NOT FOR REPRODUCTION. Expanded forward volatility. Modelling

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1 cutting edge. derivative pricing Expanded forward volatility Uing a one time-tep finite difference implementation, Jeper Andreaen and Brian Huge eliminate the arbitrage in the wing of the volatility mile that reult from mot expanion technique for local tochatic volatility model, including the widely ued SABR model Modelling the implied volatility mile uing local and tochatic volatility ha been the ubject of much reearch over the pat 2 year (ee, for example, Dupire, 1996, Hagan et al, 22, Heton, 1993, Jex, Henderon & Wang, 1999, Lewi, 2, and Lipton, 22). Interet rate option de typically need to maintain very large amount of interlined volatility data. For each currency, there might be 2 expirie and 2 tenor, that i, 4 volatility mile. Furthermore, the mile might be lined acro different currencie. Interpolation of oberved dicrete quote to a continuou curve i needed for the pricing of general cap and waption. At the ame time, extrapolation of option quote are needed for contant maturity wap (CMS) pricing. For thee purpoe, the indutry tandard ha been the SABR model uing expanion a in Hagan et al (22). The implied volatility expanion have the advantage that they are fat and imple to code. But the expanion are not very accurate, particularly not for long maturitie or low trie. Numerical example of thi can be found in Paulot (29). Thi i, however, largely irrelevant a the SABR expanion are generally only ued for the pricing of European option, and not, for example, for calibration of full dynamic model. The main practical problem with the expanion i that they imply negative denitie for low trie and occaionally alo for high trie. With the low rate we have today, thi problem i more acute than ever. Furthermore, the SABR model only ha four parameter to handle the above-mentioned ta, which i not enough flexibility to exactly fit all option quote. A in Balland (26) and Lewi (27), we extend the tochatic volatility proce to include a contant elaticity of variance (CEV) ew on the volatility of volatility. The CEV volatility proce allow u to have more explicit control of the extrapolated high-trie volatilitie, which in turn allow better control of CMS price. Further, we will ue a non-parametric volatility function for the pot proce, which enable u to have an exact fit to all the oberved quote and give u the ability to model negative option trie. Rather than going through heat ernel expanion a in Hagan et al (22), we follow Balland (26) and ue a hort-maturity expanion for the implied volatility of the option. The hortmaturity expanion alo yield reult for the hort-maturity limit of the Dupire (1996) forward volatility, that i, the hort-maturity limit of the conditional expected local variance: 2 = lim 2 / dt ( t) = ϑ E d t t The forward volatility i ued in a ingle time tep implicit finite difference dicretiation of the Dupire (1994) forward partial differential equation (PDE). Thi preclude arbitrage and o avoid negative denitie. We alo derive an adjutment of the forward volatility function to compenate for the pricing in one, rather than multiple, time tep. The ingle-tep finite difference grid generate all price in one go and thi can in turn be ued for calibrating the model directly to oberved CMS price. We provide two calibration procedure: an implicit method that wor by iteration of the connection from parameter to price in a non-linear olver, and a direct method that given an arbitrage-free continuou curve of option price directly infer the parameter of the model. We how that the implicit calibration method can be ued to fit the model to 1 dicrete trie in approximately one milliecond of CPU time. Finally, we how how we are able to control the wing of the mile by varying the functional form of the diffuion of the pot and volatility procee, and the impact thi ha on CMS pricing. Short maturity expanion Firt, we will outline the hort-maturity expanion. Our approach i imilar to that ued in Balland (26). We conider the model: d = zσ( )dw (1) dz = ε( z)dz where W and Z are Brownian motion with correlation r. The non-parametric form of the volatility function () allow u to have a perfect fit to any dicrete or continuou et of oberved arbitrage-free option quote. We can write the price of a European call option on a fixing (T) a: = E t ( ( T ) ) + c t ( ) = g t,( t),ν t where n(t) i the implied normal volatility and g i the normal (Bachelier) option pricing formula: g( τ,,ν) = Φ ν τ + ν τφ ν τ, τ = T t (2) Applying Itô lemma to (2) yield: dc = g τ dt + g d g d2 + g ν dν g νν dν2 + g ν d dν (3) ri.net/ri-magazine 11

2 cutting edge. derivative pricing where ubcript denote partial derivative. In the following, we aume n >. Define x = ( )/n. Uing Itô lemma yield: dx = 1 ν d ν 2 dν 1 d dν + 2 ν ν 3 dν2 = 1 ( d xdν ) ν + O( dt) dx 2 = 1 ( ν 2 d2 + x 2 dν 2 2xd dν) The normal option pricing function, g, ha the following propertie: g ν = ντg g νν = ν g ν = ν 2 g g = g τ ν2 g Uing the propertie in (5) we can tranform (3) to: + 2τνdν dc g d = 1 2 g ν 2 dx 2 dt (6) The left hand ide of (6) i the change in value of a hedged portfolio. Taing conditional expectation yield: = 1 2 g ν2 E t dx 2 dt + g τνe t dν (4) (5) [ ] (7) A g > for n >, and for any diffuion, E t [dx 2 dt] = i equivalent to dx 2 = dt, we obtain the condition: + 2 τ ν E t dν = dx 2 dt [ ] (8) For mall maturitie, t, we arrive at the arbitrage condition: σ 2 x dx2 = 1 (9) dt Note that thi i a diffuion condition rather than the drift condition that we normally ee in financial mathematic. A x mut be a function of the tate variable (, z), the diffuion condition (9) lead to the differential equation: 2 / dt (1) = z 2 σ( ) 2 x 2 + ε( z) 2 x 2 z + 2ρzσ( )ε( z)x x z 1 = σ x 2 = x d + x z dz Given the function, e, we need to olve thi non-linear firtorder differential equation ubject to the boundary condition x( =, z) =. Once we have the olution x(, z), we can find the implied normal volatility a: ν = x,z (11) We note that the error of the implied volatility i O(t). The reult implie that for any choice of (), e(z) any function x = x(, z) that atifie dx 2 = dt lead to an implied volatility given by n = ( )/x. We could have choen to derive the hort-maturity expanion in implied Blac-Schole (lognormal) volatility n _ intead of implied normal volatility. Intead of x we hould then have choen the tranformation x _ = ln(/)/n _. The diffuion condition would be the ame o x _ = x. Thi relate hort-maturity implied lognormal and normal volatilitie by the imple relationhip: ν ν = ln / (12) The expanion reult that we preent in the following can eaily be witched between ue in implied normal and implied lognormal volatility form by ue of the relation (12). Determinitic volatility Firt, we will conider the cae with e(z) =. In thi cae, z 1, and the differential equation (1) reduce to the ordinary differential equation (ODE): x 2 σ( ) 2 = 1 (13) Uing the boundary condition x( = ) =, we find the olution: 1 x = σ( u) du (14) with correponding implied normal and Blac volatilitie given by: ν = ν = σ u 1 du 1 du ln / σ u (15) Thee reult appear in many place, for example in Anderen & Ratcliffe (22). We note that (14) implie the following relationhip between x and the forward volatility: = 1 (16) σ( ) Suppoe we have x from a tochatic volatility model lie (1), that i, given a the olution to (1) for ome volatility function (), e(z) and correlation r. Define the function ϑ by: = ϑ and conider the determinitic volatility model: It now follow that: 1 (17) d = ϑ( )dw (18) 1 x LV ϑ u du = x (19) So the tochatic volatility model (1) and the local volatility model (18) will produce the ame hort-maturity expanion option price. The above i a hort-maturity limit verion of the general reult by Gyongy (1986) and Dupire (1996) that the model: d = a( t, )dw, ( ) = ( ) (2) 12 Ri January 213

3 1 Short maturity expanion implied volatility 7 Hagan implied volatility 6 ZABR Blac implied volatility 5 ZABR normal implied volatility Note: implied Blac volatility for our hort maturity expanion in Blac (red) or normal volatility (green) compared with Hagan formula (dotted blue) produce the ame option price a the model (1) if a( ) i choen to be: 2 = E d( t) 2 / dt ( t) = a t, (21) We conclude that in the hort-maturity limit, the conditional expected variance of the underlying i related to the tranformed variable x by: 2 = lim ϑ t E d t 2 / dt ( t) = = 2 (22) Thi contitute a way of relating the two dimenional pricing problem (1) to the impler one-dimenional pricing problem (18). We will mae ue of thi relationhip to generate arbitrage-free price later in thi article. The SABR model Here, we will rederive the main reult of Hagan et al (22) by olving the diffuion condition for the lognormal volatility proce cae, e(z) = e z. Firt, we ue the tranformation: y = 1 1 z σ( u) du (23) to get: dy = dw εydz + O( dt) = 1+ ε 2 y 2 2ρεy 1/2 db + O ( dt ) (24) J ( y)db + O( dt) where B i a new Brownian motion. A y( = ) =, we can now get x by normaliing the volatility of y, hence: x = y 1 J u ν = x ν = ln / x ρ + εy du = 1 ε ln J y 1 ρ For the CEV cae () = b, we have: (25) 2 Comparion of (.) and ϑ(.) Volatility (%) 4.5 σ 4. ϑ Note: volatility σ() = (blue) compared with the forward volatility ϑ() = J(y)zσ() (red) for the cae of ε =.47, ρ =.48 y = 1 1 β 1 β zσ 1 β (26) Thee formula are baically the reult of Hagan et al (22). Thi i extended to include maturity and variou refinement for the CEV cae. The Hagan reult doe, however, produce implied volatility mile that are virtually identical to thoe produced with formula (25). In figure 1, we compare the Hagan expanion with (25). To be precie, the Hagan expanion ued here and in the following i (2.17) in Hagan et al (22). We can ue (25) to retrieve the forward volatility function of SABR from: = = 1 J ( y) 1 1 z σ( ) (27) Hence: ϑ( ) = J ( y)zσ( ) (28) Thi reult can alo be deduced from reult in Dout (21). Figure 2 compare the function () to the forward volatility function ϑ(). The ZABR model Next, we conider the extended SABR model where the volatility proce i of the CEV type e(z) = e z g. Again, we will introduce an intermediate variable: y = z γ 2 1 σ( u) du (29) For which Itô expanion yield: dy = z γ 1 ( dw + ( γ 2)εydZ) + O( dt) (3) Define x = z 1 g f(y), for ome function f(y), and we get: dx = z 1 γ f = f ( y)dy + ( 1 γ )εf ( y)dz + O( dt) ( y)dw + ( γ 2)εy f ( y) + ( 1 γ )εf ( y) dz + O dt (31) We conclude that the diffuion condition (8) i atified if f olve the ODE: ri.net/ri-magazine 13

4 cutting edge. derivative pricing 3 Control of high-trie behaviour 4 Problem of negative denity 1 γ = 9 γ =.5 8 γ = 1. 7 γ = γ = Note: implied Blac volatility mile for different choice of γ. Other parameter the ame a in figure 1 1 = A( y) f ( y) 2 + B( y) f ( y) f ( y) + Cf ( y) 2 = 1+ ( γ 2) 2 ε 2 y 2 + 2ρ( γ 2)εy = 2ρ( 1 γ )ε + 2( 1 γ )( γ 2)ε 2 y C = ( 1 γ ) 2 ε 2 = A y B y f The ODE (32) can be rearranged a: = B( y) f + B y f y 2 f 2 4A y 2A y Cf 2 1 F y, f (32) (33) which can be olved by tandard technique for the integration of ODE. We can evaluate the olution for all trie in one weep by: = zγ 2 σ( ) 1 = f z1 γ = z1 γ f ( y) = z 1 F( y,z γ 1 x)σ x = = y( = ) = Again, we can find the forward volatility function a: = ϑ 1 = zσ f ( y) 1 = zσ 1 (34) 1 (35) Equation (34) and (35) will typically be evaluated at z = z() = 1. Rather than numerically olving the two ODE in (34) eparately, we favour olving (32) a a joint ytem. Increaing g lift the wing of the implied volatility mile wherea the implied volatility mile for trie cloe to at-themoney are virtually unaffected. Thi i illutrated in figure 3. Thi can in turn be ued to give u better control over the CMS price. It hould here be noted that the ODE repreentation (32) ha previouly been obtained by Balland (26) for the lognormal Blac-Schole implied volatility 2.1 Implied probability denity Strie Note: denity (red) and implied Blac-Schole volatilitie (blue) from the Hagan expanion. Parameter a in figure 1 Probability denity cae () =. Further, it hould be noted that Henry-Labordère (28) ha a treatment of the general non-cev cae e = e(z). For quic identification of the model parameter, the following econd-order Taylor expanion i convenient: = ν( ) + ν ( ) ( ) ν ( ) = zσ( ) = 1 2 zγ 1 ρε + z σ ( ) ν ν ν = ν 1 6zσ (ρ 2 + 2)ε 2 z2γ γ +z ( 2 2σ( ) σ ( ) σ ( ) 2 ) ( 3 ) 2 + O For the CEV cae () = w ( ) b /( ) b and z = 1 we have: = ω ν = 1 2 ρε + ω ν ν = 1 1 β ( ( 6ω 5 + 2γ )ρ2 + 2)ε 2 + ω 2 2 β( β 2) (36) (37) For a given et of dicrete quote n^( 1 ),..., n^( n ), the Taylor expanion (37) can be ued for regreing the triple n(), nʹ(), nʹʹ(). One can in turn olve (37) to get parameter etimate for b, r, e. Finite difference volatility Uing the implied volatility coming from the hort-maturity expanion, (15), (25) and (34), directly for pricing uing (2) will not give arbitrage-free option price. Our hort-maturity expanion uffer from the ame problem of potential negative implied denitie for low trie a the original Hagan expanion. In figure 4, we have plotted an example of the implied volatilitie and the implied denity coming from the Hagan expanion. To avoid thi problem, we will intead ue the forward volatilitie derived in (28) and (35) a the bai for our pricing. The forward volatility ϑ() can be ued to generate option price a the olution to the Dupire (1994) forward PDE: 14 Ri January 213

5 c t ( t,) = 1 2 ϑ( )2 c ( t,) (38) c(,) = ( ) + The uual way of olving thi numerically i to et up a time dicretiation with multiple time tep and then ue a finite difference olver. However, to gain peed we will intead ue the ingle time tep implicit finite difference approach introduced in Andreaen & Huge (211). Here we need to olve the ODE: 1 2 tθ c t, 2 c t, = ( ) + (39) In Andreaen & Huge (211) it i hown that thi approach generate a et of arbitrage-free call price for any choice of q. It i alo hown that the one-tep finite difference price i the Laplace tranform of the olution to (38). The Laplace tranform of the Gauian ditribution i the Laplace ditribution: e t/t 1 ν t φ ν t dt = T 2ν 2 e T 2ν 2 (4) which i peaed at =. Therefore if we chooe q = ϑ we will alo get a pea in the denitie. Intead, we will find an adjutment for the forward volatility function baed on our expanion reult. A option price generated by (38) and (39) hould be the ame, we can ubtitute c = 2c t /ϑ 2 from (38) into (39) and rearrange to find: 2 = ϑ( ) 2 c( t,) ( ) + tc t ( t,) ϑ( ) g( t,,ν) 2 g( t,,ν) / t = ϑ( ) ξ Φ( ξ) φ( ξ) θ ϑ ( )+ / t 2 P( x) 2, ξ = x t 1/2 (41) where the econd (approximate) equality involve approximating the option price by our expanion reult. The function P(x) 2 can conveniently be approximated with a third- or fifth-order polynomial. Specifically: / φ( x) a n u n Φ x n, u = 1 / 1+ px (42) where the contant p, a 1, a 2,... can be found in ( ) and ( ) of Abramowitz & Stegun (1972),. The finite difference dicretiation of (39) i: tθ( )2 δ c t, = + (43) where d i the econd-order difference operator. Thi equation can be repreented a a tridiagonal matrix equation on the grid {, 1,..., n }, which in turn can be olved for {c(t, i )} in linear CPU time uing the tridag() algorithm in Pre et al (1992). 5 Probability denity and forward volatility Hagan expanion implied denity Implied denity without forward volatility adjutment.3 Implied denity with forward volatility adjutment.4 Note: the denity with (green) and without (red) the forward volatility adjutment compared with the implied denity of the Hagan Probability denity A an alternative to the finite difference olution (43), one could ue the exact olution methodology for the ODE of the type (39) decribed in Lipton & Sepp (211). However, for thi methodology to be computationally effective the forward volatility function q() need to be well approximated by a piecewie linear function with few not point over the full domain of the olution. Thi i generally not the cae here, a can be een in figure 2. We have therefore choen to bae our olution on (43). In figure 5, we have plotted the denity both with and without the forward volatility adjutment. For reference, we have alo plotted the implied denity from the Hagan expanion. We ee that the finite difference generated option price have correponding implied denitie that are poitive, that i, arbitrage i precluded. We alo ee that uing our forward volatility reult, ϑ(), directly in the ingle time tep finite difference olver produce a denity that i peaed around at-the-money. Thi, however, i eliminated when uing the adjuted forward volatility q(). Calibrating the volatility function Firt conider the cae where we have a continuou curve of arbitrage-free option price. Thi could for example be produced by the Andreaen & Huge (211) interpolation cheme or come from another ZABR model. Calculate the forward volatility function by the dicrete Dupire equation: 2 = 2 c( t,) tδ c( t,) θ + Uing (35) we can calibrate the volatility function: σ θ = F y,zγ 1 x zp( x) (44) (45) where x, y are found from (34) a the olution to the ODE ytem: = z γ 1 P x θ( ) = P x θ y = = x( = ) = The ODE ytem (46) can be olved for all trie in one weep. (46) ri.net/ri-magazine 15

6 cutting edge. derivative pricing 6 Low-trie behaviour 8 Variation with g Note: implied Blac volatilitie from the ZABR model (red) that i calibrated to the Hagan expanion mile (blue) for trie in [.2,.6]. Hagan parameter a in figure 1 7 Poitive probability of negative rate Probability denity However, typically, we prefer to calibrate directly to the oberved dicrete quote. Thi i done by olving the ODE in (34) and (35) and including the one-tep finite difference adjutment (41): = zγ 2 σ ( γ 1 x) = F y,z zσ( ) θ( ) = P( x)zσ x = = y( = ) = Hagan expanion mile ZABR mile 6 Hagan denity 5 Denity, β =.5 4 Denity, β = Note: denitie for Hagan (blue) and two ZABR model with β =.5 (red) and β = 1 (green). The ZABR model are calibrated to the Hagan expanion price for trie in [.2,.6]. Parameter a in figure 1 (47) Hagan mile ZABR () mile ZABR (.5) mile ZABR (1.) mile ZABR (1.3) mile ZABR (1.6) mile Note: calibration reult for different choice of γ. All model calibrated to Hagan expanion price for trie in [.2,.6]. Parameter a in figure 1 9 Convexity adjutment dependence on g Convexity adjutment (%) Hagan convexity adjutment ZABR convexity adjutment γ Note: convexity adjutment for a CMS forward 1 1 for different choice of γ in ZABR model (red) compared with Hagan (blue). ZABR model calibrated to Hagan expanion option price for trie in [.2,.6] After numerical olution of (47), we find the option price uing the one-tep finite difference algorithm in (43). On top of thi, we ue a non-linear olver to calibrate the volatility function () to oberved dicrete option quote. A we get all option price in one weep, we can include CMS forward and option quote in the calibration without additional computational cot. Even though non-linear iteration i involved, thi procedure i very fat. Typically, we can calibrate a non-parametric volatility function with 1 not point to a given mile in roughly 5 iteration, which tae approximately one milliecond of CPU time. When it come to outright pricing peed, the ZABR model i capable of generating 1, mile each coniting of 256 trie in approximately even econd. It hould be treed that thi include both numerical ODE and finite difference olution. Thi i actually fater than direct ue of Hagan SABR expanion, which tae 1 econd for the ame ta. The reaon for thi difference i mainly that one time-tep finite difference i fater at producing price than the Blac formula. An alternative to the ZABR model for producing arbitrage-free option price i the Fourier-baed model found in, for example, Lipton (22). For a diplaced Heton (1993) model, numerical olution for 1, mile coniting of 256 trie via the fat Fourier tranform with the Blac-Schole formula ued a a control variate tae around 18 econd (ee Andreaen & Anderen, 22). It hould be noted that thi type of model i coniderably le flexible with repect to fitting dicrete quote and more difficult to implement. Though we generally ue (47) in conjunction with a non-linear olver for the calibration, the direct calibration methodology (46) i relevant a it admit direct calibration of one ZABR model to another. 16 Ri January 213

7 The tochatic proce x ha unit diffuion and thu, in the ene of the hort-maturity limit, i normally ditributed. So it i natural to ue a uniform pacing in x and a non-uniform pacing in. For thi, the ODE ytem (47) can conveniently be tranformed to: = z γ 1 = zσ θ( ) = P( x)zσ( ) y x = =, ( x = ) = (48) In our implementation, we olve (48) on a uniform x grid to generate and fix a non-uniform trie grid {, 1,..., n } that i ued in the numerical olution of (47) during calibration and pricing. A a final remar, we note that ODE in thi ection typically will be olved at z = z() = 1. Controlling the wing Here, we will give a few example to illutrate how we are able to control the behaviour of the wing of the mile. Conider a model with: = ω( ) ( ) β (49) σ where w i a non-parametric curve and i the lower bound of the pot proce. For b < 1, we have aborption at and for b 1 the barrier i unattainable. Our finite difference olution impoe aborption for the cae where the barrier i attainable. In figure 6, we fit thi model to Hagan price for b =.5 and b = 1 and = Reference Abramowitz M and I Stegun, 1972 Handboo of mathematical function Dover Publication, New Yor Anderen L and R Ratcliffe, 22 Extended Libor maret model with tochatic volatility Woring paper, General Re Financial Product Andreaen J and L Anderen, 22 Volatile volatilitie Ri December, page , available at Andreaen J and B Huge, 211 Volatility interpolation Ri March, page 86 89, available at Balland P, 26 Forward mile Preentation, ICBI Global Derivative Dout P, 21 No arbitrage SABR Woring paper, Royal Ban of Scotland Dupire B, 1994 Pricing with a mile Ri January, page 18 2, available at Dupire B, 1996 A unified theory of volatility Woring paper, Banque Pariba Fiedler M, 1986 Special matrice and their application in numerical mathematic Martinu Nijhoff, Dordrecht Gyongy I, 1986 Mimicing the one-dimenional marginal ditribution of procee having an Ito differential Probability Theory and Related Field 71, page Hagan P, D Kumar, A Leniewi and D Woodward, 22 Managing mile ri Wilmott Magazine, September, page and we ee that the fit i good for poitive trie. In figure 7, we have plotted the reulting denitie. A before, the Hagan expanion produce negative denitie for low poitive trie. For b =.5, we have aborption at the barrier and for b = 1 we ee that the denity below zero i pread out. We now ue the model with b = 1 to illutrate the effect of g. For different level of g, we have calibrated the model to the Hagan expanion price for trie between.2 and.6. In figure 8, we ee that all model are well calibrated in the ene that the model all produce the ame mile for trie between 2% and 6%. We can alo ee the bigget impact of varying g i for high trie. One way of fixing g i to chooe it to match CMS forward or option quote. In figure 9, we have hown the impact on a CMS convexity adjutment. Concluion We have ued a imple method to derive hort-maturity expanion for forward volatilitie from tochatic volatility model. The olution i an ODE that can be olved numerically for all trie in one weep including adjutment of the forward volatility function to compenate for the one-tep finite difference option pricing. Finally, we ue the one-tep finite difference cheme to generate option price. The approach i very fat and it generate arbitrage-free option price. We have added flexibility to the original SABR model to get an exact fit of all quoted option price and better control of the wing of the mile for improved CMS pricing. Alo we can add CMS price to the calibration without additional computational cot. n Jeper Andreaen i the head of and Brian Huge i chief analyt in the quantitative reearch department at Dane Maret in Copenhagen. They would lie to than colleague Morten Karlmar and Jeper Feringhoff-Borg for aitance with Taylor expanion. want.daddy@daneban.com, brian.huge@daneban.com Henry-Labordère P, 28 Analyi, geometry, and modeling in finance Chapman & Hall Heton S, 1993 A cloed-form olution for option with tochatic volatility with application to bond and currency option Review of Financial Studie 6, page Jex M, R Henderon and D Wang, 1999 Pricing exotic under the mile Woring paper, JP Morgan Lewi A, 2 Option valuation under tochatic volatility Finance Pre Lewi A, 27 Geometrie and mile aymptotic for a cla of tochatic volatility model Preentation, UC Santa Barbara Lipton A, 22 The vol mile problem Ri February, page 61 65, available at Lipton A and A Sepp, 211 Filling the gap Ri September, page 78 83, available at Paulot L, 29 Aymptotic implied volatility at the econd order with application to the SABR model Woring paper, Sophi Technology Pre W, B Flannery, S Teuoly and W Vetterling, 1992 Numerical recipe in C Cambridge Univerity Pre ri.net/ri-magazine 17