Implied volatility and state price density estimation: arbitrage analysis

Abstract

This paper deals with implied volatility (IV) estimation using no-arbitrage techniques. The current market practice is to obtain implied volatility of liquid options as based on Black-Scholes type (BS 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 smirk or smile). 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.

Implied volatility smirk in the Australian dollar market

“…There are some studies trying to improve/critique the Black and Scholes (1973) options pricing formula, due to its limiting assumptions that do not fit the data, and proposing alternative approaches for extracting the implied volatility from the market data (Kopa et al ., 2017, amongst others). Further, to address this issue of different IVs at different strike prices, in option pricing, numerous stochastic volatility models have been developed (Stein and Stein, 1991; Heston, 1993; Bates, 1996; Barndorff‐Nielsen and Shephard, 2004, amongst others).…”

Implied volatility smoothing at COVID-19 times

“…The parameter h > 0 is the so-called bandwidth and it controls the size of the local neighborhood considered for the estimation. As analysed in Kopa et al (2017), the choice of h has a relevant impact in the computation and in the results of the fitting. On the one hand, a high value of the bandwidth induces over-smoothing, i.e.…”

“…Therefore, Benko et al (2007) suggested a discrete approximation that simplifies the algorithm solving a non-linear programming problem. Kim and Lee (2013) extended and applied the notion of Benko et al (2007) to KOSPI 200 index options no-arbitrage implied volatility modelling, while Kopa et al (2017) performed an extensive analysis on DAX index and proposed a measure to capture the magnitude of the arbitrage highlighted by the SPD. Other no-arbitrage IV and SPD estimation procedures were recently presented in e.g.…”

Implied Volatility Surface Estimation via Quantile Regularization

“…Semiparametric or nonparametric option pricing approaches are commonly considered instead (see, for instance, [1,6], or [10]) while the arbitrage-free market validity is guaranteed by using some additional pre-defined shape constraints. On the other hand, the corresponding implied volatility function (or the implied volatility surface respectively) is usually obtained either similarly, in terms of some constrained optimization problem (for instance, [11,19]), or alternatively, it can be interpolated directly from the estimated option pricing model (see [6] or [9]).…”