Robust numerical methods for PDE models of Asian options

We explore the pricing of Asian options by numerically solving the the associated partial di erential equations We demonstrate that numerical PDE techniques commonly used in nance for standard options are inaccurate in the case of Asian options and illustrate mod i cations which alleviate this problem In particular the usual methods generally produce solutions containing spurious oscillations We adapt ux limiting techniques originally de veloped in the eld of computational uid dynamics in order to rapidly obtain accurate solutions We show that ux limiting methods are total variation diminishing and hence free of spurious oscillations for non conservative PDEs such as those typically encountered in nance for fully explicit and fully and partially implicit schemes We also modify the van Leer ux limiter so that the second order total variation diminishing property is preserved for non uniform grid spacing Introduction Asian options are securities with payo s which depend on the average value of an underlying stock price over some time interval Such options have proven to be much more di cult to value than regular stock options Standard techniques tend to be impractical inaccurate or slow For example traditional binomial lattice methods require such enormous amounts of computer memory owing to the necessity of keeping track of every possible path throughout the tree that they are e ectively unusable Partial di erential equation PDE methods as traditionally implemented in the nance literature are inaccurate see Barraquand and Pudet for a discussion Monte Carlo simulation works well for European style op tions see Kemna and Vorst but is relatively slow A number of approximations have appeared in the literature e g Turnbull and Wakeman Vorst Levy Levy and Turnbull which are again suitable only for European style options See also Geman and Yor who derive the Laplace transform of the European option price Unfortunately this transform is very di cult to invert With regard to American style Asian options there are even fewer alternatives Hull and White propose a modi cation of the binomial method but do not provide any proof of convergence Neave uses a frequency distribution approach on a binomial lattice to derive approximate values for arithmetic average option values but his method still requires calculations of order N where N is the number of time steps in the lattice Barraquand and Pudet describe a forward shooting grid algorithm and prove that it is unconditionally convergent We explore another possibility a modi ed nite di erence method In general the price of an Asian option can be found by solving a PDE in two space like dimensions see Ingersoll or Wilmott Dewynne and Howison This PDE has the character of a two dimensional convection di usion problem with no di usion in one of the spatial dimensions As is well known in computational uid dynamics standard centrally weighted methods for the convective term are prone to oscillatory solutions Furthermore as argued by Barraquand and Pudet standard nite di erence methods though generally faster than their proposed algorithm are inaccurate because they introduce spurious numerical di usion p In some cases the price of an Asian option can be modeled using a one dimensional PDE The two dimensional PDE for a oating strike Asian option can be reduced to a one dimensional PDE see Ingersoll or Wilmott Dewynne and Howison Re cently Rogers and Shi have formulated a one dimensional PDE that can model the price of both oating and xed strike Asian options However this PDE applies only to the case of European style options and is particularly di cult to solve numerically since the di usion term is very small for values of interest on the nite di erence grid We demonstrate modi cations to the common discretization methods which are designed to handle these problems In particular in both the two dimensional and one dimensional cases it is necessary to solve a problem with little or no di usion i e second order deriva tive term in a space dimension The traditional approach in computational uid dynamics would be to use rst order upstream weighting for the convective term to eliminate the oscil lations caused by centrally weighted schemes Roache However rst order upstream weighting results in solutions with excessive false di usion As an alternative we employ a high order non linear ux limiter for the convective terms The resulting discrete non linear algebraic equations are solved using full Newton iteration In addition we can also apply the American early exercise constraint to the algebraic system and this can be handled in an implicit fully coupled manner In cases where the model cannot be reduced to a problem in a single space dimension the full two dimensional problem must be solved For exam ple the price of a xed strike American style Asian option must be found by solving the two dimensional PDE We apply the above methods i e the ux limiter and full Newton iteration to a full two dimensional problem In this case an iterative method ILU CGSTAB D Azevedo et al van der Vorst is used to solve the resulting Jacobian matrix The outline of the paper is as follows Section describes the option pricing models to be considered Section presents a discretization analysis for nite di erence methods as applied to standard options We concentrate on situations with extremely low volatility which as noted above are analogous to the case of Asian options We illustrate the types of problems which can arise with commonly applied methods in nance and also how our modi cations mitigate these di culties both in terms of option prices and hedging parame ters Section presents applications to Asian options and the paper concludes with a brief summary which is contained in Section The Models We adopt the usual geometric Brownian motion model for the evolution of a stock price S dS rSdt SdB where r denotes the risk free interest rate is the volatility and dB is a standard Brownian motion Under the conventional assumptions of frictionless markets the value at time t of a claim contingent on the stock price at subsequent time T may be represented as V S t t e r T t Et g S T T where g S T T denotes the payo function for the claim and Et denotes expectation conditional on information available at time t Familiar examples include European calls g S T T max S T K and puts g S T max K S T where K is the strike price of the option It is well known that V solves the following PDE V t S V S rS V S rV subject to the appropriate boundary conditions for the call or put Analytic solutions for these cases were derived by Black and Scholes The early exercise feature for American put options can be incorporated by imposing the constraint V S max K S at each point in time over the life of the option The hedging arguments underlying are standard and may also be applied in the context of Asian options see Ingersoll pp for a discussion Such options depend on the arithmetic average of the stock price over some time interval If we let