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Review of Derivatives Research

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27-10-2020

A model-free approach to multivariate option pricing

Authors: Carole Bernard, Oleg Bondarenko, Steven Vanduffel

Published in: Review of Derivatives Research | Issue 2/2021

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Abstract

We propose a novel model-free approach to extract a joint multivariate distribution, which is consistent with options written on individual stocks as well as on various available indices. To do so, we first use the market prices of traded options to infer the risk-neutral marginal distributions for the stocks and the linear combinations given by the indices and then apply a new combinatorial algorithm to find a compatible joint distribution. Armed with the joint distribution, we can price general path-independent multivariate options.

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The Rearrangement Algorithm was introduced by Puccetti and Rüschendorf (2012) and Embrechts et al. (2013) as a practical tool for assessing the impact of model uncertainty on Value-at-Risk estimates for portfolios.

Examples of RND estimation techniques can be found in Jackwerth and Rubinstein (1996), Aït-Sahalia and Lo (2000), and Bondarenko (2003). See also Figlewski (2018) for the review of the pros and cons of the methods proposed in the literature.

For example, it is well-known that a portfolio of individual options is always more expensive than an option on the portfolio, where the strikes are chosen appropriately. If this property is violated for some portfolio, then an arbitrage opportunity will exist.

When \(F_{{\alpha }}\) is available for any possible choice of \(\alpha \in {\mathbb {R}}^d\), then the characteristic function of \((X_1,X_2,\ldots ,X_d)\) is also known, from which the unique joint distribution function can be obtained (Lévy 1926; Cramér 1946). When there is only a limited number of linear combinations with known distributions, then many compatible joint distributions will typically exist, but it is no longer clear a priori how to find them.

Bernard et al. (2018) considers the special case when there is only one distributional constraint and provides an extensive numerical study to demonstrate the strength of the BRA method and its convergence. However, even for this special case, the formal convergence results are not yet available. Haus (2015) has shown that even the simplest rearrangement problem, in which the only constraint is that the distribution of the sum is constant, is NP-complete and therefore cannot be solved by an algorithm that has polynomial complexity.

(Bernard and McLeish 2016; Bernard et al. 2017), and Boudt et al. (2018).

For very large applications, say, \(d=500\) stocks in the S&P 500 index, one still has an option to use the Constrained RA (CRA) instead of Constrained BRA. Although less accurate than CBRA, CRA is much faster. CRA cycles through \((d+K)\) singleton blocks, as opposed to all admissible blocks in \({\mathcal {B}}\) in CBRA. The next subsection provides further discussion of complexity.

Clearly, the marginal distribution for a single variable \(X_{j}\) may be viewed as a special case of a linear constraint.

In the context of inference of marginal RNDs from observed option prices, the principle of maximum entropy has been explored in Rubinstein (1994) and in Jackwerth and Rubinstein (1996) and has been employed in Stutzer (1996).

We are grateful to the anonymous referee for pointing out this interesting alternative formulation. The connection between OT and CBRA is rather fascinating and it opens exciting possibilities for future research. Perhaps the extensive machinery developed for OT can also help in analyzing CBRA.

In our experiments, we have successfully implemented the CBRA approach for \(d=30\) stocks in the DJIA. This case would be completely hopeless for the OT approach.

We program CBRA algorithm in Matlab and run it in parallel using 60 cores. The running time is approximately 1 minute and on average the algorithm converges to a solution after about 8500 iterations.

More precisely, our arbitrage strategy involves trading in so-called simple variance swaps, see Martin (2017).

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Title: A model-free approach to multivariate option pricing
Authors: Carole Bernard
Oleg Bondarenko
Steven Vanduffel
Publication date: 27-10-2020
Publisher: Springer US
Published in: Review of Derivatives Research / Issue 2/2021
Print ISSN: 1380-6645
Electronic ISSN: 1573-7144
DOI: https://doi.org/10.1007/s11147-020-09172-2