COS Methods for Option Pricing and Distribution Estimation
Summary
The document surveys applications of the Fourier-cosine, or COS, method in finance, focusing on option valuation and recovery of terminal price distributions. The method approximates a probability density with a cosine expansion whose coefficients come from the characteristic function of the log spot price. In models with known characteristic functions, this supports option prices and Greeks, as well as extraction of the distribution itself.
The account traces extensions from European options to early-exercise features and Asian options, including an approach called ASCOS. It also describes a data-driven variant that estimates distribution coefficients from historical observations when an analytic characteristic function is unavailable. These examples show the method’s range, but the document is a brief literature overview rather than a comparative evaluation: it supplies no implementation details, accuracy benchmarks, or practical guidance for choosing among methods.
Key ideas
- The COS method approximates a probability density using a cosine expansion based on a characteristic function.
- The method can price options and calculate Greeks when the model’s characteristic function is known.
- COS techniques have been extended to early-exercise and Asian options.
- A data-driven variant estimates distribution coefficients from historical observations.
- The document gives no accuracy comparisons or implementation guidance.
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# Cos Method in Finance / Practice # Cos Method in Finance / Practice A lot of my professors advised me on doing an undergrad thesis that has something to do with the "relatively new" cosine method (~10 years). What applications are there in Finance of the FFT/Cosine Method beside of option Pricing? What are common uses of the resulting density in practice? Thanks in advance ## Answer by Kevin (score 6, accepted) https://quant.stackexchange.com/a/49835 As ilovevolatility pointed out, the main application of the COS method is to price options. The initially proposed method simply approximates the probability density function by a cosine expansion using the characteristic function of the log spot price. So, if you know the characteristic function (say for exponential Levy models and many stochastic volatility models), you can easily price options and obtain the Greeks. As you said, you can also extract the probability distribution of the log spot price. The COS method was originally developed by Fang and Oosterlee (2008) as part of Fang's PhD thesis. This paper deals with the pricing of European-style options. Fang and Oosterlee (2009) show how the method can be adapted to price ealy-exercise features. Zhang and Oosterlee (2013) and Zhang and Oosterlee (2014) discuss the pricing of discretely monitored geometric and arithmetic Asian options (with early exercise features). This algorithm is named ASCOS and is part of Zhang's PhD thesis. The prices of continuously monitored option are obtained by Richardson extrapolation. Leitao et al. (2018) develop a model-free approach which does not require an analytically known characteristic function. Instead, the coefficients $A_n$ representing the distribution of the terminal stock price are estimated based on historical values. Thus, they name this approach data driven cosine (ddCOS) method.
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