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This paper investigates several competing procedures for computing the prices of vanilla European options, such as puts, calls and binaries, in which the underlying model has a characteristic function that is known in semi-closed form. The algorithms investigated here are the half-range Fourier cosine series, the half-range Fourier sine series and the full-range Fourier series. Their performance is assessed in simulation experiments in which an analytical solution is available and also for a simple affine model of stochastic volatility in which there is no closed-form solution. The results suggest that the half-range sine series approximation is the least effective of the three proposed algorithms. It is rather more difficult to distinguish between the performance of the half-range cosine series and the full-range Fourier series. However there are two clear differences. First, when the interval over which the density is approximated is relatively large, the full-range Fourier series is at least as good as the half-range Fourier cosine series, and outperforms the latter in pricing out-of-the-money call options, in particular with maturities of three months or less. Second, the computational time required by the half-range Fourier cosine series is uniformly longer than that required by the full-range Fourier series for an interval of fixed length. Taken together, these two conclusions make a case for pricing options using a full-range range Fourier series as opposed to a half-range Fourier cosine series if a large number of options are to be priced in as short a time as possible.

The need to price vanilla European options in a rapid manner arises in numerous activities at financial institutions. Perhaps the most well-known situation in which this need occurs in practice is model calibration, in which exotic options are priced using models with values for the (risk-neutral) parameters chosen in such a way as to ensure that the model reproduces quoted prices for liquid options. For each set of parameters considered in the search space, it is therefore necessary to evaluate the prices of all options in the calibration set using the model, before comparing these with the quoted prices. In a similar vein, there is a growing literature which uses large panels of options data for estimating model parameters ([

Various strategies have been proposed for calculating the price of option contracts from knowledge of the conditional characteristic function of the underlying model. It is an important fact that a surprisingly large number of models have a semi-closed expression for their conditional characteristic function. For example, the identification of the conditional characteristic function for multivariate affine models with/without jump processes leads to the solution of a family of ordinary differential equations, albeit in the complex plane. In view of the Levy-Khintchine theorem, the identification of the conditional characteristic function for Levy processes is expressed in terms of various integrals with respect to the Levy measure.

The most commonly used techniques for taking advantage of a known conditional characteristic function have at their core the application of the Fast Fourier Transform (FFT). The best documented approaches is due to Carr and Madan [

Rather than describing in detail the nuances of these various strategies, it is useful to point out what overarching assumptions connect them. Recall that the FFT is simply a clever piece of linear algebra that reduces the arithmetical load in implementing the Discrete Fourier Transform (DFT), namely the pair of equations connecting the coefficients of a finite Fourier series with values of the underlying function and vice versa. Therefore the decision to use the FFT implicitly makes the assumption that the underlying function is periodic over an interval of finite length, in practice determined by the range of frequencies submitted to the characteristic function, and that the function has been approximated over the interval by a finite Fourier series. The values of Fourier coefficients calculated from the characteristic function are in error by the extent to which the Finite Fourier Transform^{1} differs from the Fourier transform.

Thus techniques using the FFT and those based on the construction of Fourier series share the same common assumptions and deficiencies. However, an important difference between an implementation using the FFT approach and one using the Fourier series approach is that the latter is parsimonious in its use of arithmetic whereas the former typically performs more arithmetic than necessary, albeit in an efficient way. For example, if the FFT is used to determine the value of a probability density function, what is recovered is the value of the function at each node of the interval, whereas all that might be needed is the value of the probability density function over a sub-interval.

The focus of this work is on the algorithm proposed by Fang and Oosterlee [^{2} cosine series because the actual function to be expanded is defined only on half the interval of periodicity (or range), the function being extended to the full range as an even-valued function. Half-range cosine series usually fail to represent derivatives whereas half-range Fourier sine series usually fail to represent function values. While the use of the half-range Fourier cosine series is a solid idea, Fang and Oosterlee [

An important but subtle difference shared by the half-range Fourier cosine and full-range Fourier series approximations of density, but different from representations of density based on the half-range Fourier sine series, is that the former assign unit probability to the interval of support when in reality probability lies outside this interval, whereas the latter imposes zero probability density at the endpoints of the interval of support in contravention of reality, but on the other hand does not assign unit probability to the interval of support. Is one approach always superior to the other or is it a case of horses for courses? Intuition might suggest the latter. For example, when pricing a call option the most important component of the pricing error comes from the exclusion of contributions from asset price exterior to the finite interval of support. Because the half-range cosine and full-range Fourier series necessarily capture unit density, intuition might suggest that these approximations provide potential compensation for this component of pricing error. On the other hand intuition would suggest that the same approximations, when used to price binary options, might have a tendency to exaggerate the probability of exercise and therefore overprice this option in contrast to the half-range sine series approximation of probability density.

Suppose that

The use of the term “half-range” in describing Expressions (a) and (b) simply refers to the fact that the function

In the case of the half-range cosine and sine series in Expressions (1a) and (1b) respectively, the coefficients

in which

In terms of the exponential function the real coefficients

In the case of the full-range Fourier series in Expression (1c) the real coefficients

Suppose now that

where

while the coefficients of the full-range Fourier series can be approximated from knowledge of the characteristic function via the formula

The accuracy of approximations (6) and (7) is investigated in Section 6, where it is demonstrated that the error can be made arbitrarily small by choosing a suitably large interval.

The quality of this practical idea is now explored for three trial probability density functions with known closed-form expressions for their characteristic functions. The first choice is the Gaussian density which may be regarded as representative of distributions with super-exponentially decaying tail density. The second and third choices are the Gamma density and the Cauchy density which are treated as representative examples of distributions with exponentially decaying and algebraically decaying tail densities respectively.

It is well known that the Gaussian density with mean value

where

With as few as 4 frequencies it is clear that the approximating density still provides a good representation of the true density; with 40 frequencies the approximating density function is indistinguishable from the true density function, at least in terms of the resolution in

The Gamma density with shape parameter

The approximating density is identical to Expression (8) with

strates the quality of this approximation for

cies (dashed line,

The quality of the approximation is again due to the fact that the cumulative distribution function of the Gamma density converges exponentially to zero as

The important observation from both of these experiments is that distributions with exponentially decaying tail density can also be well described by a relatively small range of frequencies.

The Cauchy density with median

The approximating density is again Expression (8) with

which in this case converges algebraically to zero as

While some erratic behaviour is evident in the tails of the approximating density, nevertheless the quality of the approximation is remarkably good considering the small number of frequencies in use.

In general, the approximate representations of the Gaussian, Gamma and Cauchy probability density functions all share the common fact that

While the illustrations of Figures 1-4 suggest that the Fourier series approximation of density is effective, it would be useful to quantify just how well density is approximated by the various Fourier methods. Instinctively it would seem that one possible way to achieve this objective is to use the Kullback-Leibler (KL) divergence criterion

to measure the “distance” or departure of the probability density function

The central idea of each Fourier approximation is to replace the true probability density function by a function of compact support, that is,

As might be anticipated, the Gaussian distribution is most efficiently approximated by Fourier methods followed by the Gamma distribution and finally the Cauchy distribution. However, it is clear that all of these distributions are well approximated by the half range Fourier cosine and full range Fourier series for sufficiently large intervals of support and an adequate number of frequencies. The results in these tables also reinforce the

Interval | KL | Number of frequencies used in Fourier sums | |||||
---|---|---|---|---|---|---|---|

Length | Measure | 5 | 10 | 25 | 50 | 100 | 200 |

6 | D_{c} | 0.0038 | 0.0031 | 0.0031 | 0.0031 | 0.0031 | 0.0031 |

D_{f} | 0.0031 | 0.0031 | 0.0031 | 0.0031 | 0.0031 | 0.0031 | |

8 | D_{c} | 0.0498 | 0.0001 | 0.0001 | 0.0001 | 0.0001 | 0.0001 |

D_{f} | 0.0001 | 0.0001 | 0.0001 | 0.0001 | 0.0001 | 0.0001 | |

10 | D_{c} | 0.4284 | 0.0005 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

D_{f} | 0.0005 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | |

12 | D_{c} | 1.2491 | 0.0181 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

D_{f} | 0.0181 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | |

14 | D_{c} | 2.4615 | 0.1301 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

D_{f} | 0.1301 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

Interval | KL | Number of frequencies used in Fourier sums | |||||
---|---|---|---|---|---|---|---|

Length | Measure | 5 | 10 | 25 | 50 | 100 | 200 |

6 | D_{c} | 0.0793 | 0.0720 | 0.0715 | 0.0715 | 0.0715 | 0.0715 |

D_{f} | 0.0890 | 0.0874 | 0.0871 | 0.0871 | 0.0871 | 0.0871 | |

8 | D_{c} | 0.0396 | 0.0174 | 0.0158 | 0.0158 | 0.0158 | 0.0158 |

D_{f} | 0.0196 | 0.0173 | 0.0168 | 0.0168 | 0.0167 | 0.0167 | |

10 | D_{c} | 0.0669 | 0.0076 | 0.0033 | 0.0032 | 0.0032 | 0.0032 |

D_{f} | 0.0105 | 0.0039 | 0.0031 | 0.0031 | 0.0031 | 0.0031 | |

12 | D_{c} | 0.1532 | 0.0142 | 0.0008 | 0.0006 | 0.0006 | 0.0006 |

D_{f} | 0.0290 | 0.0038 | 0.0007 | 0.0006 | 0.0005 | 0.0005 | |

14 | D_{c} | 0.2944 | 0.0391 | 0.0006 | 0.0001 | 0.0001 | 0.0001 |

D_{f} | 0.0745 | 0.0111 | 0.0006 | 0.0001 | 0.0001 | 0.0001 |

Interval | KL | Number of frequencies used in Fourier sums | |||||
---|---|---|---|---|---|---|---|

Length | Measure | 5 | 10 | 25 | 50 | 100 | 200 |

14 | D_{c} | 0.2425 | 0.1148 | 0.1076 | 0.1075 | 0.1075 | 0.1075 |

D_{f} | 0.1148 | 0.1076 | 0.1075 | 0.1075 | 0.1075 | 0.1075 | |

20 | D_{c} | 0.4117 | 0.1296 | 0.0760 | 0.0753 | 0.0753 | 0.0753 |

D_{f} | 0.1296 | 0.0775 | 0.0753 | 0.0753 | 0.0753 | 0.0753 | |

26 | D_{c} | 0.6356 | 0.2188 | 0.0630 | 0.0580 | 0.0580 | 0.0580 |

D_{f} | 0.2188 | 0.0713 | 0.0580 | 0.0580 | 0.0580 | 0.0580 | |

32 | D_{c} | 0.8602 | 0.3391 | 0.0670 | 0.0472 | 0.0471 | 0.0471 |

D_{f} | 0.3391 | 0.0932 | 0.0472 | 0.0471 | 0.0471 | 0.0471 | |

38 | D_{c} | 1.0715 | 0.4885 | 0.0937 | 0.0403 | 0.0397 | 0.0397 |

D_{f} | 0.4885 | 0.1423 | 0.0403 | 0.0397 | 0.0397 | 0.0397 |

idea that the number of frequencies used in the approximation is of secondary importance to the size of the interval of support once sufficient frequencies are in use. This observation accords with intuition in the respect that larger intervals of support capture more of the true density and reduce the distortion in the approximating density, which as has already been commented, will always integrate to one. It would seem that approximations based on 50 frequencies are adequate in all of these examples. The results suggest that using more frequencies provides no meaningful improvement in accuracy. Moreover, for practical purposes there is little to choose between approximation based on the half range Fourier cosine and the full range Fourier series, although the latter has a slight edge for sufficiently large intervals and an adequate number of frequencies.

The successfully pricing of European option contracts for affine models of stochastic volatility requires knowledge of the marginal density of the asset price under the risk-neutral measure. The difficulty, however, is that no closed-form expression for this density is available for even the simplest of the multivariate affine models used in finance, although it is well known that such models have characteristic function,

where

In the case of a European call option with strike price

where

where

where

where

The price of a binary (call) option with strike price

where

where

where

where

Heston’s [

where

Suppose that

Let

By taking the Fourier transform of Equation (20) with respect to the backward variables, the characteristic function

with terminal condition

Thereafter, it is straightforward to show that the anzatz

with

The characteristic function of the marginal density of the terminal value of

On a practical note, the fact that the characteristic function of

Let

Truncation error occurs when the semi-infinite interval in Expression (13) or (16) is replaced by an integral over a finite interval, say

resulting in an error due to the loss of the contribution to the price from values of

which in turn simplifies to give

where

The inference from this observation is that approximations of

Suppose now that the transitional density

where the choice of frequencies

Factor | Call | Strike = 1200 | T = 0.083 | Binary | Strike = 1200 | T = 0.083 |
---|---|---|---|---|---|---|

Cos | Sin | Full | Cos | Sin | Full | |

5.00 | −4.160746 | −17.315659 | −7.719366 | −0.031036 | −4.731845 | −1.295507 |

6.00 | −0.000537 | −0.129887 | −0.052720 | 0.000369 | −0.017647 | −0.005300 |

7.00 | 0.008273 | 0.018872 | 0.001766 | −0.002995 | −0.000040 | −0.000010 |

8.00 | 0.058873 | 0.019228 | 0.002259 | 0.000286 | −0.000010 | 0.000003 |

9.00 | 0.015106 | 0.018981 | 0.002546 | −0.000038 | 0.009813 | 0.000007 |

Factor | Call | Strike = 1200 | T = 0.083 | Binary | Strike = 1200 | T = 0.083 |
---|---|---|---|---|---|---|

Cos | Sin | Full | Cos | Sin | Full | |

5.00 | −0.006890 | −0.084497 | −0.035497 | 0.000000 | −0.010412 | −0.003842 |

6.00 | −0.000014 | −0.000471 | −0.000204 | 0.000000 | −0.000041 | −0.000015 |

7.00 | 0.000000 | 0.000004 | −0.000001 | 0.000000 | −0.000000 | 0.000001 |

8.00 | 0.000000 | 0.000005 | −0.000001 | 0.000000 | −0.000000 | 0.000000 |

9.00 | 0.000000 | 0.000005 | −0.000001 | 0.000000 | 0.000000 | 0.000000 |

Factor | Call | Strike = 1200 | T = 0.500 | Binary | Strike = 1200 | T = 0.500 |
---|---|---|---|---|---|---|

Cos | Sin | Full | Cos | Sin | Full | |

5.00 | −0.001412 | −0.020338 | −0.009446 | 0.000000 | −0.002338 | −0.000993 |

6.00 | −0.000005 | −0.000115 | −0.000055 | 0.000000 | −0.000010 | −0.000004 |

7.00 | 0.000000 | −0.000000 | −0.000000 | 0.000000 | 0.000000 | −0.000000 |

8.00 | 0.000000 | −0.000000 | −0.000000 | 0.000000 | 0.000000 | −0.000000 |

9.00 | 0.000000 | −0.000000 | −0.000000 | 0.000000 | 0.000000 | −0.000000 |

Consequently a misspecification error arises in the Fourier coefficients. Second, the truncation of the Fourier series (27) to a finite number of terms, say

in which

which has explicit expression

The misspecification error in the coefficients

Standard properties of integral Calculus guarantee that

where

It therefore follows directly that

for all values of

In conclusion, the total error in pricing a call option, namely

is formed by connecting together Equation (26) for the error arising in the truncation of the density

Each integral is replaced by its value and the triangle inequality is used to deduce that

The contributions to the error from the first, second, third and fourth terms on the right hand side of inequality (36) are dominated by the behaviour of

The well known result that if

In conclusion, the error in pricing a call option can be made arbitrarily small by restricting the marginal probability density to a finite interval

A series of simulation experiments was undertaken in order to examine the efficacy of the half-range Fourier cosine series, half-range Fourier sine series and full-range Fourier series in respect of how accurately these approximations price European call options and binary options. The first experiment prices options in a Black- Scholes world so that a closed-form solution may be used to assess the pricing error, whereas the second experiment prices the same options using Heston’s model of stochastic volatility.

Assume that the asset price,

in which

Three major experiments are performed. In each of these experiments

Two very clear general conclusions emerge from these results.

1) For options that are deep in-the-money (

2) For options that are deep out-of-the-money (

a) The half-range Fourier sine series does not perform as well as the other two approximations and its use is therefore not recommended. This finding accords well with our previous intuition based on a consideration of the contribution to the price of a call option lost as a result of truncating marginal density.

b) The half-range Fourier cosine series and the full-range Fourier series both perform relatively well. When the size of the interval of approximation is a relatively small multiple of

On the basis of this analysis and on accuracy grounds, it is hard to ignore the claim that the full-range Fourier series is the algorithm of choice when using Fourier methods to price options. Moreover, the full-range Fourier series converges faster than either the half-range sine or cosine series and is therefore likely to price options more rapidly. These themes are explored in more detail in the pricing of call options under Heston’s model of stochastic volatility.

A total of approximately 40,000 options over ten years were generated by simulation of Heston’s model. Sixteen

Factor | Call | Strike = 1000 | T = 0.083 | Binary | Strike = 1000 | T = 0.083 |
---|---|---|---|---|---|---|

Cos | Sin | Full | Cos | Sin | Full | |

5.00 | −0.000044 | −0.000313 | −0.001045 | 0.000000 | −0.000159 | −0.000043 |

6.00 | −0.000000 | −0.000002 | −0.000005 | 0.000000 | −0.000001 | −0.000000 |

7.00 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | −0.000000 | −0.000000 |

8.00 | −0.000000 | 0.000000 | 0.000000 | 0.000000 | −0.000000 | −0.000000 |

9.00 | −0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

Factor | Call | Strike = 1000 | T = 0.250 | Binary | Strike = 1000 | T = 0.250 |
---|---|---|---|---|---|---|

Cos | Sin | Full | Cos | Sin | Full | |

5.00 | −0.000063 | −0.001375 | −0.000547 | 0.000000 | −0.000202 | −0.000074 |

6.00 | −0.000000 | −0.000007 | −0.000003 | 0.000000 | −0.000001 | −0.000000 |

7.00 | 0.000000 | −0.000000 | −0.000000 | 0.000000 | −0.000000 | −0.000000 |

8.00 | −0.000000 | −0.000000 | −0.000000 | 0.000000 | −0.000000 | −0.000000 |

9.00 | −0.000000 | −0.000000 | −0.000000 | 0.000000 | 0.000000 | 0.000000 |

Factor | Call | Strike = 1000 | T = 0.500 | Binary | Strike = 1000 | T = 0.500 |
---|---|---|---|---|---|---|

Cos | Sin | Full | Cos | Sin | Full | |

5.00 | −0.000116 | −0.002208 | −0.001059 | −0.000000 | −0.000254 | −0.000108 |

6.00 | −0.000000 | −0.000012 | −0.000006 | −0.000000 | −0.000001 | −0.000000 |

7.00 | 0.000000 | −0.000000 | −0.000000 | −0.000000 | −0.000000 | −0.000000 |

8.00 | 0.000000 | −0.000000 | 0.000000 | −0.000000 | −0.000000 | −0.000000 |

9.00 | 0.000000 | −0.000000 | 0.000000 | −0.000000 | −0.000000 | −0.000000 |

options were generated each day spread over 4 maturities ranging from 92 to 5 days and 4 strikes two of which are initialised at 20% out of and into the money and two of which are initialised at 7% out of and into the money. The half-range Fourier cosine series and the full-range Fourier series are then used to price call options and binary options with these strikes. In this instance no exact solutions are available, and so the accuracy of each method in respect of each type of option is gauged by comparison against values calculated using a large interval. The left hand and middle columns of

A clear finding from

Factor | Call | Strike = 800 | T = 0.083 | Binary | Strike = 800 | T = 0.083 |
---|---|---|---|---|---|---|

Cos | Sin | Full | Cos | Sin | Full | |

5.00 | −0.000003 | −0.000184 | −0.000049 | −0.000043 | −0.000127 | −0.000068 |

6.00 | −0.000000 | −0.000001 | −0.000000 | −0.000000 | −0.000000 | −0.000000 |

7.00 | −0.000000 | 0.000000 | −0.000000 | −0.000000 | −0.000000 | 0.000000 |

8.00 | −0.000000 | 0.000000 | −0.000000 | −0.000000 | −0.000000 | 0.000000 |

9.00 | −0.000000 | 0.000000 | 0.000000 | −0.000000 | −0.000010 | 0.000000 |

Factor | Call | Strike = 800 | T = 0.250 | Binary | Strike = 800 | T = 0.250 |
---|---|---|---|---|---|---|

Cos | Sin | Full | Cos | Sin | Full | |

5.00 | −0.000011 | −0.000343 | −0.000133 | −0.000000 | −0.000111 | −0.000041 |

6.00 | −0.000000 | −0.000002 | −0.000001 | −0.000000 | −0.000000 | −0.000000 |

7.00 | −0.000000 | 0.000000 | −0.000000 | −0.000000 | −0.000000 | 0.000000 |

8.00 | −0.000000 | 0.000000 | −0.000000 | −0.000000 | −0.000000 | 0.000000 |

9.00 | −0.000000 | 0.000000 | 0.000000 | −0.000000 | −0.000000 | 0.000000 |

Factor | Call | Strike = 800 | T = 0.500 | Binary | Strike = 800 | T = 0.500 |
---|---|---|---|---|---|---|

Cos | Sin | Full | Cos | Sin | Full | |

5.00 | −0.000022 | −0.000580 | −0.000259 | −0.000008 | −0.000147 | −0.000063 |

6.00 | −0.000000 | −0.000003 | −0.000001 | −0.000000 | −0.000001 | −0.000000 |

7.00 | −0.000000 | 0.000000 | −0.000000 | −0.000000 | −0.000000 | 0.000000 |

8.00 | −0.000000 | 0.000000 | −0.000000 | −0.000000 | −0.000000 | 0.000000 |

9.00 | −0.000000 | 0.000000 | 0.000000 | −0.000000 | −0.000000 | 0.000000 |

performance of the full-range Fourier series is poor with regard to both measures for shorter intervals. However, the quality of approximation provided by the half-range cosine series is erratic as the size of the Fourier window increases whereas that provided by the full-range Fourier series improves systematically to the extent that its performance surpasses that of the half-range Fourier cosine series for intervals of length 24 standard deviations or more. Furthermore, this level of accuracy is achieved by the full-range Fourier series in approximately 25% quicker than that required by the half-range Fourier cosine series.

Fang and Oosterlee [

Factor | Out-of-the-money call | Out-of-the-money call | Timings (sec) | |||
---|---|---|---|---|---|---|

(L_{2} Error) | (L_{2} Error) | (L_{1} Error) | (L_{1 }Error) | |||

Cos Series | Full Series | Cos Series | Full Series | Cosine | Full | |

10 | 0.000228 | 0.172011 | 0.000289 | 0.208449 | 0.140 | 0.109 |

12 | 0.000523 | 0.056627 | 0.000045 | 0.063078 | 0.156 | 0.109 |

14 | 0.000053 | 0.016726 | 0.000029 | 0.018403 | 0.187 | 0.141 |

16 | 0.000242 | 0.004654 | 0.000030 | 0.005235 | 0.203 | 0.172 |

18 | 0.000374 | 0.001251 | 0.000030 | 0.001468 | 0.234 | 0.171 |

20 | 0.000190 | 0.000330 | 0.000035 | 0.000417 | 0.265 | 0.187 |

22 | 0.000064 | 0.000086 | 0.000028 | 0.000129 | 0.281 | 0.218 |

24 | 0.000460 | 0.000023 | 0.000030 | 0.000052 | 0.296 | 0.218 |

26 | 0.000058 | 0.000010 | 0.000027 | 0.000031 | 0.328 | 0.249 |

28 | 0.000315 | 0.000008 | 0.000029 | 0.000025 | 0.343 | 0.265 |

30 | 0.000219 | 0.000007 | 0.000035 | 0.000021 | 0.359 | 0.280 |

Factor | In-the-money call | In-the-money call | Timings (sec) | |||
---|---|---|---|---|---|---|

(L_{2} Error) | (L_{2} Error) | (L_{1} Error) | (L_{1} Error) | |||

Cos Series | Full Series | Cos Series | Full Series | Cosine | Full | |

10 | 0.000027 | 0.002421 | 0.000362 | 0.215297 | 0.140 | 0.109 |

12 | 0.000008 | 0.000845 | 0.000063 | 0.061658 | 0.156 | 0.109 |

14 | 0.000008 | 0.000281 | 0.000043 | 0.017301 | 0.187 | 0.141 |

16 | 0.000008 | 0.000090 | 0.000043 | 0.004790 | 0.203 | 0.172 |

18 | 0.000008 | 0.000029 | 0.000043 | 0.001326 | 0.234 | 0.171 |

20 | 0.000024 | 0.000011 | 0.000052 | 0.000382 | 0.265 | 0.187 |

22 | 0.000008 | 0.000008 | 0.000041 | 0.000130 | 0.281 | 0.218 |

24 | 0.000008 | 0.000008 | 0.000043 | 0.000062 | 0.296 | 0.218 |

26 | 0.000007 | 0.000007 | 0.000041 | 0.000044 | 0.328 | 0.249 |

28 | 0.000008 | 0.000007 | 0.000042 | 0.000038 | 0.343 | 0.265 |

30 | 0.000025 | 0.000007 | 0.000053 | 0.000034 | 0.359 | 0.280 |

The results reported in

The results presented in

Factor | Binary out-of-the-money | Binary out-of-the-money | Timings (sec) | |||
---|---|---|---|---|---|---|

(L_{2} Error) | (L_{2} Error) | (L_{1} Error) | (L_{1} Error) | |||

Cos Series | Full Series | Cos Series | Full Series | Cosine | Full | |

10 | 0.000020 | 0.005846 | 0.000000 | 0.000328 | 0.172 | 0.094 |

12 | 0.000006 | 0.001753 | 0.000000 | 0.000081 | 0.203 | 0.125 |

14 | 0.000020 | 0.000504 | 0.000000 | 0.000020 | 0.234 | 0.125 |

16 | 0.000007 | 0.000141 | 0.000000 | 0.000005 | 0.265 | 0.141 |

18 | 0.000028 | 0.000047 | 0.000000 | 0.000001 | 0.281 | 0.156 |

20 | 0.000004 | 0.000022 | 0.000000 | 0.000000 | 0.312 | 0.172 |

22 | 0.000020 | 0.000020 | 0.000000 | 0.000000 | 0.343 | 0.187 |

26 | 0.000020 | 0.000020 | 0.000000 | 0.000000 | 0.390 | 0.203 |

30 | 0.000022 | 0.000020 | 0.000000 | 0.000000 | 0.437 | 0.250 |

Factor | Binary in-the-money | Binary in-the-money | Timings (sec) | |||
---|---|---|---|---|---|---|

(L_{2} Error) | (L_{2} Error) | (L_{1} Error) | (L_{1} Error) | |||

Cos Series | Full Series | Cos Series | Full Series | Cosine | Full | |

10 | 0.000025 | 0.000678 | 0.000004 | 0.000186 | 0.172 | 0.109 |

12 | 0.000007 | 0.000193 | 0.000000 | 0.000044 | 0.203 | 0.125 |

14 | 0.000007 | 0.000054 | 0.000000 | 0.000011 | 0.234 | 0.125 |

16 | 0.000007 | 0.000016 | 0.000000 | 0.000003 | 0.265 | 0.156 |

18 | 0.000007 | 0.000008 | 0.000000 | 0.000001 | 0.281 | 0.156 |

20 | 0.000006 | 0.000007 | 0.000000 | 0.000000 | 0.312 | 0.171 |

22 | 0.000009 | 0.000007 | 0.000000 | 0.000000 | 0.343 | 0.187 |

26 | 0.000016 | 0.000007 | 0.000000 | 0.000000 | 0.390 | 0.219 |

30 | 0.000012 | 0.000007 | 0.000000 | 0.000000 | 0.437 | 0.234 |

The results reported in Tables 7-10 are characterised by two common denominators. First, the computational time required by the half-range Fourier cosine series is uniformly longer than that required by the full-range Fourier series for an interval of fixed length. The simple explanation for this observation is that the full-range Fourier series uses larger frequencies for a given length of interval, and therefore the full-range Fourier series converges more rapidly. Second, the pricing of call options and binary options using the half-range Fourier cosine series representation of transitional density is noticeably better than the corresponding pricing using the full-range Fourier series for short intervals

For an interval of given length, the frequencies present in the half-range Fourier cosine expansion are smaller that the frequencies present in the full-range Fourier series, and therefore when the length of the interval

One clear conclusion from these calculations is that the half-range Fourier cosine series and the full-range Fourier series perform uniformly better than the half-range Fourier sine series. The half-range Fourier cosine series and the full-range Fourier series both perform with credit. When the length of the interval