General Properties of Option Prices

Abstract

When the underlying price process is a one‐dimensional diffusion, as well as in certain restricted stochastic volatility settings, a contingent claim's delta is bounded by the infimum and supremum of its delta at maturity. Further, if the claim's payoff is convex (concave), the claim's price is a convex (concave) function of the underlying asset's value. However, when volatility is less specialized, or when the underlying process is discontinuous or non‐Markovian, a call's price can be a decreasing, concave function of the underlying price over some range, increasing with the passage of time, and decreasing in the level of interest rates.

General Properties of Solutions to Inhomogeneous Black-Scholes Equations with Discontinuous Maturity Payoffs and Application

“…In particular, the property of convexity preservation is related to the monotonicity of option price with respect to the volatility and widely studied. El Karoui et al [6] obtained the same results as [2] with stochastic approach. Hobson [8] simplified the result of [6] using stochastic coupling.…”

“…Cox et al [5] generalize this result and show that, under the same proportionality assumption of the stock price process, the price function of any European contingent claim, not just a call option, inherits qualitative properties of the claim's contractual payoff function [2]. (This means that the solution to Black-Scholes equation inherits such qualitative properties as monotonicity or convexity of the maturity payoff function.…”

“…Hobson [8] simplified the result of [6] using stochastic coupling. Janson et al [11] generalized the result of [2] by replacing the differentiability of volatility with respect to current stock price into local Hölder continuity and improved the results of [8]. Ekström et al [7] provided a counter example of the solution not preserving convexity when the stock price follows a jump-diffusion process and studied the condition to preserve convexity.…”

“…(This means that the solution to Black-Scholes equation inherits such qualitative properties as monotonicity or convexity of the maturity payoff function. ) Bergman et al [2] generalized the results of [5,9,14] into the case when the stock price follows a diffusion whose volatility not only depends on time but also is a differentiable function of and current stock price. They used PDE method to study gradient estimate and convexity preservation.…”

The Valuation of Volatility Options

“…An investor wishing to hedge volatility risk can do so by buying or selling n = S = V volatility options depending on the direction of his exposure. 5 , 6 Figures 3 and 4 show h o w V c hanges as initial volatility and maturity c hange. For European volatility options of all three maturities, delta rst increases then decreases.…”