3.2 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.
Combining Theorem
3.2 and Proposition
3.4 brings us to the main observation of this paper: If
\((C(\cdot, t) )_{t \ge0}\) is a one-parameter sub-semigroup of
\(\mathcal {C}\), then
\(C(\cdot, \cdot)\) 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 \(\mathcal {C}\). 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 \(y=\sigma\sqrt{t}\) corresponding to total standard deviation.
We make use of the following notation. For a probability density function
\(f\), let
$$ C_{f}(\kappa,y) = \int_{-\infty}^{\infty} \big( f(z+y)- \kappa f(z)\big)^{+} dz = 1 - \int_{-\infty}^{\infty} f(z+y) \wedge \big( \kappa f(z) \big) dz $$
for
\(y \in\mathbb{R} \) and
\(\kappa\ge0\). Note that
$$ C_{\mathrm{BS}}(\cdot, y) = C_{\varphi}(\cdot, y) $$
for
\(y \ge0\), where
\(\varphi\) is the standard normal density.
It will be useful to distinguish a special class of densities.
We use repeatedly a useful characterisation of log-concave densities due to Bobkov [
4, Proposition A.1].
Let
\(f\) be a log-concave density supported on the closed interval
\([L,R]\), where
\(-\infty\le L < R \le+ \infty \). It is well known that a convex function
\(g:\mathbb{R} \to\mathbb{R} \cup\{ + \infty\}\) is continuous on the interior of the interval
\(\{ g < \infty\}\); see for instance the book of Borwein and Vanderwerff [
5, Theorem 2.1.2]. Consequently, the log-concave function
\(f\) is continuous on the open interval
\((L,R)\). However, if either
\(L > -\infty\) or
\(R < \infty\), it might be that
\(f\) has a discontinuity at the boundary of the support; consider for example the log-concave density
which is continuous at
\(L=0\) but discontinuous at
\(R=1\). But given a log-concave density
\(f\) supported on
\([L,R]\), we can find another log-concave function
\(\tilde{f}\) with the same support such that
\(\tilde{f} = f\) on the open interval
\((L,R)\) and such that
\(\tilde{f}\) is right-continuous at
\(L\) and left-continuous at
\(R\) by setting
\(\tilde{f}(L) = \lim_{x \downarrow L} f(x)\) and
\(\tilde{f}(R) = \lim_{x \uparrow R} f(x)\). Therefore, without loss of generality,
we make the convention that if\(f\)is a log-concave density, then\(f\)is continuous on its support\([L,R]\).
We now present a family of one-parameter sub-semigroups of \(\mathcal {C}\).
Note that Theorems
2.16 and
3.8 together say that for all
\(\kappa_{1}, \kappa_{2} > 0\) and
\(y_{1}, y_{2} > 0\), we have
$$ C_{f}(\kappa_{1} \kappa_{2}, y_{1} + y_{2}) \le C_{f}(\kappa_{1}, y_{1}) + \kappa_{1} C_{f}(\kappa_{2}, y_{2}), $$
proving Theorem
1.1.
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.
The upshot of Proposition
3.4 and Theorem
3.8 is that given a log-concave density
\(f\), the function
\(C_{f}(\kappa, \cdot)\) is nondecreasing for each
\(\kappa\ge0\). Hence, given an increasing function
\(\Upsilon\), we can conclude from Theorem
3.2 that we can define a call price surface by
$$ (\kappa,t) \mapsto C_{f}\big(\kappa, \Upsilon(t) \big). $$
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
\(f = \varphi\), the standard normal density, and the increasing function to
\(\Upsilon(t) = \sigma\sqrt{t}\), where
\(\sigma\) 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. [
19, Definition 1.3] and Ewald and Yor [
14, Definition 2.1], among others.
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.
Note that the semigroup
\(( C_{f}(\cdot, y) )_{y \ge0}\) does not correspond to a unique log-concave density. Indeed, fix a log-concave density
\(f\) and set
$$ f^{(\lambda,\mu)}(z) = |\lambda| f(\lambda z + \mu) $$
for
\(\lambda, \mu\in\mathbb{R} \),
\(\lambda\ne0\). Note that
$$ C_{f^{(\lambda,\mu)}}( \kappa, y) = C_{f}( \kappa, \lambda y ) \qquad \mbox{for all } \kappa\ge0, y \in\mathbb{R} . $$
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
\(\lambda\), we can interpolate between two possibilities. On the one hand, we have for all
\(\kappa\ge0\) and
\(y \in\mathbb{R} \) that
$$ C_{f^{(\lambda,\mu)}}(\kappa, y) \longrightarrow (1- \kappa)^{+} \qquad \mbox{as } \lambda \to0 $$
and, on the other hand, when
\(y \ne0\), that
$$ C_{f^{(\lambda,\mu)}}(\kappa, y) \longrightarrow 1 \qquad \mbox{as } |\lambda| \to\infty $$
by the dominated convergence theorem.
Recall that the call price curve \(E(\kappa) = (1-\kappa)^{+}\) is the identity element for the binary operation •. Hence the family \(C_{\mathrm{triv}}\) defined by \(C_{\mathrm{triv}}(\cdot, y) = E\) for all \(y \ge0\) is another example of a sub-semigroup of \(\mathcal {C}\).
Similarly, the call price curve \(Z(\kappa) = 1\) is the absorbing element for •. Hence the family \(C_{\mathrm{null}}\) defined by \(C_{\mathrm{null}}(\cdot, 0) = E\) and \(C_{\mathrm{null}}(\cdot, y) = Z\) for all \(y > 0\) is yet another example of a sub-semigroup of \(\mathcal {C}\).
The following theorem says that the above examples exhaust the possibilities.
The proof appears in Sect.
5.1.