There is vast empirical evidence that given a set of assumptions on the real-world dynamics of an asset, the European options on this asset are not efficiently priced in options markets, giving rise to arbitrage opportunities. We study these opportunities in a generic stochastic volatility model and exhibit the strategies which maximize the arbitrage profit. In the case when the misspecified dynamics is a classical Black-Scholes one, we give a new interpretation of the classical butterfly and risk reversal contracts in terms of their (near) optimality for arbitrage strategies. Our results are illustrated by a numerical example including transaction costs.
Deep Dive into Arbitrage Opportunities in Misspecified Stochastic volatility Models.
There is vast empirical evidence that given a set of assumptions on the real-world dynamics of an asset, the European options on this asset are not efficiently priced in options markets, giving rise to arbitrage opportunities. We study these opportunities in a generic stochastic volatility model and exhibit the strategies which maximize the arbitrage profit. In the case when the misspecified dynamics is a classical Black-Scholes one, we give a new interpretation of the classical butterfly and risk reversal contracts in terms of their (near) optimality for arbitrage strategies. Our results are illustrated by a numerical example including transaction costs.
It has been observed by several authors [1,3,12] that given a set of assumptions on the real-world dynamics of the underlying, the European options Figure 1: Historical evolution of the VIX index (implied volatility of options on the S&P 500 index with 1 to 2 month to expiry, averaged over strikes, see [10]) compared to the historical volatility of the S&P 500 index. The fact that the implied volatility is consistently above the historical one by several percentage points suggests a possibility of mispricing. on this underlying are not efficiently priced in options markets. Important discrepancies between the implied volatility and historical volatility levels, as illustrated in Figure 1, as well as substantial differences between historical and option-based measures of skewness and kurtosis [3] have been documented. These discrepancies could be explained by systematic mispricings / model misspecification in option markets, leading to potential arbitrage opportunities 1 . The aim of this paper is to quantify these opportunities within a generic stochastic volatility framework, and to construct the strategies maximizing the gain. The arbitrage opportunities analyzed in this paper can be called statistical arbitrage opportunities, because their presence / absence depends on the statistical model for the dynamics of the underlying asset. Contrary to model independent arbitrages, such as violation of the call-put parity, a statistical arbitrage only exists in relation to the particular pricing model.
The issue of quantifying the gain/loss from trading with a misspecified 1 There exist many alternative explanations for why the implied volatilities are consistently higher than historical volatilities, such as, price discontinuity [4], market crash fears [5] and liquidity effects such as transaction costs [22,19,8,9], inability to trade in continuous time [6] and market microstructure effects [24]. The literature is too vast to cite even the principal contributions here model has so far mainly been studied in the case of the Black-Scholes model with misspecified volatility [13,25]. In this paper we go one step further, and analyze the effects of misspecification of the volatility itself, the volatility of volatility and of the correlation between the underlying asset and the volatility in a stochastic volatility model. Since these parameters may be observed from a single trajectory of the underlying in an almost sure way, their misspecification leads, in principle, to an arbitrage opportunity. The questions are whether this opportunity can be realized with a feasible strategy, and how to construct a strategy maximizing the arbitrage gain under suitable conditions guaranteeing the well-posedness of the optimization problem.
While the issue of consistency between real-world and risk-neutral probability measures has been given a rigorous treatment in several papers [1,3,12], the corresponding arbitrage trading strategies are usually constructed in an ad-hoc manner [1,17,18]. For instance, when the risk-neutral skewness is greater than the historical one (which roughly corresponds to correlation misspecification in a stochastic volatility model), [1] suggest a strategy consisting in buying all OTM puts and selling all OTM calls. Similarly, if the risk-neutral kurtosis is greater than the historical one, the trading strategy consists in selling far OTM and ATM options while simultaneously buying near OTM options. In this paper we determine exactly which options must be bought and sold to maximize arbitrage gains, depending on model parameters.
Our second objective is to analyze commonly used option trading strategies, such as butterflies and risk reversals, and provide a new interpretation of these structures in terms of their performance for volatility arbitrage. A butterfly (BF) is a common strategy in FX option trading, which consists in buying an out of the money call and an out of the money put with the same delta value (in absolute terms) and selling a certain number of at the money calls/puts. A risk reversal (RR) is a strategy consisting in buying an out of the money call and selling an out of the money put with the same delta value (in absolute terms). The financial engineering folklore goes that “butterflies can be used to arbitrage misspecified volatility of volatility” and “risk reversals can be used to arbitrage misspecified correlation”. In section 4, we study these strategies and discuss their optimality for volatility trading.
During the last decade we have witnessed the appearence of a large spectrum of new products specifically designed for volatility and correlation trading, such as variance, volatility and correlation swaps. However, in most markets, European options continue to be much more liquid than exotic volatility products and still constitute the most commonly used tool for volatility arbitrage. In this paper we therefore concentrate on arbitrage strategies involving only the underlying
…(Full text truncated)…
This content is AI-processed based on ArXiv data.
