Truncation to Finite Domain¶
The Fourier cosine series and the COS pricing formula are defined on a finite interval \([a, b]\), but the log-price under most financial models has support on the entire real line (or a half-line). Restricting the density to a finite interval introduces a truncation error that must be controlled. Choosing the interval too narrow discards significant probability mass and produces inaccurate prices; choosing it too wide wastes computational effort on regions where the density is negligible and can degrade the convergence rate. This section develops the cumulant-based truncation rule of Fang and Oosterlee (2008), derives error bounds, and provides practical guidance for selecting \([a, b]\) across different models.
Prerequisites
- Cosine Coefficients via CF (the CF-based coefficient formula and its truncation error)
- Probability: cumulants, moment-generating functions, tail bounds
Learning Objectives
By the end of this section, you will be able to:
- Define cumulants and compute them from the characteristic function
- Derive the cumulant-based truncation formula for \([a, b]\)
- Bound the truncation error in terms of the tail probability
- Select appropriate truncation intervals for Black--Scholes, Heston, and heavy-tailed models
- Diagnose truncation-related errors in COS method implementations
Why Truncation Is Needed¶
The cosine expansion of a density \(f\) requires a bounded interval \([a, b]\). For a density on \(\mathbb{R}\), we replace \(f\) by its restriction:
The truncated density is not a probability density (it integrates to \(1 - \delta\) where \(\delta = P(X \notin [a, b])\)), and the cosine coefficients of \(\tilde{f}\) differ from those of \(f\) by the truncation error analyzed in the previous section. The goal is to choose \([a, b]\) so that \(\delta\) is negligible---say, below \(10^{-12}\)---while keeping \(b - a\) as small as possible to maximize the convergence rate of the cosine series.
Cumulants and the Characteristic Function¶
Cumulants provide a natural characterization of a distribution's location, spread, and shape, and they are readily computable from the characteristic function.
Definition: Cumulants
The cumulants \(\{c_n\}_{n=1}^{\infty}\) of a random variable \(X\) are defined by the cumulant-generating function:
where \(\phi(u) = \mathbb{E}[e^{iuX}]\). The first four cumulants are:
- \(c_1 = \mathbb{E}[X]\) (mean)
- \(c_2 = \text{Var}(X)\) (variance)
- \(c_3 = \mathbb{E}[(X - c_1)^3]\) (third central moment, related to skewness)
- \(c_4 = \mathbb{E}[(X - c_1)^4] - 3c_2^2\) (excess kurtosis times \(c_2^2\))
In practice, \(c_1\) through \(c_4\) are computed by differentiating \(\ln\phi(u)\) at \(u = 0\):
For models with closed-form characteristic functions, these derivatives can often be evaluated analytically or by automatic differentiation.
The Cumulant-Based Truncation Rule¶
Fang and Oosterlee (2008) proposed a truncation rule based on the first four cumulants:
Definition: Cumulant-Based Truncation Interval
Given a distribution with cumulants \(c_1, c_2, c_3, c_4\) and a truncation parameter \(L > 0\), the truncation interval is
The standard choice is \(L = 10\) or \(L = 12\).
The formula centers the interval at the mean \(c_1\) and scales the half-width using a quantity that accounts for both variance (\(c_2\)) and excess kurtosis (\(c_4\)). For a normal distribution (\(c_4 = 0\)), this simplifies to
which is wider than \([c_1 - L\sqrt{c_2}, c_1 + L\sqrt{c_2}]\) by a factor of \(\sqrt{1 + \sqrt{2}} \approx 1.554\).
Simplified Rule for Light-Tailed Distributions
For many practical purposes, the simpler rule
with \(L = 10\) works well for Black--Scholes and Heston models. The full cumulant formula provides additional safety for distributions with significant excess kurtosis.
Truncation Error Bound¶
The truncation error is the probability mass outside \([a, b]\), which we bound using tail inequalities.
Theorem: Chebyshev-Based Truncation Bound
For a random variable \(X\) with mean \(c_1\) and variance \(c_2\):
by Chebyshev's inequality. With \(L = 10\), this gives \(\delta \leq 0.01\).
This bound is extremely conservative for light-tailed distributions. Tighter bounds are available:
Proposition: Exponential Tail Bound
If \(X\) has a moment-generating function \(M(\theta) = \mathbb{E}[e^{\theta X}]\) that is finite for \(|\theta| < \theta_0\), then for the interval \([a, b] = [c_1 - L\sqrt{c_2}, c_1 + L\sqrt{c_2}]\):
For the normal distribution, this is exact: \(P(|X - \mu| > L\sigma) = \text{erfc}(L/\sqrt{2})\).
With \(L = 10\) and a normal-like tail:
This is far below machine epsilon, making the truncation error completely negligible.
Effect on the COS Pricing Error¶
The truncation error propagates to the option price through two channels:
Channel 1: Coefficient error. The CF-based cosine coefficients \(F_k\) differ from the true coefficients \(A_k\) by \(|\varepsilon_k| \leq 2\delta/(b-a)\). Over \(N\) terms, the total coefficient-induced price error is bounded by
Channel 2: Missing payoff mass. The density outside \([a, b]\) contributes to the option value. For a call option with \(b\) large enough that \(f(x)\) is negligible for \(x > b\), this contribution is bounded by \(e^{-rT}\mathbb{E}[(S_T - K)^+\mathbf{1}_{X \notin [a,b]}]\), which is exponentially small for typical truncation parameters.
In practice, with \(L = 10\), the truncation error is below \(10^{-15}\)---far smaller than the series truncation error for any reasonable \(N\).
Truncation Intervals for Standard Models¶
The following table gives typical truncation intervals for common financial models, using \(L = 10\):
Model-Specific Truncation Intervals
| Model | Cumulants | Typical \([a, b]\) |
|---|---|---|
| Black--Scholes (\(\sigma = 0.2\), \(T = 1\)) | \(c_1 = (r-\sigma^2/2)T\), \(c_2 = \sigma^2 T = 0.04\) | \([c_1 - 2.0, c_1 + 2.0]\) |
| Heston (typical) | \(c_2 \approx 0.04\), \(c_4 > 0\) | \([c_1 - 2.5, c_1 + 2.5]\) |
| Variance Gamma (\(\sigma = 0.2\), \(\nu = 0.5\)) | Higher kurtosis | \([c_1 - 4.0, c_1 + 4.0]\) |
| CGMY (\(Y = 0.5\)) | Heavy tails | \([c_1 - 6.0, c_1 + 6.0]\) |
Heavier-tailed models require wider truncation intervals, which in turn require more cosine terms \(N\) for the same accuracy (since the cosine frequencies are \(k\pi/(b-a)\) and wider intervals mean smaller frequency spacing).
Practical Diagnostics¶
When implementing the COS method, truncation errors can be diagnosed using the following checks:
-
Increase \(L\) and check stability. If the price changes when \(L\) is increased from 10 to 12, the original truncation was too narrow.
-
Check \(A_0\) normalization. Since \(A_0 = 2/(b-a)\int_a^b f(x)\,dx \approx 2/(b-a)\) for a properly truncated density, the quantity \(A_0(b-a)/2\) should be close to 1. A significant deviation indicates truncation loss.
-
Check recovered density. Reconstruct \(\hat{f}_N(x)\) using the cosine series and verify that \(\int_a^b \hat{f}_N(x)\,dx \approx 1\) and \(\hat{f}_N(x) \geq 0\) for all \(x \in [a, b]\).
Over-Truncation for Short Maturities
For very short maturities (\(T < 0.01\)), the density concentrates sharply near the forward price. The variance \(c_2 \propto T\) is small, so the truncation interval \([c_1 - L\sqrt{c_2}, c_1 + L\sqrt{c_2}]\) is narrow. However, the density's peak is very high (proportional to \(1/\sqrt{c_2}\)), requiring more cosine terms to resolve. In such cases, increasing \(N\) is more important than widening the interval.
Summary¶
Truncation to a finite domain is the first step of the COS method, and the cumulant-based rule provides a principled way to choose \([a, b]\):
| Item | Value / Formula |
|---|---|
| Truncation interval | \([c_1 - L\sqrt{c_2 + \sqrt{c_4 + 2c_2^2}},\; c_1 + L\sqrt{c_2 + \sqrt{c_4 + 2c_2^2}}]\) |
| Typical \(L\) | \(10\) to \(12\) |
| Truncation error | \(\delta = P(X \notin [a,b])\), exponentially small for light tails |
| Price error from truncation | \(O(\delta)\), negligible compared to series truncation |
| Diagnostic | Check \(A_0(b-a)/2 \approx 1\) |
The cumulant-based truncation rule adapts the interval \([a, b]\) to the distribution's shape, ensuring that the truncation error is negligible while keeping the interval as narrow as possible to maximize the convergence rate of the cosine series.
Exercises¶
Exercise 1. For \(X \sim N(0, 1)\), the cumulants are \(c_1 = 0\), \(c_2 = 1\), \(c_3 = 0\), \(c_4 = 0\). Compute the cumulant-based truncation interval \([a, b]\) using the formula \([c_1 - L\sqrt{c_2 + \sqrt{c_4 + 2c_2^2}},\; c_1 + L\sqrt{c_2 + \sqrt{c_4 + 2c_2^2}}]\) with \(L = 10\). Compare this to the simpler rule \([c_1 - L\sqrt{c_2}, c_1 + L\sqrt{c_2}]\) and explain the difference.
Solution to Exercise 1
For \(X \sim N(0, 1)\), the cumulants are \(c_1 = 0\), \(c_2 = 1\), \(c_3 = 0\), \(c_4 = 0\).
Full cumulant formula: Substituting into \([c_1 - L\sqrt{c_2 + \sqrt{c_4 + 2c_2^2}},\; c_1 + L\sqrt{c_2 + \sqrt{c_4 + 2c_2^2}}]\):
The half-width is \(L\sqrt{1 + \sqrt{2}} = 10\sqrt{2.4142} \approx 10 \times 1.554 = 15.54\).
Simplified rule: \([c_1 - L\sqrt{c_2},\; c_1 + L\sqrt{c_2}] = [-10, 10]\).
Comparison: The full cumulant formula gives an interval that is wider by a factor of \(\sqrt{1 + \sqrt{2}} \approx 1.554\). The extra width comes from the \(\sqrt{c_4 + 2c_2^2}\) term, which for the normal distribution evaluates to \(\sqrt{2}\) even though \(c_4 = 0\) (because of the \(2c_2^2\) contribution). This provides additional safety margin. However, for \(N(0,1)\), even the simpler interval \([-10, 10]\) captures all but \(P(|X| > 10) \approx 1.5 \times 10^{-23}\) of the probability, so the extra width is unnecessary in practice. The full formula is designed to accommodate distributions with significant excess kurtosis (\(c_4 > 0\)), where the simpler rule might be too narrow.
Exercise 2. Compute the first four cumulants \(c_1, c_2, c_3, c_4\) for the Variance Gamma model with parameters \(\sigma = 0.12\), \(\theta = -0.14\), \(\nu = 0.2\), \(r = 0.05\), \(T = 1\). Using the formulas \(c_2 = (\sigma^2 + \nu\theta^2)T\), \(c_4 = (3\sigma^4\nu + 12\sigma^2\theta^2\nu^2 + 6\theta^4\nu^3)T\), determine the truncation interval with \(L = 12\). How much wider is this interval compared to the Black-Scholes case with the same variance?
Solution to Exercise 2
For the Variance Gamma model with \(\sigma = 0.12\), \(\theta = -0.14\), \(\nu = 0.2\), \(T = 1\):
Cumulant \(c_2\) (variance):
Cumulant \(c_4\) (excess kurtosis related):
Computing each term:
- \(3\sigma^4\nu = 3 \times (0.12)^4 \times 0.2 = 3 \times 2.0736\times 10^{-4}\times 0.2 = 1.244\times 10^{-4}\)
- \(12\sigma^2\theta^2\nu^2 = 12\times 0.0144\times 0.0196\times 0.04 = 1.355\times 10^{-4}\)
- \(6\theta^4\nu^3 = 6\times 3.842\times 10^{-4}\times 0.008 = 1.844\times 10^{-5}\)
Truncation interval with \(L = 12\):
Half-width: \(L\sqrt{0.04913} = 12\times 0.2217 = 2.660\).
so \(b - a = 5.32\).
Comparison with Black--Scholes of the same variance: For Black--Scholes with \(c_2 = 0.01832\) and \(c_4 = 0\), using \(L = 12\):
Half-width: \(12\sqrt{0.04423} = 12\times 0.2103 = 2.524\), giving \(b - a = 5.048\).
The Variance Gamma interval (\(b - a = 5.32\)) is wider than the Black--Scholes interval (\(b - a = 5.05\)) by about 5.4%, reflecting the additional kurtosis of the VG model.
Exercise 3. The Chebyshev-based truncation bound gives \(P(X \notin [c_1 - L\sqrt{c_2}, c_1 + L\sqrt{c_2}]) \leq 1/L^2\). For \(L = 10\), this yields \(\delta \leq 0.01\). Why is this bound extremely conservative for a normal distribution? Compute the exact value of \(P(|X| > 10)\) for \(X \sim N(0, 1)\) and compare to the Chebyshev bound.
Solution to Exercise 3
The Chebyshev bound gives \(P(|X| > 10) \leq 1/10^2 = 0.01\) for \(X \sim N(0,1)\).
The exact value is:
The Chebyshev bound (\(0.01\)) overestimates the true probability by a factor of about \(6.6 \times 10^{20}\), which is roughly 21 orders of magnitude.
Why so conservative: Chebyshev's inequality uses only the mean and variance of the distribution. It must hold for all distributions with those moments, including heavy-tailed distributions that concentrate much more mass in the tails than the Gaussian. For example, a two-point distribution \(P(X = \pm 10) = 0.005\), \(P(X = 0) = 0.99\) has \(\text{Var}(X) = 0.5\), and for appropriately scaled versions, the Chebyshev bound is tight. The normal distribution has exponentially decaying tails (\(f(x) \sim e^{-x^2/2}\)), which makes the polynomial Chebyshev bound absurdly loose. The exponential tail bound \(P(|X| > L) \approx 2e^{-L^2/2}\) is much tighter and nearly exact for the Gaussian.
Exercise 4. Explain the interaction between the truncation interval width \(b - a\) and the COS convergence rate. If \([a, b]\) is widened by a factor of 2, how does this affect (a) the truncation error \(\varepsilon_1\), (b) the cosine frequencies \(\omega_k = k\pi/(b-a)\), and (c) the number of terms \(N\) needed for a given series truncation error \(\varepsilon_2\)? What is the optimal balance?
Solution to Exercise 4
(a) Effect on truncation error \(\varepsilon_1\): Widening \([a, b]\) by a factor of 2 captures more probability mass, reducing \(\delta = P(X \notin [a,b])\). For sub-Gaussian tails, \(\delta\) decreases exponentially with the interval half-width, so doubling the interval reduces \(\varepsilon_1\) dramatically (e.g., from \(10^{-20}\) to \(10^{-80}\) for a Gaussian).
(b) Effect on cosine frequencies: The frequencies are \(\omega_k = k\pi/(b-a)\). Doubling \(b - a\) halves all frequencies: \(\omega_k^{\text{new}} = k\pi/(2(b-a)) = \omega_k/2\). This means the cosine basis functions oscillate more slowly, and the series needs more terms to capture the same level of detail in the density.
(c) Effect on \(N\): The exponential convergence rate is \(\varepsilon_2 \sim e^{-\beta N\pi/(b-a)}\). Doubling \(b - a\) halves the exponent, so to maintain the same \(\varepsilon_2\), we need to double \(N\). More precisely, if \(N_1\) terms suffice for interval \(b - a\), then \(N_2 = 2N_1\) terms are needed for interval \(2(b - a)\) to achieve the same series error.
Optimal balance: The optimal \([a, b]\) minimizes the total error \(\varepsilon_1 + \varepsilon_2\). Since \(\varepsilon_1\) decreases exponentially with the interval width while \(\varepsilon_2\) increases (for fixed \(N\)), the optimum occurs where both errors are comparable. In practice, the truncation error is already negligible (\(< 10^{-15}\)) for \(L = 10\), so the series error dominates and the interval should be kept as narrow as possible while maintaining negligible truncation error.
Exercise 5. The diagnostic check "\(A_0(b-a)/2 \approx 1\)" verifies proper truncation. For a correctly truncated density, \(A_0 = 2/(b-a)\int_a^b f(x)\,dx \approx 2/(b-a)\), so \(A_0(b-a)/2 \approx 1\). If you observe \(A_0(b-a)/2 = 0.95\), what does this indicate about the truncation interval? Estimate the fraction of probability mass outside \([a, b]\).
Solution to Exercise 5
If \(A_0(b-a)/2 = 0.95\), then:
Since \(\int_{-\infty}^{\infty}f(x)\,dx = 1\) for a probability density, the fraction of probability mass outside \([a, b]\) is:
This means 5% of the probability mass lies outside the truncation interval, which is far too much for accurate pricing. The resulting truncation error bound is \(|\varepsilon_k| \leq 2\times 0.05/(b-a) = 0.1/(b-a)\), which can lead to significant price errors.
This situation indicates the truncation interval is too narrow. The remedy is to increase \(L\) (e.g., from 6 to 10 or 12) or check whether the cumulant computation is correct. For a properly truncated density with \(L = 10\), one should observe \(A_0(b-a)/2 > 1 - 10^{-15}\).
Exercise 6. For very short maturities (\(T = 0.001\)), the density concentrates sharply near the forward price. Compute \(c_2 = \sigma^2 T\) for \(\sigma = 0.20\) and determine the half-width \(L\sqrt{c_2}\) with \(L = 10\). Explain why the COS method requires many more terms \(N\) for short maturities even though the truncation interval is narrow, and suggest a practical strategy for handling this regime.
Solution to Exercise 6
For \(\sigma = 0.20\) and \(T = 0.001\):
The half-width is:
So \(b - a = 2\times 0.06325 = 0.1265\), a very narrow interval.
Why many terms \(N\) are needed: The density peak height is proportional to \(1/\sqrt{c_2} = 1/\sqrt{4\times 10^{-5}} \approx 158\), meaning the density is very tall and sharply peaked. Although the truncation interval is narrow, the cosine series on \([a, b]\) has basis functions with frequencies \(\omega_k = k\pi/(b-a) = k\pi/0.1265\). The density changes rapidly relative to \(b - a\), but the key issue is that the absolute number of oscillations needed to resolve the density shape is determined by the density's intrinsic smoothness, not the interval width.
More precisely, the analyticity strip width \(\beta\) (in absolute terms) is roughly independent of \(T\) for the Black--Scholes model, but \(b - a \propto \sqrt{T}\). The convergence rate \(e^{-\beta N\pi/(b-a)}\) improves with smaller \(b - a\) only if \(\beta\) scales the same way. For log-normal densities, the Gaussian decay of coefficients gives \(|A_k| \sim e^{-\sigma^2 T k^2\pi^2/(2(b-a)^2)}\). With \(b - a = 2L\sigma\sqrt{T}\), this becomes \(|A_k| \sim e^{-k^2\pi^2/(8L^2)}\), which is actually independent of \(T\). So in the Black--Scholes case, \(N\) does not grow with decreasing \(T\).
However, for stochastic volatility models, short maturities produce densities with sharper features (near-ATM kinks from discrete monitoring, or mixed distributions from variance process), requiring more terms. The practical strategy is:
- Use a fine grid of \(N\) values and check convergence empirically via Richardson extrapolation.
- Scale \(N\) inversely with \(\sqrt{T}\) as a starting heuristic for non-Gaussian models.
- Consider alternative methods (FFT or direct quadrature) for very short maturities where the COS method requires \(N > 500\).