# 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?

• The main application of COS method is pricing and calibration. Not sure what other potential methods you had in mind?
– user34971
Commented Nov 20, 2019 at 16:34
• My professor was talking about how it is slowly becoming popular in risk management, but for me it was hard to understand in what context (I'm only an undergrad). I think he meant that the approximated density function by the cosine expansion could be used for calculating risk measures such as VaR. Commented Nov 20, 2019 at 20:42
• COS can be used for efficient density recovery in VaR/CVaR applications too Commented Nov 21, 2019 at 2:04
• @JamesSpencer-Lavan Do you know if this is something very common, or rather something used only by very quantitative-driven (hedge) funds / institutions? Commented Nov 21, 2019 at 10:15
• I use it at an investment bank, but it is not a common technique as the lazy option of monte carlo is too appealing. Commented Nov 22, 2019 at 0:11

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.

• I wasn't aware of the Leitao paper, thanks for pointing it out.
– user34971
Commented Nov 20, 2019 at 19:43
• @ilovevolatility Perhaps you remember, but we once talked about model-free approaches to derivatives pricing. So, I try to remain up to date with pricing frameworks which do not impose a certain stock price model Commented Nov 20, 2019 at 20:01
• Thanks a lot for this clear answer. I will definitely have a look at the Leitao et al. paper. Commented Nov 20, 2019 at 20:45
• Leitao is here papers.ssrn.com/sol3/papers.cfm?abstract_id=2917536 Commented Nov 20, 2019 at 21:18