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.
Based on a criterium of mathematical simplicity and consistency with empirical market data, a stochastic volatility model has been obtained with the volatility process driven by fractional noise. Depending on whether the stochasticity generators of log-price and volatility are independent or are the same, two versions of the model are obtained with different leverage behavior. Here, the no-arbitrage and incompleteness properties of the model are studied. Some risk measures are also discussed in this framework.
In this article we propose a generalisation of the recent work of Gatheral and Jacquier on explicit arbitrage-free parameterisations of implied volatility surfaces. We also discuss extensively the notion of arbitrage freeness and Roger Lee's moment formula using the recent analysis by Roper. We further exhibit an arbitrage-free volatility surface different from Gatheral's SVI parameterisation.
In this article, we show how to calibrate the widely-used SVI parameterization of the implied volatility surface in such a way as to guarantee the absence of static arbitrage. In particular, we exhibit a large class of arbitrage-free SVI volatility surfaces with a simple closed-form representation. We demonstrate the high quality of typical SVI fits with a numerical example using recent SPX options data.
We present an explicit hedging strategy, which enables to prove arbitrageness of market incorporating at least two assets depending on the same random factor. The implied Black-Scholes volatility, computed taking into account the form of the graph of the option price, related to our strategy, demonstrates the "skewness" inherent to the observational data.
We consider a generic market model with a single stock and with random volatility. We assume that there is a number of tradable options for that stock with different strike prices. The paper states the problem of finding a pricing rule that gives Black-Scholes price for at-money options and such that the market is arbitrage free for any number of tradable options, even if there are two Brownian motions only: one drives the stock price, the other drives the volatility process. This problem is reduced to solving a parabolic equation.
University of Toronto（多伦多大学）；Riskfuel Analytics（Riskfuel分析公司）
We propose a hybrid method for generating arbitrage-free implied volatility (IV) surfaces consistent with historical data by combining model-free Variational Autoencoders (VAEs) with continuous time stochastic differential equation (SDE) driven models. We focus on two classes of SDE models: regime switching models and Lévy additive processes. By projecting historical surfaces onto the space of SDE model parameters, we obtain a distribution on the parameter subspace faithful to the data on which we then train a VAE. Arbitrage-free IV surfaces are then generated by sampling from the posterior di…
We investigate financial markets under model risk caused by uncertain volatilities. For this purpose we consider a financial market that features volatility uncertainty. To have a mathematical consistent framework we use the notion of G-expectation and its corresponding G-Brownian motion recently introduced by Peng (2007). Our financial market consists of a riskless asset and a risky stock with price process modeled by a geometric G-Brownian motion. We adapt the notion of arbitrage to this more complex situation and consider stock price dynamics which exclude arbitrage opportunities. Due to vo…
In this work, we aim to gain a better understanding of the volatility smile observed in options markets through microsimulation (MS). We adopt two types of active traders in our MS model: speculators and arbitrageurs, and call and put options on one underlying asset. Speculators make decisions based on their expectations of the asset price at the option expiration time. Arbitrageurs trade at different arbitrage opportunities such as violation of put-call parity. Difference in liquidity among options is also included. Notwithstanding its simplicity, our model can generate implied volatility (IV…
This paper gives an arbitrage-free prediction for future prices of an arbitrary co-terminal set of options with a given maturity, based on the observed time series of these option prices. The statistical analysis of such a multi-dimensional time series of option prices corresponding to $n$ strikes (with $n$ large, e.g. $n\geq 40$) and the same maturity, is a difficult task due to the fact that option prices at any moment in time satisfy non-linear and non-explicit no-arbitrage restrictions. Hence any $n$-dimensional time series model also has to satisfy these implicit restrictions at each time…
We develop a dynamic version of the SSVI parameterisation for the total implied variance, ensuring that European vanilla option prices are martingales, hence preventing the occurrence of arbitrage, both static and dynamic. Insisting on the constraint that the total implied variance needs to be null at the maturity of the option, we show that no model--in our setting--allows for such behaviour. This naturally gives rise to the concept of implied volatility bubbles, whereby trading in an arbitrage-free way is only possible during part of the life of the contract, but not all the way until expiry…
We introduce a multivariate diffusion model that is able to price derivative securities featuring multiple underlying assets. Each asset volatility smile is modeled according to a density-mixture dynamical model while the same property holds for the multivariate process of all assets, whose density is a mixture of multivariate basic densities. This allows to reconcile single name and index/basket volatility smiles in a consistent framework. Our approach could be dubbed a multidimensional local volatility approach with vector-state dependent diffusion matrix. The model is quite tractable, leadi…
We characterize absence of arbitrage with simple trading strategies in a discounted market with a constant bond and several risky assets. We show that if there is a simple arbitrage, then there is a 0-admissible one or an obvious one, that is, a simple arbitrage which promises a minimal riskless gain of ε, if the investor trades at all. For continuous stock models, we provide an equivalent condition for absence of 0-admissible simple arbitrage in terms of a property of the fine structure of the paths, which we call "two-way crossing." This property can be verified for many models by the law of…
"Fundamental theorem of asset pricing" roughly states that absence of arbitrage opportunity in a market is equivalent to the existence of a risk-neutral probability. We give a simple counterexample to this oversimplified statement. Prices are given by linear forms which do not always correspond to probabilities. We give examples of such cases. We also show that arbitrage freedom is equivalent to the continuity of the pricing linear form in the relevant topology. Finally we analyze the possible loss of martingality of asset prices with lognormal stochastic volatility. For positive correlation m…
We consider a general class of continuous asset price models where the drift and the volatility functions, as well as the driving Brownian motions, change at a random time $τ$. Under minimal assumptions on the random time and on the driving Brownian motions, we study the behavior of the model in all the filtrations which naturally arise in this setting, establishing martingale representation results and characterizing the validity of the NA1 and NFLVR no-arbitrage conditions.
In this paper we investigate general linear stochastic volatility models with correlated Brownian noises. In such models the asset price satisfies a linear SDE with coefficient of linearity being the volatility process. This class contains among others Black-Scholes model, a log-normal stochastic volatility model and Heston stochastic volatility model. For a linear stochastic volatility model we derive representations for the probability density function of the arbitrage price of a financial asset and the prices of European call and put options. A closed-form formulae for the density function …
This paper formulates a model of utility for a continuous time framework that captures the decision-maker's concern with ambiguity about both volatility and drift. Corresponding extensions of some basic results in asset pricing theory are presented. First, we derive arbitrage-free pricing rules based on hedging arguments. Ambiguous volatility implies market incompleteness that rules out perfect hedging. Consequently, hedging arguments determine prices only up to intervals. However, sharper predictions can be obtained by assuming preference maximization and equilibrium. Thus we apply the model …
This study presents new analytic approximations of the stochastic-alpha-beta-rho (SABR) model. Unlike existing studies that focus on the equivalent Black-Scholes (BS) volatility, we instead derive the equivalent constant-elasticity-of-variance (CEV) volatility. Our approach effectively reduces the approximation error in a way similar to the control variate method because the CEV model is the zero vol-of-vol limit of the SABR model. Moreover, the CEV volatility approximation yields a finite value at a zero strike and thus conveniently leads to a small-time asymptotics for the mass at zero. The …
In this work, we introduce a Monte Carlo method for the dynamic hedging of general European-type contingent claims in a multidimensional Brownian arbitrage-free market. Based on bounded variation martingale approximations for Galtchouk-Kunita-Watanabe decompositions, we propose a feasible and constructive methodology which allows us to compute pure hedging strategies w.r.t arbitrary square-integrable claims in incomplete markets. In particular, the methodology can be applied to quadratic hedging-type strategies for fully path-dependent options with stochastic volatility and discontinuous payof…
In a recent formulation of a quantum field theory of forward rates, the volatility of the forward rates was taken to be deterministic. The field theory of the forward rates is generalized to the case of stochastic volatility. Two cases are analyzed, firstly when volatility is taken to be a function of the forward rates, and secondly when volatility is taken to be an independent quantum field. Since volatiltiy is a positive valued quantum field, the full theory turns out to be an interacting nonlinear quantum field theory in two dimensions. The state space and Hamiltonian for the interacting th…
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 Lévy 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…
Gulisashvili et al. [Quant. Finance, 2018, 18(10), 1753-1765] provide a small-time asymptotics for the mass at zero under the uncorrelated stochastic-alpha-beta-rho (SABR) model by approximating the integrated variance with a moment-matched lognormal distribution. We improve the accuracy of the numerical integration by using the Gauss--Hermite quadrature. We further obtain the option price by integrating the constant elasticity of variance (CEV) option prices in the same manner without resorting to the small-strike volatility smile asymptotics of De Marco et al. [SIAM J. Financ. Math., 2017, 8…
We consider the problem of superhedging under volatility uncertainty for an investor allowed to dynamically trade the underlying asset, and statically trade European call options for all possible strikes with some given maturity. This problem is classically approached by means of the Skorohod Embedding Problem (SEP). Instead, we provide a dual formulation which converts the superhedging problem into a continuous martingale optimal transportation problem. We then show that this formulation allows us to recover previously known results about lookback options. In particular, our methodology induc…
Implied volatility is at the very core of modern finance, notwithstanding standard option pricing models continue to derive option prices starting from the joint dynamics of the underlying asset price and the spot volatility. These models often cause difficulties: no closed formulas for prices, demanding calibration techniques, unclear maps between spot and implied volatility. Inspired by the practice of using implied volatility as quoting system for option prices, models for the joint dynamics of the underlying asset price and the implied volatility have been proposed to replace standard opti…
When pricing options, there may be different views on the instantaneous mean return of the underlying price process. According to Black (1972), where there exist heterogeneous views on the instantaneous mean return, this will result in arbitrage opportunities. Behavioral finance proponents argue that such heterogenous views are likely to occur and this will not impact option pricing models proposed by rational dynamic asset pricing theory and will not give rise to volatility smiles. To rectify this, a leading advocate of behavioral finance has proposed a behavioral option pricing model. As the…
Anomalous diffusions arise as scaling limits of continuous-time random walks (CTRWs) whose innovation times are distributed according to a power law. The impact of a non-exponential waiting time does not vanish with time and leads to different distribution spread rates compared to standard models. In financial modelling this has been used to accommodate for random trade duration in the tick-by-tick price process. We show here that anomalous diffusions are able to reproduce the market behaviour of the implied volatility more consistently than usual Lévy or stochastic volatility models. We focus…
Valuing Guaranteed Minimum Withdrawal Benefit (GMWB) has attracted significant attention from both the academic field and real world financial markets. As remarked by Yang and Dai, the Black and Scholes framework seems to be inappropriate for such a long maturity products. Also Chen Vetzal and Forsyth in showed that the price of these products is very sensitive to interest rate and volatility parameters. We propose here to use a stochastic volatility model (Heston model) and a Black Scholes model with stochastic interest rate (Hull White model). For this purpose we present four numerical metho…
