A Black–Scholes inequality: applications and generalisations

Abstract

The space of call price functions 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 can be identified with a peacock. 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. The key observation is an isomorphism linking an initial call price curve to the lift zonoid of the terminal price of the underlying asset.where σ is the volatility of the stock price. In particular, the first argument of C BS plays the role of the moneyness κ = K/F 0,T 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.

Options Are Also Options on Options: How to Smile With Black-Scholes

“…Proposition 2.2 implies in particular that the slope of Calls on Calls in 0 is stricly larger than −1. This is not a problem in terms of arbitrageable prices, but it is an uncommon feature since it is linked to the presence of a positive mass of the underlier in 0 (see Theorem 2.1.2. of [11]). This is expected indeed, since the new underlier is a Call option, which has a whole region of null payoff.…”

“…As Tehranchi has deeply studied normalized Call prices in [11], we will name Tehranchi space the space C of such normalized Call prices:…”

“…At m = 0, it is easy to see d dk T k c(0) = 0. From Theorem 2.1.2 of [11], there exists a random variable S such that c(y) = 1 − E[S ∧ y] and −c ′ (y) = P (S > y). Then Note that the above proposition implies in particular…”

“…In this section, we provide a quasi-closed formula for the pricing function within the Carr-Pelts-Tehranchi family (see [2,11]), which generalizes the Black-Scholes pricing function associated to the standard normal density to any log-concave (and even, unimodal, as shown by Vladimir Lucic in [5]) density function. This includes the Black-Scholes case as a particular case.…”

Explicit no arbitrage domain for sub-SVIs via reparametrization

“…Tehranchi proved in [7] that a curve of Call prices (with unit underlyer) parametrized by the strike κ is free of Butterfly arbitrage if and only if it is convex and satisfies 1 ≥ C(κ) ≥ (1 − κ) + for every κ ≥ 0. Moreover, C has these properties if and only if C * (κ) := 1 − κ + κC 1 κ has them.…”