# Option Pricing by Transform Methods: Extensions, Unification, and Error Control

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1 Option Pricing by Transform Methods: Extensions, Unification, and Error Control Roger W. Lee Stanford University, Department of Mathematics NYU, Courant Institute of Mathematical Sciences Journal of Computational Finance (2004), 7(3):51 86 This version: February 2, 2005 Abstract We extend and unify Fourier-analytic methods for pricing a wide class of options on any underlying state variable whose characteristic function is known. In this general setting, we bound the numerical pricing error of discretized transform computations, such as DFT/FFT. These bounds enable algorithms to select efficient quadrature parameters and to price with guaranteed numerical accuracy. This work was partially supported by an NSF Mathematical Sciences Postdoctoral Fellowship. I thank Marco Avellaneda, Peter Carr, Amir Dembo, Darrell Duffie, Esben Hedegaard, George Papanicolaou, Liuren Wu, and seminar participants at Banc of America Securities, the Courant Institute, and Stanford University. 1

2 1 Introduction In a large and growing family of financial models, explicit formulas exist for the characteristic functions of the state variables. Given any such characteristic function, our project is to compute, efficiently and accurately, the prices of a wide class of options on those underlying state variables. Fourier-analytic solutions to various forms of this problem have appeared in the finance literature. They express option prices in terms of Fourier-inversion integrals, which are in practice evaluated numerically. This paper extends and unifies those ideas. In a general setting, we bound the error in the numerical evaluation of these integrals as N-point sums, of the kind that may be computed as a discrete Fourier transform (DFT) by schemes including the fast Fourier transform (FFT). Then we show how these bounds lead to algorithms that make efficient choices of quadrature parameters and compute prices with guaranteed numerical accuracy. 1.1 Outline This paper generalizes Carr-Madan (1999); unifies it with extensions of the relevant elements of Duffie- Pan-Singleton (2000), Lewis (2001), and Bakshi-Madan (2000); and develops error bounds and error minimization strategies. Carr-Madan s underlying random variable X is the logarithm of a terminal stock price, and their objective is to compute the call price, as a function of log-strike. In terms of the characteristic function of X, they calculate analytically the Fourier transform of the call price function, damped to enforce integrability. Inverting this Fourier transform by FFT and then undamping, they recover simultaneously the call prices at many strikes. We begin by setting forth the option pricing problem and defining the options to be priced. Our scope includes not only vanilla calls on variables exponential in a single state variable, but also three other classes of payoffs. These extended payoff classes contain all of the derivative structures treated in Duffie-Pan- Singleton (2000), and in particular they allow payoffs dependent on multidimensional state variables. Next, we derive upper bounds on option prices, intended for use at extreme strikes. These bounds will become relevant to discretization errors in transform-pricing of options at all strikes. In Section 4 we extend, to all four payoff classes, Carr-Madan s analytic calculation of Fourier transforms, as well as their inversion formula recovering the option price. Also, by taking as given the Bakshi- Madan (2000) discounted characteristic function, we extend Carr-Madan to allow stochastic interest rates. As Lewis (2001) observes, transform representations of option prices may be interpreted as contour integrals in the complex plane; shifts of the contours generate alternative pricing formulas. Applying this 2

3 idea, we prove a unified pricing formula encompassing not just our original four but also ten complementary formulas, including as special cases some well-known transform formulas. These formulas involve integrals over (a translate of) the real line, so approximation by an N-point sum is subject to two forms of error: sampling error because the integrand is evaluated numerically only at the grid points, and truncation error because the upper limit of numeric summation is finite. We then establish bounds for both kinds of error, in all four payoff classes. Section 7 addresses strategic issues in error bound minimization. From an error-management perspective, we apply our bounds analysis to argue in favor of the Carr-Madan one-integral approach to call pricing, and against the traditional two-integral approach. Then we make recommendations for choosing among our five one-integral call formulas. For choosing quadrature parameters, we offer a simple algorithm as a robust alternative to the specific constant parameters suggested in Carr-Madan. The first appendix facilitates truncation error calculations by providing bounds on the decay of characteristic functions in two prominent models. The second appendix gives sampling error bounds, for subcases deferred from the main text. The third appendix deals with specific DFT/FFT implementation issues. 1.2 Guiding Principles Wherever possible, we observe the following principles. First, we take as primitive the discounted characteristic function. From there, our analysis proceeds to the computation of option prices. We do not derive any characteristic functions; other papers have already taken the responsibility of finding characteristic functions given, for example, SDE or generating triplet specifications of the underlying financial dynamics; and indeed others take the characteristic function as the specification of the underlying dynamics. Duplication of research effort will be reduced, one hopes, by the emerging division of labor between, on one hand, those projects that specify or derive characteristic functions; and, on the other hand, projects such as this one, which derive option pricing formulas, given arbitrary characteristic functions. Second, we strive to maintain generality. We do not assume that the underlying state variable is, say, a jump-diffusion or Lévy process. We do not assume that its probability distribution has a density. Time and the state space may be continuous or discrete. The state variables may be one-dimensional or multidimensional. Interest rates and dividends may be deterministic or stochastic. As long as the discounted characteristic function for such dynamics is known, option prices are computable. Technical restrictions do apply, which brings us to the next point. Third, we formulate our technical conditions with the view that they should facilitate the design of 3

4 provably robust pricing algorithms. So we place a premium on expressing assumptions in a complete, concise, rigorous, and readily testable way. 2 The Option Pricing Problem Working in a filtered probability space (Ω,P,{F t }), we intend to calculate numerically the time-0 price C 0 of an option paying at time T the F T -measurable random variable C T. Let r t be the interest rate process, possibly stochastic. Let M t := exp( t 0 r sds) be the time-t value of a money market account. Let B t be the time-t value of a discount bond maturing at T. 2.1 Numeraires and Martingale Measures Assuming that the prices (of C, M, B, and any other assets under consideration) admit no arbitrage, there must exist a risk-neutral probability measure P under which asset prices, discounted by M, are martingales. See Harrison and Kreps (1979) or Delbaen and Schachermayer (1994) for technical definitions of admit no arbitrage that make this statement true. Let E denote expectation with respect to P. Then the option price and bond price satisfy C 0 = E[M 1 T C T ] B 0 = E[M 1 T ]. The positive price process M is an example of a numeraire. For any numeraire N there exists a probability measure P N, said to be risk neutral with respect to N, meaning that the N t -discounted price of any asset is a P N -martingale; see El Karoui, Geman, and Rochet (1995). The change of measure from P to P N is given by dp N dp = N T /N 0. FT M T /M 0 When the numeraire is chosen to be the price B t of a T -maturity discount bond, the risk-neutral measure P B is known as the T -forward measure. Let us write E for expectation with respect to P B. The option price satisfies, therefore, C 0 = B 0 EC T. In the case of deterministic interest rates, the forward measure is identical to the usual risk-neutral measure. In our setting, however, interest rates may be stochastic, and the measures are not necessarily identical; the forward measure has the advantage of discounting outside the expectation. 4

5 2.2 Options Let the state variable X be an F T -measurable random variable with values in R n. For a payoff function G : R n R R, define C G (k) := B 0 E(G(X,k)), which is the time-0 price of an option on X, paying G(X,k) at time T. The trigger k is some contract variable, such as a strike, or the logarithm of a strike. Our goal is accurate numerical computation of C G (k) for these cases of G: G 1 (x,k) := (exp(x) exp(k)) + G 2 (x,k) := (x k) + G 3 (x,k) := exp(b 1 x)i(b 0 x > k) G 4 (x,k) := (b 2 x)exp(b 1 x)i(b 0 x > k) b 0 := 1,b 1 := 1,x R b 0 := 1,b 1 := 0,x R x R n x R n where I is the indicator function, so I(b 0 x > k) equals 1 if b 0 x > k, but 0 otherwise. In payoffs G 3 and G 4, the b 0,b 1,b 2 R n are arbitrary constants. When it is clear what payoff(s) is/are under discussion, we may suppress the subscript of G or C. We choose these four functional forms because they include a wide family of payoffs of practical interest. For example, with payoff G 1, if one chooses X to be bond yield, or the logarithm of a stock price or FX rate, then one obtains a call on a stock, bond, or currency. With payoff G 2, if one chooses X to be an interest rate, or a time-averaged interest rate, then one obtains respectively a European or an Asian option on an interest rate. Our G 3 and G 4 are the payoff classes treated in Duffie-Pan-Singleton (2000). With payoff G 3, if one chooses b 0 and b 1 appropriately, then one can obtain asset-or-nothing, binary, equity-linked FX, and twoasset exchange/maximum options, all on the exponentials of components of X, which could be stock price logarithms or bond yields or FX-rate logarithms. With payoff G 4, if one chooses b 1 = 0 and b 0 and b 2 appropriately, then one can obtain basket or spread options on the components of X, which could be interest rates or their time-averages, for example. 3 Upper Bounds on Option Prices at Extreme Strikes For practical use in bounding numerical transform-inversion errors, it is important that C G be dominated by an expression that is easily evaluated in terms of the characteristic function of X. 5

6 For each G = G 1,...,G 4, we give two bounds; both bounds are valid for all k, but the first is intended for use with large positive k, whereas the second is intended for use with large negative k. The usual conventions about are in force, so each of Theorems holds automatically if the expectation on the right-hand side is infinite. The first of these four results is nearly identical to a bound obtained in Broadie-Cvitanic-Soner (1998). The differences, though minor, make it appropriate to present briefly a full proof. Theorem 3.1. For any p > 0, C G1 (k) B ( ) 0Eexp((p + 1)X) p p and C G1 (p + 1)exp(pk) p + 1 (k) B 0Eexp(X). Proof. For all s 0 we have s e k s p+1 ( p ) p (p + 1)exp(pk) p + 1 because the left-hand and right-hand sides, as functions of s, have equal values and first derivatives at s = (p + 1) exp(k)/p, but the right-hand side has everywhere a positive second derivative. Moreover, since the right-hand side is positive, the left-hand side can be improved to (s exp(k)) +. Now substitute s = exp(x), take expectations, and multiply by B 0 to obtain the first bound. The second bound is obvious. Remark 3.1. Therefore, if S T is a nonnegative random variable with ES p+1 T on S T must have prices that decay as O(K p ) for strikes K. A corresponding fact for puts follows from Theorem 6.4: if ES q T S T must have prices that decay as O(K q+1 ) for strikes K 0. < for some p > 0, then calls < for some q > 0, then puts on on Lee (2003) uses these bounds to derive an explicit moment formula for the growth of implied volatility at extreme strikes. Theorem 3.2. For any p > 0, For any q > 0, Proof. For all x R we have C G2 (k) B 0Eexp(pX) pexp(pk + 1). C G2 (k) B 0 (EX k) + B 0Eexp( qx) qexp(1 qk). x k exp(px) pexp(pk + 1) 6

7 because the left-hand and right-hand sides, as functions of x, have equal values and first derivatives at x = k + 1/p, but the right-hand side has everywhere a positive second derivative. Substitute X for x, take expectations, and multiply by B 0 to obtain the first bound. A similar argument shows that for all x, which implies the second bound. Theorem 3.3. For any p > 0, (x k) + = x k + (k x) + x k + exp( qx) qexp(1 qk), C G3 (k) B 0Eexp((pb 0 + b 1 ) X) exp(pk) and C G3 (k) B 0 Eexp(b 1 X). Proof. For all x R n we have I(b 0 x > k) exp(pb 0 x), exp(pk) which implies the first bound. The second bound is obvious. Theorem 3.4. For any p 0 > 0 and p 2 > 0, For any q 0 > 0 and q 2 > 0, Proof. For all x R n we have and C G4 (k) B 0Eexp((p 0 b 0 + p 2 b 2 + b 1 ) X). p 2 exp(p 0 k + 1) C G4 (k) B 0 E((b 2 X)exp(b 1 X)) + B 0Eexp(( q 0 b 0 q 2 b 2 + b 1 ) X). q 2 exp( q 0 k + 1) (b 2 x)i(b 0 x > k) exp(p 2b 2 x) exp(p 0 b 0 x) p 2 e exp(p 0 k) (b 2 x)i(b 0 x > k) = (b 2 x)(1 I(b 0 x k)) b 2 x + exp( q 2b 2 x) q 2 e implying the two bounds. exp( q 0 b 0 x), exp( q 0 k) Remark 3.2. To bound C G4, apply Theorem 3.4 to (b 0,b 1,b 2 ) and (b 0,b 1, b 2 ), and take the larger of the two bounds. To bound C G4, take the sum of those two bounds, because b 2 X is the sum of (b 2 X) + and ( b 2 X) +. 7

8 4 From Characteristic Functions to Option Prices Our starting point is the discounted characteristic function f of the state variable X. Unlike the usual characteristic functions of probability theory, the definition of f includes a discount factor inside the expectation, which is essential for pricing under stochastic interest rates. We produce formulas for prices of each of the four option classes, by expressing option price transforms in terms of f, and then inverting the transforms. 4.1 The Discounted Characteristic Function Let X be an R n -valued random variable. Let A X denote the interior of the set { v R n : Ee v X < }. The complex vectors whose negated imaginary parts are in A X form a strip or tube Λ X := {ζ C n : Im(ζ ) A X }. Adopting the terminology suggested in Bakshi-Madan (2000), define the discounted characteristic function of X, with respect to a discount factor exp( T 0 r tdt), to be the function f : Λ X C where f (ζ ) := E ( e T 0 r t dt e iζ X). Note that the expectation is with respect to P, but f is also related to the forward measure P B, because f (ζ )/ f (0) = Ee iζ X, which is (for ζ restricted to R n ) the usual characteristic function of X with respect to P B. Theorem 4.1. The discounted characteristic function f is well-defined and analytic in Λ X, which is a convex set. Partial derivatives of f may be taken through the expectation. Proof. This follows from Zemanian (1966), Theorems 4 and 5. In certain models, one can derive the discounted characteristic function from an SDE specification of state variable dynamics. For example, affine jump-diffusion specifications give rise to tractable characteristic functions, as shown in Heston (1993), Bates (1996, 2000), Bakshi-Cao-Chen (1997), Bakshi-Madan (2000), Duffie-Pan-Singleton (2000), and Chacko-Das (2002). Outside of that family, Lewis (2000), Schöbel-Zhu (1999), and Zhu (2000) obtain characteristic functions also for non-affine volatility and interest rate models. 8

9 In other models, the state variables follow Lévy processes, and one can derive the characteristic function from a specification of the generating triplet, or directly take the characteristic function to define the dynamics. Examples include the Finite Moment Log Stable model in Carr-Wu (2003), the Normal Inverse Gaussian model in Barndorff-Nielsen (1998), the Generalized Hyperbolic model in Eberlein-Prause (2002), the Variance Gamma model in Madan-Carr-Chang (1998), and the CGMY and KoBoL models in Carr-Geman-Madan-Yor (2002) and Boyarchenko-Levendorskiǐ (2002). Extensions of Lévy process models which introduce stochastic time changes also have, in certain cases, explicit solutions for characteristic functions; see Barndorff-Nielsen/Nicolato/Shephard (2002), Carr-Wu (2002), and Carr-Geman-Madan-Yor (2003). Appendix A gives details of the characteristic functions in two models one in the affine class, and one in the Lévy class. Note that if discounted characteristic functions are available not just for state variables but also for path functionals of the state variables, then our pricing and error control results will apply not just to European options, but also to path-dependent options. For example, in affine models, the availability of characteristic functions for time-averages enables us to price Asian options (on the state variables, not on their exponentials). Such availability is, however, the exception rather than the rule. Transform-based pricing of exotic options is feasible even without a readily computable characteristic function for the path-dependent quantity, provided that the dynamics are simple enough (under geometric Brownian motion, for example, see Fu-Madan-Wang (1999) or Carr-Schröder (2003) for Asian options, and Geman-Yor (1996) or Pelsser (2000) for barrier options); but this falls outside the scope of our pricing and error control results, which assume the availability of the characteristic function. 4.2 Fourier Transform of the Damped Option Price The usual Fourier transform of C G itself does not exist, because C G (k) does not decay as k. Following Carr-Madan, then, for each damping constant α > 0, we define the damped option price function c α,g : R R by c α,g (k) := exp(αk)c G (k). We will show that the damped option price c α,g does have a Fourier transform ĉ α,g : R C, well-defined by provided that α is chosen appropriately. ĉ α,g (u) := e iuk c α,g (k)dk, 9

10 Theorem 4.2. Assume that G satisfies b 1 A X. Then there exists α > 0 with αb 0 + b 1 A X. For any such α the Fourier transform ĉ α,g of c α,g exists and ĉ α,g1 (u) = f (u (α + 1)i) α 2 + α u 2 + i(2α + 1)u ĉ α,g2 (u) = f (u αi) (α + iu) 2 ĉ α,g3 (u) = f (ub 0 (αb 0 + b 1 )i) α + iu ĉ α,g4 (u) = ib 2 f (ub 0 (αb 0 + b 1 )i). α + iu Proof. There exists p > α such that pb 0 + b 1 A X. So Theorem 3.1, 3.2, 3.3, or 3.4 implies that c(k) decays exponentially for k. Also c(k) is bounded. Therefore c(k) is L 1 and has a Fourier transform; moreover, the use of Fubini in the following computation of ĉ is justified: ĉ α,g (u) := e iuk c α,g (k)dk = e iuk e αk B 0 EG(X,k)dk = f (0)E G(X,k)e (α+iu)k dk. Evaluating the integral, ĉ α,g1 (u) = f (0)Ee (α+1+iu)x α 2 + α u 2 + i(2α + 1)u ĉ α,g2 (u) = f (0)Ee(α+iu)X (α + iu) 2 ĉ α,g3 (u) = f (0)E(eb 1 X e (α+iu)b 0 X ) α + iu ĉ α,g4 (u) = f (0)E((b 2 X)e b1 X e (α+iu)b0 X ). α + iu The result follows because ub 0 (αb 0 + b 1 )i Λ X. 4.3 Fourier Inversion Option prices may be recovered via Fourier inversion. Theorem 4.3. Suppose G and α satisfy the hypotheses of Theorem 4.2. In cases G = G 1,G 2, the option price is given by C G (k) = e αk 2π e iuk ĉ α,g (u)du = e αk π 0 Re [ e iuk ĉ α,g (u) ] du. (4.1) In cases G = G 3,G 4, define the average of left and right limits C(k) := [C(k + 0) C(k 0)]/2. Then C G (k) = e αk 2π which can be strengthened to (4.1) if ĉ is L 1. R lim e iuk ĉ α,g (u)du = e αk Re [ e iuk ĉ α,g (u) ] du, (4.2) R R π 0 10

11 Proof. In all cases, the damped option price c(k) is L 1, as argued in Theorem 4.2. In cases G = G 1 and G = G 2, the damped price c(k) is continuous, by the dominated convergence theorem; and the transform is L 1, because ĉ(u) f ( (α + b 1 )i) /(u 2 + α 2 ). Therefore the usual Fourier inversion recovers c; see, for example, Champeney (1987) Theorem 8.2. Undamping with a factor of e αk yields (4.1). In cases G = G 3 and G = G 4, the damped price c(k) is locally of bounded variation, because C(k) is the difference of two monotonic functions, exp(αk) is monotonic, and both C(k) and exp(αk) are bounded on any finite interval. By, say, Champeney (1987) Theorem 8.12, we have (4.2). 5 The Pricing Formula for General α Transform representations of option prices can be viewed as contour integrals in the complex plane. Shifting the contour across a pole of the integrand changes the value of the integral, a technique which Lewis (2001) exploits, as will we. Lewis differs from our approach in that he derives formulas for the transforms of option prices with respect to the spot variable X 0 ; whereas we, like Carr-Madan and Duffie-Pan-Singleton, transform with respect to the trigger variable k. His assumptions require that the option be written on the exponential of a variable X T where the distribution of X T X 0 is not permitted to depend on X 0. Our formulas are not subject to this restriction and apply to a wider class of underlying state variables X, including those exhibiting mean-reversion. One can modify the formulas of Lewis for non-independent-increments. However, the resulting formulas in that case do not allow the direct application of FFT to calibrate parameters to the prices of options at multiple strikes. For that purpose one needs transform-in-strike formulas, which we now derive. Specifically, let Γ := Γ X,G := {z C : Im(z)b 0 + b 1 A X }, and define ˆ C G : Γ X,G C by ˆ C G1 (z) := f (z i) iz z 2 Cˆ G2 (z) := f (z) z 2 Cˆ G3 (z) := f (b 0z b 1 i) iz Theorem 4.2 proves that for positive α with αb 0 + b 1 A X, we have and hence, for z Γ such that Im(z) > 0, ĉ α,g (u) = ˆ C G (u αi), ˆ C G (z) = 11 Cˆ G4 (z) := b 2 f (b 0 z b 1 i). (5.1) z e izk C G (k)dk. (5.2)

12 Thus, in this region, Cˆ G (z) is the complex Fourier transform of the unmodified option price C G (k). Equivalently (modulo rotation by a factor of i), ˆ C G (z) is the bilateral Laplace transform of C G (k). Rewriting the conclusion of Theorem 4.3 shows that C G may be inverted by integrating along the contour Im(z) = α in the complex plane: C G (k) = 1 αi Cˆ G (z)e ikz dz = 1 2π αi π C G (k) = 1 R αi 2π lim Cˆ G (z)e ikz dz = 1 R π R αi αi 0 αi αi 0 αi Re[ ˆ C G (z)e ikz ]dz (G = G 1,G 2 ) Re[ ˆ C G (z)e ikz ]dz (G = G 3,G 4 ), (5.3) again assuming positive α with αb 0 + b 1 A X. For negative α, the transform ĉ α,g does not exist (for G = G 1,...,G 4 ); likewise, the integral in (5.2) does not exist for Im(z) < 0. Nonetheless, the definitions (5.1) do make sense, and the integrals in (5.3) do exist for α < 0, but they do not recover C G, because the integration path has shifted across the pole z = 0; instead they recover C G less the contribution of the residue of ˆ C G at z = 0. In each case G = G 1,...G 4, this generates one additional pricing formula. In case G = G 1, it generates a second additional formula, because ˆ C G1 has a second pole at z = i. For zero α (and for α = 1 in case G = G 1 ), the final integrals in (5.3) are again well-defined, but now the integration contour passes through a pole, and the contribution from the residue is cut in half. (The only exception is in the case G = G 2 which has a double pole at z = 0; this case calls for introducing into the integrand a term that tames the singularity, without affecting the value of the integral.) This generates two additional pricing formulas for payoff G 1, and one additional formula for the other payoffs. Theorem 5.1 makes this discussion precise. Note that by taking α = 0 in cases G = G 3 and G = G 4, we recover both of Duffie-Pan-Singleton s (2000, Prop 2 and Eqn 3.8) pricing formulas. Taking α > 0 in case G = G 1 recovers Carr-Madan s damped-call pricing formula. Taking α = 0 in two instances of case G = G 3 recovers the traditional two-integral call-pricing formulas, which we discuss further in Section 7.1. Our central pricing result is as follows. Theorem 5.1. Assume that b 1 A X. Let α be any real number such that αb 0 + b 1 A X. Then in all cases except (G = G 2 ;α = 0), C G (k) = R α,g + 1 π αi 0 αi Re[ ˆ C G (z)e ikz ]dz (5.4) 12

13 where f ( i) e k f (0) f ( i) e k f (0)/2 α < 1 α = 1 i f (0) k f (0) α < 0 R α,g1 := f ( i) 1 < α < 0 f ( i)/2 α = 0 0 α > 0 R α,g2 := ( i f (0) k f (0))/2 α = 0 0 α > 0 f ( b 1 i) α < 0 ib 2 f ( b 1 i) α < 0 R α,g3 := f ( b 1 i)/2 α = 0 0 α > 0 R α,g4 := ib 2 f ( b 1 i)/2 α = 0 0 α > 0 and ˆ C G is given in (5.1). In cases G = G 1 and G = G 2, the C G can be replaced by C G. We will prove simultaneously Theorem 5.1 and the following (G = G 2 ;α = 0) theorem. Theorem 5.2. Assume that b 1 A X. Then we have the (G = G 2 ;α = 0) formula Proofs. For α > 0, see Theorem 4.3. C G2 (k) = R 0,G2 + 1 π For α < 0 (except for α = 1 in case G = G 1 ): Note that each 0 Re[ Cˆ G2 (z)e ikz ] + 1 dz. (5.5) z2 ˆ C G is analytic in the strip Γ X,G, except for a pole at z = 0 (and also z = i in case G = G 1 ). The residue theorem applies to any rectangular path with horizontal segments on Im(z) = α 1 and Im(z) = α 2, and vertical segments on Re(z) = ±R. Since the integrals over the vertical segments approach 0 as R, it follows that shifting a horizontal contour across the pole changes the value of the integral by 2πi times the residue at that pole. Residue calculation is straightforward. For α = 0 (including α = 1 in case G = G 1 ): Our proof will be for α = 0; a similar argument proves the (G = G 1 ; α = 1) formula. Define the functions S G1 (z) := S G3 (z) := S G4 (z) := 0 and S G2 (z) := 1/z 2. On Γ X,G Γ X,G the function h(z) := S G (z) + 1 [ ] Cˆ G (z)e izk + Cˆ G ( z)e izk 2 (modulo a removable singularity in case G 2 ) is analytic. Choose ε > 0 such that b 1 ± εb 0 A X. Applying 13

14 Cauchy s Theorem to the appropriate rectangle, and then using the relevant α 0 results, we have as claimed. 1 R 2π lim h(z)dz = 1 R εi R R 2π lim h(z)dz = 1 R R εi = 1 [ R εi 4π lim ˆ R R εi = 1 [ C G (k) + ( C G (k) R ε,g ) 2 2π lim R C G (z)e izk dz + ] Theorem 5.1 s final assertion is by continuity of C G1 and C G2. R εi R εi R+εi R+εi h(z) S G (z)dz ] Cˆ G (z)e izk dz = C G (k) R ε,g, 2 A single piece of numerical integration code (coupled with the appropriate R α,g adjustment) can evaluate, for example, all five formulas for payoff G 1 ; the only difference is the value of α passed into the procedure. Thus, without writing additional code, one gains the flexibility to choose, say, a negative or zero α if the integrand should happen to behave better there than it does along positive α. The extent to which an integrand is well-behaved can be quantified by the error bounds that arise from that particular choice of α. This is the subject of the next section. Also in the next section we give alternative proofs for many of the formulas in Theorem 5.1. The contour-shift proof, given above, has the purpose of unifying the various Fourier pricing formulas; but for the purpose of deriving error bounds, it will be useful to reinterpret the results. For example, our α < 0 bounds will exploit the equivalence between contour shifts and parity relations, such as put/call. 6 Bounds for Sampling and Truncation Errors The Fourier inversion (5.4) can be approximated discretely via an N-point sum with a grid spacing of in the Fourier domain. This quadrature introduces two forms of error (aside from roundoff error): truncation error because the upper limit of the numeric integration is finite, and sampling error because the integrand is evaluated numerically only at the grid points. Our bounds will account for both sources of error. The total error is defined as the absolute difference between the true value C G (k) = R α,g + e αk π and the discrete approximation given by the N-point sum 0 Re [ e iuk ĉ α,g (u) ] du Σ N (k) := Σ N, α,g (k) := R α,g + e αk [ N 1 π Re ĉ α,g ((n + 1/2) )e ]. i(n+1/2)k (6.1) n=0 14

15 The total error is bounded by the sum of the sampling error and the truncation error C Σ N C Σ + Σ Σ N, where Σ is defined as Σ N is, except with an infinite upper limit of summation. Truncation errors can be bounded by a formula that applies regardless of the sign of α. Sampling errors, however, will require treatment that depends on the sign of α. Our strategy is based on Davies (1973), but he restricts attention to the inversion of characteristic functions to recover probabilities, which is not always appropriate for us; we extend his approach to the inversion of option price transforms. 6.1 Truncation Error Carr-Madan and Pan each suggest bounds on the tails of certain Fourier inversion integrals, but our specific need is to bound the tails of the infinite discrete sums that approximate our Fourier integrals. Theorem 6.1. Assume the hypotheses of Theorem 5.1. If f is such that ĉ α,g decays as a power ĉ α,g (u) Φ(u)/u 1+γ for all u > u 0, where γ γ α,g > 0 and Φ(u) Φ α,g (u) is decreasing in u, then the truncation error provided that N > u 0. Σ (k) Σ N (k) Φ(N ) πe αk γ(n ) γ, (6.2) If f is such that ĉ α,g decays exponentially ĉ α,g (u) Φ(u)e γu for all u > u 0, where γ > 0 and Φ(u) is decreasing in u, then the truncation error provided that N > u 0. Proof. In any case, In the power decay case, n=n Σ (k) Σ N (k) Σ (k) Σ N (k) e αk π ĉ((n + 1/2) ) n=n where the middle step uses the convexity of 1/x 2. In the exponential decay case, as claimed. n=n ĉ((n + 1/2) ) Φ((N + 1/2) ) Φ((N + 1/2) ) 2πe αk+γn sinh(γ /2), (6.3) ĉ((n + 1/2) ). n=n Φ(n ) Φ(N ) [(n + 1/2) ] γ+1 γ N n=n 15 e γ(n+1/2) dx Φ(N ) = xγ+1 γ(n ) γ, Φ((N + 1/2) ) e γ(n+1/2) e γ(n 1/2),

16 Remark 6.1. As observed by Carr and Madan in case G = G 1, and as one can verify also in case G = G 2, the power decay hypothesis always holds; specifically, let γ = 1 and Φ = f ( (α + b 1 )i). However, the resulting bound is typically poor. For practical purposes it is desirable to improve the power γ or to establish exponential decay, by factoring in the contribution from the large-u decay of f (u (α + b 1 )i). The nature of this decay presents itself in the explicit expression for f ; examples appear in Appendix A. Remark 6.2. The requirement that N > u 0 can be dropped, by modifying the right-hand sides of (6.2) and (6.3) as follows: first replace each N by u 0 (thus bounding the n u 0 / terms of truncation error). Then add a second term, to bound the N n < u 0 / terms of the truncation error, by integrating, over the appropriate finite interval, a bound on ĉ(u) valid for u < u 0, such as the quadratically decaying bound of Remark Sampling Error: Positive α A form of the aliasing effect is at work here; by sampling ĉ only at regular discrete intervals, one recovers not c but rather a periodic function equal to a combination of c and infinitely many shifted copies of c. The unwanted copies are shifted farther away as 0, so the extreme-strike bounds of Section 3 come into play. In the main text, our sampling error analysis will focus on the payoff classes of greatest practical interest, G 1 and G 3. For sampling error in cases G 2 and G 4, see Appendix B. Theorem 6.2. Assume that b 1 A X and αb 0 + b 1 A X with α > 0. In case G = G 1 we have C G1 Σ α,g 1 inf p>α: p+1 A X [ e 2πα/ ( ) f ( i) 1 e 4πα/ + e2π(α p)/ f ( i(p + 1)) p p ] (p + 1)e pk (1 e 4π(α p)/. ) p + 1 In case G = G 3, assume also that ĉ(u) = O(u 1 γ ) as u, where γ > 0. Then [ e C G3 Σ 2πα/ ] f ( ib 1 ) α,g 3 inf p>α: pb 0 +b 1 A X 1 e 4πα/ + e2π(α p)/ f ( i(pb 0 + b 1 )) e pk (1 e 4π(α p)/. ) Proof. For any > 0 and any positive integer j, c(k 2π j/ ) + c(k + 2π j/ ) = 1 π = 2 0 e iuk ĉ(u)cos(2π ju/ )du F(u) cos(2π ju/ )du, where F(u) := 1 2π ĉ(u + n )e i(u+n )k. n= 16

17 Since F is Lipschitz, the Fourier cosine series may be summed: [ ] c(k) + c(k 2π j/ ) + c(k + 2π j/ ) cos(2π ju/ ) = F(u). j=1 In particular, taking u = /2, we have c(k) F( /2) = j=1( 1) j[ c(k 2π j/ ) + c(k + 2π j/ )]. Multiplying by exp( αk) to undamp the call prices, C(k) Σ (k) = ( 1) j[ e 2π jα/ C(k 2π j/ ) + e 2π jα/ C(k + 2π j/ )]. j=1 Therefore, Theorems imply that and C G1 (k) Σ (k) B 0 j=1 j odd C G3 (k) Σ (k) B 0 j=1 j odd The results follow from computing the sums. [e 2π jα/ Ee X + e2π j(α p)/ Ee (p+1)x (p + 1)e kp ( p ) p ] p + 1 [e 2π jα/ Ee b 1 X + e2π j(α p)/ Ee (pb 0+b 1 ) X e kp (6.4) ]. (6.5) Note that application of these bounds does not require actual computation of infimums. For example, in case G = G 1, any choice of p > α with p + 1 A X produces a valid upper bound, which is subject to improvement by taking more trial values of p. 6.3 Sampling Error: Negative α The Theorem 6.2 error bounds assumed that α > 0, and must be modified for α < 0. We have seen that shifting a Fourier inversion contour across a pole of the integrand changes the value of the integral. Sampling error bounds will now follow from the fact that the new integral value is the price of an contract related to the original option via a parity identity, such as put/call. Specifically, for each G = G 1,...,G 4, define one complementary payoff G, and for case G 1 define a second complementary payoff G by G 1(x,k) := min(exp(x),exp(k)) b 0 := 1,b 1 := 1,x R G 1 (x,k) := (exp(k) exp(x)) + b 0 := 1,b 1 := 1,x R G 2(x,k) := (k x) + G 3(x,k) := exp(b 1 x)i(b 0 x k) b 0 := 1,b 1 := 0,x R x R n G 4(x,k) := (b 2 x)exp(b 1 x)i(b 0 x k) x R n. 17

18 These payoffs have the following time-0 values. Theorem 6.3. Assume that b 1 A X and αb 0 + b 1 A X. In cases G = G 2,G 3,G 4 with α < 0; or in case G = G 1 with 1 < α < 0, we have C G (k) = 1 π αi 0 αi Re[ ˆ C G (k)e izk ]dz. In case G = G 1 with α < 1, this holds after replacing the G with G. Proof. Subtract from each original payoff function G its complementary payoff G ; then take expectations to verify the parity relation B 0 EG(X,k) B 0 EG (X,k) = R 0,G (6.6) and similarly for G. Theorem 5.1 now implies the result. Alternatively, without using Theorem 5.1, one may adapt Theorem 4.2 and compute directly the complex Fourier transforms of each C G. Inverting as in Theorem 4.3 finishes the proof. Moreover, the negative-α formulas in Theorem 5.1 would then follow from (6.6). This equivalence between contour shifts and parity relations allows us to control the negative-α sampling error by bounding the extreme-strike values of the complementary payoffs. In particular, we state explicitly the complementary bounds for cases G = G 1 and G = G 3. Theorem 6.4. In case G = G 1 we have C G 1 (k) B 0 e k and C G 1 (k) B 0 Ee X. In case G = G 1 we have, for any q > 0, In case G = G 3 we have, for any q > 0, C G (k) B 0Ee qx ( ) q q e (1+q)k and C 1 G q 1 + q (k) B 0e k. C G 3 (k) B 0Ee ( qb 0+b 1 ) X e qk and C G 3 (k) B 0 Ee b 1 X. Proof. Adapt the reasoning in Theorems 3.1 and 3.3. We omit the details. The sampling error bounds now follow. 18

19 Theorem 6.5. Assume that b 1 A X and αb 0 + b 1 A X. In case G = G 1, for α ( 1,0): C G1 Σ α,g 1 ek 2π(α+1)/ f (0) 1 e 4π(α+1)/ + e2πα/ f ( i) 1 e 4πα/. and for α < 1: [ e C G1 Σ k+2π(1+α)/ f (0) α,g 1 inf q> (α+1): 1 e 4π(1+α)/ + e(1+q)k e 2π(1+q+α)/ ( ) f (iq) q q ] (1 + q)(1 e 4π(1+q+α)/. ) 1 + q q A X In case G = G 3, assume also ĉ(u) = O(u 1 γ ) as u, where γ > 0. Then for α < 0, [ e C G3 Σ 2π(α+q)/ ] f ( i( qb 0 + b 1 )) α,g 3 inf q> α: e qk (1 e 4π(α+q)/ + e2πα/ f ( ib 1 ) ) 1 e 4πα/. qb 0 +b 1 A X Proof. Adapt the reasoning in Theorem 6.2. We omit the details. 6.4 Sampling Error: Zero α Here we bound the sampling error along contours that pass through a pole. This means α = 0 and, in case G = G 1, also α = 1. We present results for cases G = G 1 and G = G 3. In each case the option price function can be interpreted, after normalization, as a cumulative distribution function, so bounds from the probability literature apply directly, and we avoid reinvention of the wheel. Our proof strategy yields, as a by-product, complete alternative proofs of 3 of the 13 formulas in Theorem 5.1, including the (G = G 3 ; α = 0) case, which was Duffie-Pan-Singleton s (2000, Prop 2) pricing formula; their proof influenced ours but lacks the highly convenient normalization step. Theorem 6.6. Assume that b 1 A X and αb 0 + b 1 A X. Then the (α,g) {(0,G 1 ),( 1,G 1 ),(0,G 3 )} subcases of Theorem 5.1 all hold. In case G = G 1, the α = 1 and α = 0 sampling errors are bounded by [ C G1 Σ 1,G 1 max inf q>0: q A X [ C G1 Σ 0,G 1 max f (0)e k 2π/, ( ) f (iq) q q e (1+q)k 1 + q 1 + q e 2πq/, f ( i)e 2π/ f ( i(p + 1)) inf p>0: p+1 A X (p + 1)e p(k+2π/ ) In case G = G 3 assume also that ĉ(u) = O(u 1 γ ) as u, where γ > 0. Then C G3 Σ 0,G 3 inf max p>0: pb 0 +b 1 A X q>0: qb 0 +b 1 A X ] ( p p + 1 [ f (i(qb0 b 1 )) e q(k 2π/ ), f ( i(pb ] 0 + b 1 )) e p(k+2π/ ). ) p ]. 19

20 Proof. Case G = G 1, α = 1: On some probability space (Ω 1,P 1,F ) there exists a real-valued random variable Y with density ϕ(y) := e y E[e X I(X > y)]. It is easy to verify that C G 1 (k) = f (0)ek P 1 (Y < k), and that Y has P 1 -characteristic function f (u)/[ f (0)(1 iu)]. By the Gil-Pelaez (1951) formula, C G 1 (k) = f (0)ek ( π Davies (1973) now directly implies the sampling error bound 0 [ ] ) f (u)/ f (0) Re iu(1 iu) e iuk du = f (0)ek + 1 +i [ ] f (z i) Re 2 π 0+i iz(iz + 1) e izk dz. [ ] C G1 Σ 1,G 1 f (0)e k max P 1 (Y < k 2π/ ), P 1 (Y > k + 2π/ ). So for any q > 0, [ C G1 Σ B0 Ee qx 1,G 1 max 1 + q ( ) q q e (1+q)k ] 1 + q e 2πq/, e 2π/ B 0 e X, as claimed. Case G = G 1, α = 0: On some probability space (Ω 1,P 1,F ) there exists a real-valued random variable Y with density ϕ(y) := e y P B (X > y) f (0)/ f ( i). It is easy to verify that C G1 (k) = f ( i)p 1 (Y > k), and that Y has P 1 -characteristic function f (u i)/[ f ( i)(iu + 1)]. By the Gil-Pelaez formula, By Davies (1973), ( 1 C G1 (k) = f ( i) [ ) f (z i)/ f ( i) Re e ]dz izk. π 0 iz(iz + 1) [ ] C G1 Σ 0,G 1 f ( i)max P 1 (Y < k 2π/ ), P 1 (Y > k + 2π/ ). So for any p > 0, [ C G1 Σ 0,G 1 max f (0)e k 2π/, B 0 Ee (p+1)x ( ) p p ] (p + 1)e p(k+2π/ ). p + 1 as claimed. Case G = G 3, α = 0: 20

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