Abstract
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.
Highlights
We define the Black–Scholes call price function CBS : [0, ∞) × [0, ∞) → [0, 1] by the formulaCBS(κ, y) = ∞ φ(z + y) − κ φ(z) +dz −∞ ⎧ ⎨ (− log y κ + y 2 ) − κ
With the motivation of finding a tractable family of call price surfaces, we study a family of one-parameter sub-semigroups of C
Are extremely simple to write down. It is the simplicity of these formulae that is the claim to practicality of the results presented here
Summary
We define the Black–Scholes call price function CBS : [0, ∞) × [0, ∞) → [0, 1] by the formula. As a hint of things to come, it is worth pointing out that the expression y1 + y2 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. In [31], Theorem 1.3 was used to derive upper bounds on the implied total standard deviation function YBS by selecting various values of p to insert into the inequality. 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. The isomorphism has the additional interpretation as the lift zonoid of a related random variable
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