Robust Trading of Implied Skew

Nadtochiy, Sergey; Obłój, Jan

doi:10.1142/s021902491750008x

Cited by 6 publications

(3 citation statements)

References 36 publications

(127 reference statements)

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“…It also allows incorporating the information from time series of data coherently into the option pricing setup, as no probability measure is fixed and hence no distinction between real-world and risk-neutral measures is made. The idea of such a prediction set first appeared in Mykland [39]; see also Nadtochiy and Obłój [40] and Cox et al [16] for an extended discussion.…”

Section: Beliefsmentioning

confidence: 99%

Robust pricing–hedging dualities in continuous time

Hou

Obłój

2018

Finance Stoch

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We pursue a robust approach to pricing and hedging in mathematical finance. We consider a continuous-time setting in which some underlying assets and options, with continuous price paths, are available for dynamic trading and a further set of European options, possibly with varying maturities, is available for static trading. Motivated by the notion of prediction set in Mykland (Ann. Stat. 31:1413Stat. 31: -1438Stat. 31: , 2003, we include in our setup modelling beliefs by allowing to specify a set of paths to be considered, e.g. superreplication of a contingent claim is required only for paths falling in the given set. Our framework thus interpolates between model-independent and model-specific settings and allows us to quantify the impact of making assumptions or gaining information. We obtain a general pricing-hedging duality result: the infimum over superhedging prices of an exotic option with payoff G is equal to the supremum of expectations of G under calibrated martingale measures. Our results include in particular the martingale optimal transport duality of Dolinsky and Soner (Probab. Theory Relat. Fields 160:391-427, 2014) and extend it to multiple dimensions, multiple maturities and beliefs which are invariant under time-changes. In a general setting with arbitrary beliefs and for a uniformly continuous G, the asserted duality holds between limiting values of perturbed problems.

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Section: Beliefsmentioning

confidence: 99%

Robust pricing–hedging dualities in continuous time

Hou

Obłój

2018

Finance Stoch

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“…Moreover, there is no restriction or debate on continuous or discrete time. While our framework is discrete, continuous time could perhaps show a way forward to incorporate models from mathematical finance and financial economics [103][104][105]. Jarrow [106] makes the case for continuous time, arguing that today's financial markets trade and update at high frequency.…”

Section: Connectionmentioning

confidence: 99%

Learning agents in Black–Scholes financial markets

2020

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Black–Scholes (BS) is a remarkable quotation model for European option pricing in financial markets. Option prices are calculated using an analytical formula whose main inputs are strike (at which price to exercise) and volatility. The BS framework assumes that volatility remains constant across all strikes; however, in practice, it varies. How do traders come to learn these parameters? We introduce natural agent-based models, in which traders update their beliefs about the true implied volatility based on the opinions of other agents. We prove exponentially fast convergence of these opinion dynamics, using techniques from control theory and leader-follower models, thus providing a resolution between theory and market practices. We allow for two different models, one with feedback and one with an unknown leader.

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“…If we considered a restricted version of dynamic trading in which we make further assumptions about the price dynamics of vanilla options, then this could imply that

J \circ {monospacei}^{- 1}

is not surjective and the superhedging prices might strictly decrease. Such a setup is studied in Nadtochiy and Obłój (), where the authors consider restrictions on the levels of implied volatility through time.…”

Section: Pricing–hedging Duality For American Optionsmentioning

confidence: 99%

The robust pricing–hedging duality for American options in discrete time financial markets

Aksamit

Deng

Obłój

et al. 2018

Mathematical Finance

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We investigate the pricing–hedging duality for American options in discrete time financial models where some assets are traded dynamically and others, for example, a family of European options, only statically. In the first part of the paper, we consider an abstract setting, which includes the classical case with a fixed reference probability measure as well as the robust framework with a nondominated family of probability measures. Our first insight is that, by considering an enlargement of the space, we can see American options as European options and recover the pricing–hedging duality, which may fail in the original formulation. This can be seen as a weak formulation of the original problem. Our second insight is that a duality gap arises from the lack of dynamic consistency, and hence that a different enlargement, which reintroduces dynamic consistency is sufficient to recover the pricing–hedging duality: It is enough to consider fictitious extensions of the market in which all the assets are traded dynamically. In the second part of the paper, we study two important examples of the robust framework: the setup of Bouchard and Nutz and the martingale optimal transport setup of Beiglböck, Henry‐Labordère, and Penkner, and show that our general results apply in both cases and enable us to obtain the pricing–hedging duality for American options.

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Robust Trading of Implied Skew

Cited by 6 publications

References 36 publications

Robust pricing–hedging dualities in continuous time

Robust pricing–hedging dualities in continuous time

Learning agents in Black–Scholes financial markets

The robust pricing–hedging duality for American options in discrete time financial markets

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