A very common situation is to have a correct anticipation, but resulting in a loss, because the position is not consistent with the view: At the previous time step, its value at each node gives a profile that can be written as a portfolio of three Calls with neighboring strikes expiring immediately. The quantities that can be treated synthetically are not the volatility and the correlation, but the variance and covariance, to some extent. Add a new comment. Topics Discussed in This Paper. It is also the tool that allows to exploit the differences between forward values and views, converting them into trading strategies.

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If an option price is given by the market we can invert this relationship to get the implied volatility. If the model were perfect, this implied value would be the same for all option market prices, but reality shows this is not the case.

Implied Black—Scholes vol Implied Black—Scholes volatilities strongly depend on the maturity and the strike of the European option under scrutiny. It is easy to solve this paradox by allowing volatility to be timedependent, as Merton did see Merton, We now have a single process, compatible with the two option prices. From the term structure of implied volatilities we can infer a time-dependent instantaneous volatility, because the former is the quadratic mean of the latter.

We present a model for pricing and hedging derivative securities and option portfolios in an environment where the volatility is not known precisely, but is assumed instead to lie between two extreme values oe min and oe max. These bounds could be inferred from extreme values of the implied volatil These bounds could be inferred from extreme values of the implied volatilities of liquid options, or from high-low peaks in historical stock- or option-implied volatilities.

They can be viewed as defining a confidence interval for future volatility values. We show that the extremal non-arbitrageable prices for the derivative asset which arise as the volatility paths vary in such a band can be described by a non-linear PDE, which we call the Black-Scholes-Barenblatt equation. A simple algorithm for solving the equation by finite-differencing or a trinomial tree is presented. We show that this model capture Show Context Citation Context In this paper a stochastic volatility model is presented that directly prescribes the stochastic development of the implied Black-Scholes volatilities of a set of given standard options.

Thus the model is able to capture the stochastic movements of a full term structure of implied volatilities. The conditions are derived that have to be satisfied to ensure absence of arbitrage in the model and its numerical implementation is discussed.

Sin - University of Cambridge , " We show a class of stochastic volatility price models for which the most natural candidates for martingale measures are only strictly local martingale measures, contrary to what is usually assumed in the finance literature.

We also show the existence of martingale measures, however, and give explici We also show the existence of martingale measures, however, and give explicit examples. If we want to apply results from arbitrage-pricing theory, which char Sun, P. Carr, J. Sun - Review of Derivatives Research , " Abstract We develop a new approach for pricing European-style contingent claims written on the time T spot price of an underlying asset whose volatility is stochastic.

Like most of the stochastic volatility literature, we assume continuous dynamics for the price of the underlying asset. In contrast In contrast to most of the stochastic volatility literature, we do not directly model the dynamics of the instantaneous volatility. Instead, taking advantage of the recent rise of the variance swap market, we directly assume continuous dynamics for the time T variance swap rate.

The initial value of this variance swap rate can either be directly observed, or inferred from option prices. We make no assumption concerning the real world drift of this process. We assume that the ratio of the volatility of the variance swap rate to the instantaneous volatility of the underlying asset just depends on the variance swap rate and on the variance swap maturity. Since this ratio is assumed to be independent of calendar time, we term this key assumption the stationary volatility ratio hypothesis SVRH.

The instantaneous volatility of the futures follows an unspecified stochastic process, so both the underlying futures price and the variance swap rate have unspecified stochastic volatility. Despite this, we show that the payoff to a path-independent contingent claim can be perfectly replicated by dynamic trading in futures contracts and variance swaps of the same maturity.

In contrast to standard models of stochastic volatility, our approach does not require specifying the market price of volatility risk or observing the initial level of instantaneous volatility. This paper presents a new approach to modeling the dynamics of implied distributions. Subsequently, we develop new arbitrage-free Subsequently, we develop new arbitrage-free Monte-Carlo simulation methods that model the evolution of the whole distribution through time as a diffusion process.

The out-of-sample performance within a Value-at-Risk framework is examined. Hedging variance options on continuous semimartingales. We find robust model-free hedges and price bounds for options on the realized variance of [the returns on] an underlying price process. Assuming only that the underlying process is a positive continuous semimartingale, we superreplicate and subreplicate variance options and forward-starting variance Assuming only that the underlying process is a positive continuous semimartingale, we superreplicate and subreplicate variance options and forward-starting variance options, by dynamically trading the underlying asset, and statically holding European options.

We thereby derive upper and lower bounds on values of variance options, in terms of Europeans. This paper includes extension and unification of the replica Model independent hedging strategies for variance swaps by David Hobson, Martin Klimmek , " A variance swap is a derivative with a path-dependent payoff which allows investors to take positions on the future variability of an asset. In the idealised setting of a continuously monitored variance swap written on an asset with continuous paths it is well known that the variance swap payoff can In the idealised setting of a continuously monitored variance swap written on an asset with continuous paths it is well known that the variance swap payoff can be replicated exactly using a portfolio of puts and calls and a dynamic position in the asset.

This fact forms the basis of the VIX contract. But what if we are in the more realistic setting where the contract is based on discrete monitoring, and the underlying asset may have jumps? We show that it is possible to derive model-independent, no-arbitrage bounds on the price of the variance swap, and corresponding sub- and super-replicating strategies.

Further, we characterise the optimal bounds. The form of the hedges depends crucially on the kernel used to define the variance swap. In those articles, it was shown that if we assume that the asset price process is a continuous forward price, then the continuously monitored variance sw

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