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40
Multiscale stochastic volatility asymptotics
 SIAM J. MULTISCALE MODELING AND SIMULATION
, 2003
"... In this paper we propose to use a combination of regular and singular perturbations to analyze parabolic PDEs that arise in the context of pricing options when the volatility is a stochastic process that varies on several characteristic time scales. The classical BlackScholes formula gives the pri ..."
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Cited by 50 (20 self)
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In this paper we propose to use a combination of regular and singular perturbations to analyze parabolic PDEs that arise in the context of pricing options when the volatility is a stochastic process that varies on several characteristic time scales. The classical BlackScholes formula gives the price of call options when the underlying is a geometric Brownian motion with a constant volatility. The underlying might be the price of a stock or an index say and a constant volatility corresponds to a fixed standard deviation for the random fluctuations in the returns of the underlying. Modern market phenomena makes it important to analyze the situation when this volatility is not fixed but rather is heterogeneous and varies with time. In previous work, see for instance [5], we considered the situation when the volatility is fast mean reverting. Using a singular perturbation expansion we derived an approximation for option prices. We also provided a calibration method using observed option prices as represented by the socalled term structure of implied volatility. Our analysis of market data, however, shows the need for introducing also a slowly varying factor in the model for the stochastic volatility. The combination of regular and singular perturbations approach that we set forth in this paper deals with this case. The resulting approximation is still independent of the particular details of the volatility model and gives more flexibility in the parametrization of the
Singular Perturbations In Option Pricing
 SIAM J. Applied Math
, 2002
"... After the celebrated BlackScholes formula for pricing call options under constant volatility, the need for more general nonconstant volatility models in financial mathematics has been the motivation of numerous works during the Eighties and Nineties. In particular, a lot of attention has been paid ..."
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Cited by 41 (11 self)
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After the celebrated BlackScholes formula for pricing call options under constant volatility, the need for more general nonconstant volatility models in financial mathematics has been the motivation of numerous works during the Eighties and Nineties. In particular, a lot of attention has been paid to stochastic volatility models where the volatility is randomly fluctuating driven by an additional Brownian motion. We have shown in [2, 3] that, in the presence of a separation of time scales, between the main observed process and the volatility driving process, asymptotic methods are very efficient in capturing the effects of random volatility in simple robust corrections to constant volatility formulas. From the point of view of partial differential equations this method corresponds to a singular perturbation analysis. The aim of this paper is to deal with the nonsmoothness of the payoff function inherent to option pricing. We present the case of call options for which the payoff function forms an angle at the strike price. This case is important since these are the typical instruments used in the calibration of pricing models. We establish the pointwise accuracy of the corrected BlackScholes price by using an appropriate payoff regularization which is removed simultaneously as the asymptotics is performed.
IMPLIED AND LOCAL VOLATILITIES UNDER STOCHASTIC VOLATILITY
, 2001
"... For asset prices that follow stochasticvolatility diffusions, we use asymptotic methods to investigate the behavior of the local volatilities and Black–Scholes volatilities implied by option prices, and to relate this behavior to the parameters of the stochastic volatility process. We also give app ..."
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Cited by 26 (3 self)
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For asset prices that follow stochasticvolatility diffusions, we use asymptotic methods to investigate the behavior of the local volatilities and Black–Scholes volatilities implied by option prices, and to relate this behavior to the parameters of the stochastic volatility process. We also give applications, including riskpremiumbased explanations of the biases in some naïve pricing and hedging schemes. We begin by reviewing option pricing under stochastic volatility and representing option prices and local volatilities in terms of expectations. In the case that fluctuations in price and volatility have zero correlation, the expectations formula shows that local volatility (like implied volatility) as a function of logmoneyness has the shape of a symmetric smile. In the case of nonzero correlation, we extend Sircar and Papanicolaou’s asymptotic expansion of implied volatilities under slowlyvarying stochastic volatility. An asymptotic expansion of local volatilities then verifies the rule of thumb that local volatility has the shape of a skew with roughly twice the slope of the implied volatility skew. Also we compare the slowvariation asymptotics against what we call smallvariation asymptotics, and against Fouque, Papanicolaou, and Sircar’s rapidvariation
Stochastic Volatility Corrections for Interest Rate Derivatives
, 2002
"... We study simple models of short rates such as the Vasicek or CIR models, and compute corrections that come from the presence of fast meanreverting stochastic volatility. We show how these small corrections can affect the shape of the term structure of interest rates giving a simple and efficient ..."
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Cited by 19 (7 self)
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We study simple models of short rates such as the Vasicek or CIR models, and compute corrections that come from the presence of fast meanreverting stochastic volatility. We show how these small corrections can affect the shape of the term structure of interest rates giving a simple and efficient calibration tool. This is used to price other derivatives such as bond options. The analysis extends the asymptotic method developed for equity derivatives in (Fouque, Papanicolaou and Sircar 2000b).
Stochastic Volatility, Smile & Asymptotics
, 1998
"... We consider the pricing and hedging problem for options on stocks whose volatility is a random process. Traditional approaches, such as that of Hull & White, have been successful in accounting for the much observed smile curve, and the success of a large class of such models in this respect is g ..."
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Cited by 17 (9 self)
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We consider the pricing and hedging problem for options on stocks whose volatility is a random process. Traditional approaches, such as that of Hull & White, have been successful in accounting for the much observed smile curve, and the success of a large class of such models in this respect is guaranteed by a theorem of Renault & Touzi, for which we present a simplified proof. We also present new asymptotic formulas that describe the geometry of smile curves and can be used for interpolation of implied volatility data. Motivated by the robustness of the smile effect to specific modelling of the unobserved volatility process, we present a new approach to stochastic volatility modelling starting with the BlackScholes pricing PDE with a random volatility coefficient. We identify and exploit distinct time scales of fluctuation for the stock price and volatility processes yielding an asymptotic approximation that is a BlackScholes type price or hedging ratio plus a Gaussian random variable quantifying the risk from the uncertainty in the volatility. These lead us to translate volatility risk into pricing and hedging bands for the derivative securities, without needing to estimate the market's value of risk. For some special cases, we can give explicit formulas. We outline
Stochastic volatility: option pricing using a multinomial recombining tree
, 2006
"... We treat the problem of option pricing under the Stochastic Volatility (SV) model: the volatility of the underlying asset is a function of an exogenous stochastic process, typically assumed to be meanreverting. Assuming that only discrete past stock information is available, we adapt an interacting ..."
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Cited by 17 (5 self)
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We treat the problem of option pricing under the Stochastic Volatility (SV) model: the volatility of the underlying asset is a function of an exogenous stochastic process, typically assumed to be meanreverting. Assuming that only discrete past stock information is available, we adapt an interacting particle stochastic filtering algorithm due to Del Moral, Jacod and Protter (Del Moral et al., 2001) to estimate the SV, and construct a quadrinomial tree which samples volatilities from the SV filter’s empirical measure approximation at time 0. Proofs of convergence of the tree to continuoustime SV models are provided. Classical arbitragefree option pricing is performed on the tree, and provides answers that are close to market prices of options on the SP500 or on bluechip stocks. We compare our results to nonrandom volatility models, and to models which continue to estimate volatility after time 0. We show precisely how to calibrate our incomplete market, choosing a specific martingale measure, by using a benchmark option. Key words and phrases: incomplete markets, MonteCarlo method, options market, option pricing, particle method, random tree, stochastic filtering, stochastic volatility. 1
Financial Modeling in a Fast MeanReverting Stochastic Volatility Environment
, 1998
"... We present a derivative pricing and estimation methodology for a class of stochastic volatility models that exploits the observed "bursty" or persistent nature of stock price volatility. Empirical analysis of highfrequency S&P 500 index data confirms that volatility reverts slowly to ..."
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Cited by 16 (3 self)
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We present a derivative pricing and estimation methodology for a class of stochastic volatility models that exploits the observed "bursty" or persistent nature of stock price volatility. Empirical analysis of highfrequency S&P 500 index data confirms that volatility reverts slowly to its mean in comparison to the tickbytick uctuations of the index value, but it is fast meanreverting when looked at over the time scale of a derivative contract (many months). This motivates an asymptotic analysis of the partial differential equation satis ed by derivative prices, utilizing the distinction between these time scales. The analysis yields pricing and implied volatility formulas, and the latter provides a simple procedure to "fit the skew" from European index option prices. The theory identifies the important group parameters that are needed for the derivative pricing and hedging problem for Europeanstyle securities, namely the average volatility and the slope and intercept of the implied volatility line, plotted as a function of the logmoneynesstomaturityratio. The results considerably simplify the estimation
Pricing Asian options with stochastic volatility”,
 Quantitative Finance,
, 2003
"... Abstract In this paper, we generalize the recently developed dimension reduction technique of Vecer for pricing arithmetic average Asian options. The assumption of constant volatility in Vecer's method will be relaxed to the case that volatility is randomly fluctuating and is driven by a mean ..."
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Cited by 12 (1 self)
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Abstract In this paper, we generalize the recently developed dimension reduction technique of Vecer for pricing arithmetic average Asian options. The assumption of constant volatility in Vecer's method will be relaxed to the case that volatility is randomly fluctuating and is driven by a meanreverting (or ergodic) process. We then use the fast meanreverting stochastic volatility asymptotic analysis introduced by Fouque, Papanicolaou and Sircar to derive an approximation to the option price which takes into account the skew of the implied volatility surface. This approximation is obtained by solving a pair of onedimensional partial differential equations.
Approximation and calibration of shortterm implied volatility under jumpdiffusion stochastic volatility, The review of Financial Studies
, 2007
"... We derive a closedform asymptotic expansion formula for option implied volatility under a twofactor jumpdiffusion stochastic volatility model when timetomaturity is small. Based on numerical experiments we describe the range of timetomaturity and moneyness for which the approximation is accur ..."
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Cited by 11 (1 self)
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We derive a closedform asymptotic expansion formula for option implied volatility under a twofactor jumpdiffusion stochastic volatility model when timetomaturity is small. Based on numerical experiments we describe the range of timetomaturity and moneyness for which the approximation is accurate. We further propose a simple calibration procedure of an arbitrary parametric model to shortterm nearthemoney implied volatilities. An important advantage of our approximation is that it is free of the unobserved spot volatility. Therefore, the model can be calibrated on option data pooled across different calendar dates in order to extract information from the dynamics of the implied volatility smile. An example of calibration to a sample of S&P500 option prices is provided. We find that jumps are significant. The evidence also supports an affine specification for the jump intensity and ConstantElasticityofVariance for the dynamics of the return volatility. Key words: Option pricing, stochastic volatility, asymptotic approximation, jumpdiffusion. JEL Classification: G12. 1An early version of this paper has circulated under the title "A simple calibration procedure of stochastic volatility models with jumps by short term asymptotics". We would like to thank the Editor and the referee for constructive criticism and comments which have led to a substantial improvement over the previous
Path dependent option pricing: the path integral partial averaging method
 Journal of Computational Finance
, 2002
"... In this paper I develop a new computational method for pricing path dependent options. Using the path integral representation of the option price, I show that in general it is possible to perform analytically a partial averaging over the underlying riskneutral diffusion process. This result greatly ..."
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Cited by 10 (0 self)
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In this paper I develop a new computational method for pricing path dependent options. Using the path integral representation of the option price, I show that in general it is possible to perform analytically a partial averaging over the underlying riskneutral diffusion process. This result greatly eases the computational burden placed on the subsequent numerical evaluation. For shortmedium term options it leads to a general approximation formula that only requires the evaluation of a one dimensional integral. I illustrate the application of the method to Asian options and occupation time derivatives.