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# Local volatility surfaces for the Heston (a) and Hull-White (b) models as functions of the time to maturity τ = T − t and log-moneyness log( K/S ).

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Motivated by the desire to integrate repeated calibration procedures into a single dynamic market model, we introduce the notion of a "tangent model" in an abstract set up, and we show that this new mathematical paradigm accommodates all the recent attempts to study consistency and absence of arbitrage in market models. For the sake of illustration...

## Context in source publication

**Context 1**

... is called the local volatility. For the sake of illustration, we computed and plotted the graph of this function in the case of the two most popular stochastic volatility models mentioned earlier, the Heston and the Hull-White models. These plots are given in Fig. 1. In the terminology which we develop below, the artificial financial model given by the process ( ˜ S t ) t≥0 , introduced for the sole purpose of reproducing the prices of options at time zero (in other words, the result of cal- ibration at time zero), is said to be tangent to the true model (S t ) t≥0 at t = 0. Together with the ...

## Citations

... In particular, a short net gamma position (after vega hedging) is exposed to high spot volatility (positive σ ), a short net vanna position is exposed to volatility of implied volatility that is positively correlated with the underlying (positive η), and a short net volga position is exposed to volatility of implied volatility (positive ξ ). 16 Conversely, 14 Gamma, vanna and volga are the second-order partial derivatives ∂ 2 /∂S 2 , ∂ 2 /(∂S∂ ) and ∂ 2 /∂ 2 of the Black-Scholes value of an option. 15 In contrast, if there is no liquidly traded call available as a hedging instrument, then the option's cash gamma is the only greek that appears in the probabilistic representation of the cash equivalent [28]. ...

... Here, σ is the spot volatility, and ν, η and ξ correspond to the drift of implied volatility, the 20 Other early articles on risk-neutral dynamics for stochastic implied volatility models include [41,11,39]. For more recent developments on arbitrage-free market models for (parts of or the whole) option price surface, we refer the reader to [55,54,13,35,14,15,36] and the references therein. ...

We study option pricing and hedging with uncertainty about a Black–Scholes reference model which is dynamically recalibrated to the market price of a liquidly traded vanilla option. For dynamic trading in the underlying asset and this vanilla option, delta–vega hedging is asymptotically optimal in the limit for small uncertainty aversion. The corresponding indifference price corrections are determined by the disparity between the vegas, gammas, vannas and volgas of the non-traded and the liquidly traded options.

... This has been previously done in works on the so-called market models, see e.g. [34], [14], [16], [15], [13]. While these works focused on a fixed probabilistic setting, herein, we pursue the robust approach. ...

In this paper, we present a method for constructing a (static) portfolio of co-maturing European options whose price sign is determined by the skewness level of the associated implied volatility. This property holds regardless of the validity of a specific model – i.e. the method is robust. The strategy is given explicitly and depends only on one's beliefs about the future values of implied skewness, which is an observable market indicator. As such, our method allows to use the existing statistical tools to formulate the beliefs, providing a practical interpretation of the more abstract mathematical setting, in which the belies are understood as a family of probability measures. One of the applications of the results established herein is a method for trading one's views on the future changes in implied skew, largely independently of other market factors. Another application of our results provides a concrete improvement of the model-independent super-and sub-replication strategies for barrier options proposed in [10], which exploits the given beliefs on the implied skew. Our theoretical results are tested empirically, using the historical prices of S&P 500 options.

... In this paper, we implement and test a market-based model for European-type options, based on the tangent Lévy models proposed in [4] and [3]. As a result, we obtain a method for generating Monte Carlo samples of future paths of implied volatility surfaces. ...

... Such mapping became known as a code-book mapping, and it turns out that it can be constructed by means of the so-called tangent models (cf. [2], [4], [3]). The concept of a tangent model is very close to the method of calibrating a model for underlying to the target derivatives' prices (in the present case, European options calls). ...

... Finally, one needs to characterize all possible dynamics of (θ t ) that produce no dynamic arbitrage in the associated call prices C θt . An interested reader is referred to [3], for a more detailed description of this general algorithm, and, for example, to [2], [4], [14], [29], [18], [23], for the analysis of specific choices of the families of models {M(θ)}. ...

In this paper, we implement and test a market-based model for European-type options, based on the tangent Levy models proposed in [4] and [3]. As a result, we obtain a method for generating Monte Carlo samples of future paths of implied volatility surfaces. These paths and the surfaces themselves are free of arbitrage, and are constructed in a way that is consistent with the past and present values of implied volatility. We use market data to estimate the parameters of this model and conduct an empirical study, to compare the performance of the chosen market-based model with the classical SABR model and with the method based on direct simulation of implied volatility, described in [7]. We choose the problem of minimal-variance portfolio choice as the main measure of model performance and compare the three models. Our study demonstrates that the tangent Levy
model does a better job at finding a portfolio with the smallest variance than the SABR model. In addition, the prediction of return variance, provided by the tangent Levy model, is more reliable and the portfolio weights are more stable. We also find that the performance of the direct simulation method, at the portfolio choice problem, is not much worse than that of the tangent Levy model. However, the direct simulation method of [7] is not arbitrage-free. We illustrate this shortcoming by comparing the direct simulation method and the tangent Levy model at a different problem: estimation of Value at Risk of an options' portfolio. To the best of our knowledge, this paper is the first example of empirical analysis, based on real market data, which provides a convincing
evidence of the superior performance of market-based models for European options, as compared to the classical spot models.

... In this paper, we implement and test two types of market-based models for European-type options, based on the tangent Lévy models proposed in [4] and [3]. As a result, we obtain a method for generating Monte Carlo samples of future paths of implied volatility surfaces. ...

... Such mapping became known as a code-book mapping, and it turns out that it can be constructed by means of the so-called tangent models (cf. [2], [4], [3]). The concept of a tangent model is very close to the method of calibrating a model for underlying to the target derivatives' prices prices (in the present case, European options calls). ...

... Finally, one needs to characterize all possible dynamics of (θ t ) that produce no dynamic arbitrage in the associated call prices C θt . An interested reader is referred to [3], for a more detailed description of this general algorithm, and, for example, to [2], [4], [17], [33], [22], [27], for the analysis of specific choices of the families of models {M(θ)}. ...

In this paper, we implement and test two types of market-based models for
European-type options, based on the tangent Levy models proposed recently by R.
Carmona and S. Nadtochiy. As a result, we obtain a method for generating Monte
Carlo samples of future paths of implied volatility surfaces. These paths and
the surfaces themselves are free of arbitrage, and are constructed in a way
that is consistent with the past and present values of implied volatility. We
use a real market data to estimate the parameters of these models and conduct
an empirical study, to compare the performance of market-based models with the
performance of classical stochastic volatility models. We choose the problem of
minimal-variance portfolio choice as a measure of model performance and compare
the two tangent Levy models to SABR model. Our study demonstrates that the
tangent Levy models do a much better job at finding a portfolio with smallest
variance, their predictions for the variance are more reliable, and the
portfolio weights are more stable. To the best of our knowledge, this is the
first example of empirical analysis that provides a convincing evidence of the
superior performance of the market-based models for European options using real
market data.

... Using the PDE (2.1) to substitute C t = 20 Other early articles on risk-neutral dynamics for stochastic implied volatility models include [41,11,39]. For more recent developments on arbitrage-free market models for (parts of or the whole) option price surface, we refer the reader to [55,54,13,35,14,15,36] and the references therein. 21 The parametrisation in terms of the squared volatility of implied volatility is explained in Remark 3.2. ...

We study option pricing and hedging with uncertainty about a Black-Scholes reference model which is dynamically recalibrated to the market price of a liquidly traded vanilla option. For dynamic trading in the underlying asset and this vanilla option, delta-vega hedging is asymptotically optimal in the limit for small uncertainty aversion. The corresponding indifference price corrections are determined by the disparity between the vegas, gammas, vannas, and volgas of the non-traded and the liquidly traded options.

We model the dynamics of asset prices and associated derivatives by consideration of the dynamics of the conditional probability density process for the value of an asset at some specified time in the future. In the case where the price process is driven by Brownian motion, an associated "master equation" for the dynamics of the conditional probability density is derived and expressed in integral form. By a "model" for the conditional density process we mean a solution to the master equation along with the specification of (a) the initial density, and (b) the volatility structure of the density. The volatility structure is assumed at any time and for each value of the argument of the density to be a functional of the history of the density up to that time. In practice one specifies the functional modulo sufficient parametric freedom to allow for the input of additional option data apart from that implicit in the initial density. The scheme is sufficiently flexible to allow for the input of various types of data depending on the nature of the options market and the class of valuation problem being undertaken. Various examples are studied in detail, with exact solutions provided in some cases.

This paper aims at transferring the philosophy behind Heath-Jarrow-Morton to
the modelling of call options with all strikes and maturities. Contrary to the
approach by Carmona and Nadtochiy (2009) and related to the recent contribution
Carmona and Nadtochiy (2012) by the same authors, the key parametrisation of
our approach involves time-inhomogeneous L\'evy processes instead of local
volatility models. We provide necessary and sufficient conditions for absence
of arbitrage. Moreover we discuss the construction of arbitrage-free models.
Specifically, we prove their existence and uniqueness given basic building
blocks.

We model the dynamics of asset prices and associated derivatives by
consideration of the dynamics of the conditional probability density process
for the value of an asset at some specified time in the future. In the case
where the price process is driven by Brownian motion, an associated "master
equation" for the dynamics of the conditional probability density is derived
and expressed in integral form. By a "model" for the conditional density
process we mean a solution to the master equation along with the specification
of (a) the initial density, and (b) the volatility structure of the density.
The volatility structure is assumed at any time and for each value of the
argument of the density to be a functional of the history of the density up to
that time. In practice one specifies the functional modulo sufficient
parametric freedom to allow for the input of additional option data apart from
that implicit in the initial density. The scheme is sufficiently flexible to
allow for the input of various types of data depending on the nature of the
options market and the class of valuation problem being undertaken. Various
examples are studied in detail, with exact solutions provided in some cases.