**Walter Schachermayer**

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## When does Convergence of Asset Price Processes Imply Convergence of Option Prices?

**F. Hubalek, W. Schachermayer**

Mathematical Finance, Vol. 8 (1998), No. 4, pp. 385-403.

### Abstract:

We consider weak convergence of a sequence of asset price models
*(S*^{n}) to a limiting asset price model *S*. A
typical case for this situation is the convergence of a sequence of
binomial models to the Black-Scholes model, as studied by Cox, Ross, and
Rubinstein.

We put emphasis on two different aspects of this convergence: firstly
we consider convergence with respect to the given ''physical'' probability
measures *(P*^{n}) and secondly with respect to the
''risk-neutral'' measures *(Q*^{n}) for the asset price
processes *(S*^{n}). (In the case of non-uniqueness of the
risk-neutral measures also the question of the ''good choice'' of
*(Q*^{n}) arises.) In particular we investigate under which
conditions the weak convergence of *(P*^{n}) to *P*
implies the weak convergence of *(Q*^{n}) to *Q* and
thus the convergence of prices of derivative securities.

The main theorem of the present paper exhibits an intimate relation of
this question with contiguity properties of the sequences of measures
*(P*^{n}) with respect to *(Q*^{n}) which in
turn is closely connected to asymptotic arbitrage properties of the
sequence *(S*^{n}) of security price processes.

We illustrate these results with general homogeneous binomial and some
special trinomial models.

### Preprints:

[PostScript (201 k)] [PS.gz (75 k)] [PDF (272 k)] [DOI: 10.1111/1467-9965.00060]

Publications marked with have appeared in refereed journals.

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