AN EFFICIENT TRANSFORM METHOD FOR ASIAN OPTION

AN EFFICIENT TRANSFORM METHOD FOR ASIAN OPTION PRICING J. LARS KIRKBY

Abstract. This paper introduces a novel method to price generalized arithmetic Asian options in Levy-driven models, with discrete and continuous averaging, by expanding on the approach of sequential characteristic function recovery. By utilizing frame duality and a FFT-based implementation of density projection, we obtain rapidly converging value approximations to high precision, consistently resulting in a 10- to 100-fold time reduction compared to state-of-the-art procedures. Theoretical convergence rates are confirmed by an in-depth analysis of error propagation.

1. Introduction Since their introduction in 1987, Asian options (known also as average rate or average price options) have provided a popular means of risk management in a variety of markets. For example, Eydeland and Wolyniec (2003) document their importance in mitigating the delivery risks present in gas markets. Since Asian options have payoffs that are contingent on the average price of an underlying asset (index, interest rate, exchange rate, commodity, etc.) over a given time horizon, their prices are less sensitive to price manipulations, and they become easier to hedge towards the option’s expiry. For discretely monitored Asian options, geometric or arithmetic averaging are possible, while continuously monitored contracts can be thought of as the limit of arithmetic averaging. By taking an average of the underlying, these options are typically much cheaper than standard European contracts. Moreover, their relative stability has led to the hybridization of exotic options that contain an Asian type specification towards the end of the contract, known as an “Asian tail”. As is generally the case with path-dependent contracts, robust pricing of Asian options is very challenging and computationally demanding. Even in the Black-Scholes-Merton (BSM) framework, no analytical formulas exist for the pricing of arithmetic Asian options. The computational approaches can be categorized as analytical approximations and bounds [1, 2, 25, 28], partial differential equation (PDE) methods [3, 4, 13, 35], lattices [11], Monte Carlo [22, 30], and transform methods [6, 9, 10, 12, 36], to which our approach belongs. Alternative methods include Taylor expansion [21], perturbation [37], direct iterated integration [19], and maturity randomization [18]. In terms of both speed and accuracy, the transform based approaches are generally superior for models with Levy (log) returns, including BSM. By working in the Fourier domain, we develop a fast and highly accurate method for pricing generalized Asian options in exponential Levy models, which we call APROJ. This includes discretely contracts as well as the continuously monitored options that pervade foreign exchange markets. In-progress option prices and Greeks are also determined efficiently. Compared to state-of-the-art-methods, the ARPOJ method provides a 10- to 100-fold improvement in terms of cpu time to reach the same (or better) accuracy. This is confirmed for the methods of [6, 9, 10, 26, 36], most notably the improved convolution method of Cerny and Date: 2014. 2010 Mathematics Subject Classification. 62P05, 60E10, 91G20, 91B25, 91G60, 65C20, 65T50, 65D07, 42C15, 65T40. Key words and phrases. arithmetic Asian options, fast Fourier transform, Levy processes, basis, characteristic function, Carverhill-Clewlow factorization, PROJ, COS, FFT, frame projection, option pricing, B-spline, exotic options. The author wishes to thank Shijie Deng, Richard Birge, and Mike Staunton for fruitful discussions. 1

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Kyriakou [10], the ASCOS method of Zhang and Oosterlee [36], and the inverse Fourier transform method of Levendorskii and Xie [26], which are (to our knowledge) the fastest available pricing methods for discretely monitored arithmetic Asian options under Levy dynamics. Our approach combines aspects of several different techniques, including the factorization of Carverhill-Clewlow [9], a variant of the grid shift proposed by Benhamou [6], the characteristic function recovery approach of Zhang and Oosterlee [36], and finally the frame projection methodology introduced in Kirkby [24]. Moreover, like the algorithm developed in [10], our method is able to price generalized averages of the underlying. The paper is organized as follows. Section 2 begins with a brief review of exponential Levy models, followed by an account of density projection by frame duality, which is the crucial ingredient of our efficient algorithms. The problem of arithmetic Asian option pricing is formulated in Section 3. After a change of variable, the basic recursive algorithm is presented, first at a high level, and then in careful detail. The derivation motivates key features of the method and underscores the selection of input parameters. The section concludes with a summary of the algorithm and complexity analysis. Section 4 develops extensions to inprogress option pricing and Greeks, generalized averaging, and continuous averaging. An in-depth analysis of error propagation and terminal valuation error is given in Section 5, after which Section 6 demonstrates the accuracy and efficiency of the method with a series of numerical experiments. Comparisons are made to existing methods with parameter sets from the literature. Finally, Section 7 concludes the paper. 2. Projection Method 2.1. Exponential Levy Models. Since the variance gamma (VG) model was introduced in 1990 to derivatives pricing [27], the versatility and tractability of Levy processes as generalizations of the BSM framework have generated a surge of research and modeling success. While application of the VG model itself has waned, subsequent developments such as the CGMY (KoBoL) [7,8,29] and NIG [5] models have proven to be excellent alternatives which calibrate well to market data [20, 29], and the exponential (semi-heavy) decay of their tails engenders a significant computational advantage over the VG model. Our pricing approach resides in the Fourier domain, so the characteristic functions of a process are of primary interest. In general, suppose L(t), t ≥ 0, is a Levy process with ψL (ξ) := ln E[exp(iξL(1))] its (Levy) symbol or characteristic exponent. The process has stationary and independent increments, L(t + ∆t) − L(t), and by the Levy-Khintchine theorem the characteristic function satisfies φL(t) (ξ) := E[eiL(t)ξ ] = etψL (ξ) ,

t ≥ 0.

Figure 7 in the appendix provides some of the more popular Levy symbols used in financial modeling, along with any parameter restrictions1. These processes satisfy an exponential moment condition E[e−αL(t) ], ∀t ≥ 0, where IL = (λ− , λ+ ) denotes the set of all such α. Here −∞ ≤ λ− ≤ 0 ≤ λ+ ≤ ∞ with possible inclusion of the endpoints. As a function of z = ξ + iw, ψL (z) is complex analytic in the strip D(λ− ,λ+ ) := {z ∈ C : =(z) ∈ (λ− , λ+ )}. To model the underlying randomness on which Asian options are contracted, we consider exponential Levy processes of the form S(t) = S(0)eY (t) = S(0)e(r−q+ω)t+L(t) ,

ω = −ψL (−i),

where ω is a “convexity correction” that is used to fix a risk-neutral measure from the spectrum of arbitrage-free pricing measures, and r, q ≥ 0 are the interest rate and dividend yield. To ensure that discounted asset processes (with reinvested dividends) behave as martingales once ω is fixed, we have chosen ω so that E[S(t+∆t )|S(t)] ≡ S(t)E[eR∆t ] = S(t)e(r−q)∆t , ∆t , t ≥ 0, where S(t + ∆t ) d = (r − q + ω + L(1))∆t , t, ∆t > 0. R∆t := ln S(t) 1If no restriction is given, the permissible parameter values are taken to be the real line.

AN EFFICIENT TRANSFORM METHOD FOR ASIAN OPTION PRICING

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The characteristic function of R∆t is given by φR∆t (ξ) = eiξ(r−q−ψL (−i))∆t eψL (ξ)∆t ,

∆t > 0.

With the exception of the pure jump VG (ie when σ = 0), the Levy processes of interest in finance satisfy (1)

ν

|φR∆t (ξ)| = |eψL (ξ)∆t | ≤ κe−∆t c|ξ| ,

where c, κ > 0 and ν ∈ (0, 2]. For the pure jump VG, |φR∆t (ξ)| ≤ κ|ξ|−2∆t /ν , so that φR∆t 2

fails to be integrable for ∆t ≤ ν/2. By adding a Brownian motion component, − σ2 ξ 2 , (1) is satisfied with ν = 2. 2.2. Density Recovery and Option Pricing by Frame Projection. In [24], a method of European option pricing, called PROJ, is derived from the theory of frames and Riesz bases. The insight is to project the risk-neutral log return density, given in terms of its characteristic function (chf), onto a tractable basis of compactly supported functions. The resulting method produces highly accurate approximations at low resolutions, where the number of basis elements grows with the resolution. For ease of exposition, we introduce the APROJ method within the context of piecewise linear projection. After its derivation, we provide in addition a quadratic projection algorithm which is obtained after a few simple modifications. 2.2.1. Density Projection by Duality. An especially tractable Riesz basis is the linear B-spline basis, defined in terms of translated dilations of the generator ϕ(x) := (1 − |x|)+ = (1 − |x|)1[−1,1] (x). Given a resolution a, and a fixed grid xn = x1 + (n − 1)/a, n = 1, ..., N , the piecewise linear approximation space is spanned by the functions {ϕa,n }N n=1 given by ( a3/2 (x − xn−1 ) if x ∈ [xn−1 , xn ] (2) ϕa,n (x) = a3/2 (xn+1 − x) if x ∈ [xn , xn+1 ] , where ϕa,n (x) is centered over xn . To derive finite dimensional approximations in terms of {ϕa,n }N n=1 , we will truncate the corresponding projections onto the infinite dimensional space Ma := span{ϕa,n }n∈Z , using the fact that ϕ satisfies the frame bounds X (3) Akf k2 ≤ |hf, ϕa,n i|2 ≤ Bkf k2 , ∀f ∈ L2 (R), n∈Z

for some 0 < A ≤ B (independent of a). Given a random variable X, with unknown density2 fX , the frame representation theorem states that the projection PMa fX of fX onto Ma is given by X PMa fX = hfX , ϕ ea,n iϕa,n , n∈Z

where {ϕ ea,n }n∈Z is the dual basis, which is guaranteed to exist in some form. As shown in [24], if the characteristic function φX (ξ) := E[eiXξ ] is known, the projection coefficients with respect to the piecewise linear basis satisfy for 1 ≤ n ≤ N Z ∞ a−1/2 be ξ dξ , (4) hfX , ϕ ea,n i = E[ϕ ea,n (X)] = < exp(−ixn ξ) · φX (ξ)ϕ π a 0 where

Z 12 sin2 (ξ/2) be ϕ(ξ) = 2 , and Ff (ξ) = fb(ξ) = eiξx f (x)dx. ξ (2 + cos(ξ)) R The coefficients can thus be calculated efficiently using the fast Fourier transform (FFT), as described next. 2Levy models, with the exception of the compound Poisson process, possess a continuous density [31].

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When φX (ξ) satisfies a growth estimate of the form of equation (1), the truncation error from numerically integrating (4) will decay exponentially, and polynomially otherwise. Even be so, multiplication of the chf by ϕ(ξ) in equation (4) has a damping effect which reduces aliasing caused by an otherwise insufficient choice of a (the discrete Fourier transform implies a truncation interval of 2πa in Fourier space). This is one factor which contributes to accurate approximations at low resolutions (small a). 2.2.2. Implementation. To recover the orthogonal projection of the density of a random variable X, the first step is to fix a resolution a = 2P for P ∈ N. By further specifying P¯ ∈ N, which determines the support width of the projected density, and x1 , which determines its location in log return space, a conceptual grid xn = x1 + (n − 1)/a, n = 1, ..., N , is designated where ¯ ¯ N = 2P +P = a2P := a¯ a, where the choice of parameters is discussed in Section 3.5. For example, if E[X] := µX , then N to center the grid over µX , set x1 = µX − 2 − 1 ∆x (where ∆x := 1/a), so that µX = x N . 2 The density is then recovered on a ¯ a ¯ ¯ [x1 , x1 + 2P − ∆x ] ≈ [µX − , µX + ]. 2 2 X Namely, we use the characteristic function φX (ξ) to approximate the coefficients βa,n of an expansion of fX (x) in terms of the linear basis: (5)

fX (x) ≈

N N X 24a5/2 X ˘X X βa,n ϕa,n (x) = a1/2 Ca,N β˘a,n ϕa,n (x), N n=1 n=1

X approximate the true coefficients in where Ca,N = 24a2 /N . The coefficients a1/2 Ca,N β˘a,n equation (4), and are recovered by the discrete Fourier transform as follows3. By the Nyquist frequency requirement ∆x ∆ξ = 2π/N , the grid in frequency space is set to ξj = (j − 1)∆ξ , j = 1, ..., N , where ∆ξ = 2πa/N = 2π/¯ a. We then define

(6)

H1 = 1/24a2 ,

Hj = φX (ξj )ζj exp(−ix1 ξj ),

2 ≤ j ≤ N,

where (7)

ζj :=

sin2 (ξj /2a) , ξj2 (2 + cos(ξj /a))

2 ≤ j ≤ N.

According to equation (4), the beta coefficients β˘X = {β˘nX }N n=1 are recovered by the discrete Fourier transform (DFT) (8)

β˘X := 0, trapezoidal approximations to the DFT converge exponentially with respect to the a ¯. Together with the exponentially decaying truncation error (in a), the total error in approximating hfX , ϕ ea,n i by DFT is exponential in a, a ¯, as described next. 2.2.3. Projection Coefficient Errors. We define H(Dd ) to be the set of analytic functions in the strip Dd = {z ∈ C : =(z) ∈ (−d, d)} which satisfy Z d |h(x + iy)|dy → 0, as |x| → ∞. −d

For h ∈ H(Dd ), we define the norm Z Z khkHd := lim+ |h(x + i(d − ))|dx + |h(x − i(d − ))|dx . →0

R

R

3The term a1/2 will be absorbed by an intermediate calculation to reduce the computational cost at each stage in the algorithm.


5

Given a random variable X with density fX , it is shown in [24] that the truncated true projection, N X feX (x) := hfX , ϕ ea,n iϕa,n (x), n=1

is well represented by the numerical approximation N X X f˘X (x) := ϕa,n (x), a1/2 Ca,N β˘a,n n=1

where the coefficients are calculated by the discrete Fourier transform, in the absence of characteristic function error4:   N   −1/2 X a X be ξj vj ∆ξ , < (9) a1/2 Ca,N · β˘a,n exp(−ixn ξj ) · φX (ξj )ϕ =   π a j=1

where νj := 1 − (δj,1 + δj,N )/2. As long as the numerical error is controlled, the overall convergence of the APROJ algorithm will be at least of the order of projection convergence. Assuming that φX is analytic, we have the following. Corollary 2.1. Suppose that φX (ξ) ∈ H(Dd ) for some d > 0, and let µ ¯ = µ ¯X be an apP P¯ ¯ > 1 + log2 |¯ proximation to E[X]. Fix a = 2 and N = a · a ¯ , where a ¯ = 2 for P µ|. Fix N 1 H x1 = µ ¯ + 1 − 2 a . Then for some 0 < γ ≤ d and Cγ (φX ) ≤ 24kφX k a−1/2 e−(¯a−2|¯µ|)γ/2 1/2 X ˘ sup a Ca,N · βa,n − hfX , ϕ ea,n i ≤ + τa (X) Cγ (φX ) π 1 − e−¯aγ 1≤n≤N If for some c, κ > 0 and ν ∈ (0, 2], the tail of φX satisfies (10)

|φX (ξ)| ≤ κ exp(−tc|ξ|ν ),

ξ ∈ R,

where t > 0 is some fixed time, then 6κ · a exp(−tc · (2πa)ν ). π In this case, the largest trapezoidal error converges exponentially in a ¯, while the truncation error is exponential in a. Moreover, when a > 2d, γ = d. (11)

τa (X) ≤

Proof. See appendix.

2.2.4. Pricing. For any finitely valued European payoff g, the price V ◦ g is approximated by (12)

VN ◦ g = e−rT Ca,N

N X

β˘nX gn ,

n=1

where the payoff coefficients are defined by Z xn+1 (13) gn := a1/2 ϕa,n (y)g(S0 ey )dy,

n = 1, ...N.

xn−1

For payoffs such as calls and puts, analytical formulas for gn are available. By utilizing the orthogonally projected density, PROJ obtains highly accurate approximations even at low resolutions, which is crucial for its extension to Asian options (APROJ) where the required number of inversions grows linearly with the monitoring frequency. This phenomenon is explained in [33], where for modest resolutions the least squares projection behaves like an interpolation with twice the order of accuracy. Consequently, the use of projected densities results in a substantial reduction in overall cost. Another advantage of PROJ is the narrow support of its basis elements. This feature is especially desirable in the extension to Asian options which require repeated evaluation of integrals of a form similar to equation (13). In contrast, the globally supported basis 4Error in the characteristic functions will be introduced.

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elements of a cosine series expansion required a much more expensive procedure to evaluate the analogous integrals. 3. Mean Adjusted APROJ Method 3.1. Arithmetic Asian Options. Discretely monitored arithmetic Asian options are contracts on the average over an observed set of prices of an underlying, with observations taken at a discrete set of M + 1 monitoring dates, {0 = t0 , t1 , ..., tM = T }, with S0 = S(t0 ) observed upon entering the contract. We assume a uniform spacing between observations5, PM T tm = m∆t = m M , m = 0, ..., M . If the density of UM := m=0 Sm is known, say fUM , then the initial value of an option paying g(U ) at time T must initially satisfy M R V ◦ g(S0 ) = e−rT R g(u)fUM (u; S0 )du. Fixed strike vanilla Asian options (calls and puts) are priced according to the terminal payoffs  + PM  1 , for a call, m=0 Sm − W M +1 (14) g(UM ) := +  W − 1 PM S , for a put. m=0 m M +1 By considering a change of numeraire, floating strike arithmetic options can be priced using an analogous formula, but only at inception [14]. On the other hand, frame projection can be used to efficiently obtain bounds on the prices of floating strike arithmetic options in terms of their geometrically averaged counterparts. 3.1.1. Change of Variable. The frame projection method for Asian options, APROJ, relies on a convolution scheme called the Carverhill-Clewlow-Hodges factorization [9], which has been used by [10, 18, 19, 36]. The idea is to express the terminal payoff in terms of a random variable YM , defined below, so that (15)

AM :=

M X S0 1 Sm = (1 + exp(YM )) . M + 1 m=0 M +1

Given an approximation of the density fYM , prices for call and put options are then determined by Z −rT V ◦ g(S0 ) = e g(y; S0 )fYM (y)dy, R

where

(16)

 +   S0 (1 + exp(y)) − W  , for a call,  M +1 g(y) := +  S (1 + exp(y))    W− 0 , for a put. M +1

In this way, pricing of a path-dependent Asian option is reduced to the valuation of a European option on the variable YM . As will be demonstrated, such a variable can also be found for generalized Asian options with fixed strikes, and for geometric Asian options with fixed and floating strikes (see [24] for the PROJ implementation for geometric Asian options). Following the direction of [36], we recover the characteristic function φYM by computing the characteristic functions of a sequence of intermediate random variables. Not only does this facilitate the use of PROJ, but convolution is accomplished much more efficiently in Fourier space where it reduces to the product of characteristic functions. After showing how YM is derived, we develop a procedure for the efficient computation of its characteristic function. 5This assumption is easily relaxed at a modest increase in cpu time.


7

The insight of Carverhill-Clewlow is that the arithmetic average can be expressed as S0 S1 S2 SM −1 SM AM = 1+ 1+ ··· 1+ M +1 S0 S1 SM −2 SM −1 S0 = 1 + eR1 1 + eR2 · · ·eRM −1 1 + eRM M +1 S0 (1 + exp(R1 + ln (1 + exp(R2 + ln (· · ·RM −1 + ln (1 + exp(RM ))))) = M +1 where the log return increments are defined by6 Sm Rm := log , Sm−1

m = 1, ..., M,

where we have suppressed the dependence of Rm on the time step ∆t = T /M . By introducing the sequence {Ym }M m=1 , defined recursively by (17)

Y1 := RM ,

Ym := RM +1−m + log(1 + exp(Ym−1 )),

m = 2, ..., M,

we have  (18)

Ym = log 

from which it follows that exp(YM ) =

1 S(M −m)

1 S0

PM

m=1

m X

 S(M −m)+j  ,

j=1

Sm , and so equation (15) holds.

3.2. The Basic Recursion. With Zm := log(1 + exp(Ym )), the characteristic function of YM is found recursively from Y1 = RM by the equation (19)

Ym = RM +1−m + Zm−1 ,

m = 2, ..., M.

Assuming exponential Levy dynamics, the log return increments Rm are independent, from which independence of RM +1−m and log(1 + exp(Ym−1 )) follow. Moreover, stationarity (and uniform monitoring) implies that RM +1−m = R in law for all m, where R has known characteristic function for many Levy processes. Hence, starting with φY1 (ξ) = φR (ξ), the characteristic function of Ym is derived from that of Ym−1 using equation (19): φYm (ξ) = φR (ξ)φZm−1 (ξ),

m = 2, ..., M.

Specifically, (20)

h i Z φZm−1 (ξ) := E eiξ log(1+exp(Ym−1 )) = (ey + 1)iξ fYm−1 (y)dy, R

where fYm−1 is approximated using φYm−1 . The skeleton of the APROJ algorithm is as follows: Algorithm Overview: e Step 1: From φR , obtain projected density. R y fR , the Determine φZ (ξ) ≈ (e + 1)iξ feR (y)dy. 1

R

Step 2: For m = 2, ..., M − 1: compute the projection feYm using φYm (ξ) = φR (ξ)φZm−1 (ξ). R Find φZm (ξ) ≈ R (ey + 1)iξ feYm (y)dy. Step 3: From φYM , obtain the final projection feYM . Calculate the payoff coefficients as in equation (13), for the appropriate payoff in equation (16). 6We reserve the notation R to denote the return distribution over a time increment of size ∆ , while R t m denotes the return random variable itself. To make the dependence on ∆t explicit, we will at times use R∆t to denote a generic return increment.

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5

4

3

2

1

0

−1

−2

0

50

100

150

200

250

m

Figure 1. Plot of µ ¯m , the approximated mean of Ym , as a function of m = 1, ..., 250 in the BSM model with r = .05, q = 0 and σ = .3. The bounds µ ¯m ± a¯2 are given by dashed lines, where a ¯ = 1.

In the remainder of this section, we provide a detailed derivation of the APROJ algorithm, which is an efficient implementation of this basic outline. Examples are provided to motivate the use of particular numerical techniques, and facilitate an understanding of its components as well as choices for the parameters a, a ¯. The algorithm itself is summarized in Section 3.9, along with a discussion of its complexity. The next result will ensure that the DFT errors, which are incurred at each density projection step, converge exponentially with respect to a, a ¯. Proposition 3.1. Suppose that φR (z) is analytic in the strip Dd := {z ∈ C : =(z) ∈ (−d, d)}, for some d > 0, and satisfies equation (1) for some κ, c > 0 and ν ∈ (0, 2]. If {Ym }M m=1 are defined by equation (17), then the characteristic functions satisfy (i) φYm is analytic in Dd , 1 ≤ m ≤ M , and ν (ii) |φYm (ξ)| ≤ κe−∆t c|ξ| , ξ ∈ R, 1 ≤ m ≤ M . Hence, the domain of analyticity and the decay of φYm are independent of m. Proof. See appendix.

It should also be noted that fYm (y) ∼ e−d|y| as |y| → ∞, ie the densities have exponentially decaying tails7, determined by the tail behavior of fR . This follows since analyticity of φYm in Dd implies that E[eηYm ] < ∞ for η ∈ (−d, d). In particular, we are dealing with densities of rapid decrease. As described next, a grid shift is employed to ensure that we capture to within a set tolerance the mass of fYm−1 , while the grid specific to each Ym will belong to a single enlarged grid, for m = 1, ..., M − 1. The final grid corresponding to YM will vary slightly according to the payoff to be priced. 3.3. Mean-adjusted Grid. Since the distribution of Y1 = R∆t is more or less centered about its mean, a natural starting grid in log return space is fixed by centering about E[R∆t ] = (r − q + w + E[L(1)])∆t . For example, the Black-Scholes-Merton (BSM) model satisfies E[R∆t ] = (r − q − σ 2 /2)∆t , where σ is the rate of volatility. With µ ˜1 := E[R∆t ], we can approximate µm := E[Ym ] = E[R∆t ] + E[ln(1 + eYm−1 )] by (21) µ ˜m = µ ˜1 + ln 1 + eµ˜m−1 , m = 2, ..., M, which is the approach taken by [6]. Note that by convexity of log(1 + ey ), Jensen’s inequality implies log(1 + exp(E[Ym−1 ])) ≤ E[log(1 + exp(Ym−1 ))], so the mean shift underestimates the true mean. We employ an alternative grid-shift scheme, described next. 7The rate of decay could be faster than d, but this gives a conservative estimate.


9

3.3.1. Grid Shift and Bounds. As an alternative to the grid adjustment of [6], and to bound the growth of the grid shift, we derive an upper bound on E[Ym ] by applying Jensen’s rule (for concave functions) to equation (18):   ! m T X S(M −m)+j exp (r − q) M m −1 T  = (r − q) E[Ym ] ≤ log  + log , E T S(M −m) M exp (r − q) M −1 j=1 h i h i S FM −m = E S(M −m)+j , where the first equality follows from since e(r−q)j∆t = E S(M(M−m)+j S(M −m) −m) the martingale property and the second from the fact that Levy increments are independent of the current filtration, FM −m . In particular, this upper bound is model independent and can be used without knowledge m , so of E[R∆t ] to obtain the sequence of shifts. We also find that E[Ym ] ≤ log(m) + (r − q)T M as we would expect, the growth in E[Ym ] is no faster than log(m), independently of M (the second term is always bounded by (r − q)T ). Similarly, m S(M −m)+j 1 X Ym ≥ log(m) + log , m j=1 S(M −m) from which we derive E[Ym ] ≥ log(m) + E[R∆t ] m+1 2 . This gives us (22)

E[R∆t ]

m+1 m ≤ E[Ym ] − log(m) ≤ (r − q)T , 2 M

∀m ≤ M,

T where E[R∆t ] = (r − q + w + E[L(1)]) M . Hence, E[Ym ] = log(m) + O(m(r − q)∆t ).

3.3.2. Grid Shift Algorithm. The APROJ grid shift is implemented by as follows. First we define m+1 T µ ˜m := ln(m) + µ ˜1 , m = 2, ..., M. (23) µ ˜1 := (r − q − ψL (−i) + E[L(1)]) , M 2 In order to reduce the computations required below (namely in computing a matrix Ψ), we perturb each µ ˜m slightly to obtain µ ¯m , which belongs to an extension of the initial grid defined by µ ˜1 : µ ˜m − µ ˜1 (24) µ ¯m := µ ˜1 + Nm ∆, Nm := , m = 2, ..., M − 1, ∆ and µ ¯1 ≡ µ ˜1 , N1 := 0. Hence, we define the mean-adjusted grids N m m m (25) xn = x1 + (n − 1)∆, x1 := µ ¯m + 1 − ∆, m = 1, ..., M − 1, 2 N +N

each corresponding to a subset of the linear basis {ϕa,n }n=1 M −1 with ϕa,1 centered over x11 . In particular, the density of Ym is recovered over a ¯ a ¯ [¯ µm − , µ ¯m + ], m = 1, ..., M, 2 2 which is illustrated in figure 1 for M = 250. For simplicity and ease of implementation, N is fixed for all m. The choice of xM 1 will be detailed in Section 3.7. M M To implement the algorithm, only {xm 1 }m=1 and {Nm }m=1 are needed (there is no need to actually generate the grids at each stage). With a complexity of O(M ), the cost of grid adjustment is trivial, especially compared to the cost saved by the consequent reduction in P¯ , which can be substantial. 3.4. Characteristic Function Recovery. We now derive the characteristic function recovery by successive PROJ expansions on the mean-adjusted grid. The algorithm is summarized in Section 3.9, along with a discussion of its complexity. In the algorithm description, we will denote by β¯X the discrete Fourier approximation in the presence of characteristic function error, to distinguish it from β˘X in the error analysis.

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J. LARS KIRKBY 1 xn = 0 xn = .5

0.9

xn = 1 xn = 1.5

0.8

xn = 3 θ(ξ)

0.7

¯ n) Ψ(ξ,

0.6

0.5

0.4

0.3

0.2

0.1

0

0

50

100

150

200 ξ ∈ [0, 2πa)

250

300

350

400

Figure 2. Convergence in xn of Ψ(ξ, n) to a1/2 F [ϕa,n ](ξ), a plot of the modulus.

3.4.1. Initialization. In order to start the characteristic function recursion we need h i Z φZ1 (ξ) := E eiξ log(1+exp(R)) = (ey + 1)iξ fR (y)dy. R

Since φR (ξ) is known, φZ1 (ξ) is approximated by a PROJ expansion of fR (y) to yield ! Z N 1 X y iξ b hφR (ξ), ϕ ea,n (ξ)iϕa,n (y) dy φZ1 (ξ) ≈ (e + 1) 2π n=1 R Z N 1 X b hφR (ξ), ϕ ea,n (ξ)i (ey + 1)iξ ϕa,n (y)dy = 2π n=1 In ≈

(26)

N 24a2 X ¯1 ¯ β · Ψ(ξ, n) := φ¯Z1 (ξ), N n=1 n

be (ξ) := F[ϕ where ϕ ea,n ](ξ). a,n With the initial grid implied by the choice of x11 = E[R] + (1 − N/2)∆, so that φZ1 is approximated by a projected expansion of fR about E[R], the column vector β¯1 is determined by8 (27) β¯1 := 0 fixed, the elements of Ψ 2 sin(ξ/2a) Ψ(ξ, n) ∼ eixn ξ + O(a · exp(−xn−1 )), ξ/(2a) when xn is large. Proof. See appendix.

¯ Fast implementations, especially when M is large (in which case a significant portion of Ψ will be well approximated by Lemma 3.1) should make use of this fact. 3.4.2. Recovery of φZm−1 . From the definition of Zm−1 , the characteristic function is approx¯m + a¯2 ], and imated in terms of the PROJ expansion of fYm−1 , recovered over [¯ µm − a2¯ , µ corresponding to the subset of basis elements ϕa,n , n = Nm−1 + 1, ..., Nm−1 + N : Z φZm−1 (ξ) = (ey + 1)iξ fYm−1 (y)dy R   Z Nm−1 +N X 1 be (ξ)iϕa,n (y) dy hφ¯Ym−1 (ξ), ϕ ≈ (ey + 1)iξ  a,n 2π R n=Nm−1 +1

(29)

≈ Ca,N

N X

¯ Nm−1 + n) := φ¯Z β¯nm−1 · Ψ(ξ, (ξ). m−1

n=1

As before, the grid is fixed by xm−1 , and the column vector β¯m−1 := ¯ ¯ ¯Z where Φ m−1 = (φZm−1 (ξ1 ), ..., φZm−1 (ξN )) , and for m = 2, ...M , ¯ m−1 (j, n) = Ψ(j, ¯ Nm−1 + n), j, n = 1, ..., N. Ψ ¯ m−1 is defined for notational compactness and to indicate that only a subset of Ψ ¯ Here, Ψ takes part in the matrix-vector product.

¯ Y , equation (29) yields 3.4.3. Recovery of φYm . To determine Φ m Z N X (33) φ¯Zm−1 (ξ)φR (ξ) = Ca,N β¯nm−1 (ey + 1)iξ φR (ξ)dy. n=1

INm−1 +n

In matrix form the algorithm reads ¯Y = Ψ ¯ m−1 β¯m−1 ◦ ΦC (34) ΦC Φ R := Ca,N ΦR , R, m

m = 2, ..., M,

where ◦ denotes the Hadamard product and ΦR = (φR (ξ1 ), ..., φR (ξN ))> .

12

J. LARS KIRKBY 1 m=1 m = 10 m = 20 m = 50

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 −400

−300

−200

−100

0

100

200

300

400

Figure 3. Moduls of recovered chfs φYm , m ∈ {1, 10, 20, 50} with M = 50, given the log return model (C, G, M, Y ) = (.0244, .0765, 7.5515, 1.2945), and r = .0367. The x-axis: ξ ∈ [−2πa, 2πa], ∆ξ = 2π/¯ a, where a = 26 , a ¯ = 23 .

An example of the modulus of recovered chfs for the CGMY model with M = 50 is given in Figure 3, where the line corresponding to m = 1 is just |φR (ξ)|. Notice how the chfs collapse about the origin as m approaches M . The reflects the fact that, as m increases, the density of fYm becomes less peaked (ie smoother), which translates into a more rapid decay of φYm . Consequently, truncation in Fourier space (as imposed by the DFT) will be controlled for all 1 ≤ m ≤ M if it is controlled at m = 1, which can be done precisely since φY1 = φR is given. An expression for this truncation error, as a function of a > 0, is provided in Corollary 3.5 below. At the same time, it is clear from Figure 3 that the choice of a ¯, which controls discretization error in Fourier space (∆ξ = 2π/¯ a), should be such that the discretization error of φYM is controlled. One such choice is provided by Corollary 3.5. It should also be remarked that, while the density of fYm becomes less peaked for large m, its tails exhibit faster decay than those of fR , which is intuitively clear (convolution reduces tail heaviness). Hence, the choice of a ¯, in terms of its effect on the truncation error of fYm over [¯ µm − a ¯/2, µ ¯m − a ¯/2], can also be made with respect to the density of fR . For simplicity, we proceed in the numerical section by fixing a simple set of rules which are relatively robust.9 3.5. Parameter Selection. In order to apply the APROJ method, only two parameters are ¯ required: P and P¯ , which together define the budget N = a · a ¯ = 2P · 2P . As a consequence of Corollary 2.1 and Proposition 3.1, we can bound the beta coefficient error over 1 ≤ m ≤ M , in the absence of characteristic function error. The propagation of errors in the chf is studied T in the error analysis. With M, T > 0 and ∆t = M fixed, let (35)

CM := 24

max kφYm kHd ,

m=1,...,M

be where the constant 24 accounts for the multiplication by ϕ(ξ/a). Corollary 3.2. Suppose that φR∆t (ξ) ∈ H(Dd ) for some d > 0. Fix a = 2P and N = a · a ¯, ¯ where a ¯ = 2P for P¯ > 1 + log2 |¯ µM |. Assume for some c, κ > 0 and ν ∈ (0, 2], φR∆t (ξ) 9From this discussion, one can design more sophisticated schemes based on precise decay rates of φ (ie R using its cumulants, as in [15]), and the recovered value of β¯11 . An adaptive procedure would be to initialize P¯ , a by a rule of thumb, say using Corollary 3.5, recover a · Ca,N β¯11 as an estimate of fR (x11 ), and increase a ¯ until fR (x11 ) < T OL. Since the cost to recover β¯11 from φR is trivial, robust implementations should include some procedure of this form. Alternatively, one can use the representation P[R∆t ≤ x] = 21 − 2i H(e−iξx φR∆ (ξ))(0), t

where H is the Hilbert transform operator (cf [16, 17]).


13

satisfies equation (1). Then for some 0 < γ ≤ d a−1/2 e−(¯a−2|¯µM |)γ/2 m (36) + τ (R ) sup a1/2 Ca,N · β˘a,n − hfYm , ϕ ea,n i ≤ CM a ∆t π 1 − e−¯aγ 1≤n≤N independently of 1 ≤ m ≤ M where τa (R∆t ) = O(a exp(−∆tc · (2πa)ν )) is as in equation (11). For large enough a > 0, and d < ∞, γ will approach d. The value of P¯ prescribed in Corollary 3.5 ensures a ¯ −2|¯ µM | > 0, and hence the exponential convergence in a ¯, and it is usually around P¯ = 3 for M ≤ 50, or P¯ = 4 when M = 250. Since this controls the largest coefficient error, it tends to be conservative although robust for heavy tailed returns (for BSM, P¯ = 2 is more than sufficient for M ≤ 250 and σ ≤ .5, and practical accuracy of about e-04 is achieved with P¯ = 0 ∼ 1). In practice, a conservative rule of thumb is to choose P¯ = 4 for heavy tailed distributions, and P¯ = 2 for diffusion models. This rule is applied for linear and quadratic APROJ implementations. For the BSM, KOU (double exponential), and MJD (Merton’s Jump Diffusion) models 2

from Table 7, the characteristic function of log return satisfies |φR∆t (ξ)| ≤ exp −∆t σ2 |ξ|2 , 2

so equation (1) is satisfied with ν = 2 and c = σ2 . For the CGMY model, ν = Y and c can be taken as c = 2C|Γ(−Y ) cos(πY /2)| · , for any ∈ (0, 1). With the Normal Inverse Gaussian (NIG) model, ν = 1 and c = δ. Finally, we stress that with decreasing time steps ∆t = T /M , the density fR∆t peaks while φR∆t spreads out, implying a larger value of a to control the discretization error of fR∆t and the truncation error of φR∆t . This is demonstrated in the numerical section. At the same time, while Corollary 3.5 suggests a value of a ¯ that increases (modestly) in M , this motivation can be tempered by a thinning of the tails of fR∆t , and hence of fYm−1 , which reduces the density truncation error. This finding is verified in numerical experiments, where the choice of a ¯, governed by a desired accuracy, is fairly constant for M ≤ 250 (see Figure 8 in appendix). Resolution. Once P¯ is selected, the parameter P drives convergence through the grid spacing ∆ = 1/a = 2−P . In the numerical experiments, we illustrate the convergence behavior of APROJ by starting with and initial P ≈ 1 and incrementing it until P ≈ 8. In practice, we suggest the following approach, after fixing P¯ . Starting with a value of P0 = 3, compare the change in value approximations from P0 to P1 = P0 + 1. If the difference is less than a user specified TOL, say e-03, increment P1 to P2 = P1 + 1. This process is iterated until TOL is satisfied. Given a higher initial tolerance, a larger P0 should be selected, where 2 ≤ P0 ≤ 5 for TOL≤ e − 07. 3.6. Approximation of Ψ. We now discuss the numerical approximation of the matrix Ψ. From equation (28), for j = 1, ..., N , Z 1/2 Ψ(j, n) := a (ey + 1)iξj ϕa,n (y)dy, n = 1, ..., N + NM −1 . In M −1 {xm 1 }m=1

By fixing the grids with defined by equation (25), each can be considered as a subset of x11 + (n − 1)∆, n = 1, ..., N + NM −1 , so quadrature points (and function evaluations) can be reused in subsequent integral approximations. Moreover, the induced grid overlap reduces the ¯ from N × ((M − 1)N ) elements to N × (N + NM −1 ) ≤ N × (ln(M − 1)N ) computation10 of Ψ (see section 3.9). We consider Newton-Cotes quadrature rules11, which for In = [xn−1 , xn+1 ] are applied to each of [xn−1 , xn ] and [xn , xn+1 ] and then combined. Because ϕa,n (y) is zero at each of the endpoints of In , and the point xn is common to both subintervals, an nq -point quadrature ¯ n). rule applied to each subinterval requires 2nq − 3 basic operations to evaluate Ψ(j, 10For example, when N = 211 and M = 250, the size of Ψ ¯ is reduced from 1.04×108 to 7.08×105 elements. 11Gauss-Legendre quadrature with 8 points total is comparable to the 7-point rule with 11 points total.

However, non-uniformity of the Gauss-Legendre grid complicates the implementation, and is not justified by the modest gain in accuracy.

14

J. LARS KIRKBY

¯ n , ξ)| log10 |Ψ(xn , ξ) − Ψ(x

xn = 1

xn = 10

−2

−2

−4

−4

−6

−6

−8

−8 Boole Seven Simp4

−10

−10

−12

−12

−14

−14

−16

−16 0

100

200 ξ ∈ [0, 2πa)

300

400

0

100

200 ξ ∈ [0, 2πa)

300

400

Figure 4. Comparison of integral approximation errors for Ψ(n, ξ). “Boole” is given by equation (37), “Simp4” by equation (38), and “Seven” by equation (39).

A natural starting point is Simpson’s rule: iξj iξj iξj 1 xn −∆/2 Ψ(j, n) ≈ e +1 + exn + 1 + exn +∆/2 + 1 , 3 which results from applying an nq = 3 point rule to each half of In . As ξj approaches 2πa, the approximation accuracy of Simpson’s rule deteriorates. For the BSM model, Simpson’s rule is sufficient because of the rapid decay of φR (ξ), which multiplies Ψ in the algorithm. This is illustrated in Figure 9 in the appendix, where the decay of φR (ξ) is much faster for the BSM model than the NIG model. For M = 12, a smooth return density results in sufficient decay in both cases. However, for heavy tailed return distributions (especially for large M ), the decay of φR (ξ) is generally not enough, so higher order quadrature rules are required to ensure a robust implementation. We consider two such rules, the first of which is Boole’s rule. To apply Boole’s rule to integrate a smooth function g(x) against a1/2 ϕa,n (x), Z xn+1 a1/2 g(x)ϕa,n (x)dx, xn−1

{νk }7k=1

{g(σk )}7k=1

the sample := is taken, where σk = xn−1 + k∆/4. The quadrature rules are combined with the sampled values of a1/2 ϕa,n (ηk ) to produce an approximation (37)

Qn (ν) =

1 [4(ν1 + ν7 ) + 3(ν2 + ν6 ) + 12(ν3 + ν5 ) + 7ν4 ] . 45

¯ a grid is defined by {ηk }Nη , To apply Boole’s rule to calculate an entire row in the matrix Ψ, k=1 where ηk = η1 + ∆/4, η1 = x11 − 3∆/4 and Nη = 4(N + NM −1 ) + 7. The convergence rate is O((∆/(nq − 1))7 ), although one must account for the sixth derivative of a1/2 ϕa,n (x)g(x) on each of (xn , xn+1 ), which is unwieldy.12 We find that as ξ approaches 2πa, combining four applications of Simpson’s rule (ie by splitting In into quarters) results in a more accurate (and slightly cheaper) approximation than Boole’s, but for the same number of quadrature 12We have written the convergence in terms of the step size ∆/(n − 1) since, strictly speaking, the decay q depends more on nq than on ∆.


15

points: 1 [ν1 + ν2 + ν6 + ν7 + 2ν4 + 3(ν3 + ν5 )]. 12 Applying equation (37) for j = 2, ..., 3N/4, and equation (38) for j = 3N/4+1, ..., N results in a good approximation across [0, 2πa].13 Figure 4 motivates this choice, with similar behavior as the values of a, a ¯ and xn vary. Similarly, an order O((∆/(nq − 1))9 ) approximation is given by applying a 7-point rule to each subinterval to obtain (39) 1 [36(ν1 + 5(ν5 + ν7 ) + ν11 ) + 9(ν2 + 2(ν4 + ν8 ) + ν10 ) + 136(ν3 + ν9 ) + 82ν6 ] , Qn (ν) = 840 where on In , νk = g(xn−1 + k∆/6), k = 1, ..., 11. To ensure robustness, we specify either Boole’s or the 7-point rule, and no additional parameters are required to implement the algorithm. At first glance Figure 4 would suggest the use of higher order schemes to address accuracy loss at larger values of ξ. However, as φYm (ξ) = φZm−1 (ξ)φR (ξ), rapid decay of φR (ξ) eliminates the need for additional accuracy, as is illustrated in Figure 9 in the appendix . We observe that Boole’s rule (modified by equation (38)) is sufficient for M ≤ 50, while the seven point rule is advised for M = 250. This is due to the decay of φR∆t , which is an increasing function of ∆t = T /M .

(38)

Qn (ν) =

¯ Algorithm 1 Calculation of Ψ xn ← x11 + (n − 1)∆, n = 1, . . . , N + NM −1 η ← ηk , k = 1, . . . , Nη θ ← exp (i∆ξ ln (1 + exp(ηk ))) , k = 1, . . . , Nη η←θ ¯ n) ← 1, n = 1, . . . , N + NM −1 Ψ(1, for j = 2 . . . N do for n = 1, . . . , N + NM −1 do ν ← νk , k = 1, . . . , nν ¯ n) ← Qn (ν) Ψ(j, end for η ←η◦θ end for 3.6.1. Implementation. An efficient implementation (to avoid redundant function evaluations) ¯ is summarized in Algorithm 1, which assumes a general choice of quadrature. To compute Ψ, the matrix is initialized and the first row replaced with ones. Given a choice of quadrature, and Nη a grid η = {ηk }k=1 , the quadrature rule Qn (ν) is applied over each interval In = [xn−1 , xn+1 ]. Using the fact that (ey + 1)iξj = exp (iξj ln (1 + ey )) = exp (i(ξj−1 + ∆ξ ) ln (1 + ey )) = exp (i∆ξ ln (1 + ey )) · exp (iξj−1 ln (1 + ey )) , ¯ where η ◦ θ denotes the Hadamard product, and we have the following implementation for Ψ, nν is the number of quadrature points required for each Qn (ν). Once the quadrature rule is selected (e.g. Boole or seven-point), no user-supplied inputs are required. This simplifies the implementation as compared to a procedure such as ASCOS [36], which requires a specification of nq (quadrature points for the Clenshaw-Curtis integration rule), which can vary substantially from one application to the next. 13One simply replaces the for loop in Algorithm 1 by two loops, and the same vector of samples is used and updated.

16

J. LARS KIRKBY 5 BSM NIG CGMY

4.5

4

3.5

3

2.5

2

1.5

1

0.5

0 1.5

2

2.5

3

3.5

Figure 5. Plotted densities fYM , M = 12, recovered by PROJ for the models: BSM(.17801), NIG(6.1882, -3.8941, .1622), CGMY(.6509, 5.853, 18.27, .8) in section 6.

To evaluate the complexity, η requires on the order of O(Nη ) operations to initialize (as θ), followed by N − 1 Hadamard products for a total cost of O((N − 1)Nη ) operations. Each ¯ of which there are N − 1, requires O((N − 1)(N + quadrature application across a row of Ψ, ¯ NM −1 )(2nq −3)) operations. Hence, Ψ is populated at a cost of O((N −1)(N +NM −1 )(3nq −3)) operations. ¯ Y , the final step is analogous to the 3.7. The Valuation Step. Given the approximation Φ M valuation problem for a European option. Rather than specify xM 1 as before, the valuation accuracy can be further improved by perturbing the terminal grid so that the vanilla option ”kink”, defined by W ∗ (40) y := ln (M + 1) −1 , S0 is a member. In this case, equation (16) can be expressed as  S0 (1 + exp(y))   − W 1[y ≥ y∗ ], for a call,  M +1 (41) g(y) :=  S (1 + exp(y))   W− 0 1[y ≤ y∗ ], for a put. M +1 By initially defining x ˜M ¯M + (1 − N/2)∆ and n∗ = b(y ∗ − x ˜M 1 =µ 1 )a + 1c, we set ∗ ∗ xM 1 := y − (n − 1)∆,

(42)

from which y ∗ = xM n∗ . This defines the conceptual grid in log space M xM n = x1 + (n − 1)∆,

n = 1, .., N.

N M M If we then define the terminal basis {ϕM a,n (y)}n=1 where ϕa,n (y) is centered over xn , the density is approximated by

fYM (y) ≈

N 1 X ¯ beM (ξ)iϕM (y), hφY (ξ), ϕ a,n a,n 2π n=1 M

where ϕM a,N/2 (y) is roughly centered over the mean of YM .


17

The final step is to approximate the initial value by integrating the terminal payoff against the PROJ expansion of fYM (see Figure 5): Z V ◦ g(S0 ) = e−rtM g(y; S0 )fYM (y)dy R

≈

e

≈e

(43)

−rtM

2π −rtM

N X

beM (ξ)i hφYM (ξ), ϕ a,n

Z

ϕM a,n (y)g(y)dy

n=1

Ca,N

N X

β¯nM gn ,

n=1

where β¯M := n∗ M +1 M +1 3 while

(46)

gnput

 exp(xM ) ∆ ∆ S0 S0  n  2 + e− 2 W − · − 1 + e   M +1 M +1 3   S0 1 := 2 −∆  W− −a e −1+∆   M +1 2    0

n < n∗ n = n∗ n > n∗

As for European options, put-call parity can be used to price Asian call options. This approach, reviewed in Section 4.3 within the context of Asian options, is preferred numerically since the put has a bounded payoff. 3.8. Greeks. We now demonstrate how price sensitivities are calculated at almost no additional cost from the valuation algorithm. Consider the call option payoff g(y; S0 ) defined

in equation (41), where y ∗ = y ∗ (S0 ) = ln (M + 1) SW0 − 1 . First we observe that YM is independent of S0 . Indeed, ! ! M M m M m X X X 1 X 1 X exp(YM ) = Sm = S0 exp Rk = exp Rk . S0 m=1 S0 m=1 m=1 k=1

k=1

From equation (43), Leibniz rule is used to determine the call option Delta, noting that g(y ∗ (S0 ), S0 ) = 0: Z ∞ ∂V ◦ g ∂g(y; S0 ) −rT ∆ := =e fYM (y)dy ∂S0 ∂S0 ∗ y (S0 ) Z ∞ e−rT (47) = (1 + ey )fYM (y)dy. M + 1 y∗ (S0 )

18

J. LARS KIRKBY

Using quantities that were already computed during the valuation stage, we find that ∆ ≈ Ca,N

(48)

N e−rT X ¯M ∆ β g , M + 1 n=n∗ n n

where (49)

gn∆

1 ∗   + a2 e∆ − 1 − ∆ ey 2 :=  1 + 1 + e ∆2 + e− ∆2 · exp(xn ) 3

n = n∗ n > n∗ ,

and similarly for put options (put sensitivities are also derived by put call parity). Likewise, the call option Gamma is given by Z ∞ e−rT ∂ ∂2V ◦ g = (1 + ey )fYM (y)dy Γ := ∂S02 M + 1 ∂S0 y∗ (S0 ) 2 ∗ e−rT ∂y ∗ (S0 ) W M +1 =− 1 + ey fYM (y ∗ ) (50) = e−rT fYM (y ∗ ) . M +1 ∂S0 S0 W (M + 1) − S0 Hence, using the approximation fYM (y ∗ ) ≈ a · Ca,N · β¯nM∗ , where n∗ is given in the previous subsection, Γ is computed as a byproduct of the pricing algorithm. 3.9. The Algorithm and its Complexity. We now summarize the proceeding steps which define the linear APROJ algorithm. An extension to the piecewise quadratic case follows with a few slight modifications, and is provided in Appendix B. Naturally, a single vector should ¯ Y , m = 2, ..., M , which are updated be used to store β¯m , m = 1, ..., M and similarly for Φ m within the loop.14 APROJ Algorithm (Linear): (i) Initialize: M M • Determine the left grid points {xm 1 }m=1 and {Nm }m=1 from equations (24), (25) and (23) • Fix the grid ξj = (j − 1)2π/¯ a, j = 1, . . . , N • Compute ΦC = C Φ , where ΦR = (φR (ξ1 ), ..., φR (ξN ))> a,N R R • Compute ζ in equation (7). ¯ by Algorithm 1, for a fixed quadrature rule. • Compute Ψ ¯1 ¯ ¯ ¯ • ΦY2 = Ψ1 β¯1 ◦ ΦC R , where β is given by equation (27), and Ψm is defined as the ¯ :Ψ ¯ m (j, n) = Ψ(j, ¯ Nm + n), j, n = 1, . . . , N (where N1 = 0) subset of Ψ (ii) Loop: for m=3,. . . ,M: • H m−1 is given by equation (30) • β¯m−1 = 0. Fix a = 2P and N = a · a P¯ ¯ where a ¯ = 2 for P > 1 + log2 |¯ µM |. Assume for some c, κ > 0 and ν ∈ (0, 2], the tail of φR∆t (ξ) satisfies |φR∆t (ξ)| ≤ κ exp(−∆t c|ξ|ν ),

(57)

ξ ∈ R.

The terminal characteristic function error satisfies (φ¯YM (ξ1 )) = 0 and ∆t ν (58) |(φ¯YM (ξj ))| = O ∆2 · e−˜c a¯ (j−1) a ¯1/2 kξ 2 φR∆t (ξ)k2 , 2 ≤ j ≤ N, where c˜ := (2π)ν c. The dependence on M is governed by the behavior of φR∆t , where ∆t = T /M . Proof. Fix any ξ ≥ 0, and let G := ∪m=1,..,M Gm the full truncated integration range implied ¯m + a¯2 ]. To manage notation, we will suppress the dependence by P¯ , where Gm = [¯ µm − a2¯ , µ of certain objects on m. For example, we will assume by the indexing on β¯nm that the corresponding basis elements ϕa,n have been shifted appropriately. We start by fixing m ≥ 3, for which (φ¯Z (ξ)) := φZ (ξ) − φ¯Z (ξ) m−1

m−1

Z

m−1

(ey + 1)iξ fYm−1 (y)dy − Ca,N

= R

N X

¯ n) β¯nm−1 Ψ(ξ,

n=1

Z

(ey + 1)iξ fYm−1 (y)dy

= R/Gm

Z

y

iξ

(e + 1) fYm−1 (y)dy − Ca,N

+ Gm

+ Ca,N

N X

! βnm−1 Ψ(ξ, n)

n=1

N X

¯ n) − Ψ(ξ, n)) + Ca,N βnm−1 (Ψ(ξ,

n=1

N X

¯ n)(βnm−1 − β¯nm−1 ) Ψ(ξ,

n=1

:= τ (Gm−1 ) + J1m−1 (ξ) + J2m−1 (ξ) + J m−1 (ξ), where the error term J m−1 (ξ) will be further split into two components. Here we have defined βnm−1 so that a1/2 Ca,N βnm−1 = hfYm−1 , ϕ ea,n i, from which feYm−1 (y) := a1/2 Ca,N

N X

βnm−1 ϕa,n (y)

n=1

is the true projection truncated to the set {ϕa,n }N n=1 . Since |(ey + 1)iξ | = | exp(iξ ln(1 + ey ))| = 1, the truncation error satisfies Z Z y iξ τ (Gm−1 ) = (e + 1) fYm−1 (y)dy ≤ fYm−1 (y)dy ≤ τM (G), R/Gm−1

R/Gm−1

for m = 1, ..., M , where τM (G) bounds the largest such truncation error (typically, τM (G) ≈ τ (G1 ), since fR has the heaviest tails) .The next result characterizes the convergence of J1m−1 , which is governed by the projection error. The case of quadratic projection follows analogously, with ∆2 replaced by ∆3 .


23

Lemma 5.1. For ξ ∈ R, 1 ≤ m ≤ M , and C1 (R∆t ) := C1 (ϕ) · kξ 2 φR∆t (ξ)k2 /(2π), J1m−1 satisfies √ |J1m−1 (ξ)| ≤ a ¯ · C1 (R∆t )∆2 , with the constant C1 (ϕ) defined in (60), independent of φR∆t . Proof. In particular, by Cauchy-Schwartz J1m−1 (ξ) =

Z

(ey + 1)iξ fYm−1 (y)dy − Ca,N

Gm−1

Z

βnm−1 Ψ(ξ, n)

n=1 y

=

N X

(e + 1)

iξ

1/2

fYm−1 (y) − a

Ca,N

Gm−1

N X

! βnm−1 ϕa,n (y)

dy

n=1

G G ≤ k(ey + 1)iξ k2 m−1 · kfYm−1 − feYm−1 k2 m−1 G

≤ k(ey + 1)iξ k2 m−1 · kfYm−1 − PMa fYm−1 kR 2. To characterize the convergence rate of density projections onto the linear spline basis, we note that ϕ is Riesz generator which satisfies (59)

ϕ(0) b = 1,

and for m ∈ {0, 1},

ϕ b(m) (2πk) = 0,

k ∈ Z/{0},

where ϕ b(m) denotes the mth derivative of ϕ. In particular, ϕ is a second order Riesz generator. It then follows from [34] that for any fX ∈ L2 (R), the projection error satisfies (2)

inf kfX − fa k2 ≤ kfX − PMa fX k2 ≤ C1 (ϕ)a−2 kfX k2 ,

(60)

fa ∈Ma

where C(ϕ) is independent of fX . In our particular case, we note that (61)

(2)

kfYm k2 =

1 1 2 1 (2) kF[fYm ]k2 = k(−iξ)2 φYm (ξ)k2 ≤ kξ φR∆t (ξ)k2 < ∞, 2π 2π 2π

since for ξ ∈ R, |φYm (ξ)| ≤ |φR∆t (ξ)|, and the second moment is finite by exponential decay of φR∆t (ξ). Thus if we define C1 (R∆t ) as in the statement of the Lemma, it follows that 2 kfYm−1 − PMa fYm−1 kR 2 ≤ C1 (R∆t )∆ ,

∀m ≥ 2.

Hence, for m ≥ 2 and ξ ∈ R G

|J1m−1 (ξ)| ≤ k(ey + 1)iξ k2 m−1 C1 (R∆t )∆2 ≤

√

a ¯ · C1 (R∆t )∆2 ,

since |(ey + 1)i2ξ | = 1 and |Gm−1 | ≤ a ¯.

Remark 2. We should note that, while the bound in (61) is chosen to be independent of m, the behavior of this term is truly a decreasing function of m, although is difficult to quantify. This can be seen by examining the behavior of φYm from the approximations given in figure 3 for a CGMY model. ¯ The next source of error materializes from the approximation of Ψ by Ψ. Lemma 5.2. For ξ ∈ R 1 ≤ m ≤ M , and C2 (R∆t ) := C2 (ϕ)kφR∆t k2 /2π, J2m−1 satisfies √ ¯ 2 (R∆ ), ¯ · (Ψ)C (62) |J2m−1 (ξ)| ≤ a t where the constant C2 (ϕ) is the lower frame bound defined in equation (3) for the piecewise linear basis, and ¯ := sup{|Ψ(ξ ¯ j , n) − Ψ(ξj , n)| : 1 ≤ j ≤ N, 1 ≤ n ≤ N + NM −1 }. (Ψ)

24

J. LARS KIRKBY

Proof. By the discrete version of Cauchy-Schwartz, J2m−1 (ξ) = Ca,N

N X

¯ n) − Ψ(ξ, n)) βnm−1 (Ψ(ξ,

n=1 −1/2

≤a

N X

!1/2 ¯ n) − Ψ(ξ, n) Ψ(ξ,

2

n=1

≤ ≤

√ √

N X

1/2

a

Ca,N βnm−1

2

!1/2

n=1

!1/2 ¯ · a ¯ · (Ψ)

X

2

|hfYm−1 , ϕ ea,n i|

n∈Z

¯ · C2 (ϕ)kfY a ¯ · (Ψ) k . m−1 2

The term C2 (ϕ)kfYm−1 k2 follows from Bessel’s inequality, which is the upper frame bound corresponding to the piecewise linear basis. Noting that kfYm−1 k2 = kφYm−1 k2 /2π ≤ kφR∆t k2 /2π, we have |J2m−1 (ξ)| ≤

√

¯ · C2 (ϕ)kφR k2 /2π. a ¯ · (Ψ) ∆t

¯ is made so that we obtain an overall convergence rate Remark 3. While the definition of (Ψ) in ∆ when a ¯ has been fixed and a sufficiently accurate quadrature rule has been selected, the ¯ tends to be much smaller for ξj ∈ [0, 2πa] near zero than for ξj near 2πa. If the error in Ψ we define ¯ m−1 ) := sup |Ψ(ξ ¯ j , Nm−1 + n) − Ψ(ξj , Nm−1 + n)| j (Ψ 1≤n≤N √ ¯ m−1 ) a then J2m−1 (ξj ) ≤ j (Ψ ¯ · C2 (R∆t ). This is more than offset, however, when multiplied by φR∆t (ξj ) to obtain the error in φ¯Ym , since φR∆t (ξj ) is close to one for ξj near zero, and decays exponentially for larger ξj . For the lognormal model (BSM), with rapid chf decay, ¯ is Simpson’s rule is sufficient to obtain high accuracy. In practice, the contribution of (Ψ) dominated by the projection error when using Newton-Cote’s rules on the order of Boole’s or better, at least for M ≤ 50. Given the slower decay of φR∆t (ξj ) for larger M , a safer choice is the seven-point rule, which in some cases outperforms Boole’s rule even for M = 50, although the discrepancy is of the order e-06 or less. For the final term, which reflects the discrete Fourier transform error inherent in β¯m , we have N N X X ¯ n)(βnm−1 − β¯nm−1 ) = a−1/2 ¯ n) · (β¯nm−1 ), J m−1 (ξ) := Ca,N Ψ(ξ, Ψ(ξ, n=1

n=1

where (β¯nm−1 ) := a1/2 Ca,N (βnm−1 − β¯nm−1 ). Lemma 5.3. The error source J m−1 (ξ) can be bounded by a ¯ ¯) + C(J4 ) · (φ¯Zm−2 )a−1/2 |φ¯Z1 (ξ)| (63) |J m−1 (ξ)| ≤ M (a, a π where C(J4 ) is a constant, and (64)

M (a, a ¯) := CM

e−(¯a−2|¯µM |)γ/2 + τa (R∆t ). 1 − e−¯aγ

Proof. Splitting (β¯nm−1 ) in terms of the discrete Fourier transform and characteristic function errors, where a1/2 Ca,N β˘nm−1 is the discrete Fourier transform approximation using the true φYm−1 (see equation (9)), it follows that (β¯nm−1 ) = hfYm−1 , ϕ ea,n i − a1/2 Ca,N β˘nm−1 + a1/2 Ca,N β˘nm−1 − β¯nm−1 := 1 (β¯nm−1 ) + 2 (β¯nm−1 ).


25

Hence, (φ¯Zm−1 (ξ)) = τ (Gm−1 ) + J1m−1 (ξ) + J2m−1 (ξ) + J3m−1 (ξ) + J4m−1 (ξ), where we have defined J3m−1 (ξ) := a−1/2

N X

¯ n) · 1 (β¯nm−1 ), Ψ(ξ,

J4m−1 (ξ) := a−1/2

n=1

N X

¯ n) · 2 (β¯nm−1 ). Ψ(ξ,

n=1

¯ j , n)| ≤ 1 for any 1 ≤ j, n ≤ N , and Moreover, for the Newton-Cotes quadrature rules, |Ψ(ξ by Corollary 3.5 a−1/2 M (a, a ¯), |1 (β¯nm−1 )| ≤ π where M (a, a ¯) is defined in equation (64). Hence, |J3m−1 (ξj )| ≤

N X a−1/2 M ¯ M ¯ j , n)| ≤ a (a, a ¯) · a−1/2 |Ψ(ξ (a, a ¯) π π n=1

Note that J4m−1 (ξ) alone depends on (φ¯Zm−2 (ξj )), since   N −1/2 X 0 a <  ∆ξ ha,n (ξj ) φYm−1 (ξj ) − φ¯Ym−1 (ξj )  2 (β¯nm−1 ) = π j=1   N a−1/2  X 0 (65) = < ∆ξ ha,n (ξj )φR∆t (ξj )(φ¯Zm−2 (ξj )) , π j=1 P0 where ha,n (ξ) and ha (ξ) are defined in equation (74), and indicates that the first and last terms in the sum are halved. If we define (φ¯Zm−2 ) := max1≤j≤N |(φ¯Zm−2 (ξj ))|, it follows that   N N X X 0 0 ¯ ¯ < ∆ξ  h (ξ )φ (ξ )( φ (ξ )) ≤ ( φ )∆ ha (ξj )ei·0 0

R

λ, σJ , σ > 0

R

C, G > 0, M > 1 Y ∈ (0, 1) ∪ (1, 2)

[−M, G]

α, δ > 0 β ∈ (−α, α − 1)

[β ± α]

λ, σ > 0, p ∈ [0, 1] η1 > 1, η2 > 0 (−η1 , η2 )

σ2 ln 1 − iνθξ + ν 2V ξ 2

ν, σV > 0, σ ≥ 0 q 2 ζ := σθ4 + νσ22 V

V

θ σ2

±ζ

Figure 7. Symbols, parameter restrictions and strip of analyticity IL for tractable Levy processes.

M=12, BSM

M=12, NIG P¯ P¯ P¯ P¯

log10 |err|

0 -2

2 =0 =1 =2 =3

0

log10 |err|

2

-4 -6

-2 -4

-8

-8

-10

-10 2

3

4

5

6

P¯ P¯ P¯ P¯

-6

7

2

3

M=50, BSM

5

6

7

6

7

6

7

2 0

log10 |err|

0

log10 |err|

4

M=50, NIG

2

-2 -4 -6

-2 -4 -6

-8

-8

-10

-10 2

3

4

5

6

7

2

3

M=250, BSM

4

5

M=250, NIG

2

2 0

log10 |err|

0

log10 |err|

=1 =2 =3 =4

-2 -4 -6

-2 -4 -6

-8

-8

-10

-10 2

3

4

5

6

7

2

P = log2 (a)

3

4

5

P = log2 (a)

Figure 8. Convergence in P¯ of quadratic APROJ prices for BSM and NIG models (one legend for each model). Parameters and reference values as in Table 7, strike W = 100.

generator ϕ[2] (y) for the quadratic B-spline basis is  3 9 1 2  2y + 2y + 8, [2] 3 2 ϕ (y) := −y + 4  1 2 3 9 2y − 2y + 8,

defined as y ∈ [−3/2, −1/2) y ∈ [−1/2, 1/2) y ∈ [1/2, 3/2).

AN EFFICIENT TRANSFORM METHOD FOR ASIAN OPTION PRICING M=12, BSM

0

−4 −6

2

3

4

5

6

−8

7

M=50, BSM

log10 |err|

log10 |err|

3

4

5

6

7

6

7

6

7

M=50, NIG

0

−2 −4

−2 −4 −6

2

3

4

5

6

−8

7

M=250, BSM

2

2

3

4

5

M=250, NIG

2 0 log10 |err|

0 log10 |err|

2

2

0

−2 −4 −6

−4 −6

2

−6

Seven Boole Simps

−2 log10 |err|

log10 |err|

−2

−8

M=12, NIG

0 Seven Boole Simps

35

−2 −4

2

3

4 5 P = log2(a)

6

7

−6

2

3

4 5 P = log2(a)

Figure 9. Convergence of linear APROJ prices for BSM and NIG models. Comparison of quadrature rules: seven-point, Boole’s, and Simpson’s (3-point) rules. Parameters and reference values as in table 7, strike W = 100.

With In := [xn −3∆/2, xn −∆/2]∪[xn −∆/2, xn +∆/2]∪[xn +∆/2, xn +3∆/2] := In1 ∪In2 ∪In3 , ¯ we evaluate the integrals to obtain the matrix Ψ Z Ψ(j, n) := a1/2 (ey + 1)iξj ϕ[2] n = 1, ..., N + NM −1 a,n (y)dy, In

by applying the seven point rule to each subinterval Ink , k = 1, 2, 3. Combined with the known values of ϕ[2] (y) at each quadrature point, this results in the (composite) seven-point rule29 1 n Qn (ν) = 3 ν1 + ν17 + 25(ν5 + ν13 ) + 46(ν7 + ν11 ) 840 27 + ν2 + ν16 + 4(ν4 + ν14 ) + 13(ν8 + ν10 ) 18 o + 34 [ν3 + ν15 + 6ν9 ] + 41 [ν6 + ν12 ] .

¯ n) in the matrix Ψ), ¯ νk = For each ξj and xn fixed (where (j, n) fixes the element Ψ(j, iξj σk (e + 1) are the function evaluations over the quadrature points σk = (xn − 3∆/2) + k∆/6, k = 1, ..., 17. An efficient implementation is again provided by Algorithm 1. To obtain density projections, from [24] we have 480 sin3 (ξ/2) . ξ 3 (26 cos(ξ) + cos(2ξ) + 33) For a random variable X with chf φX , the beta coefficients β¯ = {β¯n }N n=1 are obtained using β¯ = −η ln(1 + ey ). Hence, Z ∞ Z ∞ Z y e−η ln(1+e ) fYm−1 (y)dy ≤ e−˜ηy fYm−1 (y)dy ≤ e−ηy fYm−1 (y)dy < ∞, τ

τ

R

so φZm−1 (z) exists and is finite ∀z ∈ Dd . To prove continuity, fix any {zn } ∈ Dd with ¯ ⊂ Dd . zn → z ∈ Dd . Let G ⊂ Dd be a bounded open set containing the tail of {zn }, so G ¯ With η¯ := max{|η| : z = x + iη ∈ G}, it follows that Z sup |φZm−1 (z)| ≤ e−¯ηy·sign(y) fYm−1 (y)dy, ¯ z∈G

R

so by dominated convergence Z lim φZm−1 (zn ) =

zn →z

lim exp (izn ln(1 + ey )) = φZm−1 (z).

R zn →z

Analyticity is now proved as follows. Fix any positively oriented triangle Γ ∈ Dd . By Fubini’s theorem Z Z Z φZm−1 (z)dz = Γ

exp (iz ln(1 + ey )) dzdy = 0,

fYm−1 (y) Γ

R

where the final equality holds by Cauchy’s theorem. Hence, by Morera’s theorem, we conclude that φZm−1 (z) is analytic on Dd , and so too is φYm (z) = φR (z)φZm−1 (z). The growth estimate (ii) follows immediately from |φZm−1 (ξ)| ≤ 1 for ξ ∈ R. Proof of Lemma 3.1. For ξ ∈ [0, 2πa), where a > 0 is fixed, we have Z xn+1 1/2 ¯ n) ≤ a1/2 ea,n ](ξ) − Ψ(ξ, ϕa,n (y) eiξy − eiξ ln(1+exp(y)) dy a F[ϕ xn−1

Z

1

= −1

y y y ϕ(y) eiξ(xn + a ) 1 − eiξ(ln(1+exp(xn + a ))−(xn + a ))

Z

1

y y ϕ(y) ln 1 + exp xn + − xn + dy a a −1 Z 1 ≤ 2πa (ln(1 + exp(xn−1 )) − xn−1 ) ϕ(y)dy

≤ |ξ|

−1

= 4πa (ln(1 + exp(xn−1 )) − xn−1 ) , where the next to last line follows since ln(1+exp(x))−x is strictly decreasing. An asymptotic expansion yields ln(1 + exp(xn−1 )) − xn−1 ∼ e−xn−1 −

e−2xn−1 + O(e−3xn−1 ), 2

and the result follows from F[ϕ ea,n ](ξ). Proof of Corollary 2.1. We first define (74)

ha,n (ξ) := 12

sin2 (ξ/2a) exp(−ixn ξ) := ha (ξ) exp(−ixn ξ), (ξ/a)2 (2 + cos(ξ/a))

38

J. LARS KIRKBY

X X and ξj = (j − 1)∆ξ where ∆ξ = 2πa/N . We have that (β˘a,n ) := a1/2 Ca,N · β¯a,n − hfX , ϕ ea,k i satisfies   Z ∞ N −1/2 X a X (β˘a,n )= < ∆ξ φX (ξ)ha,n (ξ)dξ  νj φX (ξj )ha,n (ξj ) − π 0 j=1  Z ∞ ∞ a−1/2  X = < ∆ξ φX (ξ)ha,n (ξ)dξ ν˜j φX (ξj )ha,n (ξj ) − π 0 j=1  ∞ X a−1/2 (trap (a, a ¯) + τa (X)) , +∆ξ ν¯j φX (ξj )ha,n (ξj ) := π j=N

where νj := 1 − (δj,1 + δj,N )/2, ν˜j = 1 − δj,1 /2, and ν¯j = 1 − δj,N /2. To apply Theorem 3.2.1 in [32], we must show that the presence of ha (ξ) does not affect the integrand’s analyticity or the finiteness of the Hardy norm, both of which will follow if we can bound ha (ξ) in a strip contained within Dd (note that Proposition 3.1 of [24] demonstrates the existence of a sin2 (ξ/2) be bound). Consider ϕ(ξ) = ξ12 2 (2+cos(ξ)) = ha (aξ), and let z = x + iy. Note first that 1 4 + e−y (cos(x) + i sin(x)) + ey (cos(x) − i sin(x)) 2 1/2 = sinh2 (y) sin2 (x) + (cosh(y) cos(x) + 2)2 .

|2 + cos(x + iy)| =

For |y| ≤ 1/2, cosh(y) ≤ 3/2, from which (cosh(y) cos(x) + 2)2 ≥ 1/4, and |2 + cos(x + iy)| ≥ 1/2, uniformly in x. Similarly, for |y| ≤ 1, sin x+iy 2 sinh2 y2 cos2 x2 + cosh2 y2 sin2 x2 2 ≤ 1, = x + iy y 2 + x2 be + iy)| ≤ 24, so for |y| ≤ a/2, |ϕ((x be uniformly in x. Hence, ∀|y| ≤ 1/2, |ϕ(x + iy)/a)| ≤ 24, ∀x ∈ R. Thus, φX · ha,n ∈ H(Dγ ) where γ = γ(a) = d ∧ a/2, and Cγ (φX ) := kφX · ha,n kHγ ≤ 24kφX kHγ . For a sufficiently large, the integrand is bounded within Dd (for any finite d > 0). Moreover, since P¯ > 1 + log2 |¯ µ|, it holds that a ¯/2 > |¯ µ| and so |xn | ≤ |¯ µ| + a ¯/2 < a ¯, ∀1 ≤ n ≤ N . Thus by Theorem 3.2.1 in [32], trap (a, a ¯) converges exponentially in a ¯, according to the bound given. 2 The truncation error depends on the tail behavior of φX . Since |ha,n (ξ)| ≤ 12a ξ 2 , and |φX (ξ)| satisfies equation (10), the truncation error is bounded by Z ∞ −tc|ξ|ν Z ∞ ∞ X e 1 2 −tc|2πa|ν ∆ξ ν¯j φX (ξj )ha,n (ξj ) ≤ 12κa2 dξ ≤ 12κa e dξ, 2 2 ξ 2πa 2πa ξ j=N

and the result follows after simplifying.

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