Notsocomplex Logarithms in the Heston Model


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1 Notsocomplex Logarithms in the Heston Model Christian Kahl * Department of Mathematics, University of Wppertal, Gaßstraße, Wppertal, D49, Germany, and Qantitative Analytics Grop, ABN AMRO, 5 Bishopsgate, London ECM 4AA, UK Peter Jäckel Head of credit, hybrid, commodity, and inflation derivative analytics, ABN AMRO, 5 Bishopsgate, London ECM 4AA,UK Abstract: In Heston s stochastic volatility framework [Heston 993], semianalytical formlæ for plain vanilla option prices can be derived. Unfortnately, these formlæ reqire the evalation of logarithms with complex argments dring the involved inverse Forier integration step. This gives rise to an inherent nmerical instability as a conseqence of which most implementations of Heston s formlæ are not robst for moderate to long dated matrities or strong mean reversion. In this article, we propose a new approach to solve this problem which enables the se of Heston s analytics for practically all levels of parameters and even matrities of many decades. Introdction The Heston stochastic volatility model is given by the system of stochastic differential eqations ds t = µs t dt + V t S t dw S (t), () dv t = κ(θ V t )dt + ω V t dw V (t) () with correlated Brownian motions dw S (t)dw V (t) = ρdt. [Heston 993] fond a semianalytical soltion for pricing Eropean calls and pts sing Forier inversion techniqes. The price for a Eropean Call with strike K and time to matrity τ can be expressed very similarly to the BlackScholes one, namely C(S, K, V,τ)= P(τ ) [ (F K) + π ] (F f K f ) d (3) where f and f are ( ) ( ) e i ln K ϕ( i) e i ln K ϕ() f := Re and f := Re, (4) if i with F = Se µτ, and P(τ ) is the discont factor to the option expiry date. The fnction ϕ( ) is defined as the logcharacteristic fnction of the nderlying asset vale S τ at expiry: Note that by virte of definition (5) we have ϕ() := E[e i ln Sτ ]. (5) ϕ() = and ϕ( i) = F. (6) Eqation (3) for the price of a call option given the logcharacteristic fnction of the nderlying asset at expiry is generic and applies to any model, stochastic volatility or otherwise. It can be derived directly by the 94 Wilmott magazine
2 TECHNICAL ARTICLE 6 aid of the general reslt from fnctional analysis that the Forier transform of the Heaviside fnction is given by a Dirac and a hyperbolic component: e π ix h(x) dx = δ() + πi. (7) Specifically for the Heston model, we have ϕ() = e C(τ,)+D(τ,)V +i ln F. (8) The coefficients C and D are soltions of a twodimensional system of ordinary differential eqation of Riccatitype. They are C(τ, ) = κθ ( ( )) c()e d()τ (κ ρωi + d()) τ ln, (9) ω c() κ ρωi + d() D(τ, ) = ω with the axiliary fnctions ( e d()τ ) c()e d()τ κ ρωi + d() c() = κ ρωi d(), d() = (ρωi κ) + iω + ω. () What remains to be done for the valation of plain vanilla options is the nmerical comptation of the integral in eqation (3). This calclation is made somewhat complicated by the fact that the integrands f j are typically of oscillatory natre. Still, the integration can be done in a reasonably simple fashion by the aid of GassLobatto qadratre [Gander and Gaschi ]. The real problem, however, starts when the fnctions f j are evalated as part of the qadratre scheme since the calclation of the embedded complex logarithm on the right hand side of eqation (9) is not as straightforward as it may look at first sight. Two Types of Complex Discontinity The first obvios problem with the inverse Forier transformation (3) is the choice of branch of the mltivaled complex logarithm embedded in C in eqation (9). This is a generic problem with option pricing calclations based on the inverse Forier transformation (3) of an expression derived from the logcharacteristic fnction. One sggestion in the literatre to remedy this is to careflly keep track of the branch [Schöbel and Zh 999, Lee 5] along a discretised path integral of f j () as goes from to. Let s recall that the logarithm of a complex variable z = a + ib = re i(t+π n) with t [ π, π), n Z, and r R + can be written as () ln z = ln r + i(t + πn). () If we restrict or choice to the principal branch, the fnctions f j as defined in eqation (4) incr discontinities as is shown in figre for f. Integrating this fnction with an adaptive qadratre scheme will not only reslt in the wrong nmber (becase the area nder the discontinos crve shown in figre is not the same as the area nder the continitycorrected crve), it will also take nnecessarily long de to the fact f () Figre : The fnction f defined in (4). Underlying: ds t = µs t dt + V t S t dw S (t) with S = and µ =. Variance: dv t = κ(θ V t ) dt + ω V t dw V (t) with V =., κ =, θ =., ω =.5 and ρ =.3, time to matrity: τ = 3, strike: K =, red crve: implementation sing only the principal branche, green dashed crve: adjsted crve sing formla (3). that the adaptive rotine refines the nmber of sampling points excessively near the discontinity. In order to nderstand the mechanisms that gives rise to the discontinity, we now examine more closely the component C defined in eqation (9). To simplify the notation frther on, we divide this term as follows: C(τ, ) = R(τ, ) α ln G(τ, ) (3) R(τ, ) := α(κ ρωi + d)τ (4) α := κθ ω (5) G(τ, ) := cedτ (6) c Looking at the whole fnction ϕ() in (8), one cold arge that it is not necessary to evalate a complex logarithm at all since ϕ() = G(τ, ) α e R(τ,)+D(τ,)V +i ln F. (7) This is tre, thogh, we jst shifted the problem from the logarithm to the evaltation of G() α as this is exactly the part of the fnction where the jmp arises. Indeed, if we try this formla in or implementation, the discontinities do not go away (yet). The branch switching of the complex logarithm is in fact not the main problem that gives rise to the jmps of f (). A second effect is at work here and we will have to investigate frther. ^ Wilmott magazine 95
3 In the following, we make repeated se of the fact that, for any complex variable, we can choose either a real/imaginary part or a radis/ phase representation: z = a z + ib z = r z e itz, t z [ π, π). (8) The fact that we restrict the phase t z [ π, π) means that we ct the complex plane along the negative real axis. Taking z to the power α gives z α = r α z eiαtz. (9) Whenever G() crosses the negative real axis along its path (as varies) the sign of the phase of G() changes from π to π and therefore the phase of G() α changes from πα to πα. This leads to a jmp since { e iπ = e iπ e iαπ = e iαπ if α/ Z () e iαπ = e iαπ else In other words, what appeared earlier as a problem with the branch choice of the mltivaled complex logarithm in the evalation of f given by eqation (4) is in fact a problem with the branch switching of the complex power fnction which is related, thogh slightly different. To demonstrate this frther, consider the case when, by coincidence, the parameter setting leads to α Z. In this scenario, there is no jmp at all as we can see in figre. In general, thogh, we have α / Z. This woldn t be a problem if G() didn t cross the negative real axis. However, as it trns ot, the trajectory of G() in the complex plane as is varied from to, tends to start initially with a rapidly otwards f () Figre : The fnction f defined in (4) in the case that α Z. Underlying: ds t = µs t dt + V t S t dw S (t) with S = and µ =. Variance: dv t = κ(θ V t ) dt + ω V t dw V (t) with V =.5, κ =, θ =.5, ω =.5 and ρ =.3, exponent: α =, time to matrity: τ = 3, strike: K =. moving spiral, prior to entering an asymptotic escaping behavior. This is displayed in figre 3 for G(). Note that the shown trajectory is that of ln ln G() γ() := G(), i.e. we have rescaled the coordinate system doblelogarithmically in radis to compensate for the rapid otward move G() ment of the spiralling trajectory of G(). The he h [, ) of the crve is given by h = log ( + ) mod which means that segments of slowly varying color represent rapid movement of G() as a fnction of. Another aspect why the correct treatment of the phase jmps of G() is so important is the fact that it does not reqire nsal or particlarly large parameters for the nmerically indced discontinity to occr. We show in figre 4 how the discontinities in the complex phase of G() arise simply as time to matrity is increased. What we can learn from all of the above is that the only way to avoid the discontinity of the integrands f j () altogether is to ensre that the phase of G() is continos. Varios athors [Schöbel and Zh 999, Lee 5, Sepp 3] propose the idea to keep track of the nmber of jmps by comparing G( k ) and G( k+ ) where k and k+ are adjacent points in their respective nmerical integration scheme. This is an involved technical procedre leading to the need for a particlarly complicated algorithm if we want to se an adaptive qadratre method. Frthermore, we have to make sre that we accont for all jmps which means that, for this to work reliably, we oght to have an estimate for the total nmber of discontinities. Fortnately, however, there is a mch simpler procedre to garantee the continity of G() which we present in the following. 96 Wilmott magazine Im[γ()] 4 ln ln G() γ() := G() G() he: log ( + )mod Re[γ()] Figre 3: Part of the trajectory of the fnction G() in the complex plane has the strctral shape of a spiral which gives rise to repeated crossing of the negative real axis. Here: ρ =.8, κ =, ω =, time to matrity τ =.
4 TECHNICAL ARTICLE 6 First, we introdce the notation c =: r c e itc, () d =: a d + ib d. () The next step is to have a closer look at the denominator of G which was defined in eqation (6): c = r c e itc (3) =: r e iχ +π m where [ ] tc + π m := int (4) π χ := arg(c ) (5) r := c (6) and with int[ ] denoting Gass s integer brackets. Note that we have assmed arg(z) [ π, π) z. The key observation is that the sbtraction of the nmber from the complex variable c is simply a shift parallel to the real axis and therefore the phase of (c ) [ π, π) if t c [ π, π), or in general, both χ and t c can be assmed to be on the same phase interval. In other words, taking a complex nmber of arbitrary phase, and adding the real nmber, cannot possibly give rise to this operation moving the complex nmber across the negative real axis since this wold reqire the addition of an imaginary component. We now do the same calclation with the nmerator of G ln G() = ln( r / r ) + i [χ χ + π(n m)]. (33) In figre (5), we show the rotation cont corrected angle arg (G()) compared with the reslt by sing jst the principal branch of G(). We see that, withot the correction, de to the sheer nmber of discontinities, it can be very difficlt indeed to keep track of the jmps by simply comparing neighboring integration points. Note that it does not reqire the choice of nsal parameters: the discontinities arise qite natrally for practically all choices of stochastic volatility configrations simply as time to matrity is increased, as was shown in figre 4. The procedre for the discontinitycorrected evalation of fnction f is smmarized in algorithm. The evalation of f () is to be done by the same procedre with replaced by ( i) in all bt the last step of algorithm, with a sbseqent division by the forward as one wold expect from eqation (4). Remark. Whilst algorithm is mathematically correct, it cannot actally be implemented directly in its presented form. For long times to matrity τ, the nmerator term G N can become very large which leads to nmerical overflow. In order to avoid this, we can monitor the expression Re(d) τ : when this term becomes greater than ln(dbl EPSILON) we can neglect 3 the sbtraction of in steps 6 and 9 and the whole procedre of estimating the term ln(g()) shold be done in logarithmic coordinates. Defining ce dτ = r c e itc e (ad +ibd )τ = r c e ad τ e i(tc+bd τ) [ ] (tc + b d τ + π) n := int π (7) (8) χ := arg ( ce dτ ) (9) r := ce dτ, (3) we have ce dτ = r e i(χ +π n). (3) Again, we sed the fact that the sbtraction of does not affect the rotation cont of the phase of a complex variable. Combining these reslts we obtain G() = cedτ c = r r ei[χ χ +π(n m)] (3) wherein the innocos bt crcial rotation cont nmbers m and n are given by eqations (4) and (8). We are now in the position to compte the logarithm of G() qite simply as Figre 4: Withot special handling, the phase of G() incrs more and more jmps as time to matrity is increased. Here: ρ =.8, κ =, ω =, time to matrity τ [, ]. ^ Wilmott magazine 97
5 Algorithm Evalate f () Reqire: F, K,κ,θ,ω,τ >,V and ρ (, ) : C d := (ρωi κ) + ω (i + ) : C c := κ ρωi+d κ ρωi d 3: R t c := arg(c) 4: C G D := c {Denominator of G()} 5: Z m := int [ ] tc+π π 6: C G N := ce dτ [ ] {Nmerator of G()} 7: Z n := int (tc+im(d)τ +π) π 8: C lng := ln(abs(g N )/abs(g D )) + i (arg(g N ) {see (33)} arg(g D ) + π(n m)) ) 9: C D := κ ρωi+d e dτ ω ( ce dτ : C C := κθ [(κ ρωi + d)τ lng] {see (3)} ω : C ϕ := exp(c + D V + i ln(f)) [ ] : R f := Re exp( i ln(k))ϕ 3 Integration scheme i The choice of the right integration scheme is another crcial point for a robst implementation of the semianalytical Heston soltion. De to the fact that the integrand can vary in its shape from almost simply exponentially decaying to highly oscillatory depending on the choice of parameters, most simple qadratre or nmerical integration schemes are bond to fail significantly for some relevant scenarios. More advanced schemes sch as the adaptive GassLobatto algorithm [Gander and Gatschi ], however, are capable of handling the wide range of fnctional forms attainable by the integrand defined in (4). Since the GassLobatto algorithm is designed to operate on a closed interval [a, b], we show below how one can transform the original integral bondaries [, ) to the finite interval [, ], taking into accont or knowledge of the analytical strctre of the integrand for large. Adapting the transformation to the asymptotic strctre for not only aids the stability of the adaptive qadratre scheme, it also makes the integration scheme significantly more efficient in the sense that far fewer evalation points are needed. Preposition 3. Assming that κ, θ, ω, τ > and ρ (, ) we obtain the following asymptotics: d() lim = ω ρ =: d (34) lim c() = + ρ + iρ ρ = D() ρ + iρ lim = ω ( i ρ + ρ) (35) (36) C() lim = iαωρτ αd τ (37) arg (G()) arg (G()) (a) (b) Figre 5: The phase of G() as defined in eqation (6) with (dashed) and withot (solid) rotation cont correction given by eqation (33), for κ = and τ = 3. (a) ρ =.8, ω =.5. (b) ρ =.95, ω = Wilmott magazine
6 TECHNICAL ARTICLE 6 This leads to ( ) e it lim f j() f j () e C Re = f j () e C sin(t ) (38) i with ( ) ρ ρ C = αd τ + V = (V + κθτ) (39) ω ω as well as t = αωρτ ρv + ln(f/k). (4) ω The initial vales f j () are given in propositions 3. and 3.3. Proof: The proof can be fond in appendix A. Eqation (38) shows that the asymptotic decay of the integrand is at least exponential. In particlar, C > garantees the existence of the integral. A simple and reliable transformation is to translate the integration according to with f j () d = f j ((x)) x C dx (4) (x) = ln x. (4) C The transformation is only possible when C >. Fortnately, this is given as long as ρ =. One last step is needed before we can proceed with the implementation. Looking at the fnctions f j in (4), we can see that f j () is, strictly speaking, not defined at =. The continity of the fnction at zero in figres and gives s hope, thogh, that we can find the vale analytically by the aid of l Hospital s rle. Preposition 3. The fnction ( ) e i ln K ϕ( i) f () = Re if has the following property at zero: lim f () = ln(f/k) + Im ( C ( i) ) + Im ( D ( i) ) V (44) with (43) Im ( C ( i) ) = e(ρ ω κ)τ θκ + θκ((κ ρω)τ ) (κ ρω) (45) Im ( D ( i) ) ρω)τ e (κ = (κ ρω), (46) provided that (κ ρω) =. When (κ ρω) =, we obtain Im ( C ( i) ) = κθτ 4 (47) Proof: The proof can be fond in Appendix A. Preposition 3.3 The fnction ( ) e i ln K ϕ() f () = Re i has the following property at zero: with Im ( D ( i) ) = τ. (48) (49) lim f () = ln(f/k) + Im ( C () ) + Im ( D () ) V (5) Im ( C () ) = e κτ θκ + θκ(κτ ) κ (5) Im(D ()) = e. (5) κ Proof: The proof can also be fond in Appendix A. The above analysis enables s to implement the reqired Forier inversion as a GassLobatto integration over the interval [, ] sing the transformation given by (4), (4), and (39). Since GassLobatto qadratre evalates its integrand at the bondary vales, we need to se the analytical limit vales (44) and (5) for the evalation of f j ((x)) x=. The limit vales at x = seem at first sight to be more complicated. However, it is straightforward to see from eqation (38) that those limit vales are zero as long as C > which in trn is given as long as ρ <. In practice, we combine all calclations in (3) into the single nmerical integration for the ndisconted call option price with C(S, K, V,τ)/P(τ ) = κτ y(x) dx (53) y(x) := (F K) + F f ( ln x ) K f C ( ln x ) C (54) x π C and F = S e µτ, as before. The limits of y(x) at the bondaries of the integral are lim y(x) = (F K) (55) x and lim y(x) = x (F K) + F lim f () K lim f () (56) π C with lim f () and lim f () given by (44) and (5), respectively. An example for the set of sampling points chosen by the adaptive Gass Lobatto scheme for a target accracy of 6 is shown in figre 6. ^ Wilmott magazine 99
7 ..8 y(x) GassLobatto sampling (accracy =.E6) Figre 6: The sampling points selected by the adaptive GassLobatto scheme for the integration of eqation (53). S = F =, µ =, V = θ, θ=.6, κ =, ω =, ρ =.8, τ =, K =. The total nmber of evalation points was 98, and the reslting integral vale was 4.95%. 7% 6% 5% 4% 3% % % K/F τ Figre 7: Black implied volatilities from the Heston model. Underlying: ds t = µs t dt + V t S t dw S (t) with S = F = and µ =. Variance: dv t = κ(θ V t ) dt + ω V t dw V (t) with V = θ =.6, κ =, ω =, ρ =.8, τ [/4, 5], and K [/, 4]. Wilmott magazine
8 TECHNICAL ARTICLE 6 The stability of GassLobatto integration over the interval x [, ] of the transformed integrand f j ((x)) / (x C ) as described is sch that even extremely far otofthemoney option prices can be compted, as well as very long dated matrities. We demonstrate this in figre 7, where implied Black volatilities are displayed for the same parameters as previosly sed in figre 4. 4 Conclsion The problem we addressed was that Forier inversion integrals of the form (3) sed for the calclation of option prices based on analytical knowledge of the logcharacteristic fnction of the nderlying stochastic process are particlarly prone to nmerical instabilities de to their involved complex logarithms and complex power expressions. We fond that the difficlty is ltimately not, as was previosly reported, cased by the mltivaled natre of the complex logarithm, bt instead by the mltivaled natre of the complex power fnction. We analysed this isse for the Heston stochastic volatility model, thogh, the key insight is readily transferable to the calclation of option prices in other models that are amenable to the Forier inversion integral approach [Carr 999, Carr and Madan 999] based on a logcharacteristic fnction. We also presented a detailed analysis how the Forier integral itself can be compted reliably by the aid of adaptive Gass Lobatto qadratre, sing asymptotic analytical methods to identify the most sitable transformation from the halfopen integration domain [, ) to the interval [, ]. These reslts, too, are not only sefl for the prpose of compting option prices for the Heston model, bt also to illstrate how similar methods can be devised for other models that are to be evalated with Forier integration techniqes. A Proofs In this section we present the missing proofs of section 3. Proof:[Prop.3.] The proof of eqation (34) is basic analysis: d() lim = lim ω ( ρ ) iω (ρκ )/ ω + κ / (57) Note that lim c() =. In order to show eqation (36), we need the calclation [ ] [ ( e d()τ lim = lim + c() )] c()e d()τ c() c()e d()τ = lim c() c() = lim c() (59) = + ρ iρ ρ. (6) D() lim = ω ( + ρ iρ ρ )( ρωi + ω ρ ) = iρ ρ. (6) ω Finally, we prove (37) lim ln [ c()e d()τ c() From here, we obtain ]/ [ ]/ = lim ln e d()τ c() c() (6) ( = lim ln [ e ] [ ]) / c() d()τ + ln c() (63) = lim d()τ/ = d τ. (64) C() lim = ατ (d + iωρ). (65) Combination of the limit reslts for D() and C() makes it straightforward to arrive at the statements (38) and (39) for f j (). Proof:[Prop.3.] Considering the fnction f as given in (4) we see that ], i)+d(τ, i) V +ln(f) i ln(f/k) ec(τ lim f () = lim Im [e. (66) F = ω ρ Using this reslt, we are able to confirm eqation (35) ρωi + ω ρ lim c() = ρωi ω ρ = (ω ρ ρωi) ω ( ρ ) + ρ ω = ω ( ρ ) iω ρ ρ ρ ω ω = + ρ + iρ ρ = ( i ρ + ρ). (58) The expressions that need attention for the evalation of this limit are lim C(τ, i) as well as lim D(τ, i). We start with lim d( i) = κ ρω. (67) The limit lim c( i) depends on the sign of κ ρω. We first consider the case where κ ρω <. We obtain κ iρω( i) + d( i) lim c( i) = lim =. (68) κ iρω( i) d( i) This allows s to dedce lim D( i) =, lim C( i) =. (69) ^ Wilmott magazine
9 The case κ ρω > is a little bit more complicated. First we see that κ iρω( i) + d( i) lim c( i) =lim = (7) κ iρω( i) d( i) as the denominator goes to zero. Using this we can see that ( ) κ iρω( i) + d( i) e d( i)τ lim D( i) = lim ω c( i)e d( i)τ Frthermore, we have (7) =. (7) lim C( i) = lim α ((κ iρω( i) + d( i))τ [ ]) ln e d( i)τ + ed( i)τ c( i) (73) =. (74) Last we have to check the case κ ρω =. First we have to notice that d( i) lim = lim ω ( ρ ) iω = iω. (75) We have to mention that this limit is not zero as ω = if κ ρω = and κ>. Using this we obtain lim c( i) = + lim = + lim Again we get lim D( i) = and ρωi ρωi d( i) (76) ρωi ρωi + =. iω (77) lim C( i) = α lim ((κ ρω + d( i))τ [ ]) c( i)e d( i)τ (78) ln c( i) [ ] = α ln = (79) Hence in all different cases lim e i ln(f/k)+c( i)+d( i)v =. (8) Now we decompose the exponent into a real and imaginary part i ln(f/k) + C(τ, i) + D(τ, i) V =: H() + ij() (8) with fnctions H, J : R R. De to (), we also know that lim H() =, lim J() =. (8) Now we can calclate [ ] e H()+iJ() f () = lim Im [ ] H() cos(j()) + i sin(j()) = lim Im e H() sin(j()) = lim e = lim e H() lim sin(j()) (83) (84) (85) (86) = lim J () cos(j()) = lim J (), (87) where we applied the rle of l Hospital in the last step. Ths, f () = lim J () = J () = ln(f/k) + Im ( C ( i) ) + Im ( D ( i) ) V. (88) The comptation of C and D is tedios bt straightforward. We obtain Im ( C ( i) ) = e(ρ ω κ)τ θκ + θκ((κ ρω)τ ) (κ ρω), (89) Im ( D ( i) ) ρω)τ e (κ = (κ ρω), (9) if (κ ρω) =, and otherwise which completes the proof. Im ( C ( i) ) = κθτ 4, (9) Im ( D ( i) ) = τ, (9) Proof:[Prop.3.3] The proof of the limit lim f () can be done in complete analogy to the proof of lim f (). FOOTNOTE & REFERENCES. Note that the phase, as we refer to it here, is also known as the argment and some mathematical software packages se the expression arg for the calclation of the phase of a complex nmber.. The he is a nmber in the range [, ) which varies the color from red at throgh the entire rainbow spectrm to prple (arond.8), and then reconnecting to red at. 3. The C header <float.h> provides the macro DBL_EPSILON which is defined as the smallest floating point nmber that, when added to the nmber, still reslts in a nmber that is distinct from in the compter s floating point nmber representation. P. Carr. Option Pricing sing Integral Transforms, 3. carrp/papers/pdf/integtransform.pdf. P. Carr and D.B. Madan. Option valation sing the fast Forier transform. The Jornal of Comptational Finance, :6 73, 999. Wilmott magazine
10 TECHNICAL ARTICLE 6 W. Gander and W. Gatschi. Adaptive GassLobatto Qadratre. S. L. Heston. A closedform soltion for options with stochastic volatility with applications to bond and crrency options. The Review of Financial Stdies, 6:37 343, 993. R. Lee. Option Pricing by Transform Methods: Extensions, Unification, and Error Control. Jornal of Comptational Finance, 7(3):5 86, 5. S. Mikhailov and U. Nögel. Heston s Stochastic Volatility Model Implementation Calibration and Some Extensions. Wilmott magazine, 3. A. Sepp. Pricing EropeanStyle Options nder Jmp Diffsion Processes with Stochastic Volatility: Application of Forier Transform. Working paper, Institte of Mathematical Statistics, Faclty of Mathematics and Compter Science, University of Tart, J. Liivi, 549 Tart, Estonia, September 3. R. Schöbel and J. Zh. Stochastic Volatility With an Ornstein Uhlenbeck Process: An Extension. Eropean Finance Review, 3:3 46, 999. W Wilmott magazine 3
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