A Black–Scholes inequality: applications and generalisations

The space of call price curves has a natural noncommutative semigroup structure with an involution. A basic example is the Black–Scholes call price surface, from which an interesting inequality for Black–Scholes implied volatility is derived. The binary operation is compatible with the convex order, and therefore a one-parameter sub-semigroup gives rise to an arbitrage-free market model. It is shown that each such one-parameter semigroup corresponds to a unique log-concave probability density, providing a family of tractable call price surface parametrisations in the spirit of the Gatheral–Jacquier SVI surface. An explicit example is given to illustrate the idea. The key observation is an isomorphism linking an initial call price curve to the lift zonoid of the terminal price of the underlying asset.


Introduction
We define the Black-Scholes call price function C BS : [0, ∞) × [0, ∞) → [0, 1] by the formula where ϕ(z) = 1 √ 2π e −z 2 /2 is the density and (x) = x −∞ ϕ(z)dz is the distribution function of a standard normal random variable. Recall the financial context of this definition: a market with a risk-free zero-coupon bond of unit face value, maturity T and initial price B 0,T ; a stock with initial price S 0 that pays no dividend; and a European call option written on the stock with maturity T and strike price K. In the Black-Scholes model, the initial price C 0,T ,K of the call option is given by the formula where σ is the volatility of the stock price. In particular, the first argument of C BS plays the role of the moneyness κ = KB 0,T /S 0 and the second argument plays the role of the total standard deviation y = σ √ T of the terminal log stock price. The starting point of this note is the following observation. While it is fairly straightforward to prove Theorem 1.1 directly, the proof is omitted as it is a special case of Theorem 3.8 below. Indeed, the purpose of this paper is to try to understand the fundamental principle that gives rise to such an inequality. As a hint of things to come, it is worth pointing out that the expression y 1 + y 2 appearing on the left-hand side of the inequality corresponds to the sum of the standard deviations-not the sum of the variances. From this observation, it may not be surprising to see that a key idea underpinning Theorem 1.1 is that of adding comonotonic-not independent-normal random variables. These vague comments will be made precise in Theorem 2.16 below.
Before proceeding, we re-express Theorem 1.1 in terms of the Black-Scholes implied total standard deviation function, defined for κ > 0 to be the inverse function In particular, the quantity Y BS (κ, c) denotes the implied total standard deviation of an option of moneyness κ whose normalised price is c. We find it notationally convenient to set Y BS (κ, c) = ∞ for c ≥ 1. With this notation, we have the following interesting reformulation which requires no proof: Corollary 1.2 For all κ 1 , κ 2 > 0 and (1 − κ i ) + < c i < 1 for i = 1, 2, we have with equality if and only if where y i = Y BS (κ i , c i ) for i = 1, 2.
To add some context, we recall the following related bounds on the functions C BS and Y BS ; see [31,Theorem 3.1].
In [31], Theorem 1.3 was used to derive upper bounds on the implied total standard deviation function Y BS by selecting various values of p to insert into the inequality.
The function ( −1 (·) + y) has appeared elsewhere in various contexts. For instance, it is the value function for a problem of maximising the probability of hitting a target considered by Kulldorff [27,Theorem 6]. (Also see the book of Karatzas [23,Sect. 2.6].) In insurance mathematics, the function is often called the Wang transform and was proposed in [32, Definition 2] as a method of distorting a probability distribution in order to introduce a risk premium. In a somewhat unrelated context, Kulik and Tymoshkevych [26, Proof of Theorem 2: Statement I] observed, while proving a certain log-Sobolev inequality, that the family of functions ( −1 (·) + y) indexed by y ≥ 0 forms a semigroup under function composition. We shall see that this semigroup property is the essential idea of our proof of Theorem 1.1 and its subsequent generalisations.
The rest of this paper is arranged as follows. In Sect. 2, we introduce a space of call price curves and explore some of its properties. In particular, we shall see that it has a natural noncommutative semigroup structure with an involution. The binary operation has a natural financial interpretation as the maximum price of an option to swap one share of the first asset for a fixed number of shares of a second asset. In Sect. 3, we introduce a space of call price surfaces and provide in Theorem 3.2 equivalent characterisations in terms of either a single supermartingale or two martingales. Furthermore, it is shown that the binary operation is compatible with the decreasing convex order, and therefore a one-parameter sub-semigroup of the semigroup of call price curves can be associated with an arbitrage-free market model. A main result of this article is Theorem 3.11: each such one-parameter sub-semigroup corresponds to a unique (up to translation and scaling) log-concave probability density, generalising the Black-Scholes call price surface and providing a family of reasonably tractable call surface parametrisations in the spirit of the SVI surface. In Sect. 4, further properties of these call price surfaces are explored, including the asymptotics of their implied volatility. In addition, an explicit example is given to illustrate the idea, and is calibrated to real-world call price data. In Sect. 5, the proof of Theorem 3.11 is given.
The key observation is that the Legendre transform is an isomorphism converting the binary operation on call price curves to function composition. The isomorphism has the additional interpretation as the lift zonoid of a related random variable.

The space of call price curves
For motivation, consider a market with two (non-dividend-paying) assets whose prices at time t are A t and B t . We assume that both prices are always nonnegative and that the initial prices A 0 and B 0 are strictly positive. We further assume that there exists a martingale deflator Y = (Y t ) t≥0 , that is, a positive adapted process such that the processes Y A and Y B are both martingales. The assumption of the existence of a martingale deflator ensures that there is no arbitrage in the market. (Conversely, one can show that no-arbitrage in a discrete-time model implies the existence of a martingale deflator, even if the time horizon is infinite and if the market does not admit a numéraire portfolio; see [30,Theorem 2.10].) Now introduce an option to swap one share of asset A with K shares of asset B at a fixed time T > 0, so the payout is (A T − KB T ) + . If the asset B is a risk-free zero-coupon bond of maturity T and unit face value, then the option is a standard call option. It will prove useful in our discussion to let asset B be arbitrary, but we still refer to this option as a call option.
There is no arbitrage in the augmented market if the time-t price of this call option is In particular, setting the initial price of this option, normalised by the initial price of asset A, can be written as where the moneyness is given by κ = KB 0 A 0 . The above discussion motivates the following definition.
is a call price curve if there exist nonnegative random variables α and β defined on some probability space such that for all κ ≥ 0, in which case the ordered pair (α, β) of random variables is called a basic representation of C. The set of all call price curves is denoted C.
From a practical perspective, the normalised call price C(κ) is directly observed, while the law of the pair (α, β) is not. Therefore, a theme of this paper is to try to express notions in terms of the call price curve. Here is a first result of this type. 2) There exists a nonnegative random variable S with E[S] ≤ 1 such that and more generally that where C denotes the right derivative of C.
Proof The implications 1) ⇒ 3) and 2) ⇒ 3) are straightforward, so their proofs are omitted. Furthermore, the claim that the distribution of S can be recovered from C is essentially the Breeden and Litzenberger [3, Eq. (1)] formula.
3) ⇒ 2): By convexity, the right derivative C is defined everywhere and is nondecreasing and right-continuous. Furthermore, since (1 − κ) + ≤ C(κ) ≤ 1 for Fig. 1 The graph of a typical function C ∈ C. The random variable S is introduced in Definition 2.11 all κ, we have −1 ≤ C (κ) ≤ 0 for all κ. Let S be a random variable such that by Fubini's theorem and the absolute continuity of the convex function C.
It remains to show that either 2) ⇒ 1) or 3) ⇒ 1). That is, we must construct a basic representation (α, β) from either the random variable S or the function C. We give a construction showing 2) ⇒ 1) in the proof of Theorem 2.15, and a rather different construction showing 3) ⇒ 1) in the proof of Theorem 3.2. To avoid repetition, we omit a construction here.
By definition, a call price curve C is determined by two random variables α and β. However, the distribution of the pair (α, β) cannot be inferred solely from C. In contrast, Theorem 2.2 above says that a call price curve C is also determined by a single random variable S, and furthermore, the law of S is unique and can be recovered from C. This observation motivates the following definition. Definition 2.3 Given a call price curve C ∈ C, suppose that S is a nonnegative random variable such that C(κ) = 1 − E[S ∧ κ] for all κ ≥ 0. Then S is called a primal representation of C. Remark 2.4 As hinted by the name primal, we shall shortly introduce a dual representation. Figure 1 plots the graph of a typical element C ∈ C.
Remark 2.5 An example of an element of C is the Black-Scholes call price function C BS (·, y) for any y ≥ 0. A primal representation is where Z ∼ N (0, 1) has the standard normal distribution.
Remark 2. 6 We note that there are alternative financial interpretations of call price curves C ∈ C in the case C(∞) > 0. One popularised by Cox and Hobson [12] is to model the primal representation as the terminal price S of an asset experiencing a bubble in the sense that the price process discounted by the price of the risk-free T -zero-coupon bond is a strict local martingale under a fixed T -forward measure. For this interpretation, the option payout must be modified: rather than the payout of a standard (naked) call option, the quantity C(κ) in this interpretation models the normalised price of a fully collateralised (covered) call with payout One could argue that there are two related shortcomings of this interpretation. Firstly, this type of bubble phenomenon can only arise in continuous-time models since in discrete time (even over an infinite horizon), a nonnegative local martingale with an integrable initial condition is necessarily a true martingale. Secondly, in the case E[S] < 1 where the underlying stock is not priced by expectation, it is not clear from a modelling perspective why the market should then assign the price C(κ) = E[(S − κ) + + 1 − S] to the call option. Both shortcomings highlight the subtlety of continuous-time arbitrage theory, in particular, the sensitive dependence on the choice of numéraire on the definition of arbitrage (and related arbitrage-like conditions). See the paper of Herdegen and Schweizer [17] for a discussion of these issues in the context of a semimartingale market model where prices may vanish.

The involution
There is a natural involution on the space of call prices: Definition 2.7 Given a call price curve C ∈ C with basic representation (α, β), the function C is the call price curve with basic representation (β, α).
This leads to a straightforward financial interpretation of the involution. As described above, we may think of C(κ) as the initial price, normalised by A 0 , of the option to swap one share of asset A for K shares of asset B, where KB 0 = κA 0 . Then C (κ) is the initial price, normalised by B 0 , of the option to swap one share of asset B for K shares of asset A, where K A 0 = κB 0 .
We now record a fact about this involution , expressed directly in terms of call prices. The proof is a straightforward verification and hence omitted.

Remark 2.9
As an example, notice that for the Black-Scholes call function, we have C BS (·, y) = C BS (·, y) for all y ≥ 0 by the classical put-call symmetry formula.

Remark 2.10
The function C is related to the well-known perspective function of the convex function C defined by (η, κ) → ηC(κ/η); see for instance the book of Boyd and Vanderberghe [6, Sect. 3.2.6].
As hinted above, we can define another random variable in terms of this involution: Definition 2.11 Given a call price curve C ∈ C, a nonnegative random variable S is a dual representation of C if S is a primal representation of the call price curve C .
That this dual random variable should be called a representation of a call price is due to the following observation. Again the proof is straightforward and hence omitted. Theorem 2.12 Given a call price C ∈ C with dual representation S , we have for all κ ≥ 0.
In particular, we have P[

The binary operation
We have introduced one algebraic operation, the involution , on the set of call price curves. We now come to the second algebraic operation which will help to contextualise the Black-Scholes inequality of Theorem 1.1. To motivate it, consider a market with three assets with time-t prices A 1,t , A 2,t and B t . We know the initial cost of an option to swap one share of asset A 1 with H 1 shares of asset B, as well as the initial cost of an option to swap one share of asset B with H 2 shares of asset A 2 , for various values of H 1 and H 2 , where all the options mature at a fixed date T > 0. Our goal is to find an upper bound on the cost of an option to swap one share of asset A 1 for K shares of asset A 2 , for the same maturity date T .
Definition 2.14 For call price curves C 1 , C 2 ∈ C, define a binary operation • on C by where the supremum is taken over nonnegative random variables α 1 , β, α 2 defined on the same probability space such that (α 1 , β) is a basic representation of C 1 and (β, α 2 ) is a basic representation of C 2 .
At this stage, it is not immediately clear that given two call price curves C 1 and C 2 , one can find a triple (α 1 , β, α 2 ) satisfying the definition of the binary operation •, and in principle, we should complete the definition with the usual convention that sup ∅ = −∞. Fortunately, this caveat is not necessary, as can be deduced from the following result.

Theorem 2.15
For call price curves C 1 , C 2 ∈ C, we have where the supremum is taken over random variables S 1 and S 2 defined on the same space, where S 1 is a primal representation of C 1 and S 2 is a dual representation of C 2 .
Proof First, let S 1 be a primal representation of C 1 and S 2 a dual representation of C 2 , defined on the same probability space. We shall exhibit random variables For the construction, we introduce Bernoulli random variables γ 1 , γ 2 , δ 1 , δ 2 , independent of (S 1 , S 2 ) and each other, with If P[S 1 = 0] < 1, then set and if S 1 = 0 almost surely, set a 1 = 2δ 1 and if S 2 = 0 almost surely, set a 2 = 2δ 2 and b 2 = 2(1 − δ 2 ). Finally, set It is easy to check that the triplet (α 1 , β, α 2 ) is the desired representation. This shows For the converse inequality, given a basic representation (α 1 , β) of C 1 and a basic representation (β, α 2 ) of C 2 defined on the same probability space ( , F, P), we let and define an absolutely continuous measure P β by dP β dP = β. It is easy to check that S 1 is a primal representation of C 1 and S 2 is a dual representation of C 2 under P β and Given the laws of two random variables X 1 and X 2 and a convex function g, it is well known that the quantity E[g(X 1 + X 2 )] is maximised when X 1 and X 2 are comonotonic. See for instance the paper of Kaas et al. [22,Theorem 6] for a proof. By rewriting the expression we see that the supremum defining the binary operation • is achieved when S 1 and S 2 are countermonotonic. We recover this fact in the following result, which also continues our theme of expressing notions directly in terms of the call prices. In this case, the binary operation • can be expressed via a minimisation problem: Theorem 2.16 Let S 1 be a primal representation of C 1 ∈ C and S 2 a dual representation of C 2 ∈ C, where S 1 and S 2 are defined on the same probability space. Then There is equality if the following hold true: 1) S 1 and S 2 are countermonotonic, and In particular, we have Proof Recall that for real a, b, we have Also assume that S 1 and S 2 are countermonotonic so that Notice that in this case, we have and hence there is equality in (2.1) above.

Remark 2.18
Given the conclusion of Theorem 2.16, we caution that the operation • is not the well-known inf-convolution ; however, we shall see in Sect. 5.2 below that • is related to the inf-convolution via an exponential map.
In light of the formula for the binary operation • appearing in Theorem 2.16, the Black-Scholes inequality of Theorem 1.1 amounts to the claim that for y 1 , y 2 ≥ 0, we have This is a special case of Theorem 3.8, stated and proved below.
We now come to the key observation of this paper. To state it, we distinguish two particular elements E, Z ∈ C defined by Note that the random variables representing E and Z are constant, with S = 1 = S representing E and S = 0 = S representing Z. The following result shows that C is a noncommutative semigroup with respect to • with involution , where E is the identity element and Z is the absorbing element. The proof is straightforward, and hence omitted.
We conclude this section by introducing two useful subsets of the set of call price curves.

Definition 2.20 Let
That is, fix a call price curve C ∈ C with primal representation S and dual representation S . The call price curve C is in

Remark 2.21
As an example, notice that for the Black-Scholes call function, we have The subsets C + and C 1 are closed with respect to the binary operation.

Proposition 2.22
Given C 1 , C 2 ∈ C, we have: Proof By Theorem 2.16, we have where S 1 is a primal representation of C 1 , S 2 is a dual representation of C 2 and S 1 and S 2 are countermonotonic. For implication 1), note that For implication 2), apply Theorem 2.19, 3) and the fact that (C + ) = C 1 .

The space of call price surfaces
With the motivation at the beginning of Sect. 2, we consider the family of prices of options when the maturity date is allowed to vary. We now introduce the following definition.
such that there exists a pair of nonnegative martingales (α t , β t ) t≥0 such that α 0 = 1 = β 0 and Our goal is to understand the structure of the space of call price surfaces, and relate this structure to the binary operation • introduced in the last section. 1) C is a call price surface.
2) There exists a nonnegative supermartingale S such that S 0 = 1 and 3) There exists a nonnegative supermartingale S such that S 0 = 1 and for all (κ, t).

4)
For all ε > 0, there exist bounded nonnegative martingales α and β such that for all (κ, t) and such that α t Proof The implications n) ⇒ 5) for 1 ≤ n ≤ 4 are easy to check by the conditional version of Jensen's inequality.
The implications 5) ⇒ 2) and 5) ⇒ 3) are proved as follows. By Theorems 2.2 and 2.12, there exist families of random variables (S t ) t≥0 and (S t ) t≥0 such that for all κ ≥ 0 and t ≥ 0. The assumption that C(κ, ·) is nondecreasing implies that both families of random variables (or more precisely, both families of laws) are nondecreasing in the decreasing-convex order. The implications then follow from Kellerer's theorem [24,Theorem 3]. Implication 4) ⇒ 1) is obvious. So it remains to show the implication 5) ⇒ 4). Fix ε > 0 and let It is straightforward to verify thatC satisfies 5). Hence there exists a nonnegative supermartingale α such that But sinceC(κ, t) = 0 for all κ ≥ 1 + ε, we can conclude that for all t, we have both E[α t ] = 1 and α t ≤ 1 + ε a.s. In particular, α is a true martingale so that or equivalently Remark 3.3 The implication 5) ⇒ 2) is well known, especially in the case where C(∞, t) = 0 for all t ≥ 0 where the supermartingale S is a martingale. See for instance the paper of Carr and Madan [8,Sect. 3]. However, the implication 5) ⇒ 4) seems new.

One-parameter semigroups
Returning to the topics of Sect. 2, we note that the operation • interacts well with the natural partial ordering on the space of call price curves.

Proposition 3.4 For any
Proof Let S 1 be a primal representation of C 1 and S 2 a dual representation of C 2 . Suppose S 1 and S 2 are independent. Then by Theorem 2.15, we have by first conditioning on S 2 and applying the conditional Jensen inequality, and then using the bound E[S 1 ] ≤ 1. The other implication is proved similarly.
Combining Theorem 3.2 and Proposition 3.4 brings us to the main observation of this paper: If (C(·, t)) t≥0 is a one-parameter sub-semigroup of C, then C(·, ·) is a call price surface. Fortunately, we shall see that all such sub-semigroups can be explicitly characterised and are reasonably tractable.
With the motivation of finding a tractable family of call price surfaces, we now study a family of one-parameter sub-semigroups of C. We use the notation y, rather than t, to denote the parameter since we think of y not literally as the maturity date of the option, but rather as an increasing function of that date. For instance, in the Black-Scholes framework, we have y = σ √ t corresponding to total standard deviation. We make use of the following notation. For a probability density function f , let for y ∈ R and κ ≥ 0. Note that for y ≥ 0, where ϕ is the standard normal density. It will be useful to distinguish a special class of densities.

Definition 3.5 A probability density function
We use repeatedly a useful characterisation of log-concave densities due to Bobkov [ Remark 3.7 In Bobkov's formulation of Proposition 3.6 above, condition 2) was replaced with 2 ) F (F −1 (·) + y) is concave on (0, 1) for all y ≥ 0. Note that conditions 2) and 2 ) are equivalent. To see why, note that the derivative of . Since the function F −1 is strictly increasing, the claim amounts to the statement that g is concave if and only g is nonincreasing.
Let We now present a family of one-parameter sub-semigroups of C.

Theorem 3.8 Let f be a log-concave probability density function. Then
for all y 1 , y 2 ≥ 0.
While Theorem 3.8 is not especially difficult to prove, we offer two proofs with each highlighting a different perspective on the operation •. The first is below and the second is in Sect. 5.1.

Proof of Theorem 3.8
Letting Z be a random variable with density f , note that f (Z +y)/f (Z) is a primal representation of C f (·, y). Note also that by log-concavity of f , the function z → f (z + y)/f (z) is nonincreasing when y ≥ 0. Similarly, f (Z − y)/f (Z) is a dual representation of C f (·, y) and z → f (z − y)/f (z) is nondecreasing. In particular, the random variables f (Z + y 1 )/f (Z) and f (Z − y 2 )/f (Z) are countermonotonic, and hence by Theorem 2.16, we have The conclusion follows from changing variables in the integral on the right-hand side.
The upshot of Proposition 3.4 and Theorem 3.8 is that given a log-concave density f , the function C f (κ, ·) is nondecreasing for each κ ≥ 0. Hence, given an increasing function ϒ , we can conclude from Theorem 3.2 that we can define a call price surface by The above formula is reasonably tractable and could be seen to be in the same spirit as the SVI parametrisation of the implied volatility surface given by Gatheral and Jacquier [15]. Note that we can recover the Black-Scholes model by setting the density to f = ϕ, the standard normal density, and the increasing function to where σ is the volatility of the stock. We provide another worked example in Sect. 4.2.
At this point, we explain the name of this section. We recall the definitions of terms popularised by Hirsch et al. [

Definition 3.9
A lyrebird is a family X = (X t ) t≥0 of integrable random variables such that there exists a submartingale Y = (Y t ) t≥0 defined on some (possibly different) probability space such that the random variables X t and Y t have the same distribution for all t ≥ 0. A peacock X is a family of random variables such that both X and −X are lyrebirds, i.e., there exists a martingale with the same marginal laws as X.
The term peacock is derived from the French acronym PCOC, processus croissant pour l'ordre convexe, and lyrebird is the name of an Australian bird with peacock-like tail feathers.
Combining Proposition 3.4 and Theorem 3.8 yields the following tractable family of lyrebirds and peacocks.

The family of random variables −S = (−S t ) t≥0 is a lyrebird. If the support of f is of the form (−∞, R], then S is a peacock.
Note that the semigroup (C f (·, y)) y≥0 does not correspond to a unique logconcave density. Indeed, fix a log-concave density f and set for all κ ≥ 0, y ∈ R.
However, we shall see below that the semigroup does identify the density f up to arbitrary scaling and centring parameters. Also, note that by varying the scale parameter λ, we can interpolate between two possibilities. On the one hand, we have for all κ ≥ 0 and y ∈ R that and, on the other hand, when y = 0, that Recall that the call price curve E(κ) = (1 − κ) + is the identity element for the binary operation •. Hence the family C triv defined by C triv (·, y) = E for all y ≥ 0 is another example of a sub-semigroup of C.
Similarly, the call price curve Z(κ) = 1 is the absorbing element for •. Hence the family C null defined by C null (·, 0) = E and C null (·, y) = Z for all y > 0 is yet another example of a sub-semigroup of C.
The following theorem says that the above examples exhaust the possibilities.
3) C = C f for a log-concave density f . In case 3), the density f is uniquely defined by the semigroup, up to centring and scaling.
The proof appears in Sect. 5.1.

Remark 3.12
One could certainly consider other binary operations on the space C which are also compatible with the partial order. For instance, we could let where the primal representation S 1 of C 1 is independent of the dual representation S 2 of C 2 . Note that this binary operation ♣ is commutative, and indeed we have where S 2 is a primal representation of C 2 , again independent of S 1 . In fact, the binary operation ♣ can be expressed (somewhat awkwardly) in terms of the call price curves C 1 and C 2 as As described above, one could construct call price surfaces by studying oneparameter semigroups for this binary operation ♣. Indeed, such semigroups are easy to describe since their primal representations are essentially exponential Lévy processes. Unfortunately, the call prices given by an exponential Lévy process are not easy to write down in general. However, we have seen that the one-parameter semigroups of call prices for the binary operation • are extremely simple to write down. It is the simplicity of these formulae that is the claim to practicality of the results presented here.

Calibrating the surface 4.1 An exploration of C f
We have argued that if f is a log-concave density and ϒ an increasing function, then the family {C f (κ, ϒ(t)) : κ ≥ 0, t ≥ 0} is a call price surface as defined in Sect. 3.1, where the notation C f is defined in Sect. 3.2. The motivation of this section is to bring this observation from theory to practice. In particular, to calibrate the functions f and ϒ to market data, it is useful to have at hand some properties, including asymptotic properties, of the function C f .
In what follows, we assume that the density f has its support of the form [L, R] for some constants −∞ ≤ L < R ≤ +∞. Recall that we assume f is continuous on [L, R]. Now let Z be a random variable with density f . For each y ∈ R, define a nonnegative random variable by Note that S (y) is well defined since L < Z < R almost surely, and hence f (Z) > 0 almost surely. In this notation, we have for all y ∈ R that so that by Theorem 2.2, we have C f (·, y) ∈ C and S (y) is a primal representation of C f (·, y). Note that S (y) = 0 almost surely for |y| ≥ R − L, while for |y| < R − L, we have In particular, for y > 0, we have where the sets C + and C 1 are defined in Sect. 2.3. By changing variables, we find that a dual representation of C f (·, y) is given by and therefore C f (·, y) = C f (·, −y).
It is interesting to observe that the call price surface C f satisfies the put-call symmetry formula C f (·, y) = C f (·, y) if the density f is an even function. Put-call symmetry has found applications for instance in the semi-static hedging of certain barrier-type contingent claims; see the paper of Carr and Lee [7, Theorem 5.10]. Due to the implication 1) ⇒ 2) in Proposition 3.6, the map z → f (z+y) f (z) is nonincreasing for y ≥ 0. Let with the convention that sup ∅ = L. Note that f (z+y) f (z) ≥ κ if and only if d f (y, κ) ≥ z. With this notation, we have Theorem 4.2 below might seem like a curiosity, but in fact is a useful alternative formula for computing C f numerically, given the density f . In particular, the following formula does not require the evaluation of the function d f defined above. This is a generalisation of Theorem 3.1 of [31]. The proof is essentially the same, but included here for completeness. We use the notation Proof Fix κ, p, y and let z = F −1 (p). Note that Given κ, there is equality when z = d f (κ, y).
The next result gives an asymptotic expression for call prices at short maturities and close to the money. In what follows, we use the notation where f is the right derivative of f . Recall that f always exists on the interval (L, R).

Theorem 4.3
As ε ↓ 0, we have that We only consider the case L < a < R, as the cases a = L and a = R are similar. Fix x and a − L < ε < R − a and write 1 ε C f (e εx , ε) = I 1 + I 2 + I 3 , where Note that for a ≤ z ≤ R − ε, we have so that as ε ↓ 0. In particular, we have I 2 → 0. Finally, for the third term, apply the put-call parity formula to get again by the dominated convergence theorem. The conclusion follows from another application of put-call parity and recombining the integrals.
There are two interesting consequences of Theorem 4.3. The first is that the density f can be recovered from short-time asymptotics. We use the notation for 0 ≤ p ≤ 1. The proof follows the same pattern as that of Theorem 4.2, so it is omitted.

Remark 4.5
Given the functionĤ f , we can recover f , up to centring, as follows. Fix 0 < p 0 < 1 and set F (0) = p 0 . Then we have We note in passing that the call price function C f satisfies a nonlinear partial differential equation featuring the functionĤ f when f is suitably well behaved as specified in the following proposition. Then Proof By log-concavity, we have for all z ∈ R and y > 0 that and hence Also, by the strict log-concavity of f , the map z → f (z+y) f (z) is strictly decreasing. This shows that d f (κ, y) is finite for all κ > 0 and that By the differentiability of f and the implicit function theorem, the function d f is differentiable for (κ, y) ∈ (0, ∞) × (0, ∞).
One checks that The conclusion follows.
We now comment on a second interesting consequence of Theorem 4.3. Note that for ε ↓ 0, the limit for the Black-Scholes call function is The function H ϕ has an interesting financial interpretation. Recall that in the Bachelier model, assuming zero interest rates, the initial price of a call option of maturity T and strike K is given by where S 0 is the initial price of the stock and σ is its arithmetic volatility. Hence H ϕ can be interpreted as a normalised call price function in the Bachelier model. Following the motivation of this section, we are interested not only in the call price surface itself, but also in the corresponding implied volatility surface. Recall that the function Y BS is defined by We use the notation Recall that if the normalised price of a call of moneyness κ and maturity t is given by C f (κ, ϒ(t)), then the option's implied volatility is given by 1  With this setup, we now present a result that gives the asymptotics of the implied volatility surface in the short-maturity, close-to-the-money limit.

Theorem 4.8 We have as
and more generally that where Ba (x, c) is defined by Proof Fix x ∈ R and δ > 0. Let λ = Ba (x, H f (x) + δ). By Theorem 4.3, there exists ε 0 > 0 such that for all 0 < ε < ε 0 . Hence C f (e εx , ε) ≤ C BS (e εx , λε) and therefore lim sup A lower bound is established similarly. The conclusion follows from the continuity of Ba .
For the final theorem of this section, we fix the maturity date and now compute extreme strike asymptotics of the implied volatility. In what follows, we say that an eventually positive function g varies regularly at infinity with exponent α if Regular variation at zero is defined analogously.

Theorem 4.9
Suppose that f is a log-concave density such that − log •f • log varies regularly at infinity with exponent a > 0 and varies regularly at zero with exponent −b < 0. Then for y > 0, we have The key observation is that f θ is Lebesgue-integrable if and only if θ > 0.
Indeed, since f is integrable and log-concave, there exist constants A, B with B > 0 such that f (z) ≤ e A−B|z| , and hence f θ is bounded from above by an integrable function for θ > 0 and bounded from below by a non-integrable function for θ ≤ 0. Fix y > 0. The moment-generating function of log S (y) is calculated as By assumption, or equivalently f (z + y) = f (z) e −by +δ(z) , where δ(z) → 0 as z → −∞, yielding the expression The exponent of f on the right-hand side is eventually positive, implying I 1 is finite, if and the exponent is eventually negative, implying I 1 is infinite, if On the other hand, f (z + y) = f (z) e ay +ε(z) , where ε(z) → 0 as z → ∞. Writing we see that for any p ≥ 0, the exponent of f on the right-hand side is eventually positive, implying I 2 is finite.
Therefore, we have shown that

A parametric example
In this section, we consider a parametrised family of log-concave densities in which several interesting calculations can be performed explicitly. We then try to fit this family to real call price data as a proof of concept.
Consider the family of densities of the form for strictly positive parameters a, b, r and real c, with normalising constant where¯ (x, θ) = ∞ x z θ−1 e −z dz is the complementary incomplete gamma function. It is straightforward to check that f is a log-concave probability density.
Letting a = b = r −1/2 and c = 0 and then sending r → ∞ recovers the Black-Scholes model f → ϕ. Roughly speaking, a controls the left wing, b the right wing, c the at-the-money skew and r the at-the-money convexity. Although there are four parameters, recall from Sect. 3.2 that we have C f (λa,λb,λc,r) (κ, y) = C f (a,b,c,r) (κ, λy) for κ ≥ 0, y ≥ 0 and λ > 0. Hence there is no loss of generality if we impose for instance that abr = 1, leaving us with only three free parameters.
The distribution function is given explicitly by The function d f can be calculated explicitly when the absolute log-moneyness | log κ| is sufficiently large; we get which can be calculated numerically for instance by the bisection method. The call price curve can be calculated by the formula Note that this formula is rather explicit when the absolute log-moneyness is sufficiently large, and furthermore, it is numerically tractable in all cases.
This choice of f has the advantage that call prices can be calculated very quickly. Also, for other vanilla options, numerical integration is very efficient since the density function f is smooth and decays quickly at infinity. Alternatively, rejection sampling is available, since the density is bounded for instance by a Gaussian density.
When it comes to calibrating the model, we must find parameters a, b, c, r and an increasing function ϒ such that for all (κ, t) ∈ S, where C obs (κ, t) is the observed normalised price of a call option of moneyness κ and maturity t and S is the set of pairs (κ, t) for which there is available market data. Equivalently, we fit the parameters a, b, c, r and the function ϒ to try to approximate the observed implied volatility surface. For this exercise, E-mini S&P MidCap 400 call options call and put option price data were downloaded from the CME Group FTP site [11], for maturities t 1 = 0.2, t 2 = 0.4, t 3 = 0.7 years for all available strikes. Letting be the set of available strikes for maturity t i , we have |S 1 | = 251, |S 2 | = 248 and |S 3 | = 232 observations. There are six parameters to find, namely a, c, r and ϒ(t 1 ) = y 1 , ϒ(t 2 ) = y 2 , ϒ(t 3 ) = y 3 , to fit 251 + 248 + 232 = 731 observations.
To speed up the calibration, we can use the asymptotic implied total standard deviation calculations of Sect. 4.1. In particular, we can apply Theorem 4.9 by noting that However, we can do better and replace each lim sup with a proper limit by applying standard asymptotic properties of the complementary incomplete gamma function and the tail-wing formula of Benaim-Friz [2] and Gulisashvili [16] to find as κ → 0. Figure 2 shows a calibration of this family of call prices to real market data. It is important to stress that there is no a priori reason why this data should resemble the call surfaces generated by this family of models. Nevertheless, although the fit is not perfect, it does seem to indicate that this modelling approach is worth pursuing further.

A nonparametric calibration
In this section, we take a somewhat different approach. Rather than assuming that the log-concave density f is a fixed parametric family, we use the results of Sect. 4.1 to Fig. 3 log f estimated nonparametrically (blue), versus the parametric fit (red). Both are centred so that their maxima are at the origin estimate f nonparametrically. In particular, we assume that where now the function f is unknown. Since the fit of the parametric model was reasonably good, we set ϒ(t 1 ) to be the same value y 1 found in Sect. 4.2.
Recall that Theorem 4.3 says that It is straightforward to check that if f satisfies a mild regularity condition as in the hypothesis of Proposition 4.6, then we have the slightly improved asymptotic formula Hence we assume that since y 1 is small. Theorem 4.4 tells us that An estimate of the density f can now be computed numerically. Figure 3 compares log f when estimated nonparametrically with the calibrated parametric example from the last section. Considering the fact that the nonparametric density is estimated from the earliest maturity date while the parametric density is calibrated using all three maturity dates, the agreement is uncanny.
Given that the calibrated parametric density seems to recover market data reasonably well, and that the nonparametric density agrees with the parametric one reasonably well, it is natural to compare the market implied volatility to that predicted by the nonparametric model. Recall that the model call surface is determined by the density f and the increasing function ϒ . We have estimated f from the short-maturity call prices and the assumption that ϒ(t 1 ) = y 1 , where y 1 was found from the parametric Fig. 4 Implied volatility vs. log-moneyness from market data (blue), versus the nonparametrically estimated density (red) calibration. However, we still need to estimate the function ϒ(t i ) for i = 2, 3. For lack of a better idea, we let ϒ(t i ) = y i for i = 2, 3 as well. Figure 4 compares the market implied volatility (the same as in Fig. 2) with the implied volatility computed from the nonparametric model. Since the estimated density f is not given by an explicit formula, the formula in Theorem 4.2 was used to compute the call prices. Again, given that the density f is estimated using only the call price curve for t 1 , it is interesting that the model-implied volatility for maturities t 2 and t 3 should match the market data at all.

The isomorphism
In this section, to help understand the binary operation • on the space C, we show that there is a nice isomorphism of C to another function space which converts the somewhat complicated operation • into simple function composition •.
We introduce a transformationˆon the space C which will be particularly useful. For C ∈ C, we define a new functionĈ on [0, 1] by the formulâ We quickly note that the notationˆintroduced here is in fact consistent with the prior occurrence of this notation in Sect. 4.1. Indeed, the connection between the transformationˆ: C →Ĉ defined here and the conclusion of Theorem 4.4 is explored in Sect. 5.2 below. Given a call price curve C ∈ C, we can immediately read off some properties of the new functionĈ. The proof is routine and hence omitted.

Proposition 5.1
Fix C ∈ C with primal representation S and dual representation S . 1)Ĉ is nondecreasing and concave.
Then C ∈ C and g =Ĉ.
Proof By the continuity at κ = 0 of a function C ∈ C, we have the equivalent expres-sionĈ Hence for any 0 ≤ p ≤ 1, we have In light of Theorem 5.3, Theorem 2.19 says that the setĈ of conjugate functions is a semigroup with respect to function composition •, with identity elementÊ(p) = p and absorbing elementẐ(p) = 1. The involution onĈ induced by is identified in Theorem 5.6 below.
In preparation for reproving Theorem 3.8 and proving Theorem 3.11, we identify the image of the set C f of functions under the isomorphismˆ. As the notation introduced in Sect. 4.1 suggests, we have C f (·, y)(p) = F F −1 (p) + y by Theorems 4.2 and 5.2. For notational ease, we continue to use the notation Another proof of Theorem 3.8 The family of functions (Ĉ f (·, y)) y≥0 forms a semigroup with respect to function composition. The result follows from applying Theorems 5.2 and 5.3.
We now come to the proof of Theorem 3.11. for a positive concave functionĤ and fixed 0 < p 0 < 1. Note that for 0 < p < 1, the integral is well defined and finite asĤ is positive and continuous by concavity. Let L = G(0) and R = G(1), define a function F : [L, R] → [0, 1] as the inverse function F = G −1 and extend F to all of R by F (x) = 0 for x ≤ L and F (x) = 1 for x ≥ R. Note that we can compute the derivative as for all x ∈ R.

Remark 5.4
An earlier study of the translation equation without the concavity assumption can be found in the book of Aczél [1, Chap. 6.1].

Infinitesimal generators and the inf-convolution
In this section, we briefly and informally discuss the connection between the binary operation • defined in Sect. 2.3 and the well-known inf-convolution . Let f be a log-concave density with distribution function F and set The content of Theorem 3.11 is that aside from the trivial and null semigroups, the only sub-semigroups ofĈ with respect to composition are of the above form. The infinitesimal generator is given by Note that this equation also holds for the trivial sub-semigroup (Ĉ triv (·, y)) y≥0 with the functionĤ = 0, whereĈ triv (p, y) = p for all 0 ≤ p ≤ 1 and y ≥ 0.
The key property of the functionĤ is that it is nonnegative and concave. Let Note that for every element ofĤ, aside fromĤ = 0, one can assign a unique (up to centring) log-concave density f by the discussion of Sect. 4.1.
The spaceĤ is closed under addition. Furthermore, we have for every non-null one-parameter sub-semigroup (Ĉ(·, y)) y≥0 ofĈ that for small ε > 0 for someĤ ∈Ĥ. Let (Ĉ 1 (·, y)) y≥0 and (Ĉ 2 (·, y)) y≥0 be two such sub-semigroups. Note thatĈ implying that function composition near the identity element ofĈ amounts to addition in the space of generatorsĤ. Similarly, let For H ∈ H, letĤ One can check thatˆis a bijection between the sets H andĤ by a version of the Fenchel biconjugation theorem. In particular, the space H can be identified with the generators of one-parameter semigroups in C.
Recall that the inf-convolution of two functions f 1 , f 2 : R → R is defined by The basic property of the inf-convolution (see for example [5,Exercise 2.3.15]) is that it becomes addition under conjugation, i.e., in analogy with Theorem 5.3. Since there is an exponential map lifting function addition + to function composition • inĈ, we can apply the isomorphismˆto conclude that there is an exponential map lifting inf-convolution to the binary operation • in C. Indeed, let (C(·, y)) y≥0 be a one-parameter sub-semigroup of C with generator H so that C(e εx , ε) ≈ εH (x) for small ε > 0.

Lift zonoids
Finally, to see why one might want to compute the Legendre transform of a call price with respect to the strike parameter, we recall that the zonoid of an integrable random d-vector X is the set ]. The precise shape of this set is intimately related to call and put prices as seen in the following theorem.
Theorem 5.5 Let X be an integrable random variable. Its lift zonoidẐ X is given by Note that if we let We now show that if 0 ≤ p ≤ 1 and then (p, q) ∈Ẑ X . Since the lift zonoidẐ X is clearly convex, it is sufficient to show that q can attain the upper and lower bounds for fixed p. Fix p ∈ [0, 1]. There exists an x such that For this choice of x, there exists a θ ∈ [0, 1] such that Note that then p = E[g(X)], where Since for this choice of g, we have the almost sure equality computing expectations on both sides shows that the upper bound is attained, i.e., That the lower bound is also attained follows from a similar calculation.
We remark that the explicit connection between lift zonoids and the price of call options has been noted before, for instance in the paper of Molchanov and Schmutz [29, proof of Theorem 2.1]. In the setting of the present paper, given C ∈ C represented by S, the lift zonoid of S is given by the set We recall that a random vector X 1 is dominated by X 2 in the lift zonoid order if Z X 1 ⊆Ẑ X 2 . Koshevoy and Mosler [25,Theorem 5.2] noticed that in the case d = 1, the lift zonoid order is exactly the convex order.
We conclude this section by exploiting Theorem 5.5 to obtain an interesting identity. A similar formula can be found in the paper of Hiriart-Urruty and Martínez-Legaz [18, Theorem 2].
where we have used the observation that the optimal g in the final maximisation problem assigns zero weight to the event {S = 0}.

An extension
Let F be the distribution function of a log-concave density f which is supported on all of R, so that L = −∞ and R = +∞ in the notation of Sect. 3.2. Let C f (p, y) = F F −1 (p) + y for all 0 ≤ p ≤ 1, y ∈ R.
It is interesting to note that the family of functionsĈ f (·, y) indexed by y ∈ R is a group under function composition, not just a semigroup. Indeed, we havê C f (·, −y) =Ĉ f (·, y) −1 for all y ∈ R.
Note thatĈ f (·, y) is increasing for all y, is concave if y ≥ 0, but is convex if y < 0. In particular, when y < 0, the functionĈ f (·, y) is not the concave conjugate of a call function in C. Unfortunately, the probabilistic or financial interpretation of the inverse is not readily apparent. For comparison, note that for y ≥ 0, we have by Theorem 5.6 that C f (·, −y)(p) = C f (·, y) (p) = 1 − F F −1 (1 − p) − y for all 0 ≤ p ≤ 1.