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By Karlheinz Spindler
A complete presentation of summary algebra and an in-depth therapy of the purposes of algebraic recommendations and the connection of algebra to different disciplines, equivalent to quantity thought, combinatorics, geometry, topology, differential equations, and Markov chains.
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Additional info for Abstract Algebra with Applications: Rings and Fields
F ∈ Pa if 0 ≤ f < a is convex. f ∈ Pa ⇒ erτ Sτ f ∈ Pa . We are now going to introduce slightly larger classes of payoff functions than the classes Pa , a > 0. To this end, let a > 0 be given and suppose f : R m + → [0, a[ is a continuous function and set for fixed τ > 0, g = gτ = − 1 √ ln(1 − f /a) . σm τ (23) GEOMETRIC INEQUALITIES IN OPTION PRICING 49 We shall say that the the function f belongs to the class Pa,m if the function Φ−1 (P[gτ (x1 eσ1 √ τ c1 ,G , . . , xm eσm √ τ cm ,G ) ≤ s]) − s, s>0 is non-decreasing for every x ∈ R m + and τ > 0.
In fact, already the early paper  by Merton treats a variety of convexity properties of puts and calls, sometimes without any distributional assumptions on the underlying stock prices. Here, however, it will always be assumed that the price process X(t) = (X1 (t), . . , Xm (t)), t ≥ 0, of the underlying risky assets X1 , . . , Xm is governed by a so called joint geometric Brownian motion. Furthermore, all options will be of European type and so, from now on, option will always mean option of European type.
Then f ∈ C if and only if f > 0 and f (xeξ ) ≤ f (x)e ξ ∞ m x ∈ Rm +, ξ ∈ R . , (12) Moreover , f ∈ Ca if and only if a + f (xeξ ) ≤ (a + f (x))e ξ ∞ m x ∈ Rm +, ξ ∈ R . , In particular , any f ∈ Ca is a payoff function. Proof. Suppose first that f > 0 and set g(ξ) = ln f (eξ ), ξ ∈ Rm. Clearly, the inequality (12) just means that g ∈ Lip∞ (R m ; 1). Now let f ∈ C. e. e. and, hence, g ∈ Lip∞ (R m ; 1). Conversely, suppose g ∈ Lip∞ (R m ; 1). Then f ∈ Liploc (R m + ) and (13) holds. Accordingly, the inequality (11) must be true.
Abstract Algebra with Applications: Rings and Fields by Karlheinz Spindler