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Turn a characteristic function into a cosine series15 min read
COS Method for Option Pricing Explained
Learn how the COS method combines a truncated density, Fourier-cosine coefficients, payoff coefficients, and convergence controls to price options
Prepared by Mark · Primary sources below
Direct answer
The COS method truncates a log-price distribution to a finite interval, expands its density in a cosine series, and obtains coefficients from the characteristic function. Payoff coefficients then turn the expansion into an option price
COS is a Fourier-series pricing method
Suppose a terminal log state X has a known characteristic function and an option pays v(X) at maturity
After discounting under the matching pricing measure, the desired value is an integral of v(x) against the conditional density of X
COS approximates that integral with a finite cosine series rather than simulating paths or solving a price grid backward in time
The infinite state range is truncated first
Because log prices usually live on the real line, the method selects a finite interval [a,b] containing most economically relevant probability mass
Probability outside [a,b] creates domain-truncation error before any cosine terms are omitted
An interval that is too narrow misses tails; one that is unnecessarily wide requires more terms to resolve the same local features
Characteristic functions supply density coefficients
On [a,b], the density is expanded in cos(kπ(x−a)/(b−a)) for nonnegative integer k
Its cosine coefficients are approximated from Re{φ(kπ/(b−a))e^{-ikπa/(b−a)}}, avoiding explicit recovery of the full density
The zero-frequency term receives half weight in the usual primed sum, a small convention that materially affects the level of the price
Payoff coefficients complete the price formula
Each density coefficient is multiplied by the corresponding integral of v(x) against the cosine basis over [a,b]
For European calls and puts, these payoff coefficients have analytical forms; other payoffs may need stable numerical integration
Once model coefficients and payoff coefficients are separated, many strikes or contracts can reuse much of the same transform work
Rapid convergence is conditional
Smooth densities with fast-decaying cosine coefficients can deliver very rapid, often exponential, convergence as the term count N increases
Short maturities, sharp densities, atoms, heavy tails, or discontinuous payoffs can slow convergence and produce ringing near nonsmooth points
Therefore the method’s reputation for speed is a regularity-dependent result, not a fixed accuracy guarantee for every model and payoff
Interval choice is part of the error budget
A common heuristic uses cumulants, such as c₁±L√(c₂+√c₄), when those cumulants exist and the convention is applicable
The scale L is not universal, and unreliable or nonexistent higher cumulants can make a familiar interval rule misleading
Tail probability bounds, wider reference intervals, and stability across [a,b] choices provide stronger evidence than one heuristic setting
Verification separates range from series error
Increase [a,b] while holding N sufficiently large to diagnose domain truncation, then increase N on a stable interval to test series convergence
Check the characteristic function at zero, the half-weight convention, discounting, payoff coefficients, parity, bounds, monotonicity, and convexity
Compare selected prices with direct Fourier quadrature or another independent method before using COS inside calibration loops
Common questions
What is the COS method?
It is an option-pricing method that approximates a terminal density with a Fourier-cosine series whose coefficients come from its characteristic function
Is COS the same as FFT pricing?
No. Both use transforms, but COS evaluates a cosine expansion on a chosen state interval rather than requiring the same linked FFT strike grid
How is the interval [a,b] chosen?
Cumulant heuristics are common, but the interval should be validated with tail information and price stability because no single width is universal
Why can COS pricing be inaccurate?
A narrow interval, too few terms, weak density regularity, payoff discontinuities, coefficient mistakes, or invalid transform inputs can all cause error
Sources and further reading
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