7.
In appendices B and C, the interested reader can find proofs and additional results.

New relationship between Weibull and Dagum distributions
Suppose that a positive random variable X > 0 is exponentially distributed with SF e −λx , x > 0, where λ > 0 is a positive intensity parameter.For p > 0, let (6.1) W = X 1 p be a positive power of this exponential random variable.It is well-known that the distribution of W is Weibull with SF e −(θw) p , w > 0, where θ = λ 1 p is the rate parameter [13,14].Now let K = 1 W .Then, the distribution of K is called inverse Weibull with CDF equal to (6.2) royalsocietypublishing.org/journal/rsos R. Soc.Open Sci.11: 231690 contingent-claim value.In both cases, we are then able to connect two distinct probability distributions in a novel way.In our view, future research should extend the use of pseudo-analytic arguments in pricing and financial modelling, as well as in probability and statistics.
For instance, a further use of pseudo-analysis, with great potential in both financial mathematics and probability, is linked to the observations below, following straight from insights in Aczél [1], Castagnoli [3] and Pap [6].
Let us first give a definition.
Definition 7.1 (G-normality).Let G be a generator satisfying the requirements in §2, and G −1 its inverse.A random variable Y is said to be G-normal, if G −1 (Y ) is normally distributed, so that its probability density function is In other words, if Φ(x; μ, σ) indicates CDF of a normal random variable with mean μ and standard deviation σ, then Y is G-normal if its CDF is Φ(G −1 (y); μ, σ), and the density (probability density function, PDF) is the one given in equation (7.1).
An immediate example is represented by the lognormal random variable, which one obtains by exponentiating a normal random variable [26].In fact, if X is normal, then Y = G(X) = e X is clearly lognormal.However, this implies that G −1 (y) = log(y) and that the CDF of Y is Φ(log(y); μ, σ), with the corresponding PDF: 1 as expected.Now, let X 1 , X 2 , . . ., X n be a collection of i.i.d.random variables and define the pseudo-partial-sum: Then, for n → ∞, the G-standardized quantity will converge to a standard G-normal random variable with CDF Φ(G −1 (x); 0, 1), where σ[G −1 (Y n )] is the finite standard deviation of If the i.i.d.random variables X 1 , X 2 , . . ., X n are positive and their density function is square-integrable, and we choose G(X) = e X , we can immediately recognize the so-called Gibrat's law [27], in the convergence of equation (7.2) to its lognormal limit.A natural question is therefore to ask what do we get for different generators; and a very interesting case emerges for G(X) = log(X).In fact, we immediately get that where log ∑ i = 1 n e x i is the LogSumExp function LSE(x 1 , . . ., x n ) commonly used in machine learning as a smooth approximation to the maximum function max (x 1 , . . ., x n ) [16].
If, again, we assume that the random variables X 1 , X 2 , . . ., X n are positive, i.i.d., and with a square-integrable density, then the limit law of the pseudo-partial-sum S n strongly recalls a Gumbel (Type I) distribution [13], suggesting a non-trivial but yet-to-be-investigated connection with fundamental results in extreme value theory, for example, the Fisher-Tippett-Gnedenko theorem [25].
The alleged connection strengthens, if we look back at the distributions of §6, and recognize both the Gumbel and the Weibull as two of the three extreme value distributions, and we see that both are linked to the GMP.Curiously, the only extreme value distribution missing in the list seems to be the Fréchet but also here we can notice a simple fact.As we wrote in §4, the Pareto random variable is the variable showing the GMP when x ⊕ y = x × y, and the Pareto is a fat-tailed random variable linked to the Fréchet [25].All in all, it seems that there are still a large amount of things to be clarified and possibly discovered using pseudo-sums as we suggested.

(4. 6 )
G(a) = − blog a, a > 0, b > 0. The right-hand side of equation (4.6) can be written as the negative of a logarithm whose base is e 1 b .As a result, we treat the parameter b > 0 in equation (4.6) as the base controller.
royalsocietypublishing.org/journal/rsos R. Soc.Open Sci.11: 231690 2 > 0, and E[D b ] = 1, as per appendix A. Now observe that the multiplier of x 2 is the value of a put written on D b and struck at x 1 x 2