Pricing Options Under Hestons Stochastic Volatility Model Via Accelerated Explicit Finite Differencing Methods
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| Author | : Conall O'Sullivan |
| Publisher | : |
| Total Pages | : 41 |
| Release | : 2010 |
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We present an acceleration technique, effective for explicit finite difference schemes describing diffusive processes with nearly symmetric operators, called Super-Time-Stepping (STS). The technique is applied to the two-factor problem of option pricing under stochastic volatility. It is shown to significantly reduce the severity of the stability constraint known as the Courant-Friedrichs-Lewy condition whilst retaining the simplicity of the chosen underlying explicit method. For European and American put options under Heston's stochastic volatility model we demonstrate degrees of acceleration over standard explicit methods sufficient to achieve comparable, or superior, efficiencies to a benchmark implicit scheme. We conclude that STS is a powerful tool for the numerical pricing of options and propose them as the method-of-choice for exotic financial instruments in two and multi-factor models.
| Author | : |
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| Release | : 2001 |
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The Heston stochastic volatility model is one extension of the Black-Scholes model which describes the money markets more accurately so that more realistic prices for derivative products are obtained. From the stochastic differential equation of the underlying financial product a partial differential equation (p.d.e.) for the value function of an option can be derived. This p.d.e. can be solved with the finite difference method (f.d.m.). The stability and consistency of the method is examined. Furthermore a boundary condition is proposed to reduce the numerical error. Finally a non uniform structured grid is derived which is fairly optimal for the numerical result in the most interesting point.
| Author | : Dominique Joubert |
| Publisher | : |
| Total Pages | : 144 |
| Release | : 2013 |
| Genre | : Electronic dissertations |
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| Author | : Dominique Joubert |
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| Total Pages | : 0 |
| Release | : 2013 |
| Genre | : Electronic dissertations |
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Early exercise boundary -- Free boundary value problem -- Linear complimentary problem -- Crank-Nicolson finite difference method -- Projected Over-Relaxation method (PSOR) -- Stochastic volatility -- Heston stochastic volatility model -- Vroeë uitoefengrens -- Vrye grenswaardeprobleem -- Linêere komplimentêre probleem -- Crank-Nicolson eindige differensiemetode -- Geprojekteerde oorverslappingsmetode (PSOR) -- Stogastiese volatiliteit -- Heston stogastiese volatiliteitsmodel.
| Author | : Bertram Düring |
| Publisher | : |
| Total Pages | : 18 |
| Release | : 2015 |
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We propose a new high-order alternating direction implicit (ADI) finite difference scheme for the solution of initial-boundary value problems of convection-diffusion type with mixed derivatives and non-constant coefficients, as they arise from stochastic volatility models in option pricing. Our approach combines different high-order spatial discretisations with Hundsdorfer and Verwer's ADI time-stepping method, to obtain an efficient method which is fourth-order accurate in space and second-order accurate in time. Numerical experiments for the European put option pricing problem using Heston's stochastic volatility model confirm the high-order convergence.
| Author | : Bertram Düring |
| Publisher | : |
| Total Pages | : 21 |
| Release | : 2017 |
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We derive a new high-order compact finite difference scheme for option pricing in stochastic volatility jump models, e.g. in Bates model. In such models the option price is determined as the solution of a partial integro-differential equation. The scheme is fourth order accurate in space and second order accurate in time. Numerical experiments for the European option pricing problem are presented. We validate the stability of the scheme numerically and compare its efficiency and hedging performance to standard finite difference methods. The new scheme outperforms a standard discretisation based on a second-order central finite difference approximation in all our experiments. At the same time, it is very efficient, requiring only one initial LU-factorisation of a sparse matrix to perform the option price valuation. It can also be useful to upgrade existing implementations based on standard finite differences in a straightforward manner to obtain a highly efficient option pricing code.
| Author | : Carl Chiarella |
| Publisher | : |
| Total Pages | : 19 |
| Release | : 2009 |
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A compound option (the mother option) gives the holder the right, but not obligation to buy (long) or sell (short) the underlying option (the daughter option). In this paper, we demonstrate a partial differential equation (PDE) approach to pricing American-type compound options where the underlying dynamics follow Heston's stochastic volatility model. This price is formulated as the solution to a two-pass free boundary PDE problem. A modified sparse grid approach is implemented to solve the PDEs, which is shown to be accurate and efficient compared with the results from Monte Carlo simulation combined with the Method of Lines.
| Author | : Susanne Griebsch |
| Publisher | : |
| Total Pages | : 29 |
| Release | : 2010 |
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We focus on closed-form option pricing in Heston's stochastic volatility model, where closed-form formulas exist only for a few option types. Most of these closed-form solutions are constructed from characteristic functions. We follow this closed-form approach and derive multivariate characteristic functions depending on at least two spot values for different points in time. The derived characteristic functions are used as building blocks to set up (semi-) analytical pricing formulas for exotic options with payoffs depending on finitely many spot values such as fader options and discretely monitored barrier options. We compare our result with different numerical methods and examine accuracy and computational times.
| Author | : Bertram Düring |
| Publisher | : |
| Total Pages | : 21 |
| Release | : 2014 |
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We derive high-order compact finite difference schemes for option pricing in stochastic volatility models on non-uniform grids. The schemes are fourth-order accurate in space and second-order accurate in time for vanishing correlation. In our numerical study we obtain high-order numerical convergence also for non-zero correlation and non-smooth payoffs which are typical in option pricing. In all numerical experiments a comparative standard second-order discretisation is significantly outperformed. We conduct a numerical stability study which indicates unconditional stability of the scheme.
| Author | : Jason Fink |
| Publisher | : |
| Total Pages | : 23 |
| Release | : 2005 |
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Estimation of option pricing models in which the underlying asset exhibits stochastic volatility presents complicated econometric questions. One such question, thus far unstudied, is whether the inclusion of information derived from hedging relationships implied by an option pricing model may be used in conjunction with pricing information to provide more reliable parameter estimates than the use of pricing information alone. This paper estimates, using a simple least-squares procedure, the stochastic volatility model of Heston (1993), and includes hedging information in the objective function. This hedging information enters the objective function through a weighting parameter that is chosen optimally within the model. With the weight appropriately chosen, we find that incorporating the hedging information reduces both the out-of-sample hedging and pricing errors associated with the Heston model.
