Fuzziness in valuing financial instruments by certainty equivalents
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 () be a probability space, where is a set containing all the eventual rates of interest and all the future prices of an asset, is a σ-field representing all the random events and P is a probability defined on .
In this paper, we take equal to an interval [a,b] of real positive numbers; is equal to the σ-field of Borel .
The choices of 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 accumulateat the end of a period of duration t. We call m the compound factor.
Symmetrically,is the price to pay now for having 1 at the end of the period of duration t. v is called discount factor.
If represents a fuzzy rate of interest, then the discount and the compound factors are f.n. defined byrespectively, 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 equationwhere 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.
Proposition 4 concerning the fuzzy financial c.e.
If the rate of interest r is an element
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