Universal contingent claims in a general market environment and multiplicative measures: Examples and applications

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Abstract

We present and further develop the concept of a universal contingent claim introduced by the author in 1995. This concept provides a unified framework for the analysis of a wide class of financial derivatives.

A universal contingent claim describes the time evolution of a contingent payoff. In the simplest case of a European contingent claim, this time evolution is given by a family of nonnegative linear operators, the valuation operators. For more complex contingent claims, the time evolution that is given by the valuation operators can be interrupted by discrete or continuous activation of external influences that are described by, generally speaking, nonlinear operators, the activation operators. For example, Bermudan and American contingent claims represent discretely and continuously activated universal contingent claims with the activation operators being the nonlinear maximum operators.

We show that the value of a universal contingent claim is given by a multiplicative measure introduced by the author in 1995. Roughly speaking, a multiplicative measure is an operator-valued (in general, an abstract measure with values in a partial monoid) function on a semiring of sets which is multiplicative on the union of disjoint sets. We also show that the value of a universal contingent claim is determined by a, generally speaking, impulsive semilinear evolution equation.

Introduction

We present and further develop the concept of a universal contingent claim introduced by the author in [7], [9]. This concept provides a unified framework for the analysis of a wide class of financial derivatives in a general market environment.

A universal contingent claim describes the time evolution of a contingent payoff defined as a real-valued function of several positive variables, the prices of the underlying securities. In the simplest case of a European contingent claim, this time evolution is given by a family of nonnegative linear operators, the valuation operators introduced by the author in [7]. For more complex contingent claims, the time evolution that is given by the valuation operators can be interrupted by discrete or continuous activation of external influences that are described by, generally speaking, nonlinear operators, the activation operators. For example, Bermudan and American contingent claims represent discretely and continuously activated universal contingent claims with the activation operators being the nonlinear operators of the pointwise maximum of two contingent payoffs.

We show that the value of a universal contingent claim in a general market environment is given by a multiplicative measure introduced by the author in [7], [9]. Roughly speaking, a multiplicative measure is an operator-valued (in general, an abstract measure with values in a partial monoid) function on a semiring of sets which is multiplicative on the union of disjoint sets. We also show that the value of a universal contingent claim is determined by a, generally speaking, impulsive semilinear evolution equation introduced by the author in [7], [9], [4].

The article is organized as follows. In Section 2 we present the framework of a market environment. In Section 3 we present the definition of the generators of a market environment. In Section 4 we consider an example of one of the major market environments, the Black–Scholes market environment. In Sections 5–9 we motivate the concept of a universal contingent claim by considering practically important examples of a European contingent claim with a discretely paid dividend, a European contingent claim with a jump in the price of the underlying security, a European compound contingent claim, and Bermudan and American contingent claims. In Section 10 we present the definition of a universal contingent claim in a general market environment. In Section 11 we present the definition of a multiplicative measure and show that the value of a universal contingent claim in a general market environment is given by a multiplicative measure.

Section snippets

Market environment

We present the framework of a market environment that was introduced by the author in [7]. This framework formalizes the concept of a market environment in which contingent claims are being priced. For the sake of financial clarity we only consider the case of a single underlying security.

Consider an economy without transaction costs in which trading is allowed at any time in a trading time set T, an arbitrary subset of the real numbers R. Denote by st>0 the unit price of the underlying

Generators of a market environment

A market environment V such that its trading time set T is an interval, either finite or infinite, of nonnegative real numbers and its evolution operators V(t,T) are sufficiently smooth functions of time, admits the following characterization introduced by the author in [7].

Definition

We say [7] that the one-parameter family of linear operators L={L(t):ΠΠ|tT}generates a market environment V if for each t and T in the trading time set T with tT and for each admissible payoff vT in Π the function V(t,T)vT

The Black–Scholes market environment

Following [7] we present an example of one of the major market environments, the Black–Scholes market environment, that corresponds to the Black–Scholes model. (For the Black–Scholes model and related terminology see, for example,[18].)

Definition

We call a market environment VBS={VBS(t,T):ΠΠ|t,TT,tT}generated by the family LBS={LBS(t):ΠΠ|tT}with the generator LBS(t) at time t defined by LBS(t)=12σ2(s,t)s22s2+(r(s,t)-d(s,t))ss-r(s,t)a Black–Scholes market environment, where σ(s,t) is the

European contingent claims with a discretely paid dividend

In order to motivate the introduction of the concept of a universal claim, we first consider the following practically important examples of a European contingent claim with a discretely paid dividend, a European contingent claim with a jump in the price of the underlying security, a European compound contingent claim, and Bermudan and American contingent claims.

Suppose that the underlying security pays a single dividend during the lifetime of the European contingent claim with inception time t

European contingent claims with a jump in the price of the underlying security

Suppose that there is expected to be a release of information at time τ resulting in the jump of the price of the underlying security at this time τ with the associated risk-neutral transition probability P(sτ,dsτ).

Denote by Bτ(sτ) the price at the inception time τ of the pure discount bond with face value of one unit of account that matures an instant later.

Let P:ΠΠ be a linear operator defined by (Ph)(s)=Bτ(s)0h(s)P(s,ds),for each admissible h in Π.

Then the value E(t,T,g) of a European

European compound contingent claims

Denote by Π(R++×R) the vector space of all real-valued functions on the set R++×R. Equipped with the partial order generated by the nonnegative cone Π+(R++×R) consisting of all nonnegative real-valued functions on R++×R, Π(R++×R) is a partially ordered vector space. Moreover, equipped with the lattice operations of supremum and infimum defined as pointwise maximum and minimum, Π(R++×R) is a vector lattice.

Definition

A European compound option with inception time t, expiration time τ, and payoff G in Π+(R++

Bermudan contingent claims

Assume that for t and T in the trading time set T with tT, the exercise time set E={ti:i=0,1,,n} with tt0<t1<tn-1<tn=T is contained in T.

Definition

A Bermudan option with inception time t, expiration time T, exercise time set E, and a (time-dependent) payoff g:EΠ+ is a contract that gives the right, but not the obligation, to receive the payoff gτ(sτ) at any time τ in the exercise set E, where the price of the underlying security at time τ is sτ.

For a Bermudan option with inception time t, expiration

American contingent claims

Assume that for t and T in the trading time set T with tT, the interval [t,T] is contained in T.

Definition

An American option with inception time t, expiration time T, and a (time-dependent) payoff g:[t,T]Π+ is a contract that gives the right, but not the obligation, to receive the payoff of gτ(sτ) at any time τ in the interval [t,T], where the price of the underlying security at time τ is sτ.

For an American option with inception time t, expiration time T, and payoff g denote by A(t,T,g)=A(t,T,g)(st)

Universal contingent claims

We present the definition of a universal contingent claim introduced by the author in [7], [9], and show that the value of a universal contingent claim is determined by a, generally speaking, impulsive semilinear evolution equation introduced by the author in [7], [9], [4] (see also [11], [12]).

Assume that for t and T in the trading time set T with tT, the activation time set J={ti:i=0,1,,n} with tt0<t1<tn-1<tn<T is contained in T. Let O=O(Π) be the set of all, not necessarily linear,

Multiplicative measure

We present the definition of a multiplicative measure introduced by the author in [7], [9], and show that the value of a universal contingent claim in a general market environment is given by a multiplicative measure.

Definition

A family of sets S is called a semiring of sets if

  • (i)

    the empty set is in S,

  • (ii)

    if E and F are in S then EF is in S, and

  • (iii)

    if E and FE are in S then there exists a decomposition {E1,E2,,En} of E in S such that E1=F.

Definition

Let M be the set of all maps from a nonempty set M to itself and let S be

Acknowledgements

I thank my wife Larisa and my son Nikita for their love, patience and care.

References (18)

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