Bergische Universität Wuppertl Fchbereich Mthemtik und Nturwissenschften Lehrstuhl für Angewndte Mthemtik und Numerische Mthemtik Preprint BUW-AMNA 6/3 Cthrin vn Emmerich Modelling Correltion s Stochstic Process June 6 http://www.mth.uni-wuppertl.de/org/num/
Modelling correltion s stochstic process Cthrin vn Emmerich Deprtment of Mthemtics, University of Wuppertl, Gußstrße, Wuppertl, D-49, Germny This version: /6/6 Abstrct Although mrket observtions show tht correltion between stocks, interest rtes, e. g., is not deterministic, correltion is usully modelled s fixed number. This rticle provides new pproch of modelling correltion s stochstic process which hs pplictions in mny fields. We will show n exmple from finncil mrkets where the stochsticity of correltion is fundmentl source of risk: the qunto. Keywords: Correltion, stochstic processes, men-reverting processes, finnce, Fokker- Plnck eqution. Introduction Stochstic modelling is n essentil tool in pplictions s different s finnce, biology, medicl science etc. As soon s there is more thn one fctor to consider, the question rises how to mp the reltionship between these fctors. A widely used pproch is to use correlted stochstic processes where the mgnitude of correltion is mesured by single number ρ [, ], the correltion coefficient. In the cse of two Brownin motions W nd W correlted with ρ, one cn express this concept by the symbolic nottion dw t dw t = ρ dt. If ρ = the Brownin motions re uncorrelted. The usul ssumption bout filtrtion etc. my hold. See for exmple Øksendl[5]. We will concentrte in the following on correlted Brownin motion. Further work must show how the ide cn be extended to other distributions. In finnce correlted Brownin motions pper for exmple in the Heston model[] ds t = µs t dt + V t S t dw t, dv t = (θ V t )dt + σ V t dz t, where µ >, >, θ >, σ > re constnt. The first process is used to describe the movement of n underlying sset S nd the second process describes the ssumed behviour of the voltility V. Another exmple of coupled stochstic processes is the following Blck-Scholes model for quntos (see lso section 6): ds t = µ S t dt + σ S t dw t, ds t = µ S t dt + σ S t dz t,
with positive constnts µ, µ, σ, σ. Here one of the stochstic differentil equtions is supposed to mp the performnce of trded object (stock or index, for instnce) in currency A. The second stochstic differentil eqution describes the exchnge rte between currency A nd nother currency B. Within both exmples one usully ssumes the Wiener processes to be correlted () dw t dz t = ρ dt, with constnt correltion fctor ρ [, ]. There re few pproches to extend this concept, see for exmple Burtschell et l. []. We will include rndomness nd time fctor in the following. Moreover the process we propose possesses n intuitive interprettion which mkes it vluble modelling tool. For motivting our model we look on the mrket behviour of correltion. It indictes tht correltions re even more unstble thn voltilities s mentioned by Wilmott[7]. Figure shows the estimted correltions between Dow Jones nd Euro/US-Dollr exchnge rte on dily bsis. Hereby it is ssumed tht both processes follow lognorml distribution. Figure (A) clerly shows tht correltion is not constnt over time. Moreover, correltion seems.8.8.6.4. -. -.4.7.6.5.4.3 Historicl correltion -.6. -.8. - Jn-98 Apr-98 Jul-98 Oct-98 Jn-99 Apr-99 Jul-99 Oct-99 Jn- Apr- Jul- Oct- Jn- Apr- Jul- Oct- Jn- Apr- Jul- Oct- Jn-3 Apr-3 Jul-3 Oct-3 Jn-4 Apr-4 Jul-4 Oct-4 Jn-5 Apr-5 Jul-5 Oct-5 - -.8 -.6 -.4 -...4.6.8 Figure : Correltion between Dow Jones nd Euro/US-Dollr exchnge rte, process nd density. (A): Estimted historicl correltion over time. (B): Empiricl distribution. even to be nondeterministic. In figure (B) we receive first impression how the density of stochstic process modelling correltion could look like. Tht is the motivtion for this report. We wnt to model correltion such tht it is concentrted on [, ], it vries round men, the probbility mss pproches zero in the boundry vlues, there is suitble number of prmeters to clibrte the model to mrket dt. The outline of the remining sections is s follows. First we present different pproches for modelling correltions s stochstic processes (section nd 3). In section 4 nd 5 we exmine these mthemticl models nlyticlly. Finlly we nlyse how theses processes cn be used for pricing nd pply the results exemplrily to prticulr finncil derivtive, qunto (section 6). Construction of Fully Stochsticlly Correlted Brownin Motions Before we introduce suitble processes to model correltion, we show how to construct Brownin motions correlted with stochstic process. These will be clled fully stochsticlly correlted
processes. As this is n extension of the constnt correltion cse, it is nturl requirement tht the new concept should cover the constnt cse. First we ssume tht dρ t = (t, ρ t )dt + b(t, ρ t )dk t, ρ [, ], is n Ito process with Brownin motion K nd suitble functions nd b. independent Brownin motions V nd W (lso independent of K), we define Bsed on two () Z t = ρ s dw s + ρ s dv s, Z =. For proving tht Z () is Brownin motion, we remrk tht Z = nd show tht E [Z t ] = t. Becuse of the Ito-isometry, we cn clculte: E [ ] Zt = E ρ s dw s + ρ s dw s + M t = t. with M t := ρ sdw s ρ s dv s. We used tht E[M t ] = which follows from M = nd dm t = ρ t dw t ρ s dv s + ρ t dv t t ρ s dw s + ρ t ρ t dw t dv t } {{ } = In similr wy, one verifies tht E [Z t F s ] = Z s, t s. Thus we hve two Brownin motions W nd Z correlted by the stochstic process ρ. Especilly it holds for W nd Z tht E [Z t W t ] = E ρ s ds which grees for constnt ρ with the symbolic expression dz t dw t = ρ dt. This construction should be borne in mind throughout the remining prts of this rticle.. 3 Models for Fully Stochstic Correltion We present two wys to construct stochstic processes which remin in the intervl [, ]. For the first pproch the new stochstic process is directly formulted s function of Brownin motion. It stnds out becuse of the ese of construction nd the high degree of nlyticl trctbility, but lcks intuitive interprettion. This is the reson why we focus on the second pproch lter on where the new stochstic process is described by stochstic differentil eqution driven by Brownin motion. The first nd most importnt property of process for modelling correltion is tht the process must sty within the intervl [, ]. Bsed on Brownin motion W (3) X t = π rctn (α(w t + γ)), 3
surely is in [, ] since f(x) = rctn (α(x + γ)), π mps (, ) to (, ). Following this trin of thought, we cn lso choose (4) f(x) = + x, or in generl every trnsformtion f : (, ) [, ], f C (R). Obviously not ll of these possible processes re suitble. Exemplrily we nlyse the expecttion for (3). Note tht the initil vlue is given by X = π rctn (α (W + γ)) = rctn (αγ), π since W denotes Brownin motion. Using Ito s formul, we obtin the following representtion for (3) ( dx t = α π ) ( π ) π sin X t cos 3 X t dt + α ( π ) π cos X t dw t. Thus the expecttion is E [X t ] = X E α π sin( π X s) cos 3 ( π X s)ds. This expecttion depends on t nd cnnot be computed in generl. Aprt of this technicl problem the pproch lcks intuitive interprettion which is necessry s we wnt the process to mp correltion. Becuse of its insufficient justifiction which lso holds for similr trnsformtions, we cnnot consider these trnsformed Brownin motions s pproprite modelling tools. Therefore we present second pproch. The process (5) dx t = ( X t )( + X t )dw t, with initil vlue X = x (, ) gurntees vlues in [, ] (see lso section 4). It is strightforwrd to generlise (5) to (6) dx t = α ( X t )( + X t )dw t, α R +, X = x (, ), which lso remins in [, ], but llows more freedom. The expecttion of (6) cn be computed s E [X t ] = X + E α ( X)( + X)dW = x. which is constnt over time. Thus there re two prmeters to clibrte the model to mrket dt: α nd x. We cn extend (6) by dding drift term. Motivted by the prcticl exmple in the introductory section we choose men-reverting process with - for simplicity - deterministic men: (7) dx t = (θ X t )dt + α X t dw t, X = x (, ), with constnts, α >, θ (, ). The ssignment = yields model (6). At this point we do not know much bout (6) nd (7). In the following section, we discuss nlyticl properties with focus on the boundry behviour t nd. Afterwrds we study the process with the id of the Fokker-Plnck eqution in section 5. 4
4 Anlyticl Properties of (7) - Boundries First we clssify the boundries nd. Hereby we follow the nottion of Krlin nd Tylor[3]. We strt with generl Ito diffusion (8) dx t = (x t )dt + b(x t )dw t, x R. Moreover we denote the left boundry by l nd the right boundry by r in the sense l x r. Without loss of generlity we concentrte on the left boundry in the following. The nlysis of the right one works nlogously. Now we introduce some nottions, thereby we ssume x (l, r), v (w) s(v) = exp b (w) dw, v (l, x), S(x) = v x s(v)dv, x (l, x), S[c, d] = S(d) S(c) = d s(v)dv, (c, d) (l, r), S(l, x] = lim l S[, x]. c We lredy indicte tht x nd v will be of no relevnce in the following. We use tht ds(x) = s(x)dx. S is clled the scle mesure wheres M is the speed mesure: m(x) = M[c, d] = b (x)s(x), d m(x)dx. Anlogously we cn use the reltion dm(x) = m(x)dx. Lstly we need the expression Σ(l) = lim M[v, x]ds(v). l Using these nottions we cn clssify the left boundry. The boundry l is clled ttrctive if nd only if there is n x (l, r) such tht (9) S(l, x] <. Furthermore l is clssified s n ttinble boundry if nd only if () Σ(l) <. Otherwise it is unttinble. We re going to determine S(, x] for the men-reverting process (7). We compute s(x) s first step: log (s(v)) = v v c (θ w) ( ) α ( w ) dw = log ( + v) α (θ+) ( v) α ( θ) c 5
with c = ( +v v ) θ α ( v ) α. Consequently s(v) = c ( + v) α (θ+) ( v) α ( θ). For the behviour ner the left boundry l = we hve to investigte S(l, x] = lim l c ( + v) α (θ+) ( v) α ( θ) dv. We note tht ( v) > for v [, x] (, ) nd α ( θ). Therefore the term ( v) α ( θ) is bounded for v [, x]: ( v) α ( θ) ( ) α ( θ) >, ( v) α ( θ) ( x) α ( θ) < α ( θ). Applying the first inequlity we cn estimte s follows S(, x] lim ( ) α ( θ) c ( + v) α (θ+) dv. This expression converges to constnt C R if (θ + ) <, otherwise it diverges. Hence the α boundry l = is ttrctive if (θ + ) <. By pplying the upper bound one shows tht it α is not ttrctice if (θ + ). Anlogously we cn deduce tht the upper bound r = is not α ttrctive if ( θ). This boundry behviour meets intuition. Incresing the prmeter α concentrtes the process round the men. Furthermore the boundry behviour is symmetric with respect to θ. For θ close to the prmeter needs to be decisively lrger to mke the boundries respectively not ttrctive. Now we re going to figure out if the left boundry is ttinble. We remrk tht only n ttrctive boundry cn be ttinble (see definition for Σ). For tht reson we ssume (θ + ) < in the following. For nottionl simplicity, let c denote suitble constnt in the α remining prt of this section. We consider Σ( ) = lim M[v, x]ds(v) = lim S[, v]dm(v) = lim First we clculte Σ( ) for = : v Σ( ) = c α log s(w)dw b (v)s(v) dv. ( ) 4 ( x) <. Hence the boundry l = is ttinble if =. Moreover we cn deduce tht v Σ( ) c lim ( + w) α (θ+) dw β (v)s(v) dv c lim c lim ( + v) α (θ+) ( + ) α (θ+) ( v )( + v) α (θ+) ( v) α dv lim ( + ) α (θ+) 6 ( θ) dv ( + v) ( (θ+)) α dv.
This lst integrl converges if (θ +) < which clerly holds s θ (, ). The expression α ( + ) α (θ+) converges becuse of the ttrctiveness condition (θ + ) <. Thus the left α boundry is ttinble (nd ttrctive s shown bove) if (θ + ) <. Otherwise the left α boundry is not ttrctive nd unttinble. Following the sme line of thought one derives tht the right boundry r = is ttrctive nd ttinble if ( θ) <. Otherwise it is neither ttrctive nor ttinble. α As we wnt the boundries of our correltion process to be unttinble we will consider process (7) with () α ± θ in the following. The clssifiction of the boundries is first step for deeper understnding of the underlying processes. We investigte the trnsition densities for further insight. 5 Anlyticl Properties of (7) - Trnsition Density We use the Fokker-Plnck eqution for determintion of the trnsition density. For more informtion bout the Fokker-Plnck eqution we refer to the book by Risken[6]. For simplicity we set α =. Assuming tht the stochstic differentil eqution () dx t = (t, X t )dt + b(t, X t )dw t, X = x, possesses trnsition density p(t, y x ), then p stisfies the Fokker-Plnck eqution (3) p(t, x) + t x ((t, x)p(t, x)) x ( b(t, x) p(t, x) ) =. We wnt to derive the trnsition density of (7) for t. We demnd the solution p to fulfill two structurl conditions. Firstly p is required to be density (4) p(t, x)dx =. Moreover we postulte tht p preserves the expecttion (5) x p(t, x)dx θ. t For the men reversion process (7) with () one cn show tht every two solutions of (3) re equl for t under certin conditions, see Risken[6], section 6.. As consequence it suffices to show tht sttionry solution exists to know tht it is the unique solution. This is how we proceed in the following: We will derive sttionry solution which fulfills (4) nd (5). This solution suffices the conditions stted in Risken[6]. To obtin the sttionry solution p(x) = lim p(t, x), we consider the sttionry Fokker-Plnck t eqution. Firstly we exmine the simplified cse θ = : (6) ( )p(x) + x( )p (x) ( x )p (x) =. 7
We receive s solution to (6) (7) p(x) = ( x ) ( ) c + b x ( z ) dz with constnts b, c R. If b =, p is symmetric round x =. Hence x p(x)dx = = θ, nd (5) holds. Thus we choose b =. The still free vrible c must be chosen such tht condition (4) is fulfilled. Since the ntiderivtive of p cn be described vi the hyper-geometric function we obtin: F (, b, c, y) = + b c ( + )b(b + ) x + x +...!c(c + ) ( x ) ( ) dx = x F (,, 3, x ). Thus we must choose c s With this choice c = F (,, 3, ). (8) p(x) = c( x ) ( ) becomes density. As next step we consider the full men-reverting process (7) with θ. The density p is solution of (9) ( )p(x) + x( )h (x) + θp (x) ( x )p (x) =. Lengthy clcultions show tht p(x) = ( ) θ x ( ) x + x c + b x e ( (θ tnh (z)+log(z ))) dz solves (9) with b, c R. For θ = we wnt p to be equl to the bove solution (8). This is only possible if b =. Thus the finl solution is () p(x) = c ( ) θ x ( ) x, + x with c such tht p(x) = holds. Figure shows how the trnsition densities my look like. As we could not compute the constnt c nlyticlly, it is determined numericlly. Thus p(x)dx holds for the plots. We cn observe tht p is concentrted on [, ]. p is symmetric with respect to θ in the sense of p θ (x) = p θ ( x). If θ = the globl mximum is ttined t x =. 8
.5.5 -.5 x.5 5 5 -.5 x.5 -.5 θ.5 (A) (B) Figure : (A): Trnsition density () for vrying [3, ], θ =., p(x)dx, (B): Trnsition density () for vrying θ [.75,.75], =, p(x)dx. In prticulr, it p- The probbility mss vnishes when getting wy from the men. proches zero in the boundry vlues. These properties gree with intuition nd wht we demnded in the introductory section. Especilly they correspond to figure. Therefore we consider the men-reverting process (7) s suitble for modelling correltion. Remrk I: Numericl clcultions indicte tht () lso fulfills which is still to be proven nlyticlly. xp(x)dx = θ. Remrk II: The trnsition density cn be used for clibrting the prmeters, see for exmple Wilmott [8]. 6 Prcticl Exmple - Quntos A qunto is cross-currency option. It is combintion of regulr option (Europen, Americn, Asin etc.) which pys off in currency A nd currency option which hedges ginst the exchnge risk between A nd nother currency B. The qunto pys off in B. Thus the pyoff is defined with respect to n underlying noted in currency A. Then this pyoff is converted to currency B. As n exmple we cn think of cll on the Dow Jones whose pyoff is pid in Euro (Dow T Strike) + exchnge rte. In the following we ssume Blck-Scholes world with lognorml distribution for the underlying sset ds t = µ S S t dt + σ S S t dw S t nd lognormlly distributed exchnge rte dx t = µ X X t dt + σ X X t dw X t. 9
If the correltion between W S nd W X is constnt, dw S t dw X t = ρdt, n nlyticl solution for pricing qunto cll exists. It turns out tht pricing qunto reduces to pricing cll in the Blck-Scholes setting with the djusted constnt dividend yield r f r h + ρσ S σ X. See for exmple Wilmot[7]. Hereby r f denotes the risk-free interest rte in the currency in which S is trded (foreign currency). The risk-free interest rte of the other currency is r h (home). The remining question is how to incorporte stochstic correltion in this frmework. We ssume the model to be rbitrge free. Thus the expected return of one unit of home currency, exchnged, risk-free invested in the foreign country nd re-exchnged must equl the risk-free return on one unit: () X exp(r f T )E [X T ] = exp(r h T ). The exchnge rte X t follows geometric Brownin motion nd thus E [X T ] = X exp(µ X T ). Insertion in () yields the restriction () µ X = r h r f. In ddition the rbitrge rgument leds to second restriction. Anlogous rgumenttion s bove mkes cler tht (3) E [S T X T ] = exp(r h T ). X S must hold. Hereby the left side describes the re-exchnged expecttion of n investment of one home currency unit into the underlying S. For clculting E [S(T )X(T )] we need to compute d(s t X t ) = S t dx t + X t ds t + ds t dx [ t = S t X t (µs + µ X )dt + σ S dwt S + σ X dwt X + ρ t σ S σ X dt ]. Ito s formul implies d(ln(x t )) = x t dx t (dx t ). x t Appliction to S t X t leds to T d(s t X t ) = (µ S + µ X )T + σ S WT S + σ X WT X (( ) ) σ S t X t S + σx T. A further ppliction of Ito s formul leds to (4) E [S T X T ] = S X E exp (µ S + µ X )T + σ S σ X Thus with the choice (5) µ S = r h µ X σ S σ X T T ρdt, T ρdt.
.8.7.6 Historicl correltion Trnsition density.5.4.3.. - -.8 -.6 -.4 -...4.6.8 Figure 3: Correltion between Dow Jones nd Euro/US-Dollr exchnge rte, density; empiricl distribution vs. density () computed with θ =., =.6. the no-rbitrge condition (3) is fulfilled. Note tht the choice T µ S = r h µ X ln E σ S σ X ρdt, T would lso be consistent with (3), but in tht cse the stochsticity would hve been lost. Now we cn use conditionl Monte Crlo pproch. We simulte pths for ρ. Then we obtin constnt expression for µ S for every pth. Interpreting (5) s return minus continuous dividend r h (r h r f + σ S σ X T ρ t dt), we cn simply use the Blck-Scholes price for stocks with continuous dividend pyments nd get solution for every pth. Afterwrds we compute the fir price of the qunto s the men over ll Blck-Scholes prices. We test if stochstic correltion leds to different prices for quntos compred to the constnt correltion cse. We consider the following process for stochstic correltion (6) dρ t = (θ ρ t )dt + ρ t dw t. As bove W denotes Brownin motion. If not mentioned otherwise θ =., =.6. The prmeter choice results from lest-squre fitting of (7) to the historicl dt from figure. This fitting is surprisingly good, see figure 3. Underlying nd exchnge rte re supposed to follow lognorml distributions with σ S =. nd σ X =.4, respectively. The risk free interest rtes mounts to r h =.5 nd r f =.3. The
underlying strts in, the strting vlue of the exchnge rte is. The strike mounts to. For numericl integrtion the Milstein scheme is used, see for exmple Kloeden nd Plten[4]. The number of pths mounts to, nd the step size is set to.. In none of the cses the boundries l = nd r = hve been exceeded thus we consider the Milstein scheme s n pproprite choice. This effect perfectly grees with the result tht the boundries re not ttrctive for this prmeter configurtion. Figures 4 nd 5 show prices computed for constnt ρ = θ using the nlytic formul (continuous line). The crosses show the prices determined by simultion using our correltion process (7). 4 4 35 3 Constnt correltion Stochstic correltion 35 3 Constnt correltion Stochstic correltion 5 5 Qunto price 5 Qunto price 5 5 5 4 6 8 4 Time to mturity 4 6 8 4 Time to mturity Figure 4: Comprison of prices for qunto with fixed nd stochstic ρ, (A): ρ ρ =. =., (B): Qunto price 35 3 5 5 Qunto price 9 8 7 6 5 Constnt correltion Stochstic correltion 5 4 3 Constnt correltion Stochstic correltion 3 4 5 6 7 8 9 Time to mturity 3 4 5 6 7 8 9 Time to mturity Figure 5: Comprison of prices for qunto with fixed nd stochstic ρ; (A): ρ =., =.6, (B): ρ =.4, θ =.4, =.6. In figure 4 (A) there is no decisive difference between the prices clculted with constnt correltion nd the prices clculted with (5) where ρ t follows (6). The reson is tht we do not use the whole pth of correltion but eventully only the distribution t T. One cn esily verify tht with ρ = θ the Milstein scheme genertes rndom vribles t T with men θ. As the men-reverting fctor is reltively high this distribution is strongly concentrted round the men. In contrst to tht, in figure 4 (B) we cn observe difference: The prices for stochstic correltion re higher s for constnt correltion. This is due to the fct tht the expecttion of
correltion chnges over time. Consequently the dditionl freedom we win by using stochstic processes for modelling correltion does hve n influence on pricing. In figure 5 (A) initil vlue nd men re equl but here the men reverting fctor is smller compred to figure 4. We observe tht stochstic correltion leds to higher prices thn constnt correltion lthough the expecttion is equl. Thus pricing with constnt correltion mens neglecting correltion risk. The effect is even stronger if θ is higher s seen in figure 5 (B). 7 Conclusion Bsed on independent Brownin motions V, W, K nd given stochstic process we constructed further Brownin motion dρ t = (t, ρ t )dt + b(t, ρ t )dk t, ρ [, ], Z t = ρ t dw s + ρ t dv s. Interpreting ρ t s correltion process Z nd W re stochsticlly correlted. We presented suitble functions nd b for modelling the correltion process. We focussed on (7) dρ t = (θ ρ t )dt + α ρ t dw t, ρ = r. Anlysis of boundry behviour nd trnsition density showed tht it fulfills the nturl fetures we expect correltion to possess. Finlly we showed how to incorporte the concept of fully stochstic correltion into the modelling of finncil product, the qunto. We observed tht correltion risk is neglected if we choose constnt correltion insted of the here presented fully stochstic correltion. Acknowledgments The uthor thnks the working group Numericl Anlysis t the University of Wuppertl. Prticulrly she is grteful for comments of nd discussion with Andres Brtel, Michel Günther nd Christin Khl. Especilly section 5 gretly benefitted from coopertion with Christin Khl. References [] X. Burtschell nd J. Gregory nd J.-P. Lurent, Beyond the Gussin Copul: Stochstic nd Locl Correltion, working pper (5). [] S. Heston, A closed-form solution for options with stochstic voltility with ppliction to bond nd currency options, Rev. Finnc. Studies 6 (993) 37 343. [3] S. Krlin nd H. M. Tylor, A second course in stochstic processes (Acdemic Press, 98). [4] P. E. Kloeden nd E. Plten, Numericl Solution of Stochstic Differentil Equtions (Springer-Verlg, 99). [5] B. Øksendl, Stochstic Differentil Equtions (Springer-Verlg, ). [6] H. Risken, The Fokker-Plnck eqution (Springer-Verlg, 989). [7] P. Wilmott, Quntittive Finnce, volume (John Wiley & Sons, ). [8] P. Wilmott, Quntittive Finnce, volume (John Wiley & Sons, ). 3