Efficient, Almost Exact Simulation of the Heston Stochastic Volatility Model

Transcription

1 A. van Haatrecht A.A.J. Peler Efficient, Almot Exact Simulation of the Heton Stochatic Volatility Model Dicuion Paper 09/ September 12, 2008

2 Efficient, almot exact imulation of the Heton tochatic volatility model A. van Haatrecht 1 2 and A.A.J. Peler 3. Firt verion: September 9, 2007 Thi verion: September 12, 2008 Abtract We deal with everal efficient dicretization method for the imulation of the Heton tochatic volatility model. The reulting cheme can be ued to calculate all kind of option and correponding enitivitie, in particular the exotic option that cannot be valued with cloed-form olution. We focu on to the (computational) efficiency of the imulation cheme: though the Broadie and Kaya (2006) paper provided an exact imulation method for the Heton dynamic, we argue why it practical ue might be limited. Intead we conider efficient approximation of the exact cheme, which try to exploit certain ditributional feature of the underlying variance proce. The reulting method are fat, highly accurate and eay to implement. We conclude by numerically comparing our new cheme to the exact cheme of Broadie and Kaya, the almot exact cheme of Smith, the Kahl-Jäckel cheme, the Full Truncation cheme of Lord et al. and the Quadratic Exponential cheme of Anderen. Keyword: Stochatic volatility, Simulation, Heton, Non-central chi-quared inverion, Control variate. 1 Introduction The behavior of financial derivative i uually modeled by tochatic differential equation that (jointly) decribe the movement of the underlying financial aet uch a the tock price, tock variance, interet rate or currencie. Though ome model yield cloed-form olution for certain derivative, the fat majority of the exotic option cannot be priced in cloed-form. Epecially for (forward) path-dependent option, the Monte Carlo approach yield a popular and flexible pricing alternative. Becaue of the increaingly computational power combined with the ue of modern day variance reduction technique, we expect Monte Carlo technique to become even more widely applicable in the near future. Since the introduction of the Black and Schole (1973) model and in particular ince the equity crah of the late eightie a battery of complex model ha been propoed to relax ome mipecification of the model. Though the Black and Schole (1973) model ha theoretical and practical appealing 0 The author would like to thank Frank de Jong and other participant at the Univerity of Tilburg/Netpar eminar erie for their comment and uggetion. 1 Netpar/Univerity of Amterdam, Dept. of Quantitative Economic, Roetertraat 11, 1018 WB Amterdam, The Netherland, a.vanhaatrecht@uva.nl 2 Delta Lloyd Inurance, Rik Management, Spaklerweg 4, PO Box 1000, 1000 BA Amterdam 3 Netpar/Univerity of Amterdam, Dept. of Quantitative Economic, Roetertraat 11, 1018 WB Amterdam, The Netherland, a.a.j.peler@uva.nl 1

3 propertie, mot of it aumption, like contant volatility or contant interet rate, do not find jutification in the financial market; one cla of model relaxe the contant volatility aumption and incorporate a financial phenomena know a volatility clutering, i.e. they make volatility tochatic. Within thi cla are the tochatic volatility model of Hull and White (1987), the Stein and Stein (1991) and the Schöbel and Zhu (1999) model. However by far the mot popular model tochatic volatility model i the Heton (1993) model, mainly caued by the fact that thi model, until the introduction of the Schöbel and Zhu (1999) model, wa the only tochatic volatility model that allowed for flexibility over the leverage effect, yet alo yielded a cloed-form olution for call/put option in term of one numerical integral 4. With uch a cloed-form olution the computation of vanilla European option price can be done in fat and table fahion, hence allowing for efficient calibration to market option data. Literature review Depite the fact that the Heton model wa already introduced in 1993, there ha been relatively little reearch on efficient dicretization method of it continuou time dynamic. Thi i in particularly remarkable if one conider that mot practical application of uch model, e.g. the pricing and hedging of path-dependent derivative, practically alway involve Monte carlo method. Only recently a few paper on efficient dicretization method appeared; firt of all a bia-free (dicretization) method wa introduced in Broadie and Kaya (2006), who developed a cheme that could imulate the Heton proce (i.e. tock and variance) from it exact ditribution. Though the paper i elegant, it practical ue i unfortunately rather limited: firt the variance proce i imulated baed on an acceptance-rejection method, and econdly the algorithm require Fourier inverion of the conditional characteritic function of the integrated variance proce. Next to the fact the inverion i time-conuming, it i alo complex and can lead to numerical error (e.g. truncation). Moreover the ue of acceptance and rejection ampling will cramble random path when parameter are perturbated, the algorithm reult in a low correlation in pre- and pot perturbation path and hence introduce large monte carlo bia in enitivity analyi (e.g. ee Glaerman (2003)). For thi reaon (popular) low-dicrepancy number cannot not be applied in conjunction with the Broadie and Kaya (BK) cheme. Lord et al. (2008) conider different Euler cheme, in particular they invetigate how to deal with negative value of the variance proce that occur when one ue a direct dicretization. The fix that empirically eem to work bet i denoted by the Full Truncation (FT) cheme. Though the fix i highly heuritic and ue no known analytical propertie of the variance proce, the cheme eem to work urpriingly well in comparion to mot other cheme. The author even how that the computational efficiency of thi imple Euler cheme outperform the more advanced BK and the higher order Miltein cheme. However it hould be noted that for many relevant parameter configuration 5 the dicretization error i till quite high for a practical number of time tep. Hence though the computational efficiency of the FT cheme i better than the BK cheme, the dicretization grid till need to be rather mall to obtain an accurate cheme. Some approximation to the exact cheme are conidered in Smith (2008) and Anderen (2007). 4 The method of the original Heton paper required the calculation of two numerical integral, wherea ome more recent method require only the evaluation of one numerical integral, e.g. ee Carr and Madan (1999), Lord and Kahl (2008) or Lee (2004). 5 Heton model which are calibrated to main derivative market, uually have parameter configuration uch that variance proce ha a relatively high probability of reaching the origin. Thi i often needed a level of kew or kurtoi that i often preent in market option price. 2

4 Smith approximate the Fourier inverion required to imulate the integrated variance proce. Anderen however focue on the variance proce and develop two efficient cheme which are baed on moment-matching technique. Eentially he approximate the non-centrally chi-quared ditribution by a related ditribution whoe moment are (locally) matched with thoe of the exact ditribution. Since the ampling from the approximated ditribution only require (affine) tranformation to uniform and normal draw, hi cheme can be implemented quite efficiently and without crambling random number. Next to thi approximation Anderen alo ue drift interpolation (intead of Fourier inverion) to approximate the integrated variance proce. The reulting moment-matched cheme baically outperform all exiting cheme in term of computational efficiency. Epecially the Quadratical Exponential (QE(-M)) cheme, turn out to be fat and highly accurate for mot practical ize of the time tep. Though we are aware of the fact that the cheme preented o far certainly do not contain a comprehenive lit of the all the available cheme, we feel that the cheme mentioned o far tand out for particular reaon: the BK cheme for it exactne, the Euler cheme with FT fix for it implicity and the QE-M cheme for it efficiency. For ome alternative cheme we refer to Anderen and Brotherton-Ratcliffe (2001) and Glaerman (2003) and the reference therein. Our contribution to the exiting literature i twofold. Firt, we introduce a new and efficient imulation cheme for the Heton tochatic volatility model. We how that the ampling from the (exact) variance proce can be done accurately and efficiently: in particular we how that ampling from the non-centrally chi-quared ditribution (i.e. the variance proce) can effectively be reduced from a three to a one-dimenional inverion procedure. Hence in uch a cae the invere ditribution function can efficiently be precomputed before the monte carlo run. After dicuing the cheme and the benchmark approach, we perform ome numerical tet againt exact (emi-analytical) option value. Though reult can be well compared to thoe reported in Broadie and Kaya (2006), Anderen (2007) and Lord et al. (2008), we lightly modify the benchmark approach by removing a ignificant part of the Monte Carlo noie with a imple variance reduction technique, i.e. the aet price a control variate. Thi enable u to trengthen the numerical reult about the accuracy of the imulation cheme. The etup of the paper i a follow: we firt dicu ome of the exiting Heton imulation cheme. We then introduce a new cheme, baed on an efficient ampling method from Heton variance proce. We how numerical tet in which we compare our cheme with exiting cheme and finally we conclude. 2 Heton imulation cheme: Euler, Miltein and exact method To be clear about notation, we hortly formulate the Heton dynamic: ds (t) S (t) = r(t)dt + v(t) dw S (t), S (0) := S 0 0, (1) dv(t) = κ(θ v(t))dt + ξ v(t) dw V (t), v(0) := v 0 0, (2) with (W 1, W 2 ) a two-dimenional Brownian motion under the rik-neutral meaure Q with intantaneou correlation ρ, i.e. dw S (t)dw V (t) = ρdt. (3) Hence the model parameter are the initial variance v 0 > 0, the long run variance θ 0, the mean reverion rate κ 0, the volatility of the variance ξ 0 and the leverage parameter ρ Sv 1. Typically, one find 1.0 < ρ SV < 0.6 implying that the Heton dynamic correlate a down move in the tock 3

5 with an up move in the volatility (a phenomenon known a leverage effect). For implicity we here aume that r(t) i non-tochatic, hence from now on we will write r(t) r. Since the characteritic function of the log-aet price i known in cloed-form for the Heton model, the calibration to vanilla call option can be done efficiently uing Fourier inverion, e.g. ee Carr and Madan (1999). Pleae note that in the literature there exit two (theoretically equivalent) formulation of the Heton characteritic function, however a hown in Albrecher et al. (2005) one formulation (a in Heton (1993)) lead to a numerical difficulty called branch cutting, while the other formulation doe not have thi problem. 2.1 Analytical propertie of the variance proce The quare root variance dynamic of the Heton wa firt introduced in a finance (i.e. interet rate) context in Cox et al. (1985); there exit everal analytical reult for the Feller/CIR/quare-root proce of (2), for example the variance proce i guaranteed to alway be greater or equal to zero. Specifically, if 2κθ ξ the Feller condition tate that the proce can never reach zero (a condition which i however hardly ever atified in calibration to real market data) and for 2κθ < ξ we have that the origin i acceible and trongly reflecting. The ditribution of the variance proce i alo known; conditional on v() ( < t), we have that the variance proce i ditributed a a contant C 0 time a non-central chi-quared ditribution with d degree of freedom and non-centrality parameter λ, i.e. IP ( v(t) x v() ) ( x ) = F χ 2 d (λ), (4) C 0 where F x ) χ 2 d (λ)( C 0 repreent the cumulative ditribution of the non-central chi-quared ditribution, i.e. ( ) e λ 2 ( λ z F χ 2 d (λ) z = 2 )i 0 z d 2 e u 2 du i! i=0 Γ(i + k 2 ), (5) with C 0 := ξ2 (1 e κ t ), d := 4κθ 4κ ξ 2, λ := 4κe κ t v() ξ 2 (1 e κ t and: t := t. (6) ) Hence not from (5) that the non-central chi-quared ditribution i equivalent to an ordinary chiquared G with d + 2N degree of freedom, where N i a Poion-ditribution with mean 1 2λ. The cumulative ditribution of (5) thu can be written in the following form F χ 2 d (λ) (z) = IP(N = i)g χ 2(z, d + 2i), (7) i=0 which will be an important expreion in the remainder of thi paper. From known propertie of the non-central chi-quared ditribution (e.g. ee Cox et al. (1985) or Abramowitz and Stegun (1964)) we then have that the mean m and variance 2 of v(t) conditional on v() are given by m :=θ + ( v() θ)e κ t, (8) 2 := v()ξ2 e κ t ( 1 e κ t ) + θξ2 ( 1 e κ t )2. (9) κ 2κ While ome dicretization cheme of the Heton dynamic heavily rely on thee propertie (e.g. ee Broadie and Kaya (2006),Anderen (2007) and Smith (2008)), other cheme do not incorporate the pecific ditributional propertie (e.g. ee the Euler and Miltein cheme of Lord et al. (2008) and Kahl and Jäckel (2006)). 4

6 2.2 (Log-)Euler cheme Probably the implet way to dicretize the variance dynamic i by uing a firt-order Euler cheme. One hould however take care on how to fix negative value of the variance proce; the handling negative value in the wrong way lead to extremely biaed cheme, wherea uing the right fix lead to an Euler cheme that outperform almot all exiting cheme in term of computational efficiency, e.g. ee Lord et al. (2008). Since not all literature ource ue the proper fix when comparing their cheme with an Euler cheme and the cheme provide a good intuition behind the difficultie of the imulation of the Heton model, we explicitly dicu the Euler cheme here. Conditional on time a naive Euler dicretization of the variance proce for t > (with t := t ) read v(t) = v() κ t ( θ v() ) + ξ v() Z v t, (10) with Z v tandard normal ditributed. The main ource of difficulty in above cheme i that the variance can become negative, explicitly the probability of v(t) becoming negative i IP ( v(t) < 0 ) = IP (Z v < v() + κ t( θ v() ) ) ( ( ) v() + κ t θ v() ) ξ v() = Φ t ξ. (11) v() t Notice that though thi probability decay to zero a t become maller, it will be trictly poitive for any choice of the time tep t (unle ξ = 0). Hence if one doe not want the variance proce to cro over to the imaginary domain, one ha to decide what to do if the variance proce turn negative in an Euler cheme. Several ad-hoc fixe for thi exit in the literature, for example by making zero an aborbing or reflecting boundary for the variance proce. Lord et al. Lord et al. (2008) unify everal Euler cheme in the following framework: v(t) = f 1 ( v() ) κ t ( θ f2 ( v() )) + ξ f 3 ( v() ) Zv t, (12) where all cheme hould atify f i (x) = x for x 0 and f 3 (x) 0 for all x R. Thi tranlate into the natural condition that for poitive value of the variance the regular Euler cheme hould be employed and that trictly negative value hereof are tranformed into poitive one. The mot enible choice for f i (x) are the identity function ( f (x) = x), aborption ( f (x) = x + ) or reflection ( f (x) = x ). Since all cheme coincide and are be bia-free a t 0, the choice of the fix eem innocent and almot indifferent. The contrary i true: while ome cheme are extremely biaed for practical ize of the time tep, other turn out to be almot bia-free not too extreme parameter configuration. The fix that eem to work the bet i produced by the o-called Full Truncation (2007) cheme and chooe f 1 (x) := x, f 2 (x) = f 3 (x) := max(x, 0) = x +, ee Lord et al. (2008). The reulting Euler cheme read v(t) = v() κ t ( θ v() +) + ξ v() + Z V t. (13) Hence provided with a dicretization cheme for the variance proce, we alo need to pecify the imulation cheme of the correponding aet price proce. The mot traightforward choice would be to either directly apply an Euler dicretization cheme to the tock price proce of equation (1) or to imulate the tock price from it exact (conditional) ditribution. Direct dicretization yield the following Euler cheme S (t) = S () ( ( ) ) 1 + r t + f 5 v(t) ZS t (14) and doe entail ome dicretization error of the exact proce. Here Z S i a normal ditributed random variable (with correlation ρ to Z V ) and f 5 (x) hould be choen non-negative. 5

7 Alternatively one can alo ue the exact olution of the tock price dynamic (1), which by an application of Ito lemma i given by S (t) = S () exp [ t [ 1 t r 2 v(u)] du + v(u) dws (u) ]. (15) Hence taking logarithm and dicretizing in an Eulerly fahion, one obtain the following log-euler cheme log ( S (t) ) = log ( S () ) + [ r 1 2 f ( )] ( ) 4 v() t + f 5 v() ZS t. (16) The above decribed log-euler cheme doe not entail any dicretization error in the tock price direction, of coure the cheme uually doe how biae in the Euler dicretization the variance proce (and thu in reulting tock price). Following Lord et al. (2008) we chooe to et f 3 (x) = f 4 (x) = f 5 (x) = x +, which eem to be the mot logical choice, ince the Ito correction term of equation (16) i then conitent with the correponding the volatility of the tock price, hence thi implie the martingale condition of the tock price proce i preerved in the dicretization. In an implementation the correlated tandard normal random variable Z V and Z S can (for example) be generated with the ue of a Choleky decompoition: with a (intantaneou) correlation of ρ between the driving Brownian motion thi can be done by etting Z V := Z 1 and Z S := ρz V + 1 ρ 2 Z 2, where Z 1 and Z 2 are two independent draw from the tandard normal ditribution. Note that the pure Euler cheme (14) can be een a a firt order approximation of above log-euler cheme. Since the log-euler cheme entail no dicretization error in the tock price direction, we prefer to work under thi log tranformation when employing an Euler cheme, e.g. ee alo Lord et al. (2008)). Additionally ince the full truncation cheme eem to have the mallet bia among all Euler cheme, we adopt thi fix for poible negative value of the variance proce when uing an Euler cheme. The main advantage of the Euler cheme lie it implicity and peed: little code and computing time i needed to compute one iteration in the cheme. Additionally the ue of the cheme i not retricted to the Heton model, but can alo be applied to all kind model, for example to the family of CEV-procee Lord et al. (2008). It generality alo implie it weakne: the Euler cheme doen t ue any information of known analytical propertie of the quare root variance proce. Full truncation algorithm Uing a log-euler cheme for the tock price proce, the full truncation cheme for the Heton can be ummarized by the following algorithm: 1. Generate a random ample Z 1 from the tandard normal ditribution 6 and et Z v := Z Given v(), compute v(t) from equation (13). 3. Generate a random ample Z 2 from the tandard normal ditribution and et Z S := ρ SV Z V + 1 ρ 2 SV Z 2. (17) 4. Given log ( S () ), compute log ( S (t) ) uing equation (16). 6 It may be adviable to ue an inverion method for generating of normal ample, ince then alo a quai random generator can be ued. Thi inverion over an uniform random variable with the ( approximated ) invere tandard normal ditribution function can for example be done uing Wichura method, ee Wichura (1998). 6

8 2.3 Kahl-Jäckel Scheme A generic implicit Miltein cheme for the variance proce in combination with an alternative dicretization for the tock price wa uggeted in Kahl and Jäckel (2006), i.e. the following dicretization cheme wa propoed: v(t) = v() + κθ t + ξ v() Z V t ξ2 t(zv 2 1) (18) 1 + κ t log ( S (t) ) = log ( S () ) + [ ( ) v() + v(t) ] r t + ρ v() ZV t ( )( ) ρξ t v() + v(t) ZS + ρz V t (Z2 V 1) (19) Kahl-Jäckel how that thi cheme reult in poitive path for the variance proce for 2κθ > ξ, a condition which i hard to meet in practice. Hence in many practical implementation of the above dynamic, one ha to decide on how to fix negative value of the variance proce. Since Kahl and Jäckel (2006) do not pecifically tackle thi iue, we follow Anderen (2007) who adopt the ame fix a Lord et al. (2008) ue in the full truncation Euler cheme. That i, whenever the variance proce drop below zero, we ue (13) rather than (18) and take v() + and v(t) + rather than uing v() and v(t) in (19). The reulting algorithm i imilar to the ft-algorithm 2.2: one jut replace the variance and aet proce from (2) and (4) with the above defined dicretization for the variance and aet proce. 2.4 Exact cheme of Broadie and Kaya In an elegant paper, Broadie and Kaya (2006) worked out an exact imulation cheme for the Heton model. Though theoretically the method i exact, it practical ue i limited; the cheme uffer from a lack of peed, it i complex and enitivity calculation (often ued for rik management) are hard ince the cheme relie on acceptance and rejection ampling technique. For example, the numerical tet in Lord et al. (2008) how that for mot practical ituation even a imple Euler cheme outperform the exact cheme in term of computational efficiency. 7 Though in mot practical ituation a direct implementation of the exact cheme i probably not the bet available option (ee ection 2.5), there are ome approximation or computational trick that can be made to improve upon the computational efficiency. For example, Anderen (2007) and Smith (2008) both ue the exact cheme a tarting point and from there on try to improve upon ome of the incorporated bottleneck. We will firt dicu the exact method and it incorporated difficultie: by uing the explicit olution (15) of the aet price proce and conecutively uing Ito lemma and uing a Choleky decompoi- 7 Note that in the numerical reult of Broadie and Kaya (2006) and Smith (2008), an Euler cheme i ued that handle negative value of the variance in a uboptimal way. However a hown in Lord et al. (2008) the choice on how to cope with negative value of the variance proce i extremely important for the quality (i.e. bia) of the cheme. Becaue the (emi-)exact cheme in Broadie and Kaya (2006) and Smith (2008) are benchmarked againt a uboptimal Euler cheme, thi lead them to a fale concluion in comparing their cheme againt the Euler cheme. Thi wa point wa firt noted in Lord et al. (2008) and later on in Anderen (2007). From their reult in can for example be een that the Euler cheme (with the right fix) outperform the exact and Kahl-Jäckel cheme in term of computational efficiency, wherea in Broadie and Kaya (2006) and in Smith (2008) (who ue uboptimal fixe) thi i jut the other way around. 7

9 tion one obtain t log ( S (t) ) = log ( S () ) 1 v(u)du 2 t t +ρ v(u) dwv (u) + 1 ρ 2 v(u) dw(u), (20) where W(u) i a Brownian motion independent of W v (u). Integrating the quare-root variance proce of equation (2) give the following olution: or equivalently t v(t) = v() + t t κ(θ v(u))du + ξ v(u) dwv (u), (21) v(u) dwv (u) = ξ 1( t v(t) v() κθ t + κ v(u)du ). (22) In Broadie and Kaya (2006), it i then noticed that one can ubtitute equation (22) into the olution (20) to arrive at t log ( S (t) ) = log ( S () ) + κρ v(u)du 1 v(u)du ξ 2 + ρ ( ) t v(t) v() κθ t + 1 ρ ξ 2 v(u) dw(u), (23) hence an exact imulation involve ampling from the following three tochatic quantitie: 1. v(t) v(): from (4) and (6) one ue that v(t) v() i ditributed a a contant C 0 time a non-central chi-quared ditribution with d degree of freedom and non-centrality parameter λ. t 2. v(u)du v(), v(t): Broadie and Kaya (2006) derive the characteritic function Ψ ( a, v(), v(t) ) = IE [ exp ( ia t t v(u)du ) v(), v(t) ] = γ(a)e 1 2 (γ(a) κ)(t )) (1 exp( κ(t )) κ(1 e γ(a)(t ) ) [ v() + v(t) ( κ(1 + e κ(t ) exp ξ 2 1 e κ(t ) γ(a)(1 + e γ(a)(t ) ))] 1 e γ(a)(t ) [ I 1 2 d 1 v()v(t) 4γ(a)e γ(a) (t )/ 2 ξ 2 (1 e γ(a)(t ) ) ] [ v()v(t) 4γ(a)e 2 κ (t )/ ξ 2 (1 e κ(t ) ) ], (24) I 1 2 d 1 with γ(a) := κ 2 2ξ 2 ia, d := 4κθ and where I ξ 2 ν (x) i the modified Beel function of the firt kind. Hence the characteritic function (24) can numerically be inverted to obtain the value of the ditribution function G(x) for a certain point x Ω, i.e. G(x, v(), v(t)) = 2 π 0 in(ax) Re [ Ψ ( a, v(), v(t) ) ] da. (25) x 8

10 3. Finally to generate ample from t v(u)du v(), v(t) one can ue G ( t v(u)du v(), v(t) ) = U, (26) and invert G over a uniform random ample U to find x i : x i = G 1( U, v(), v(t) ), e.g. by a Newton-Raphon root earch of G ( x i, v(), v(t) ) U = 0. Note that uch a root finding procedure involve multiple Fourier inverion: one for each evaluation of G ( x i, v(), v(t) ). t t v(u) dw(u) v(u)du: ince v(u) i independent of W(u), it directly follow that the thi expreion i ditributed a N ( 0, t v(u)). Hence thi ampling can be done eaily and efficient by ampling from a normal ditribution. Broadie and Kaya algorithm Exact imulation of (23) i feaible and can be performed by the following algorithm: 1. Conditional on v(), ue the definition of (6) to generate a ample of v(t) by ampling from a contant time a non-central chi-quared ditribution with d degree of freedom and noncentrality parameter λ. 2. Conditional on v() and v(t), generate a ample of t v(u)du by a numerical inverion of the ditribution function G of ( t v(u)du ) v(), v(t) over a uniform ample U, for example by a root earch G ( x i, v(), v(t) ) U = 0. Since the ditribution function G i not known i cloed form, G ( x i, v(), v(t) ) ha to be obtained by Fourier inverting the characteritic function of v(u)du v(), v(t). t 3. Ue (22) to et: t v(u) dwv (u) = ξ 1( t v(t) v() κθ t + κ v(u)du ) (27) 4. Generate an independent random ample Z S from the tandard normal ditribution and ue the fact that t t V(u) dw(u) i normally ditributed with mean zero and variance V(u)du and thu can be generated a t t V(u) dw(u) ZS t V(u)du, (28) 5. Given log ( S () ), t log ( S (t) ). v(u) dwv (u), t t V(u) dw(u) and V(u)du ue (23) to compute 2.5 Diadvantage of the exact cheme Though the Broadie and Kaya cheme i theoretically appealing (and thi wa probably alo the primary objective of their paper), we will dicu in the following ection why it practical ue might be limited. That i, we dicu ome practical implementation iue that incorporated with the ue of the exact cheme; 9

11 firt of all, (2.4-1) require that the variance proce v(t) v() ha to be ampled from a contant C 0 time a non-central chi-quared ditribution with d degree of freedom and non-centrality parameter λ (ee ): v(t) = d C 0 χ 2 d (λ), (29) For imulation purpoe one can ue the following repreentation of the non-central chi-quared ditribution (ee Johnon et al. and Glaerman (2003)): χ 2 d (λ) d = { (Z + λ ) 2 + χ 2 d 1 for d > 1, χ 2 d+2n for d > 0, (30) with Z N(0, 1), χ 2 ν an ordinary chi-quared ditribution with ν degree of freedom and where N i Poion ditributed with mean µ := 1 2λ. Since in mot practical application d << 1, one i uually forced to work with the econd repreentation of the non central chi-quared ditribution 8 ; hence exact ampling from the variance proce can be done by firt conditioning on a Poion variate and conecutively generating a ample from a chi-quared or gamma ditribution 9. Since direct inverion of the gamma ditribution i relatively low, Broadie and Kaya (2006) ugget to ue an acceptance and rejection method to generate gamma variate. Though uch ampling can be done fairly quick (e.g. by making ue of ome recent advance of Maraglia and Tang (2000)), the method are till relatively low in comparion to ampling method for normal variate. Moreover the main diadvantage of acceptance and rejection technique i that the (number of) ample depend on the pecific Heton parameter. A a conequence the total drawing of random number cannot be predetermined and ample path will how a rather mall correlation coefficient for different parameter input. Thee propertie are uually inconvenient in financial application, ince both perturbation analyi 10 (to calculate model enitivitie with repect to different parameter) a well a the ue of quai random number generator become extremely hard, not to ay practically almot impoible. Another practical difficulty of the cheme lie in tep (2.4-1), where one ha to generate a ample of t v(u)du v(), v(t) by numerically inverting the ditribution function of ( t v(u)du v(), v(t)) over an uniform random variable u, by a root earch of G ( x i, v(), v(t) ) U = 0. However becaue the ditribution function G i not known i cloed form, it ha to be obtained by Fourier inverting the characteritic function (24), which contain two modified Beel function that each repreent an infinite erie. The root-finding procedure (and the involved Fourier inverion) ha to repeated everal time until a tolerance level ε i reached for a gue x i, uch that G ( x i, v(), v(t) ) U < ε. Next to the fact that both in the evaluation of (24) a well a the required Fourier inverion require a great computational effort, the implementation of thi tep alo ha to be done with great care to avoid noticeable biae from the numerical inverion. 8 Otherwie, if d > 1, one might want to ue the firt repreentation, ince ampling from the normal ditribution i uually more efficient than ampling from a Poion ditribution. 9 The Chi-quared ditribution i a pecial cae of the gamma ditribution, χ 2 d ν = gamma( ν, 2), where gamma(k, θ) i a 2 gamma ditribution with hape k and cale θ. 10 The efficiency in the calculation of model enitivitie crucially depend on the ize of the correlation coefficient between pre- and pot perturbation path. 10

12 3 Approximation to the exact cheme A elaborated in ection 2.5 the exact cheme ha ome practical diadvantage. However it doe provide an extremely well bae to contruct ome approximate cheme which might be more practically and computationally more efficient. A few author have already tried to improve the bottleneck in imulating the variance and/or integrated variance proce, e.g. ee Anderen (2007) and Smith (2008). In the remainder of thi ection we will unify and dicu the two methodologie that can improve upon the performance of the Broadie and Kaya cheme. That i, we conider approximation of: 1. The integrated variance proce. 2. The variance proce itelf. Moreover we will look at cheme that combine the latter approximation. Approximating the integrated variance ditribution A elaborated in ection 2.5, a huge bottle neck of the imulation cheme i the ampling of the conditional integrated variance proce. There are however everal way to approximate a ample from the integrated variance proce t v(u)du v(), v(t). 1. Drift interpolation: Without uing any pecific information of the integrated variance proce, one can ue a drift interpolation method to approximate the integrated variance proce, i.e. t v(u)du v(), v(t) γ1 v() + γ 2 v(t), (31) for ome contant γ 1, γ 2, which can be et in everal way: an Euler-like etting would read γ 1 = 1, γ 2 = 0, while a mid-point rule correpond to the predictor-corrector etting γ 1 = γ 2 = Approximate the Fourier inverion: One can alo try to approximate the Fourier inverted ampling of the integrated variance proce. For example Smith Smith (2008) trie to peed up the inverion of the characteritic function (24) by caching value of a projected verion hereof. Though uch a method might peed the inverion, one till ha to ue a rather time-conuming Fourier inverion combined with a root finding procedure to draw a ample of the integrated proce. Alternatively one can try to ue the firt moment of the conditional integrated variance proce (which can be obtained by differentiating the cf. of (24) ) to develop a moment-matched ampling method. Approximating the variance proce Another (practical) diadvantage of the exact cheme i the ue of acceptance rejection ampling method for the non-central chi-quared ditributed variance proce (ee ection 2.5). Hence we conider two method that can be ued to approximate the variance ditribution. 1. Moment-matching: Anderen (2007) ugget to approximate the variance proce by related ditribution whoe firt two moment are (locally) matched with thoe of the true variance ditribution. Moreover, ince the ditribution can be analytically inverted, the method can be directly ued in conjunction with perturbation and low-dicrepancy method by traightforward inverion a uniform random variate. 11

13 2. Direct inverion: To overcome the acceptance and rejection ampling method, one can alo ue direct inverion of the non-central chi-quared ditribution to generate a ample of the variance proce. However ince no analytical expreion exit for thi invere, one ha to ue a (timeconuming) root finding procedure to numerical invert the ditribution. We will how however, that can efficiently create a cache of invere from which a ample of the non-central chi-quared ditribution can be generated (i.e. looked up from a table) in two imple tep. 3.1 Broadie and Kaya Drift Interpolation cheme Probably the eaiet way to give the exact cheme a performance boot i to approximate the Fourier inverted ampling of the integrated variance proce by the imple drift interpolation method of equation (31). Moreover ince the ampling of the integrated variance proce i mot time-conuming tep of the exact cheme, one can expect a large efficiency gain. The imulation of the Broadie and Kaya Drift Interpolation (BK-DI) cheme i traightforward; in the exact cheme of 2.4, one only ha to replace the ampling of the integrated variance proce in tep 2 by the drift interpolation rule (31). Hence though the reulting method i imple and reaonable efficient, ampling from the variance proce i till performed by an acceptance-rejection method, which (a dicued in ection 2.5) i rather inconvenient for mot financial application. We alo like to note that though for reaonable timepacing the drift approximation error i uually rather mall, one doe lightly violate the dicretetime no-arbitrage condition, i.e. the dicretized tock price i not exactly a martingale. In ection 4.1 we how how one can enforce thi condition with the above dicued dicretization method. 3.2 Almot Exact Fourier inverion cheme Smith (2008) trie to peed up the inverion of the characteritic function (24) by caching value of a projected verion hereof. The core of the almot exact imulation method (AESM) in Smith (2008) i to project the exact characteritic function Ψ ( a, v(), v(t) ), which depend on v() and v(t) via the arithmetic and geometric mean 1 2( v() + v(t) ) and v()v(t), onto a function Ψ ( a, z ) in which the dependency on the mean i approximated by the combination z = ω 1 2( v() + v(t) ) + ( 1 ω) v()v(t), 0 ω 1, (32) for a uitable choice of ω. Hence the arithmetic and geometric mean, which are imilar in expectation, are replaced by a weighted average of the two. In thi way the three-dimenional function Ψ ( a, v(), v(t) ) i approximated by the two-dimenional function Ψ ( a, z ) = γ(a)e 2 1 (γ(a) κ)(t )) (1 exp( κ(t )) κ(1 e γ(a)(t ) ) [ 2z ( κ(1 + e κ(t ) exp ξ 2 1 e κ(t ) γ(a)(1 + e γ(a)(t ) ))] 1 e γ(a)(t ) [ I 1 2 d 1 z4γ(a)e γ(a) (t )/( 2 ξ 2 (1 e γ(a)(t ) ) ) ] [ z4γ(a)e 2 κ (t )/( ξ 2 (1 e κ(t ) ) ) ], (33) I 1 2 d 1 which can then be cached on a ufficiently mall dicrete (two-dimenional) grid of a and z-point. Though Smith claim that the approximation work well, the implementation till require a timeconuming root earch of Fourier inverion for each time tep. Hence though the evaluation of the 12

14 characteritic can be approximated in an computationally efficient way, the root earch and inverion are till rather time-conuming in comparion with a imple drift interpolation method. Additionally the total algorithm ha to be implemented with great care to avoid numerical truncation and dicretization error. 3.3 Quadratic Exponential cheme In the Quadratic Exponential (QE) cheme, Anderen (2007) ugget to approximate the ampling from the non-central chi-quared ditribution i approximated by a draw from a related ditribution, which i moment-matched with the firt two (conditional) moment of non-central chi-quared ditribution. The choice of ditribution i plit up into two part, which are baed on the following obervation with repect to the ize of the non-centrality parameter (e.g. ee Abramowitz and Stegun (1964)): 1. For a moderate of high non-centrality parameter the non-central chi-quared can be well repreented by a power function applied to a Gauian variable (which i equivalent to a non-central chi-quared ditribution with one degree of freedom). For ufficiently high value of v(), a ample of v(t) hence can be generated by v(t) = a(b + Z v ) 2, (34) where Z v i tandard normal ditributed random variable and a and b are contant to be determined by moment-matching. 2. For ufficiently low value of v(), the denity of v(t) can (aymptotically) be approximated by IP ( v(t) [x, x + dx] ) ( pδ(0) + β(1 p)e βx) dx, x 0, (35) where δ repreent Dirac delta function, and p and β are non-negative contant. The contant a,b,p,β can (locally) be choen uch that the firt two moment of the approximate ditribution matche thoe of the exact one. Thee contant depend on the ize of the time-tep t,v(), a well a on Heton model parameter. Sampling from thee ditribution can be done in a imple and efficient way: From the firt ditribution one only ha to draw a tandard normal random variable and apply the quadratic tranformation of equation (34). Sampling according to equation (35) can be done by inverion of the ditribution function; The ditribution function i obtained by integrating the probability denity function, and can conecutively be inverted to obtain the following invere ditribution function: { 0 if 0 u p, L 1 (u) = β 1 log ( 1 p) 1 u if p < u 1. (36) Uing the invere ditribution function ampling method, one obtain an eay and efficient ampling cheme by firt generating a uniform random number U v and then etting v(t) = L 1 (U v ) (37) 13

15 Together, the equation (34) and (37) define the Quadratic Exponential (QE) dicretization cheme. What yet remain i the determination of the moment-matching contant a,b,p and β, a well a a rule that define high and low value of the non-centrality parameter, i.e. a rule that determine when to witch from (34) and (37). We firt dicu the latter: recalling that the conditional mean and variance of the quare-root proce are given by m and 2 a defined in equation (8) and (9). Anderen then bae the witching rule on the value of ψ with ψ := 2 m 2 = v()ξ 2 e κ t κ ( 1 e κ t ) ( + θξ2 1 e κ t ) 2 2κ ( θ + ( v() θ)e κ t ) 2 = C 1v() + C 2 ( C3 v() + C 4 ) 2, (38) C 1 = θ, C 2 = θ(1 e κ t ), C 3 = ξ2 e κ t ( 1 e κ t ), C 4 = θξ2 ( 1 e κ t )2. κ 2κ Note that ψ i inverely related to the ize of the non-centrality parameter. It can be hown that for ψ 2 the quadratic tranformation (34) can be moment-matched with the exact ditribution and for ψ 1 the exponential one of (37). Thu for ψ 2, we can moment match the quadratic ampling cheme (34) and for ψ 1 and we can moment match the exponential cheme (37). Since thee domain overlap, at leat one of the two method i applicable. A natural procedure i then to introduce ome critical level ψ c [1, 2], and ue (34) if ψ ψ c and (37) otherwie. Following Anderen, who note that the exact choice of ψ c ha a relatively mall impact on the quality of the overall imulation cheme, we ue ψ c = 1.5 in the numerical tet. Notice though ψ (locally) ha to be calculated for every tep in a imulation and contain computational expenive component (e.g. the exponent exp( κ t)) many of thee term only depend on the ize of time tep. From efficiency conideration it i therefore adviable to pre-cache the tatic contant C 1,..., C 4 before the Monte Carlo imulation tart. In the cae one ue a non-equiditant time grid different contant of coure need to be cached for every applicable ize of the time tep. The moment-matching contant a,b,p and β of the jut defined ampling cheme till have to be pecified, and hould be choen uch that the firt two (conditional) moment are matched with the firt and econd central moment m and 2 of the exact non-central chi-quared ditribution. The following tatement hold regarding the conditional moment of the cheme (34) and (37) 1. For ψ 2, etting b 2 =2ψ ψ 1 2ψ 1 1 0, (39) a = m 1 + b 2, (40) aure that the firt two moment of the ampling cheme (34) are matched with the exact moment non-central chi-quared ditribution, ee Anderen (2007), propoition 5, pp For ψ 1, etting p = ψ 1 [0, 1), ψ + 1 (41) β = 1 p m = 2 > 0, m(ψ + 1) (42) aure that the firt two moment of the ampling cheme (37) are matched with the exact moment non-central chi-quared ditribution, ee Anderen (2007), propoition 6, p

16 QE Algorithm Auming that ome critical witching level ψ c [1, 2] and value for γ 1, γ 2 [0, 1] have been elected, the Quadratic Exponential variance ampling can be ummarized by the following algorithm: 1. Given v(), compute m and 2 and ψ = m2 2 uing equation (8) and (9). 2. If ψ ψ c : (a) Compute a and b from equation (40) and (39). (b) Generate a random ample Z v from the tandard normal ditribution. (c) Ue (34), i.e. et v(t) = a(b + Z v ) 2. Otherwie, if ψ > ψ c : (a) Compute β and p according to equation (41) and (42). (b) Draw a uniform random number U v. (c) Ue (37), i.e. et v(t) = L 1 (U v ) where L 1 i given in (36). 3.4 Non-central Chi-quared Inverion cheme Intead of uing moment-matched cheme, another way to circumvent the acceptance and rejection technique i to ue a direct inverion of the Non-central Chi-Squared ditribution. We will call thi new cheme the Non-central Chi-quared Inverion (NCI) cheme; ince direct inverion i too low, another olution could be to deign a three-dimenional cache of the invere of the non-centrally chi-quared ditribution function F 1 (x, v, λ), which can be created by a root finding procedure of the ditribution function. Though thi method wa already uggeted by Broadie and Anderen, they comment that becaue the parameter pace i potentially extremely large in a three-dimenional cache, that uch brute-force caching will have it own challenge, like it dimenion and the deign of inter- and extrapolation rule. Therefore Anderen doe not purue thi way, but continue on the development of moment-matched cheme. In the following we however how that the three-dimenional parameter pace can effectively be projected onto an one dimenional earch pace. Thi one-dimenional cache can then be created and ued in an efficient fahion; the overhead of the one-dimenional cache i low, while the imulation of the variance proce can be done fat and by imple (linear) interpolation of two value of the cache over an uniform random variable. Moreover ince the total number of uniform draw i fixed (and independent of the Heton parameter), thi new method can directly be ued in conjunction with perturbation analyi and low-dicrepancy number. A Poion conditioned caching method Recall from (2.5) and (6) that the exact ditribution of the variance proce i a contant time a non-central chi-quared ditribution, for which repreentation (30) can be ued, i.e. v(t) v() d = C 0 χ 2 d+2n for d > 0, (43) with and N a Poion ditribution 11 with mean µ = 1 2λ and with (ee (6)): 11 Recall: IP(N = n) = µn e µ n!, n = 0, 1, 2,.... d = 4κθ ξ 2, µ = 1 2 C 5 v(), and C 5 := 15 2κe κ t ξ 2 (1 e κ t ). (44)

17 Thu ampling from the non-centrally chi-quared ditributed variance proce i equivalent to ampling from a Poion-conditioned chi-quared ditribution. Though thi obervation wa already being ued in the Broadie and Kaya cheme, our (yet to be decribed) ampling method i different. We claim our method i more efficient and better applicable in financial application; moreover our ampling method for the variance cheme can either be ued on it own or can be ued a drop in for the variance ampling of the exact or almot exact cheme of Broadie and Kaya (2006) or Smith (2008). In the following ection we firt decribe the Poion-ampling method and conecutively how how one can exploit a property of thi ditribution, which can enable you to create an efficient cache (and correponding ampling method) of the non-central chi-quared ditribution. Poion ampling Notice that the mean µ of the Poion ditribution depend on the ize of the time tep (through C 5 ) a well a on the current tate of the variance proce v(); for almot all practical model configuration one find IE[µ] << 10, for which the correponding Poion-ditribution decay quite rapidly and ha rather thin tail. 12 Thi implie that we can (efficiently) draw a ample N j from a Poion ditribution with a relatively mall mean µ by jut inverting it ditribution function over an uniform random variable (e.g. ee Knuth Knuth (1981) and Ahren and Dieter (1982)): 1. Draw a uniform random number U P, et N j = 0 and P N j = P C = exp( µ). 2. While P C U P : µ N j N j + 1, P N j P N j N j and P C P C + P N j. Hence for mall µ the above inverion algorithm i very efficient, ince mot of probability ma lie within the firt value of the ditribution, i.e. ee figure Poion probability denity function μ = 0.1 μ = 1 μ = P(N=n) n Figure 1: Poion probability denity function for different µ. Next to the fact that the thin tail of the Poion ditribution enable u to efficiently invert the Poion ditribution, it implie that we can create a cache for the non-central chi-quared ditribution by precomputing chi-quared ditribution for a truncated et Poion-outcome; ince there i a little 12 For all µ < 10, IP(N > 35) <

18 probability ma in the tail of the Poion-ditribution that one encounter during a imulation, the truncation error uually i negligible. Caching the Chi-quared ditribution We firt introduce ome notation: let N max repreent a certain threhold level (e.g. uch that IP(N > N max ) < ε) and let N := { } 0,..., N j,..., N max (45) repreent the et of Poion-value for which we will cache the invere of the correponding conditional chi-quared ditribution (i.e. according to 43). Since the invere live on the uniform domain, we let U N j repreent a correponding orted et of uniform variable for which we the invere (χ 2 d+2n j ) 1 ( ) i calculated 13, i.e. U N j := {0,..., 1 δ}. (46) Thu we ugget to create a cache of the value of the invere of the non-central chi-quared ditribution function by mean of conditioning on a truncated range of Poion-value and precomputing the correponding chi-quared ditribution function, i.e. we precompute H 1 N j (U i ) := G 1 χ 2 d+2n j ( Ui ), N j N U i U N j, (47) with G 1 χ 2 d+2n j the invere chi-quared ditribution with d + 2N j degree of freedom. Generating a ample from the variance proce From (6) we know that v(t) v() i ditributed a a contant C 0 time a non-central chi-quared ditribution, we can ue the reult of the previou ubection and ample from the variance proce by firt conditioning on a Poion variable N j and conecutively inverting the correponding chi-quared ditribution. To invert the chi-quared ditribution for N j N max, we jut draw a uniform variate and interpolate between the two value of the invere ditribution cache that urround the uniform number. In cae N j > N max we ue the ditribution correponding to N max and moment-matching technique which we explain below. The caching method (47) and the following ampling rule form the core of the NCI cheme. That i, draw a Poion number N j and a uniform random number U V (e.g. U i < U V < U i+1 ) then a ample of v(t) v() i generated by v(t) = F 1 N j (U V ), (48) with FN 1 C 0 J(U V ) for N j N max, j (U V ) := C 0 F 1 ( ) UV for N χ 2 j > N max, (49) d+2n j with C 0 a defined in (6) and where J( ) repreent an interpolation rule. The NCI ampling cheme thu conit of an inverion of the non-central chi-quared ditribution for the low and mot frequent Poion-outcome and of a moment-matching cheme baed on the chi-quared ditribution for the 13 Since lim U 1 G 1 χ 2 d+2n j (U) =, one hould ue 1 δ intead of 1 to avoid numerical difficultie. Here δ i defined a a mall machine number: in C one can for example et δ = DBL EPSILON which i defined in the header float.h 17

19 rare and high Poion outcome: though the probability of {N > N max } i uually mall, it will be trictly greater than zero for all N max and we decide to ue direct inverion 14. Deign of the cache: a practical example A example we will work out a way to implement the Non-Central Chi-Squared Inverion (NCI) cheme, pecifically we will explore how to deign the cache. A few detail till ha to be filled in: which value hould be choen for N max in (45), how hould the uniform number for a U N j in (46) be aligned and which interpolation rule J hould be choen to interpolate between two value of the invere chi-quared ditribution. For expoitional purpoe we ue the parameter value v(0) = 0.09, θ = 0.09, κ = 1.0, ξ = 1.0 for the variance proce and we ue an equiditant time grid with t = 0.25 and maturity T = 5. Uing (43), (44) and (48) thi then implie that the exact ditribution of the variance proce equal a contant C 0 ordinary chi-quared ditribution with d + 2N = 4κθ + 2N = N degree of freedom, where ξ 2 N (cf. (44)) i Poion ditributed with mean v()c 5 = v() Uing thi etting a example we comment on the choice of N max. A hown in table 1, thi choice mainly depend on the mean of Poion ditribution: for the cae = 0, we have v(0) = 0.09, hence the cheme implie that we need to ample from a Poion-ditribution with mean µ = and we could eaily ue thi mean to et a bound for N max. Unfortunately one then ignore the randomne of the mean: even though the mean of the tationary and non-tationary ditribution can be approximately equal, the randomne ignificantly increae the ma in the tail of non-tationary ditribution function (i.e. baed on all Poion-draw in the imulation) when compared to the tationary Poion-ditribution at time 0. An example of thi behavior can be een in the empirical ditribution function a reported in table 1. n IP(N > n) n IP(N > n) n IP(N > n) Table 1: Empirical ditribution function baed on the Poion ample that were drawn in 10 7 imulation with the parameter: v(0) = 0.09, θ = 0.09, κ = 1.0, ξ = 1.0, t = 0.25 and maturity T = 5. One then ha to decide which ize, alignment and interpolation rule one ue on the grid of uniform number in the cache() of the invere of the chi-quared ditribution with d + 2N j degree of freedom; firt notice that, a hown in table 1, the number of draw from the correponding chi-quared ditribution differ ignificantly acro the bin, e.g. in the example more than 74% of the drawing end up in the firt bin. Another point to take into account for interpolation rule, i that the invere ditribution function i a monotone function, hence we would like the interpolation rule to preerve the monotonicity in the cached data point. A third point might be that ome area (i.e. in the tail of 14 Since for mot parameter configuration and a reaonable choice of N max, the probability of N max we only need to ue a direct inverion a very limited amount of time, the computational overhead of direct inverion will be relatively mall. Alternatively one can alo opt to ue an approximation for a chi-quared ditribution with moderate to large degree of freedom, e.g. ee Abramowitz and Stegun (1964). 18