Introduction to the cos method for option pricing

Abstract

The main goal of this work is to introduce the COS method, a Fourier–cosine series based technique for the fast numerical pricing of financial options. We start from the basic financial and mathematical ideas needed to understand option pricing, including derivative contracts, risk-neutral valuation, the Fundamental Theorem of Asset Pricing, Brownian motion, and the Black-Scholes framework. After that, we introduce Fourier methods and explain why characteristic functions are useful when the distribution of the asset price is not handled directly.

The project then focuses on the derivation and implementation of the COS method. The method is presented through the truncation interval, the model characteristic function, the payoff coefficients, and the finite cosine sum. We apply it first to the Black-Scholes model, where a closed-form formula is available and can be used as a benchmark. Then we extend the numerical study to the Heston and Bates models, where the characteristic function approach is especially useful.

Finally, the method is implemented in Python and tested through several numerical experiments. The results show that the COS method achieves very high accuracy with a relatively small number of terms, while Monte Carlo simulation needs many more paths to reduce the error. This makes the COS method a powerful and efficient alternative for option pricing, especially in models where closed-form prices are not directly available.

Notation: Throughout this work, $i$ denotes the imaginary unit, $\mathbb{R}$ the real line, and FTAP the Fundamental Theorem of Asset Pricing. We write $S_t$ for the price of the underlying asset at time $t$, $S_0$ for its initial value, $S_T$ for its value at maturity, $K$ for the strike price, and $T$ for the time to maturity. The risk-free interest rate is denoted by $r$, the dividend yield by $q$, and the volatility by $\sigma$. Brownian motion is denoted by $W_t$.

When logarithmic variables are used in the COS method, we write

$$x=\log\left(\frac{S_0}{K}\right), \qquad y=\log\left(\frac{S_T}{K}\right).$$

The characteristic function of the model is denoted by $\phi$, and its Fourier variable is denoted by $\omega$ or $u$. The interval $[a,b]$ is the truncation interval used in the COS approximation, $c_n$ denotes the cumulants of the log-price, and $N$ is the number of terms in the finite cosine expansion. The payoff coefficients are denoted by $V_k$ or $U_k$, depending on the notation used in the corresponding section.