Introductory Course on Financial Mathematics by M V Tretyakov

Author:M V Tretyakov
Language: eng
Format: azw3
ISBN: 9781908977403
Publisher: ICP
Published: 2013-07-23T04:00:00+00:00

We note that this is one of infinitely many EMMs we can find for this market model. The existence of infinitely many EMMs implies that there can be infinitely many arbitrage prices for a particular claim in this market. Note that our conclusion here does not contradict Proposition 9.3 since there is no replicating strategy for the call on the market model from Example 11.1.

Remark 11.2. In this chapter we considered examples of markets which are incomplete because of a mismatch between the number of traded assets and the number of possible future states of the market. Incomplete markets also appear in other situations, for instance in markets with constraints (e.g., constraints on short selling) and friction (transaction costs). In general, complete markets are an idealistic model of real markets which, in many cases, can be considered as a good approximation for reality while incomplete markets are considered to be closer to reality.

Remark 11.3. In this course we are dealing only with market models which are arbitrage-free and complete, except for the current chapter. If the arbitrage-free assumption is rather natural from the real applications point of view and widely used in Finance, the completeness assumption is merely an approximation of reality because real markets are generically incomplete for various reasons (see Remark 11.2). We observed that in an incomplete market, if a claim f is not attainable, different martingale measures can give different prices. Then the natural question is how to choose the price with which both the buyer and seller might agree. The most commonly accepted ways of treating incomplete markets are via the use of utility functions or of strategies with consumption (see, e.g., Follmer and Schied, 2004; Melnikov, 2004; Pliska, 1997), which we do not consider in this course.

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1Due to the compatibility criterion for a system of linear equations (also known as Kronecker–Capelli or Rouché–Capelli theorem), the linear system Dϑ = f has a solution if and only if the rank of the coefficient matrix D is equal to the rank of the augmented matrix [D|f] obtained by appending the column of free terms f to the right of D. Because we require that Dϑ = f has a solution for any f, we should require that M ≥ L (i.e. that the number of unknowns is not less than the number of equations) and the rows of D are linearly independent, hence the rank of D is equal to L.

2It can be found, e.g., in Bingham and Kiesel (2004); Björk (2004); Melnikov (2004) (see also Harrison and Pliska (1981)).

3See in Filipovic (2009, p. 77).

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