# Implied volatility and state price density estimation: arbitrage analysis

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## Abstract

This paper deals with implied volatility (IV) estimation using no-arbitrage techniques. The current market practice is to obtain IV of liquid options as based on Black–Scholes (BS type hereafter) models. Such volatility is subsequently used to price illiquid or even exotic options. Therefore, it follows that the BS model can be related simultaneously to the whole set of IVs as given by maturity/moneyness relation of tradable options. Then, it is possible to get IV curve or surface (a so called smile or smirk). Since the moneyness and maturity of IV often do not match the data of valuated options, some sort of estimating and local smoothing is necessary. However, it can lead to arbitrage opportunity if no-arbitrage conditions on state price density (SPD) are ignored. In this paper, using option data on DAX index, we aim to analyse the behavior of IV and SPD with respect to different choices of bandwidth parameter *h*, time to maturity and kernel function. A set of bandwidths which violates no-arbitrage conditions is identified. We document that the change of *h* implies interesting changes in the violation interval of moneyness. We also perform the analysis after removing outliers, in order to show that not only outliers cause the violation of no-arbitrage conditions. Moreover, we propose a new measure of arbitrage which can be considered either for the SPD curve (arbitrage area measure) or for the SPD surface (arbitrage volume measure). We highlight the impact of *h* on the proposed measures considering the options on a German stock index. Finally, we propose an extension of the IV and SPD estimation for the case of options on a dividend-paying stock.

## Keywords

Option pricing Implied volatility State price density No-arbitrage conditions Local polynomial smoothing## Notes

### Acknowledgements

The research was supported by the Czech Science Foundation under Project 402/12/G097 and 16-09541S and the European Social Fund (CZ.1.07/2.3.00/20.0296). The third author was supported also through an SP2017/32, an SGS research project of VSB-TU Ostrava. The support is greatly acknowledged. All computations were done in MatLab 2009 and GAMS 23.5.

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