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[null,"Description\n\nThis workbook uses NAG Library routines to perform a simple Quasi Monte Carlo simulation.  A Call option is priced in the Black Scholes model, i.e. using Geometric Brownian motion, but where random numbers are drawn from quasi-random generators and not pseudorandom generators.\n\nQuasi random generators (see the G05 chapter introduction) are not random at all, rather they are lattice-based numerical quadrature routines which aim to sample the underlying multidimensional space as evenly as possible (based on discrepancy and star-discrepancy metrics).  They are thus closer in spirit to classical quadrature techniques (see chapter D01) than to Monte Carlo, and can give very good integral estimates with relative small sample sizes.\n\nThe main drawback of quasi Monte Carlo is that there is no built-in error measure: one has no idea how good a QMC integral estimate is.  In addition, many QMC sequences (such as Sobol) have an inherent bias due to the way they fill up multi-dimensional space.  In this example, the QMC integral esimate tends to under-estimate the call price (it would over-estimate a put price) as can be seen in the plot.\n\nScrambled QMC is a way to address both these issues.  Randomness is injected into the underlying lattice in a very special way, so that the uniformity properties are maintained.  Different QMC samples of a given size are therefore independent.  This allows statistics to be done, and we regain the central limit theorem and error bounds it provides.   By drawing a small number m of successive (and independent) SQMC samples of a smallish size, and from each one calculating the call option price, one can treat the set of m prices as iid realisations and regain the classic \"square root\" convergence bounds.  Note that the bounds are much tighter than for regular Monte Calro.  In addition, as can be seen from the plot, the SQMC estimate is not biased : the point estimate spends as much time above the true estimate as below.  SQMC is thus a more robust and more quantifiable alternative to QMC while maintaining the good discrepancy properties of the underlying QMC sequences.\n\nAlthough this is a simple example, the ideas it illustrates are very general.  Using this example as a template, customers can embed NAG Library functionality into Excel to solve advanced and computationally intensive problems while keeping all the benefits of using Excel as a front end."]

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[]
[]
[null,null,null,null,null,null,null,null,"Quasi Monte Carlo Option Pricing"]
[]
[null,null,null,"Inputs",null,null,null,"We price a call option with Quasi Monte Carlo (QMC) and scrambled Quasi Monte Carlo (SQMC).  The QMC is clearly biased low and has no concept of variance.  The SQMC has less bias and allows confidence intervals to be formed for the true price.  Total number of SQMC samples is SampleSize * NumSamples"]
[]
[null,null,null,"S0 :",100]
[null,null,null,"Strike :",80]
[null,null,null,"Sigma :",0.3]
[null,null,null,"Interest rate :",0.05]
[null,null,null,"Dividend yield :",0.02]
[null,null,null,"Time to maturity :",1]
[]
[null,null,null,"Quasi Monte Carlo Inputs"]
[]
[null,null,null,"QMC sample size :",600]
[null,null,null,"Plot interval :",15,null," "]
[]
[null,null,null,"Scrambled QMC Inputs"]
[]
[null,null,null,"QMC sample size :",60]
[null,null,null,"Num QMC samples :",10]
[null,null,null,"Plot interval :",2]
[]
[]
[null,null,null,"Output"]
[]
[null,null,null,"QMC Price :",24.709716290633754]
[null,null,null,"True price :",24.783318682677873]
[null,null,null,"Conf. Interval Up :",25.476578052729153]
[null,null,null,"Conf. Interval Dwn :",23.942854528538355]
[null,null,null,"Conf. Intval Width :",1.5337235241907992]