The COS method

To present the COS method, suppose that at time t_j we want to find an approximation of the expectation ap-proximate I(x₀), we first truncate the integration region to an interval [a, b] and then approximate the density f by using its Fourier cosine series expansion truncated to a finite number of terms The above cosine series coefficients can be approximated using the characteristic function φ(·; x_o) of x as

where we have only an approximation not an equality in (5.30), which is due to the trunca-tion of the integratrunca-tion region in the definitrunca-tion of the characteristic functrunca-tion. If we replace A_k with F_k in (5.28) and substitute the resulting sum into (5.27), we obtain the following approximation of I(x₀) used in the COS method:

I(xˆ ₀) := F₀

2 + b − a 2

L

X

k=1

F_k(x₀) · V_k, (5.31)

where

V_k := 2 b − a

Z b a

U (x) cos(kπx − a

b − a)dx. (5.32)

We can notice that V_k, k = 1, 2, . . . , are the cosine series coefficients of U (x) on [a, b], which suggests a close connection between the COS method and the PV method. Indeed, it is easy to verify that if we use the cosine series expansion of U (x) on [−l_x, l_x] in (S4–1) of the PV method , then (S4–2) produces a representation of the conditional expectation that has the same form as the expansion (5.31) when a = −l_x and b = l_x.

Although the PV approach based on cosine functions and the COS method produce the same approximations, they have different starting points in expansions. This fact implies that the truncation error in both approaches can be controlled by approximating more closely either the value function or the density function. This result can also be derived from the common representation (5.31), where the rate at which the product F_k(x₀) · V_k decays to zero is faster than either F_k(x₀) or V_k. However, the PV method is a broader framework than the COS method, since it offers alternative ways of approximating expectations.

To understand better how the PV approach can be used to improve the COS method, we first need to know some properties of the COS method when applied to the problem of pricing Bermudan options (Fang and Oosterlee (2008, 2009)):

(P1) Assume that the density of log-returns is smooth enough, typically we need only a small number of terms in its series expansion.

(P2) Assume that the series coefficients V_k, k = 1, 2, ..., of the option values at the first early-exercise date are known, they can be calculated, for some options, very ef-ficiently for other exercise dates through an induction formula combined with the FFT algorithm. In such cases, the computational complexity of the method is O((M − 1)L log L), where M is the number of early-exercise dates and L is the number of terms in the series expansion of the density.

The first property (P1) depends on the smoothness of the density function of log-prices.

In many financial applications, this density is often infinitely differentiable, and therefore its Fourier-cosine series expansion converges exponentially in the number of terms L on bounded intervals. As a result, in practice, we do not need to use large values of L to get accurate estimates of prices of European and Bermudan options. As we discuss below, however, for ratchet options the coefficients in Fourier-cosine series expansions show only algebraic convergence.

The second property (P2) of the COS method also leads to a significant reduction of its overall computational cost. To derive the induction formula, however, we need to know the form of the boundary that separates the continuation region from the exercise region at each exercise time. For some options and models, finding this boundary is equivalent to finding the point that separates the two regions (Fang and Oosterlee (2009)). However, the exact shape of the exercise region may be difficult to determine, especially for path-dependent options under a stochastic volatility model (for Heston model, see Ruijter and Oosterlee (2012)). Therefore, the feasibility of an induction formula and its effectiveness must be assessed on a case-by-case basis.

As mentioned earlier, for ratchet options, the exponential rate of convergence of the COS approach is typically not true. For these contracts, the density f corresponds to a truncated random variable, and it can be written in the form f^t(x; x₀) := c_nf (x; x₀)1_[F,C](x), where 1A is the indicator function of a set A, cn is a normalizing factor, and the constants F and C with F, C ∈ (a, b), F < C, determine the truncation levels. In this case, the density function has discontinuities inside the expansion region, and hence the truncation error in the cosine expansions of f^t will decay only algebraically.

Figure5.1 shows some plots for the analysis of truncation errors (y-axis) with respect to the number of terms in expansions (x-axis). The left panel in Figure 5.1 depicts the truncation error, which is measured by the L¹-distance between functions, to demonstrate the difference between these two convergence rates. In this graph, the lower line represents the truncation error of the case where we approximate the standard normal density function on the interval (−10, 10) using Fourier cosine expansions with L terms, while the upper line represents a similar error but for the normal density function truncated on the interval (−2, 2). The difference is quite large, since for the former case we need only 16 terms to ensure that the error is less than 0.007, while for the latter case the distance is still larger than 0.01 even with 500 terms.

Figure 5.1: Truncation Error Analysis with respect to L

0 20 40 60 80 100 120 140 160 180 200

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

L

Truncation Error

0 50 100 150 200 250 300 350 400

0 1 2 3 4 5 6 7 8 x 10⁻³

L

Truncation Error

The PV method allows us to improve the convergence rates of the pricing methods based on characteristic functions by constructing better approximations of value functions. The justification for such a strategy is twofold. First, the value function has better smoothness properties than the density for ratchet options with surrender risk, since it is typically continuous and infinitely differentiable everywhere except for the points that separate the exercise and the continuation regions. Second, the PV method provides a more flexible framework for approximations than the COS method, since we are not constrained to cosine functions only. In particular, we can combine polynomials and trigonometric functions to construct more accurate approximations of the value function. Since these efficiency enhancing techniques usually depend on the problems at hand, below we only describe a general method that is applicable.

The truncation error when approximating U (x) by the finite sum of trigonometric functions (5.15) is determined by the rate at which the coefficients a_l, l = 0, 1, . . . , L, and b_l, l = 1, . . . , L, defined in (5.16)–(5.17) converge to zero. Suppose that the function U is (k − 2)-times continuously differentiable on [−l_x, l_x] and its k-th derivative, U^(k), is integrable on this interval. If, in addition, we also have

U (−l_X) = U (l_X), U⁽¹⁾(−l_X) = U⁽¹⁾(l_X), . . . , U^(k−2)(−l_X) = U^(k−2)(l_X), (5.33)

then it can be shown that

|a_n| ≤ F

n^k and |b_n| ≤ F n^k

for some sufficiently large constant F , which is independent of n. Thus, under the above assumptions, the algebraic index of converge is at least as large as k. A proof of this result based on simple integration by parts is presented in Boyd (1989) (Theorem 2.4). It is known that for American options the continuation value meets the payoff function smoothly, as long as the latter is smooth too (see, for example, Wilmott et al. 1993). Therefore, if we assume that the payoff function G is continuously differentiable for in-the-money region, then the first derivative of U will be continuous on (−l_X, l_X), and hence the algebraic index of convergence will be at least 3 if the boundary conditions (5.33) hold for k = 3.

The value function ˆV^e(t_j+1, ·, y) does not usually satisfy these conditions, but a simple remedy for this problem is to replace it with a function of the form:

Vˆ_M^e(t_j+1, x, y) := ˆV^e(t_j+1, x, y) − p_Lb(x), x ∈ [−l_X, l_X], (5.34) where the polynomial pL_b(x) := PLb

l=0dlx^l is selected so that the resulting function ˆV_M^e satisfies (5.33) for a prespecified value of k. A similar approach is proposed in Boyd (1989) for solving ordinary differential equations. In the context of the PV method, this technique is feasible since by (5.19) the integral of pL_b(x) can be expressed in terms of the characteristic function of f . In particular, we can ensure that the periodic extension of Vˆ^e(t_j+1, x, y) is continuous by using a linear function with

d1 =

Vˆ^e(t_j+1, l_X, y) − ˆV^e(t_j+1, −l_X, y)

2l_X .

To illustrate the advantages of the proposed method over the cosine method, assume that U represents the log-prices of a Bermudan put option at time t = 10, where the option expires at T = 11 and its strike price is 1. We obtain U on the interval (−5, 5) by using the Black-Scholes model and formula (5.10), where we assume that σ = 30% and r = 5%.

To ensure that the boundary conditions (5.33) are satisfied, we use (5.34) with either a linear function or a polynomial of order 3. Then we approximate U by using finite sums of trigonometric functions with 2L + 1 terms for varying values of L. For comparison, we also approximate U using cosine expansions with L terms.

The right panel of Figure5.1 shows the truncation errors when approximating a value function of a Bermudan option using either cosine expansions (top line), trigonometric expansions combined with a linear function (middle line), or trigonometric expansions combined with a polynomial of order three (bottom line). It shows that we can significantly

improve the efficiency of the approximation based on trigonometric functions by combining it with a linear function. However, combining higher order polynomials with trigonometric functions does not improve the truncation error significantly over the case by combining a linear function. This can be explained by the fact that the continuation value does not have to meet the payoff function smoothly for Bermudan options, so the first derivative of U does not need to be continuous. We propose to combine trigonometric functions with polynomials of order one only when pricing options.

Regarding the second property (P2), the complexity of the COS method can be dra-matically reduced for options where an efficient induction algorithm can be derived. In such cases the PV method is equally attractive, since it can be verified that the computa-tional complexity remains unchanged if we combine cosine functions with a finite number of polynomials. Except for the cases discussed in the literature, it is unclear that whether the induction algorithm can be efficiently utilized or not for other options and/or models. For example, in Section 5.6.2 we demonstrate that for ratchet options under a regime switch-ing model, the exercise region can be a union of two subsets. In the cases when the form of the exercise region prevents us from using an induction algorithm, the computational complexity of the COS method is only quadratic in the number of terms L. Therefore, in these situations we may consider any method of reducing L, such as the proposed PV approach.