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30
Penalty Methods For American Options With Stochastic Volatility
, 1998
"... The American early exercise constraint can be viewed as transforming the two dimensional stochastic volatility option pricing PDE into a differential algebraic equation (DAE). Several methods are described for forcing the algebraic constraint by using a penalty source term in the discrete equations. ..."
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Cited by 63 (18 self)
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The American early exercise constraint can be viewed as transforming the two dimensional stochastic volatility option pricing PDE into a differential algebraic equation (DAE). Several methods are described for forcing the algebraic constraint by using a penalty source term in the discrete equations. The resulting nonlinear algebraic equations are solved using an approximate Newton iteration. The solution of the Jacobian is obtained using an incomplete LU (ILU) preconditioned PCG method. Some example computations are presented for option pricing problems based on a stochastic volatility model, including an exotic American chooser option written on a put and call with discrete double knockout barriers and discrete dividends.
Numerical Convergence Properties of Option Pricing PDEs with Uncertain Volatility
 IMA Journal of Numerical Analysis
, 2003
"... The pricing equations derived from uncertain volatility models in finance are often cast in the form of nonlinear partial differential equations. Implicit timestepping leads to a set of nonlinear algebraic equations which must be solved at each timestep. To solve these equations, an iterative approa ..."
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Cited by 26 (15 self)
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The pricing equations derived from uncertain volatility models in finance are often cast in the form of nonlinear partial differential equations. Implicit timestepping leads to a set of nonlinear algebraic equations which must be solved at each timestep. To solve these equations, an iterative approach is employed. In this paper, we prove the convergence of a particular iterative scheme for one factor uncertain volatility models. We also demonstrate how nonmonotone discretization schemes (such as standard CrankNicolson timestepping) can converge to incorrect solutions, or lead to instability. Numerical examples are provided.
Shout Options: A Framework For Pricing Contracts Which Can Be Modified By The Investor
 J. Comp. Appl. Math
, 1999
"... A shout option may be broadly defined as a financial contract which can be modified by its holder according to specified rules. In a simple example, the holder could have the right to set the strike of an option equal to the current value of the underlying asset. In such a case, the holder effective ..."
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Cited by 12 (9 self)
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A shout option may be broadly defined as a financial contract which can be modified by its holder according to specified rules. In a simple example, the holder could have the right to set the strike of an option equal to the current value of the underlying asset. In such a case, the holder effectively has the right to select when to take ownership of an atthemoney option. More generally, the holder could have multiple rights along these lines, in some cases with a limit placed on the number of rights which may be exercised within a given time period (e.g. four times per year). The value of these types of contracts can be estimated by solving a system of interdependent linear complementarity problems. This paper describes a general framework for the valuation of complex types of shout options. Numerical issues related to interpolation and choice of timestepping method are considered in detail. Some illustrative examples are provided.
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 10 (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
A General Finite Element Approach For PDE Option Pricing Models
, 1998
"... . This paper presents a general approach for solving twofactor (twodimensional) option pricing problems. The finite element method provides greater flexibility over that of the finite difference schemes (or equivalently, lattice methods) which are often employed in finance. This paper will demonst ..."
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Cited by 8 (3 self)
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. This paper presents a general approach for solving twofactor (twodimensional) option pricing problems. The finite element method provides greater flexibility over that of the finite difference schemes (or equivalently, lattice methods) which are often employed in finance. This paper will demonstrate how various twodimensional pricing problems can all be solved using the same approach. The generality of the approach is in part due to the fact that changes caused by different model specifications are localized. Constraints on the solution are treated in a uniform manner using a penalty method. This uniform approach can readily accommodate constraints such as earlyexercise opportunities and barriers. Keywords: Finite element, option pricing, local extremum diminishing Running Title: Finite element option pricing Acknowledgment: This work was supported by the National Sciences and Engineering Research Council of Canada, the Social Sciences and Humanities Research Council of Canada,...
Numerical Methods for Nonlinear Equations in Option Pricing
, 2003
"... This thesis explores numerical methods for solving nonlinear partial di#erential equations (PDEs) that arise in option pricing problems. The goal is to develop or identify robust and e#cient techniques that converge to the financially relevant solution for both one and two factor problems. To illust ..."
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Cited by 5 (0 self)
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This thesis explores numerical methods for solving nonlinear partial di#erential equations (PDEs) that arise in option pricing problems. The goal is to develop or identify robust and e#cient techniques that converge to the financially relevant solution for both one and two factor problems. To illustrate the underlying concepts, two nonlinear models are examined in detail: uncertain volatility and passport options.
Abstract The Black Scholes Barenblatt Equation for Options with Uncertain Volatility and its Application to Static Hedging
, 2004
"... The Black Scholes Barenblatt (BSB) equation for the envelope of option prices with uncertain volatility and interest rate is derived from the Black Scholes equation with the maximum principle for diffusion equations and shown to be equivalent to a readily solvable standard Black Scholes equation wit ..."
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Cited by 4 (1 self)
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The Black Scholes Barenblatt (BSB) equation for the envelope of option prices with uncertain volatility and interest rate is derived from the Black Scholes equation with the maximum principle for diffusion equations and shown to be equivalent to a readily solvable standard Black Scholes equation with a nonlinear source term. Analogous arguments yield the envelope for the delta of option prices. We then interpret the concept of static hedging for narrowing the envelope in terms of partial differential equations and define the optimal static hedge and computable approximations to it. We apply the BSB equation to find numerically some optimally hedged portfolios of representative European and American options. 1.
Pricing Methods and Hedging Strategies for Volatility Derivatives
, 2003
"... In this paper we investigate the behaviour and hedging of discretely observed volatility derivatives. We begin by comparing the e#ects of variations in the contract design, such as the di#erences between specifying log returns or actual returns, taking into consideration the impact of possible jumps ..."
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Cited by 4 (0 self)
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In this paper we investigate the behaviour and hedging of discretely observed volatility derivatives. We begin by comparing the e#ects of variations in the contract design, such as the di#erences between specifying log returns or actual returns, taking into consideration the impact of possible jumps in the underlying asset. We then focus on the di#culties associated with hedging these products. Naive deltahedging strategies are ine#ective for hedging volatility derivatives since they require very frequent rebalancing and have limited ability to protect the writer against possible jumps in the underlying asset. We investigate the performance of a hedging strategy for volatility swaps that establishes small, fixed positions in straddles and outofthemoney strangles at each volatility observation.
An ObjectOriented Framework For Valuing Shout Options On HighPerformance Computer Architectures
 Journal of Economic Dynamics and Control, forthcoming, http://www.scicom.uwaterloo.ca
, 2000
"... A shout option is a financial contract which allows the holder to change the payoff during the lifetime of the contract. For example, the holder could have the right to set the strike price to the current value of the underlying asset. Complex versions of these options are embedded in financial p ..."
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Cited by 3 (3 self)
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A shout option is a financial contract which allows the holder to change the payoff during the lifetime of the contract. For example, the holder could have the right to set the strike price to the current value of the underlying asset. Complex versions of these options are embedded in financial products which offer various types of maturity guarantees such as segregated funds marketed by Canadian insurance companies. The value of these options can be determined by solving a collection of coupled partial differential equations (PDEs). In this work we develop an extensible, objectoriented framework for valuing these contracts which is capable of exploiting modern, highperformance supercomputing architectures. We use this framework to study and illustrate practical aspects of valuing and hedging these contracts. Keywords: PDE option pricing, shout options, objectoriented, highperformance architecture Running Title: Shout Options Acknowledgement: This work was supported by ...
Calibration and hedging under jump diffusion
 Review of Derivatives Research
, 2006
"... A jump diffusion model coupled with a local volatility function has been suggested by Andersen and Andreasen (2000). This model is attractive in that it shows promise in terms of being able to capture observed market crosssectional implied volatilities, without being unduly complex. By generating a ..."
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Cited by 3 (2 self)
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A jump diffusion model coupled with a local volatility function has been suggested by Andersen and Andreasen (2000). This model is attractive in that it shows promise in terms of being able to capture observed market crosssectional implied volatilities, without being unduly complex. By generating a discrete set of American option prices assuming a jump diffusion with known parameters (i.e. in a synthetic market), we investigate two crucial challenges intrinsic to this type of model: calibration of parameters and hedging of jump risk. Our investigation suggests that it can be difficult to estimate the model parameters that govern the jump size distribution. However, the local volatility function is easier to estimate when an appropriate regularization (e.g. splines) is used to avoid overfitting. In general, even though the estimation problem is illposed, it appears that combining jump diffusion with a local volatility function produces a model which can be calibrated with sufficient accuracy to prices of liquid vanilla options. With regard to hedging jump risk, two different hedging strategies are explored: a semistatic approach which uses a portfolio of the underlying and traded short maturity options to hedge a long maturity option, and a dynamic technique which involves frequent trading of options and the underlying. Simulation experiments in the synthetic market suggest that both of these methods can be used to sharply reduce the standard deviation of the hedging portfolio relative profit and loss distribution.