JFE Online Appendix: The QUAD Method

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1 JFE Online Appendix: The QUAD Method Prt of the QUAD technique is the use of qudrture for numericl solution of option pricing problems. Andricopoulos et l. (00, 007 use qudrture s the only computtionl engine ; in this concluding pper second type of engine is dded, using closed-form pproximtions for trnsition density functions. However, in ll three ppers, qudrture is used in the sme wy. In this ppendix we outline the technique. The intuition behind QUAD is tht lthough Eqution ( is not vlid cross ll points in time, the vlution problem cn be sliced into consecutive time intervls, during which it is loclly pplicble. V (x, t = A(x B(x, yv (y, t + tdy, ( Imposition of the pproprite option fetures t their corresponding observtion times provides link between these consecutive intervls, nd solution of complex problems becomes possible nd is, by the nture of the technique, extremely fst. Any of the common qudrture methods is pplicble, including higher-order schemes nd Gussin qudrture. The lgorithm is redily modulrized to permit esy interchnge of qudrture engines. Clcultion is no more complicted thn clculting the option vlue using tree or finite difference but in QUAD the contribution t the previous timestep comes from mny nodes representing different levels of the underlying sset not just two or three nd, unlike trees nd finite difference grids, clcultions re not required in regions between observtion times. Qudrture methods solve n integrl eqution of the form shown in Eqution (. These re widely described in Mthemtics textbooks. We demonstrte for Simpson s rule, which is frequently the superior choice s the numericl engine due to its robustness, simplicity nd fst convergence. Simpson s Rule for some integrtion b f(xdx is: b f(xdx = +h f(xdx h(f 0 + 4f + f where h = b

2 This cn be derived (nd the error bounds nd rte of convergence determined by using qudrtic interpoltion. Consider Lgrnge interpolting polynomil: (x h(x P (x = ( + h h( + h f 0 (x h(x + ( + h h( + h f + (x h(x h ( h( h f Arrnging nd simplifying the polynomil, we find: P (x = Integrting nd simplifying: ( x h f 0 f + f + x ( h ( + hf 0 + ( + hf ( + hf + ( h ( + h( + hf 0 ( + hf + ( + hf for some ɛ [, b]. b f(xdx = +h P (xdx = h(f 0 + 4f + f 90 h5 f (4 (ɛ We cn lwys split the region of integrtion into n smller regions, pproximte the integrnd in ech of the n regions nd sum them bck together. This gives the composite Simpson s rule: +h f(xdx = h(f 0 + 4(f + f f n + (f + f f n + f n n 90 h5 f (4 (ɛ ( where n = b h

3 We observe tht the error is bounded by: h 4 (b mx f (4 (ɛ 80 ɛ [,b] nd hs convergence rte of order h 4. Other numericl schemes nd their error bounds cn be derived in similr fshion. A generl formul for QUAD under one dimensionl integrtion is expressed s follows: Given step size δx = b N, b f(xdx δx q f(x where f(x is {N + }-dimensionl vector with its ith term defined s: f i (x =f( + (i δx i [, N + ] nd q is {N + }-dimensionl weighting vector. For exmple, q S under Simpson s rule for even N is: 4 q S =. 4 (

4 For comprison, under the Trpezoidl rule (see Mthemtics texts, q T =. (4 In order to integrte vi qudrture, the integrnd should hve continuous derivtives. The first derivtive of the pyoff on cll option is clerly discontinuous t the strike price. If this discontinuity is voided, convergence will be smooth. We split the integrtion rnge into two smller prts with boundry defined on the discontinuity; the derivtives of the integrnd re continuous in both prts of the integrtion rnge: < y < 0 nd 0 < y < +. For Europen cll option, clerly the vlution problem is reduced to evluting the integrl corresponding to the rnge 0 < y < +. For other options this discontinuity my occur in other wys; for brrier option the discontinuity is in the pyoff function itself t the brrier, creting degree of nonlinerity error, nd for n Americn put it occurs in the second derivtive t the free boundry. If the discontinuities re not tken into ccount, then convergence will not be smooth (though resonble solutions my still be hd if discontinuities re not precisely locted, in quick nd dirty progrmming!. The loction of discontinuities in the pyoff function will be known priori for some clsses of options (such s vnill cll or put option or discrete brrier option. For other clsses of options (such s Bermudn option the position of the discontinuity is clculted t every observtion time vi Newton-Rphson itertion, which converges rpidly. Options tht re wkwrd to del with using other lttice/grid techniques re hndled vi QUAD with reltive ese (exmples: lookbck options in three dimensions nd moving brrier options becuse qudrture points cn be esily nd precisely plced. Prcticlly, the infinite integrtion rnge needs to be truncted but trunction error quickly tends to zero with incresing multiples of stndrd devition of the underlying sset. If the integrtion rnge of y is set corresponding to the movement of sset price within ten stndrd devitions wy from initil price, the trunction error is smll enough to be neglected under 4

5 even the most extreme σ setups. The integrtion rnge is then reduced to 0 < y < ξσ τ + x 0 where ξ denotes the number of stndrd devitions considered. With the integrnd nd integrtion rnge properly set up, s bove, the vlution problem cn be solved by Eqution ( s Nδy V (x, t A(x 0 A(xδy f(x, ydy ( N f(x, i=0 N + f (x, iδy + f(x, Nδy where number of steps, N, nd step size, δy, re defined s: i= f (x, (i + δy N = ξσ τ + x 0 δy (5 Consequently, the error term is bounded by: δy 4 80 (ξσ τ + x 0 mx ɛ [0,Nδy] f (4 (ɛ which quickly converges to zero s δy becomes smll, with rte of convergence of four. Richrdson Extrpoltion cn be used to improve the rte of convergence further: V ext = δyd V δy d V δy d δyd where d = 4 is the rte of convergence of Simpson s rule, nd V nd V re prices evluted with corresponding step sizes δy nd δy. 5

6 The Greeks Clcultion of Greeks is strightforwrd, vi finite differencing on option vlues for severl neighboring vlues of x nd v; for exmple: V (x + δx, v V (x δx, v V x (x, v δx V (x, v + δv V (x, v δv ν V v (x, v δv V (x + δx, v V (x, v + V (x δx, v Γ V xx (x, v δx V (x, v + δv V (x, v + V (x, v δv Vomm V vv (x, v δv Vnn V xv (x, v V (x + δx, v + δv V (x + δx, v δv V (x δx, v + δv + V (x δx, v δv 4 δx δv for smll δx nd δv ( such tht δx x nd δv v. Other first nd second order Greeks cn lso be clculted in the sme fshion. QUAD with Pth-Dependent Options A discrete pth-dependent option is n option whose pyoff function is dependent not only on the price of the underlying t the exercise dte but lso on the price of the underlying t discrete points in time before exercise. Consider pth-dependent option tht mtures t time T. Suppose tht the option nd its underlying re monitored M+ times with M intervls t m from present time t to mturity, such tht: T = t + M t i The vlution of this option is divided into evlution of M seprte options with mturity t + k+ i= t i t time t + k i= t i integer k [0, M. Denote these observtion points before mturity s: t k = t + i= k t i i= 6

7 Working bckwrds in time with known finl conditions t mturity, the vlue of option V M (x, t M with mturity T t t M cn be priced for the entire rnge of underlying prices, x. Together with conditions imposed t this observtion point, these option prices cn then be used s finl conditions for the vlution of option V M (x, t M with mturity t M t t M. This evlution process is continued until the vlue of option price V (x, t is found. The pricing of the option is effectively n M step multinomil tree with number of brnches for ech node equl to N + subject to discontinuous boundries, nd N is the number of QUAD steps defined in Eqution 5. Note tht in this multinomil tree the number of nodes does not chnge fter initiliztion t the first time step nd is not subject to discontinuity of the pyoff function. Therefore it does not give rise to the sw-tooth effect observed with tree-bsed models. In prctice, pth-dependent options re discretely monitored t discrete points in time nd the pyoff function of the option depends on discrete set of underlying prices rther thn the continuous pricing function. If the price of discretely monitored option V (M converges t rte d to the price of its theoreticl continuously monitored counterprt V s the number of observtion point M tends to infinity, then: V M = V + M d for some constnt. Richrdson extrpoltion cn be used for clculting option prices with higher observtion frequency: V M = M d (M d M d V M + M d (M d M d V M M d (M d M d M >M, M (6 Choice of K In Section (.4 of the min text, we introduce proxy for ccurcy, K, to llow comprisons between the vrious computtions. Here, we justify our pproch in more detil by deriving strictly correct mthemticl pproch which then leds to resonble proxy. Ech potentil QUAD engine hs n error upper bound O(δy d for n rbitrry order d. The computtionl complexity of the QUAD method for discretely pth dependent option 7

8 with M observtions to mturity, is of order O(M N b where N is the number of QUAD steps such tht: N = y mx y min δy b is defined by the dimension of the underlying of the option D. The number of plin QUAD methods used when D = is: N+ (N + + M ( i (N + N + This expression is polynomil of N up to second order. It is then useful to define vector s: ( g = N + N +... N N + N... N + N + We cn next formulte the computtionl complexity of D-dimensionl QUAD method s: (N + D + M ( G i,i,,i D G i,i,,i D where G i,i,,i D = g G i,i,,i D nd this expression is polynomil of N up to the order of D. For exmple, the QUAD method using locl voltility model on single sset would hve computtion complexity of O(M N, nd for stochstic voltility models would hve O(MN 4. If we construct the grid such tht the overll error term is independent of M, we re to force the following expression to be independent of M: f(n = ( (N + D + M ( G i,i,,i D G i,i,,i D O(δy d ( (N + D + M ( G i,i,,i D G i,i,,i D O(N d 8

9 we therefore choose our time step K s function, of M, S(M such tht M = S (K = constnt ( G i,i,,i D G i,i,,i D O(K d such tht M is polynomil of K with mximum order of D d nd minimum order of d. Now, this method of choosing our time step, lthough providing precisely built, uniformly dense grid cross the whole rnge of observtion M, dds unnecessry computtion cost to the lgorithm nd so we use proxy which mimics the dynmics of S(M. Note tht, for Simpson s rule with locl voltility model: S constnt (K = 4 O(δy + O(δy + O(δy 4 Conveniently, it then turns out in prctice tht there is no need to clculte explicitly the function S(M. Both log(m nd M cn be good proxy for S(M. QUAD in Multiple Dimensions Consider three dimensionl integrl in R cuboid spce y, y, y over domin y mx i y min i : I = = y min y mx Define following the functions: f(y, y, y dy dy dy ( ( y min y min y mx y mx f(y, y, y dy dy y i dy (7 nd G(y, y = y min y mx f(y, y, y dy (8 H(y = y min y mx G(y, y dy (9 9

10 such tht, I = y min y mx H(y dy (0 The integrtion problem (7 is equivlent to the three integrtions 8 to 0. QUAD cn be used recursively for ech integrtion to find G, H nd I. The D-dimensionl QUAD over D- orthotope domin is formulted s follows: Given qud steps in ech dimension s δy k = ymx k yk min N ( D I δy k Q i...i D F i..i D (y,, y D k= where F is (N+ D tensor of function vlue t qud points defined s:, the integrl cn be pproximted by: F i..i D (y,, y D =f(y + (i δy,, y D + (i D δy D i, i,, i D [, N + ] nd Q is (N +-dimension weighting tensor of order D such tht, if the weights is defined in one dimension s (N +-dimension vector q with its ith entry denoted by q i, the {i..i D }-th entry of Q is defined s: D Q i..i D = k= q ik i, i,, i D [, N + ] 0

11 For the purpose of illustrtion, the weighting mtrix Q S of Simpson s rule in -d cn be clculted from its -d weighting vector q S defined in Eqution ( s: Q S = q S q S = nd weighting mtrix Q T of Trpezoidl rule in two-dimension cn be clculted from its onedimension weighting vector q T defined in Eqution (4 s: Q T = q T q T = In fct, the numbers of QUAD steps N k for ech dimension k do not need to be consistent s long s they stisfy the conditions defined by the QUAD scheme, thus: N k is even for Simpson s qudrture. In this work, N k is set universlly cross dimensions in order to simplify the trcking of convergence rte. Agin, in order to remove ny non-linerity error, the integrtion domin is segmented long discontinuities of function f nd the remining error term of D-dimensionl QUAD tkes the form: D o(δyk d k= where d is the rte of convergence of the chosen QUAD scheme in one dimension. It is redily observed tht, for n rbitrry dimension D, the rte of convergence of ny QUAD scheme inherits its one-dimensionl vlue. Therefore (nd importntly, Richrdson s extrpoltion

12 cn lso be dpted in higher dimensions to improve the convergence of QUAD method. 04 Ding Chen, Hnnu J. Härkönen & Dvid P. Newton