Using spectral decomposition techniques and singular perturbation theory, we develop a systematic method to approximate the prices of a variety of options in a fast mean-reverting stochastic volatility setting. Four examples are provided in order to demonstrate the versatility of our method. These include: European options, up-and-out options, double-barrier knock-out options, and options which pay a rebate upon hitting a boundary. For European options, our method is shown to produce option price approximations which are equivalent to those developed in [5].
[5] Jean-Pierre Fouque, George Papanicolaou, and Sircar Ronnie. Derivatives in Financial Markets with Stochas- tic Volatility. Cambridge University Press, 2000.
We introduce a class of randomly time-changed fast mean-reverting stochastic volatility models and, using spectral theory and singular perturbation techniques, we derive an approximation for the prices of European options in this setting. Three examples of random time-changes are provided and the implied volatility surfaces induced by these time-changes are examined as a function of the model parameters. Three key features of our framework are that we are able to incorporate jumps into the price process of the underlying asset, allow for the leverage effect, and accommodate multiple factors of…
We propose a multi-scale stochastic volatility model in which a fast mean-reverting factor of volatility is built on top of the Heston stochastic volatility model. A singular pertubative expansion is then used to obtain an approximation for European option prices. The resulting pricing formulas are semi-analytic, in the sense that they can be expressed as integrals. Difficulties associated with the numerical evaluation of these integrals are discussed, and techniques for avoiding these difficulties are provided. Overall, it is shown that computational complexity for our model is comparable to …
This paper investigates problems associated with the valuation of callable American volatility put options. Our approach involves modeling volatility dynamics as a mean-reverting 3/2 volatility process. We first propose a pricing formula for the perpetual American knock-out put. Under the given conditions, the value of perpetual callable American volatility put options is discussed.
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 w…
In this paper, we study stochastic volatility models in regimes where the maturity is small, but large compared to the mean-reversion time of the stochastic volatility factor. The problem falls in the class of averaging/homogenization problems for nonlinear HJB-type equations where the "fast variable" lives in a noncompact space. We develop a general argument based on viscosity solutions which we apply to the two regimes studied in the paper. We derive a large deviation principle, and we deduce asymptotic prices for out-of-the-money call and put options, and their corresponding implied volatil…
The literature on volatility modelling and option pricing is a large and diverse area due to its importance and applications. This paper provides a review of the most significant volatility models and option pricing methods, beginning with constant volatility models up to stochastic volatility. We also survey less commonly known models e.g. hybrid models. We explain various volatility types (e.g. realised and implied volatility) and discuss the empirical properties.
Most models for barrier pricing are designed to let a market maker tune the model-implied covariance between moves in the asset spot price and moves in the implied volatility skew. This is often implemented with a local volatility/stochastic volatility mixture model, where the mixture parameter tunes that covariance. This paper defines an alternate model where the spot/volatility correlation is a separate mean-reverting stochastic variable which is itself correlated with spot. We also develop an efficient approximation for barrier option and one touch pricing in the model based on semi-static …
We derive the implied volatility estimation formula in European power call options pricing, where the payoff functions are in the form of $V=(S^α_T-K)^{+}$ and $V=(S^α_T-K^α)^{+}$ ($α>0$)respectively. Using quadratic Taylor approximations, We develop the computing formula of implied volatility in European power call option and extend the traditional implied volatility formula of Charles J.Corrado, et al (1996) to general power option pricing. And the Monte-Carlo simulations are also given.
Volatility modelling has become a significant area of research within Financial Mathematics. Wiener process driven stochastic volatility models have become popular due their consistency with theoretical arguments and empirical observations. However such models lack the ability to take into account long term and fundamental economic factors e.g. credit crunch. Regime switching models with mean reverting stochastic volatility are a new class of stochastic volatility models that capture both short and long term characteristics. We propose a new general method of pricing options for these new clas…
We discuss the pricing and hedging of volatility options in some rough volatility models. First, we develop efficient Monte Carlo methods and asymptotic approximations for computing option prices and hedge ratios in models where log-volatility follows a Gaussian Volterra process. While providing a good fit for European options, these models are unable to reproduce the VIX option smile observed in the market, and are thus not suitable for VIX products. To accommodate these, we introduce the class of modulated Volterra processes, and show that they successfully capture the VIX smile.
In this paper new analytical and numerical approaches to valuating path-dependent options of European type have been developed. The model of stochastic volatility as a basic model has been chosen. For European options we could improve the path integral method, proposed B. Baaquie, and generalized it to the case of path-dependent options, where the payoff function depends on the history of changes in the underlying asset. The dependence of the implied volatility on the parameters of the stochastic volatility model has been studied. It is shown that with proper choice of model parameters one can…
The aim of this study was to develop methods for evaluating the American-style option prices when the volatility of the underlying asset is described by a stochastic process. As part of this problem were developed techniques for modeling the early exercise surface of the American option. These methods of present work are compared to the complexity of modeling and computation speed. The paper presents the semi-analytic expression for the price of American options with stochastic volatility. The results of numerical computations and their calibration are also presented. The obtained results were…
Two novel closed-form formulas for the price of barrier options in stochastic volatility models with zero interest rate and dividend yield but nonzero correlation between the asset and its instantaneous volatility are derived. The first is a Hull and White type formula, and the second is a decomposition formula similar in form to the Alòs decomposition for vanilla options. A model-free approximation is also given.
In this paper we present some results on Geometric Asian option valuation for affine stochastic volatility models with jumps. We shall provide a general framework into which several different valuation problems based on some average process can be cast, and we shall obtain close-form solutions for some relevant affine model classes.
In previous works Avellaneda et al. pioneered the pricing and hedging of index options - products highly sensitive to implied volatility and correlation assumptions - with large deviations methods, assuming local volatility dynamics for all components of the index. We here present an extension applicable to non-Markovian dynamics and in particular the case of rough volatility dynamics.
In the current literature, the analytical tractability of discrete time option pricing models is guaranteed only for rather specific types of models and pricing kernels. We propose a very general and fully analytical option pricing framework, encompassing a wide class of discrete time models featuring multiple-component structure in both volatility and leverage, and a flexible pricing kernel with multiple risk premia. Although the proposed framework is general enough to include either GARCH-type volatility, Realized Volatility or a combination of the two, in this paper we focus on realized vol…
We consider stochastic control systems affected by a fast mean reverting volatility $Y(t)$ driven by a pure jump Lévy process. Motivated by a large literature on financial models, we assume that $Y(t)$ evolves at a faster time scale $\frac{t}{\varepsilon}$ than the assets, and we study the asymptotics as $\varepsilon\to 0$. This is a singular perturbation problem that we study mostly by PDE methods within the theory of viscosity solutions.
Multiscale stochastic volatility models have been developed as an efficient way to capture the principle effects on derivative pricing and portfolio optimization of randomly varying volatility. The recent book Fouque, Papanicolaou, Sircar and Sølna (2011, CUP) analyzes models in which the volatility of the underlying is driven by two diffusions -- one fast mean-reverting and one slow-varying, and provides a first order approximation for European option prices and for the implied volatility surface, which is calibrated to market data. Here, we present the full second order asymptotics, which ar…
The aim of this paper is to present a simple stochastic model that accounts for the effects of a long-memory in volatility on option pricing. The starting point is the stochastic Black-Scholes equation involving volatility with long-range dependence. We consider the option price as a sum of classical Black-Scholes price and random deviation describing the risk from the random volatility. By using the fact the option price and random volatility change on different time scales, we find the asymptotic equation for the derivation involving fractional Brownian motion. The solution to this equation …
Based on empirical market data, a stochastic volatility model is proposed with volatility driven by fractional noise. The model is used to obtain a risk-neutrality option pricing formula and an option pricing equation.
In this paper, we derive closed-form formulas of first-order approximation for down-and-out barrier and floating strike lookback put option prices under a stochastic volatility model, by using an asymptotic approach. To find the explicit closed-form formulas for the zero-order term and the first-order correction term, we use Mellin transform. We also conduct a sensitivity analysis on these formulas, and compare the option prices calculated by them with those generated by Monte-Carlo simulation.
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 o…
We present an adaptive approach for valuing the European call option on assets with stochastic volatility. The essential feature of the method is a reduction of uncertainty in latent volatility due to a Bayesian learning procedure. Starting from a discrete-time stochastic volatility model, we derive a recurrence equation for the variance of the innovation term in latent volatility equation. This equation describes a reduction of uncertainty in volatility which is crucial for option pricing. To implement the idea of adaptive control, we use the risk-minimization procedure involving random volat…
The Local Volatility model is a well-known extension of the Black-Scholes constant volatility model whereby the volatility is dependent on both time and the underlying asset. This model can be calibrated to provide a perfect fit to a wide range of implied volatility surfaces. The model is easy to calibrate and still very popular in FX option trading. In this paper we address a question of validation of the Local Volatility model. Different stochastic models for the underlying can be calibrated to provide a good fit to the current market data but should be recalibrated every trading date. A goo…
In this paper, we study the asymptotic behavior of Asian option prices in the worst case scenario under an uncertain volatility model. We give a procedure to approximate the Asian option prices with a small volatility interval. By imposing additional conditions on the boundary condition and cutting the obtained Black-Scholes-Barenblatt equation into two Black-Scholes-like equations, we obtain an approximation method to solve the fully nonlinear PDE.
We investigate whether it is possible to formulate option pricing and hedging models without using probability. We present a model that is consistent with two notions of volatility: a historical volatility consistent with statistical analysis, and an implied volatility consistent with options priced with the model. The latter will be also the quadratic variation of the model, a pathwise property. This first result, originally presented in Brigo and Mercurio (1998, 2000), is then connected with the recent work of Armstrong et al (2018, 2021), where using rough paths theory it is shown that impl…
Using Malliavin Calculus techniques, we derive closed-form expressions for the at-the-money behaviour of the forward implied volatility, its skew and its curvature, in general Markovian stochastic volatility models with continuous paths.
This paper proposes the sample path generation method for the stochastic volatility version of CGMY process. We present the Monte-Carlo method for European and American option pricing with the sample path generation and calibrate model parameters to the American style S\&P 100 index options market, using the least square regression method. Moreover, we discuss path-dependent options such as Asian and Barrier options.
According to the volatility feedback effect, an unexpected increase in squared volatility leads to an immediate decline in the price-dividend ratio. In this paper, we consider the properties of stock price dynamics and option valuations under the volatility feedback effect by modeling the joint dynamics of stock price, dividends, and volatility in continuous time. Most importantly, our model predicts the negative effect of an increase in squared return volatility on the value of deep-in-the-money call options and, furthermore, attempts to explain the volatility puzzle. We theoretically demonst…
This paper studies subordinate Ornstein-Uhlenbeck (OU) processes, i.e., OU diffusions time changed by Lévy subordinators. We construct their sample path decomposition, show that they possess mean-reverting jumps, study their equivalent measure transformations, and the spectral representation of their transition semigroups in terms of Hermite expansions. As an application, we propose a new class of commodity models with mean-reverting jumps based on subordinate OU process. Further time changing by the integral of a CIR process plus a deterministic function of time, we induce stochastic volatili…
In this paper, we derive the price of a European call option of an asset following a normal process assuming stochastic volatility. The volatility is assumed to follow the Cox Ingersoll Ross (CIR) process. We then use the fast Fourier transform (FFT) to evaluate the option price given we know the characteristic function of the return analytically. We compare the results of fast Fourier transform with the Monte Carlo simulation results of our process. Further, we present a numerical example to understand the normal implied volatility of the model.
Recent years have seen an emerging class of structured financial products based on options linked to dynamic asset allocation strategies. One of the most chosen approach is the so-called target volatility mechanism. It shifts between risky and riskless assets to control the volatility of the overall portfolio. Even if a series of articles have been already devoted to the analysis of options linked to the target volatility mechanism, this paper is the first, to the best of our knowledge, that tries to develop closed-end formulas for VolTarget options. In particular, we develop closed-end formul…
We present a rigorous study of the short maturity asymptotics for Asian options with continuous-time averaging, under the assumption that the underlying asset follows a local volatility model. The asymptotics for out-of-the-money, in-the-money, and at-the-money cases are derived, considering both fixed strike and floating strike Asian options. The asymptotics for the out-of-the-money case involves a non-trivial variational problem which is solved completely. We present an analytical approximation for Asian options prices, and demonstrate good numerical agreement of the asymptotic results with …
In this paper we derive an easily computed approximation to European basket call prices for a local volatility jump-diffusion model. We apply the asymptotic expansion method to find the approximate value of the lower bound of European basket call prices. If the local volatility function is time independent then there is a closed-form expression for the approximation. Numerical tests show that the suggested approximation is fast and accurate in comparison with the Monte Carlo and other approximation methods in the literature.
In the context of stochastic volatility models, we study representation formulas in terms of expectations for the power series' coefficients associated to the call price-function. As in a recent paper by Antonelli and Scarlatti the expansion is done w.r.t. the correlation between the noises driving the underlying asset price process and the volatility process. We first obtain expressions for the power series' coefficients from the generalized Hull and White formula obtained by Elisa Alòs. Afterwards, we provide representations turning out from the approach for the sensitivity problem tackled b…
We revisit the problem of pricing options with historical volatility estimators. We do this in the context of a generalized GARCH model with multiple time scales and asymmetry. It is argued that the reason for the observed volatility risk premium is tail risk aversion. We parametrize such risk aversion in terms of three coefficients: convexity, skew and kurtosis risk premium. We propose that option prices under the real-world measure are not martingales, but that their drift is governed by such tail risk premia. We then derive a fair-pricing equation for options and show that the solutions can…
We prove that the perpetual American put option price of level dependent volatility model with compound Poisson jumps is convex and is the classical solution of its associated quasi-variational inequality, that it is $C^2$ except at the stopping boundary and that it is $C^1$ everywhere (i.e. the smooth pasting condition always holds).
In the paper, we characterize the asymptotic behavior of the implied volatility of a basket call option at large and small strikes in a variety of settings with increasing generality. First, we obtain an asymptotic formula with an error bound for the left wing of the implied volatility, under the assumption that the dynamics of asset prices are described by the multidimensional Black-Scholes model. Next, we find the leading term of asymptotics of the implied volatility in the case where the asset prices follow the multidimensional Black-Scholes model with time change by an independent increasi…
Recent empirical studies suggest that the volatility of an underlying price process may have correlations that decay slowly under certain market conditions. In this paper, the volatility is modeled as a stationary process with long-range correlation properties in order to capture such a situation, and we consider European option pricing. This means that the volatility process is neither a Markov process nor a martingale. However, by exploiting the fact that the price process is still a semimartingale and accordingly using the martingale method, we can obtain an analytical expression for the op…
Some expansion methods have been proposed for approximately pricing options which has no exact closed formula. Benhamou et al. (2010) presents the smart expansion method that directly expands the expectation value of payoff function with respect to the volatility of volatility, then uses it to price options in the stochastic volatility model. In this paper, we apply their method to the stochastic volatility model with stochastic interest rates, and present the expansion formula for pricing options up to the second order. Then the numerical studies are performed to compare our approximation for…
The article is devoted to models of financial markets with stochastic volatility, which is defined by a functional of Ornstein-Uhlenbeck process or Cox-Ingersoll-Ross process. We study the question of exact price of European option. The form of the density function of the random variable, which expresses the average of the volatility over time to maturity is established using Malliavin calculus.The result allows calculate the price of the option with respect to minimum martingale measure when the Wiener process driving the evolution of asset price and the Wiener process, which defines volatili…
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…
Matrix Approximation of Bachelier Option Prices and Greeks under Stochastic Volatility models
Department of Economics and Business, Universitat Pompeu Fabra（庞培法布拉大学经济与商业系）；Barcelona School of Economics（巴塞罗那经济学校）；Departament de Matemàtica Econòmica, Financera i Actuarial, Universitat de Barcelona（巴塞罗那大学经济、金融与精算数学系）
We introduce a multi-factor stochastic volatility model based on the CIR/Heston stochastic volatility process. In order to capture the Samuelson effect displayed by commodity futures contracts, we add expiry-dependent exponential damping factors to their volatility coefficients. The pricing of single underlying European options on futures contracts is straightforward and can incorporate the volatility smile or skew observed in the market. We calculate the joint characteristic function of two futures contracts in the model in analytic form and use the one-dimensional Fourier inversion method of…
In this work, I address the issue of forming riskless hedge in the continuous time option pricing model with stochastic stock volatility. I show that it is essential to verify whether the replicating portfolio is self-financing, in order for the theory to be self-consistent. The replicating methods in existing finance literature are shown to violate the self-financing constraint when the underlying asset has stochastic volatility. Correct self-financing hedge is formed in this article.
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 i…
We discuss the possibility of obtaining model-free bounds on volatility derivatives, given present market data in the form of a calibrated local volatility model. A counter-example to a wide-spread conjecture is given.
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 app…
We present a path integral method to derive closed-form solutions for option prices in a stochastic volatility model. The method is explained in detail for the pricing of a plain vanilla option. The flexibility of our approach is demonstrated by extending the realm of closed-form option price formulas to the case where both the volatility and interest rates are stochastic. This flexibility is promising for the treatment of exotic options. Our new analytical formulas are tested with numerical Monte Carlo simulations.
