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Register a new userOption pricing under Ornstein-Uhlenbeck stochastic volatility: a linear model
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We consider the problem of option pricing under stochastic volatility models, focusing on the linear approximation of the two processes known as exponential Ornstein-Uhlenbeck and Stein-Stein. Indeed, we show they admit the same limit dynamics in the regime of low fluctuations of the volatility process, under which we derive the exact expression of the characteristic function associated to the risk neutral probability density. This expression allows us to compute option prices exploiting a formula derived by Lewis and Lipton. We analyze in detail the case of Plain Vanilla calls, being liquid instruments for which reliable implied volatility surfaces are available. We also compute the analytical expressions of the first four cumulants, that are crucial to implement a simple two steps calibration procedure. It has been tested against a data set of options traded on the Milan Stock Exchange. The data analysis that we present reveals a good fit with the market implied surfaces and corroborates the accuracy of the linear approximation.
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In the classical model of stock prices which is assumed to be Geometric Brownian motion, the drift and the volatility of the prices are held constant. However, in reality, the volatility does vary. In quantitative finance, the Heston model has been successfully used where the volatility is expressed as a stochastic differential equation. In addition, we consider a regime switching model where the stock volatility dynamics depends on an underlying process which is possibly a non-Markov pure jump process. Under this model assumption, we find the locally risk minimizing pricing of European type vanilla options. The price function is shown to satisfy a Heston type PDE.
Recent empirical studies suggest that the volatilities associated with financial time series exhibit short-range correlations. This entails that the volatility process is very rough and its autocorrelation exhibits sharp decay at the origin. Another classic stylistic feature often assumed for the volatility is that it is mean reverting. In this paper it is shown that the price impact of a rapidly mean reverting rough volatility model coincides with that associated with fast mean reverting Markov stochastic volatility models. This reconciles the empirical observation of rough volatility paths with the good fit of the implied volatility surface to models of fast mean reverting Markov volatilities. Moreover, the result conforms with recent numerical results regarding rough stochastic volatility models. It extends the scope of models for which the asymptotic results of fast mean reverting Markov volatilities are valid. The paper concludes with a general discussion of fractional volatility asymptotics and their interrelation. The regimes discussed there include fast and slow volatility factors with strong or small volatility fluctuations and with the limits not commuting in general. The notion of a characteristic term structure exponent is introduced, this exponent governs the implied volatility term structure in the various asymptotic regimes.
In this paper, we study the valuation of American type derivatives in the stochastic volatility model of Barndorff-Nielsen and Shephard (2001). We characterize the value of such derivatives as the unique viscosity solution of an integral-partial differential equation when the payoff function satisfies a Lipschitz condition.
We consider option pricing using a discrete-time Markov switching stochastic volatility with co-jump model, which can model volatility clustering and varying mean-reversion speeds of volatility. For pricing European options, we develop a computationally efficient method for obtaining the probability distribution of average integrated variance (AIV), which is key to option pricing under stochastic-volatility-type models. Building upon the efficiency of the European option pricing approach, we are able to price an American-style option, by converting its pricing into the pricing of a portfolio of European options. Our work also provides constructive guidance for analyzing derivatives based on variance, e.g., the variance swap. Numerical results indicate our methods can be implemented very efficiently and accurately.
This paper focuses on the pricing of continuous geometric Asian options (GAOs) under a multifactor stochastic volatility model. The model considers fast and slow mean reverting factors of volatility, where slow volatility factor is approximated by a quadratic arc. The asymptotic expansion of the price function is assumed, and the first order price approximation is derived using the perturbation techniques for both floating and fixed strike GAOs. Much simplified pricing formulae for the GAOs are obtained in this multifactor stochastic volatility framework. The zeroth order term in the price approximation is the modified Black-Scholes price for the GAOs. This modified price is expressed in terms of the Black-Scholes price for the GAOs. The accuracy of the approximate option pricing formulae is established, and the model parameter is also estimated by capturing the volatility smiles.
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