Fuzziness in valuing financial instruments by certainty equivalents

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Abstract

Our aim is to use the utility theory of von Neumann and Morgenstern and the approach of certainty equivalent (c.e.) in a fuzzy framework for pricing a future uncertain amount. The fuzzy theory is more adaptable to represent this problem because the financial evaluations depend on perspectives which are fuzzy known. Then we define a seller and buyer fuzzy financial c.e.

Introduction

In this paper, we build fuzzy-random models linking the theory of fuzzy numbers (f.n.) used in [3] with the utility theory of von Neumann and Morgenstern, in particular with the notion of certainty equivalent (c.e.) to obtain immediate prices of a future uncertain sum, for example, a future price of a stock or a price of a derivative contract on a security.

The c.e. is a classical crucial tool for pricing when standard finance theory does not apply as in the case of incomplete hedging and in absence of the standard hypotheses concerning perfection and efficiency of financial markets. In these cases, the pricing of options cannot be based on the Black and Scholes formula [4].

Obviously, our pricing takes into account the discrepancy between the moment at which the price of an asset is paid and that at which the asset produces future uncertain money. This is made representing the rate of interest as a convex fuzzy number and defining the fuzzy present value (p.v.) of the c.e. in according with [3], [6], [9].

Making expected utility with either Lebesgue's, or Choquet's integral, we have different pricing functionals (see [12]).

We distinguish between the positions of the seller and the buyer: the seller cedes a risk and the buyer assumes it. This important distinction is used in actuarial literature, e.g. [5] and in finance, e.g. [10].

The paper is organized as follows: in Section 2, we set the preliminaries; in Section 3, we give present and future value models; in Section 4, we define the financial c.e. with random interest rate and in Section 5 the financial c.e. with fuzzy interest rate and proposals for further researches.

Section snippets

Preliminaries

Let (Ω,A,P) be a probability space, where Ω is a set containing all the eventual rates of interest and all the future prices of an asset, A is a σ-field representing all the random events and P is a probability defined on A.

In this paper, we take Ω equal to an interval [a,b] of real positive numbers; A is equal to the σ-field of Borel A=B([a,b]).

The choices of Ω and P depend on our knowledge. In general, P is linked to the observed frequencies, for example, it may be a probability which

Fuzzy present and future value model

In the classical case and using a compound interest, at rate r, an investment of 1 will accumulatem=(1+r)tat the end of a period of duration t. We call m the compound factor.

Symmetrically,v=(1+r)−tis the price to pay now for having 1 at the end of the period of duration t. v is called discount factor.

If r∈N represents a fuzzy rate of interest, then the discount and the compound factors are f.n. defined by(1⊕r)−t,(1⊕r)t,respectively, where ⊕, is defined with Zadeh's extension principle (see, for

Pricing certainty equivalents under random conditions

We remember that in [11] Pratt gives the classical definition of certain equivalent (c.e.) of a random variable X.

Definition 3 certainty equivalents of a random variable

The c.e. of X is the solution, x, of the equationE[u(X)]=u(x),where u denotes an agent's von Neumann and Morgenstern utility function and E denotes the expectation of the random variable u(X).

For having an immediate price we may take a discount factor, v, then, the classical present value (p.v.) of the c.e. x is given by y=xv.

While, the solution x of the classical equation

Pricing certainty equivalents under fuzzy conditions

A financial pricing functional, H, is a functional from the set of the risks to the positive real numbers. In this paper, we consider the following forms:

  • (i) the random financial c.e. defined above;

  • (ii) the fuzzily financial c.e., that we define taking a fuzzy rate of interest for having a fuzzy present value and making a defuzzification as we see below.

Let us introduce the fuzzy financial c.e., necessary for defining (ii), with the following Proposition.

Proposition 4 concerning the fuzzy financial c.e.

If the rate of interest r is an element

References (15)

  • J.J. Buckley

    The fuzzy mathematics of finance

    Fuzzy Sets and Systems

    (1987)
  • M. Li Calzi

    Towards a general setting for the fuzzy mathematics of finance

    Fuzzy Sets and Systems

    (1990)
  • L.A. Zadeh

    Fuzzy sets

    Information and Control

    (1965)
  • P. Artzner et al.

    Coherent measures of risk

    Mathematical Finance

    (1999)
  • L. Biacino et al.

    Equations with fuzzy numbers

    Information Sciences

    (1989)
  • L. Biacino et al.

    The internal rate of return of fuzzy cash flow

    Stochastica

    (1992)
  • F. Black, M. Scholes, The pricing of options and corporate liabilities, Journal of Finance, June...
There are more references available in the full text version of this article.

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