Computing Greeks using multilevel path simulation

Transcription

1 Computing Greeks using mutieve path simuation Syvestre Burgos, Michae B. Gies Abstract We investigate the extension of the mutieve Monte Caro method [4, 5] to the cacuation of Greeks. The pathwise sensitivity anaysis [8] differentiates the path evoution and effectivey reduces the smoothness of the payoff. This eads to new chaenges: the use of naive agorithms is often impossibe because of the inappicabiity of pathwise sensitivities to discontinuous payoffs. These chaenges can be addressed in three different ways: payoff smoothing using conditiona expectations of the payoff before maturity [8]; an approximation of the above technique using path spitting for the fina timestep [1]; the use of a hybrid combination of pathwise sensitivity and the Likeihood Ratio Method [6]. We discuss the strengths and weaknesses of these aternatives in different mutieve Monte Caro settings. 1 Introduction In mathematica finance, Monte Caro methods are used to compute the price of an option by estimating the expected vaue EP. P is the payoff function that depends on an underying asset s scaar price St which satisfies an evoution SDE of the form dst = as,tdt + bs,tdw t, 0 t T, S0 given. 1 This is just one use of Monte Caro in finance. In practice the prices are often quoted and used to caibrate our market modes; the option s sensitivities to market param- Syvestre Burgos Oxford-Man Institute of Quantitative Finance, University of Oxford, e-mai: syvestre.burgos@maths.ox.ac.uk Michae B. Gies Oxford-Man Institute of Quantitative Finance, University of Oxford, e-mai: mike.gies@maths.ox.ac.uk 1

2 2 Syvestre Burgos, Michae B. Gies eters, the so-caed Greeks, refect the exposure to different sources of risk. Computing these is essentia to hedge portfoios and is therefore even more important than pricing the option itsef. This is why our research focuses on getting fast and accurate estimates of Greeks through Monte Caro simuations. 1.1 Mutieve Monte Caro Let us first reca important resuts from [4] and [5]. We consider a standard Monte Caro method using a discretisation with first order weak convergence e.g. the Mistein scheme. Achieving a root-mean square error of Oε requires a variance of order Oε 2, hence Oε 2 independent paths. It aso requires a discretisation bias of order Oε, thus Oε 1 timesteps, giving a tota computationa cost Oε 3. Gies mutieve Monte Caro technique reduces this cost to Oε 2 under certain conditions. The idea is to write the expected payoff with a fine discretisation using 2 L uniform timesteps as a teescopic sum. Let P be the simuated payoff with a discretisation using 2 uniform timesteps, E P L = E P 0 + L =1 E P P 1 2 We then use Monte Caro estimators using N independent sampes E P P 1 Ŷ = 1 N N P i P i 1 3 The sma corrective term P i P i 1 comes from the difference between a fine and a coarse discretisation of the same driving Brownian motion. Its magnitude depends on the strong convergence properties of the scheme used. Let V be the variance of a singe sampe P i P i 1. The next theorem shows that what determines the efficiency of the mutieve approach is the convergence rate of V as. To ensure a better efficiency we may modify 3 and use different estimators of P on the fine and coarse eves of Ŷ, E P L = E P 0 + L =1 E P f P 1 c 4 P f, P 1 c are the estimators using respectivey 2 and 2 1 steps in the computation of Ŷ. The teescoping sum property is maintained provided that E P f = E P c. 5

3 Computing Greeks using mutieve path simuation 3 Theorem 1. Let P be a function of a soution to 1 for a given Brownian path Wt; et P be the corresponding approximation using the discretisation at eve, i.e. with 2 steps of width h = 2 T. If there exist independent estimators Ŷ of computationa compexity C based on N sampes and there are positive constants α 2 1,β,c 1,c 2,c 3 such that { E P 1. EŶ = 0 if = 0 E P P 1 if > 0 2. E P P c 1 h α 3. VŶ c 2 h β N 1 4. C c 3 N h 1 Then there is a constant c 4 such that for any ε < e 1, there are vaues for L and N resuting in a mutieve estimator Ŷ = L =0 MSE = EŶ EP 2 < ε 2 with a compexity C bounded by Proof. See [5]. c 4 ε 2 if β > 1 C c 4 ε 2 ogε 2 if β = 1 c 4 ε 2 1 β/α if 0 < β < 1 Ŷ with a mean-square-error We usuay know α thanks to the iterature on weak convergence. Resuts in [9] give α = 1 for the Mistein scheme, even in the case of discontinuous payoffs. β is reated to strong convergence and is in practice what determines the efficiency of the mutieve approach. Its vaue depends on the payoff and may not be known a priori Monte Caro Greeks Let us briefy reca two cassic methods used to compute Greeks in a Monte Caro setting: the pathwise sensitivities and the Likeihood Ratio Method. More detais can be found in [2], [3] and [8]. Pathwise sensitivities Let Ŝ = Ŝ k k [0,N] be the simuated vaues of the asset at the discretisation times and Ŵ = Ŵ k k [1,N] be the corresponding set of independent Brownian increments. The vaue of the option V is estimated by V defined as [ ] V = E[PS] V = E PŜ = PŜpθ,ŜdŜ

4 4 Syvestre Burgos, Michae B. Gies Assuming that the payoff PŜ is Lipschitz, we can use the chain rue and write V = PŜθ,Ŵ pŵdŵ = PŜ Ŝ Ŝθ,Ŵ pŵdŵ where dŵ = N dŵ k and pŵ = N pŵ k is the joint probabiity density function k=1 k=1 of the normay distributed independent increments Ŵ k k [1,N]. We obtain Ŝ by differentiating the discretisation of 1 with respect to θ and iterating the resuting formua.the imitation of this technique is that it requires the payoff to be Lipschitz and piecewise differentiabe. Likeihood Ratio Method The Likeihood Ratio Method starts from [ ] V = E PŜ = PŜpθ,ŜdŜ 7 The dependence on θ comes through the probabiity density function pθ, Ŝ; assuming some conditions discussed in [3] and in section 7 of [8], we can write [ ] V = PŜ pŝ dŝ = og pŝ og pŝ PŜ pŝdŝ = E PŜ 8 with dŝ = N k=1 dŝ k and pŝ = N k=1 p Ŝk Ŝ k 1 The main imitation of the method is that the estimator s variance is ON, increasing without imit as we refine the discretisation. 1.3 Mutieve Monte Caro Greeks By combining the eements of sections 1.1 and 1.2 together, we write V = EP E P L = E P 0 As in 3, we define the mutieve estimators Ŷ 0 = N 1 0 M P i 0 and + Ŷ = N 1 L =1 N E P P 1 P i P i

5 Computing Greeks using mutieve path simuation 5 where P 0, P 1, P are computed with the techniques presented in section European ca We consider the Back-Schoes mode: the asset s evoution is modeed by a geometric Brownian motion dst = r Stdt + σ StdW t. We use the Mistein scheme for its good strong convergence properties. For timesteps of width h, Ŝ n+1 = Ŝ n 1 + r h + σ W n + σ 2 2 W n 2 h := Ŝ n D n 11 The payoff of the European ca is P = S T K + = max0,s T K. We iustrate the techniques by computing Deta and Vega, the sensitivities to the asset s initia vaue S 0 and to its voatiity σ. We take a time to maturity T = Pathwise sensitivities Since the payoff is Lipschitz, we can use pathwise sensitivities. The differentiation of equation 11 gives Ŝ 0 S 0 = 1, Ŝ n+1 S 0 = Ŝ n S 0 D n Ŝ 0 σ = 0, Ŝ n+1 σ = Ŝ n σ D n + Ŝ n Wn + σ Wn 2 h To compute Ŷ we use a fine and a coarse discretisation with N f = 2 and N c = 2 1 uniform timesteps respectivey. Ŷ = 1 N N P i i 1 ŜN f P ŜN c 12 S Nf S Nc We use the same driving Brownian motion for the fine and coarse discretisations: we first generate the fine Brownian increments Ŵ = W 0, W 2,..., W Nf 1 and then use Ŵ c = W 0 + W 1,..., W Nf 2 + W Nf 1 as the coarse eve s increments. To assess the order of convergence of VŶ L, we take a sufficient number of sampes so that the Monte Caro error of our simuations wi not infuence the resuts. We pot ogvŷ as a function of ogh and use a inear regression to measure the sope for the different estimators. The theoretica resuts on convergence are asymptotic ones, therefore the coarsest eves are not reevant: hence we per-

6 6 Syvestre Burgos, Michae B. Gies form the inear regression on eves [3, 8]. This gives a numerica estimate of the parameter β in Theorem 1. Combining this with the theorem, we get an estimated compexity of the mutieve agorithm. This gives the foowing resuts : Ŝ P Nf S Nf Estimator β MLMC Compexity Vaue 2.0 Oε 2 Deta 0.8 Oε 2.2 Vega 1.0 Oε 2 ogε 2 Gies has shown in [4] that β =2 for the vaue s estimator. For Greeks, the convergence is degraded by the discontinuity of P S = 1 S>K: a fraction Oh of the paths has a fina vaue Ŝ which is Oh from the discontinuity K. For these paths, there is a O1 probabiity that Ŝ N f and Ŝ 1 N c are on different sides of the strike K, impying 1 is O1. Thus VŶ = Oh, and β = 1 for the Greeks. P Ŝ Nc S Nc 2.2 Pathwise sensitivities and Conditiona Expectations We have seen that the payoff s ack of smoothness prevents the variance of Greeks estimators Ŷ from decaying quicky and imits the potentia benefits of the mutieve approach. To improve the convergence speed, we can use conditiona expectations as expained in section 7.2 of [8]. Instead of simuating the whoe path, we stop at the penutimate step and then for every fixed set Ŵ = W k k [0,N 2], we consider the fu distribution of ŜN Ŵ. With a n = aŝn 1 Ŵ,N 1h and b n = bŝn 1 Ŵ,N 1h, we can write Ŝ N Ŵ, W N 1 = Ŝ N 1 Ŵ + a n Ŵh + b n Ŵ W N 1 13 We hence get a norma distribution for ŜN Ŵ. pŝ N Ŵ = σŵ 1 exp 2π 2 ŜN µŵ 2σ 2 Ŵ 14 with µŵ = Ŝ N 1 + aŝn 1,N 1h h σŵ = bŝn 1,N 1h h

7 Computing Greeks using mutieve path simuation 7 [ P ŜN Ŵ If the payoff function is sufficienty simpe, we can evauate anayticay E Using the tower property, we get [ ] [ ]] V = E PŜ N = EŴ [E WN PŜ N Ŵ 1 M M [ E P m=1 Ŝm ] N Ŵ m 15 In the particuar case of geometric Brownian motion and a European ca option, we get 16 where φ is the norma probabiity density function, Φ the norma cumuative distribution function, α = 1 + rhŝ N 1 Ŵ and β = σ hŝ N 1 Ŵ. α K α K EPŜ N Ŵ = β φ + α KΦ 16 β β This expected payoff is infinitey differentiabe with respect to the input parameters. We can appy the pathwise sensitivities technique to this smooth function at time N 1h. The mutieve estimator for the Greek is then Ŷ = 1 N N P i f P c i ]. At the fine eve we use 16 with h = h f and Ŵ f = W 0, W 2,..., W Nf 2 to get EPŜ Nf Ŵ f. We then use P f = Ŝ Nf 1 EPŜ Nf Ŵ f + EPŜ Nf Ŵ f S Nf 1 At the coarse eve, directy using EPŜ Nc Ŵ c eads to an unsatisfactoriy ow convergence rate of VŶ. As expained in 4 we use a modified estimator. The idea is to incude the fina fine Brownian increment in the computation of the expectation over the ast coarse timestep. This guarantees that the two paths wi be cose to one another and heps achieve better variance convergence rates. Ŝ sti foows a simpe Brownian motion with constant drift and voatiity on a coarse steps. With Ŵ c = W 0 + W 1,..., W Nf 4 + W Nf 3 and given that the Brownian increment on the first haf of the fina step is W Nf 2, we get 1 pŝ Nc Ŵ c, W Nf 2 = exp σŵc 2π 2 ŜNc µŵc 2σ 2 Ŵ c with µŵc = Ŝ Nc 1Ŵ c + a ŜNc 1,N c 1h c h c + b ŜNc 1,N c 1h c W Nf 2

8 8 Syvestre Burgos, Michae B. Gies σŵc = bŝnc 1,N c 1h c hc /2 [ From this distribution we derive E PŜ Nc ] Ŵc, W Nf 2, which for the particuar appication being considered, eads to the same payoff formua as before with α c = 1 + r h c + σ W Nf 2Ŝ Nc 1Ŵ c and β c = σ h c Ŝ Nc 1Ŵ c. Using it as the coarse eve s payoff does not introduce any bias. Using the tower property we check that it satisfies condition 5, [ [ ] ] ] E PŜ Nc Ŵ c, W Nf 2 Ŵc = E [PŜ Nc Ŵ c E WNf 1 Our numerica experiments show the benefits of the conditiona expectation technique on the European ca: Estimator β MLMC Compexity Vaue 2.0 Oε 2 Deta 1.5 Oε 2 Vega 2.0 Oε 2 A fraction O h of the paths arrive in the area around the strike where the conditiona expectation EPŜ N Ŵ is neither cose to 0 nor 1. In this area, its Ŝ Nf 1 sope is Oh 1/2. The coarse and fine paths differ by Oh, we thus have O h difference between the coarse and fine Greeks estimates. Reasoning as in [4] we get VŴ E WN 1... Ŵ = Oh 3/2 for the Greeks estimators. This is the convergence rate observed for Deta; the higher convergence rate of Vega is not expained yet by this rough anaysis and wi be investigated in our future research. The main imitation of this approach is that in many situations it eads to compicated integra computations. Path spitting, to be discussed next, may represent a usefu numerica approximation to this technique. 2.3 Spit pathwise sensitivities This technique is based] on the [ previous one. The idea ] is to avoid the tricky computation of E [PŜ Nf Ŵ f and E PŜ Nc Ŵ c, W Nf 2. Instead, as detaied in section 5.5 of [1], we get numerica estimates of these vaues by spitting every path simuation on the fina timestep. At the fine eve: for every simuated path Ŵ f = W 0, W 2,..., W Nf 2, we simuate a set of d fina increments W i N f 1 i [1,d] which we average to get ] E [PŜ Nf Ŵ f 1 d d PŜ Nf Ŵ f, W i N f 1 20

9 Computing Greeks using mutieve path simuation 9 At the coarse eve we use Ŵ c = W 0 + W 1,..., W Nf 4 + W Nf 3. As before sti assuming a constant drift and voatiity on the fina coarse step, we improve the convergence rate of VŶ by reusing W Nf 2 in our estimation of E [PŜ Nc ] Ŵc. We can do so by constructing the fina coarse increments as W i N c 1 i [1,d] = W Nf 2 + W i N f 1 i [1,d] and using these to estimate [ ] EPŜ Nc Ŵ c = E PŜ Nc Ŵ c, W Nf 2 1 d d P ŜNc Ŵ c, W i N c 1 To get the Greeks, we simpy compute the corresponding pathwise sensitivities. We now examine the infuence of d the number of spittings on the estimated compexity. Estimator d β MLMC Compexity Vaue Oε Oε 2 Deta Oε 2 ogε Oε 2 Vega Oε Oε 2 As expected this method yieds higher vaues of β than simpe pathwise sensitivities: the convergence rates increase and tend to the rates offered by conditiona expectations as d increases and the approximation gets more precise. Taking a constant number of spittings d for a eves is actuay not optima; for Greeks we can write the variance of the estimator as 1 VŶ = 1 VŴf E P f P c Ŵ f N + 1 N d E Ŵ V P f f P c 1 Ŵ f 21 As expained in section 2.2 we have VŴf E... Ŵ f = Oh 3/2 for the Greeks. We aso have EŴf V... Ŵ f = Oh for simiar reasons. We optimise the variance at a fixed computationa cost by choosing d such that the two terms of the sum are of simiar order. Taking d = Oh 1/2 is therefore optima.

10 10 Syvestre Burgos, Michae B. Gies 2.4 Vibrato Monte Caro Since the previous method uses pathwise sensitivity anaysis, it is not appicabe when payoffs are discontinuous. To address this imitation, we use the Vibrato Monte Caro method introduced by Gies [6]. This hybrid method combines pathwise sensitivities and the Likeihood Ratio Method. We consider again equation 15. We now use the Likeihood Ratio Method on the ast timestep and with the notations of section 2.2 we get V = E Ŵ [ ]] og pŝn Ŵ [E WN 1 P ŜN Ŵ 22 We can write pŝ N Ŵ as pµŵ,σŵ. This eads to the estimator V 1 N N m=1 µŵ [ ] m og p E WN 1 P Ŝ N µŵ Ŵ m + σ ] Ŵ m og p E WN 1 [P ŜN σŵ Ŵ m 23 We compute µ Ŵ m and σ Ŵ m with pathwise sensitivities. With Ŝ m,i N = Ŝ N Ŵ m, W i N 1, we substitute the foowing estimators into 23 ] og p E WN 1 [P ŜN µŵ Ŵ m 1 d ] og p E WN 1 [P ŜN σŵ Ŵ m 1 d d d P P Ŝm,i N Ŝm,i N Ŝ m,i N µŵ m σ 2 Ŝm,i Ŵ m 2 N µŵ m σ 3 Ŵ m 1 σŵ m In a mutieve setting: at the fine eve we can use 23 directy. At the coarse eve, for the same reasons as in section 2.3, we reuse the fine brownian increments to get efficient estimators. We take Ŵ c = W 0 + W 1,..., W Nf 4 + W Nf 3 W i N c 1 i [1,d] = W Nf 2 + W i N f 1 i [1,d] 24 We use the tower property to verify that condition 5 is verified on the ast coarse step. With the notations of equation 19 we derive the foowing estimators

11 Computing Greeks using mutieve path simuation 11 ] og pc E WNc 1 [P ŜNc Ŵ c m µŵc [ [ ] og pc = E E P ŜNc Ŵ c m, W Nf 2] Ŵ c m µŵc 1 d Ŝ m,i d N c µŵ m c N c σ 2 m Ŵ ] c og p E WNc 1 [P ŜNc σŵ Ŵ c m [ [ ] ] og p = E E P ŜNc σŵ Ŵ c m, W Nf 2 Ŵ c m 2 1 d d P N c 1 N c µŵ m c + σŵ m σ 3 c Ŵ c m 25 Our numerica experiments show the foowing convergence rates for d = 10: Estimator β MLMC Compexity Vaue 2.0 Oε 2 Deta 1.5 Oε 2 Vega 2.0 Oε 2 As in section 2.3, this is an approximation of the conditiona expectation technique, and so the same convergence rates was expected. 3 European digita ca The European digita ca s payoff is P = 1 ST >K. The discontinuity of the payoff makes the computation of Greeks more chaenging. We cannot appy pathwise sensitivities, and so we use conditiona expectations or Vibrato Monte Caro. With the same notation as in section 2.2 we compute the conditiona expectations of the digita ca s payoff. α K αc K EPŜ Nf Ŵ = Φ EPŜ Nc Ŵ c, W Nf 2 = Φ β β c The simuations give Estimator β MLMC Compexity Vaue 1.4 Oε 2 Deta 0.5 Oε 2.5 Vega 0.6 Oε 2.4

12 12 Syvestre Burgos, Michae B. Gies The Vibrato technique can be appied in the same way as with the European ca. We get Estimator β MLMC Compexity Vaue 1.3 Oε 2 Deta 0.3 Oε 2.7 Vega 0.5 Oε 2.5 The anaysis presented in section 2.2 expains why we expected β = 3/2 for the vaue s estimator. A fraction O h of a paths arrive in the area around the payoff where EPŜ N Ŵ/ Ŝ N 1 is not cose to 0 ; there its derivative is Oh 1 and we have Ŝ Nf Ŝ Nc = Oh. For these paths, we thus have O1 difference between the fine and coarse Greeks estimates. This expains the experimenta β 1/2. 4 European ookback ca The ookback ca s vaue depends on the vaues that the asset takes before expiry. Its payoff is PT = S T min S t. t [0,T ] As expained in [4], the natura discretisation P = Ŝ N minŝ n is not satisfactory. To regain good convergence rates, we approximate the behaviour within each fine timestep [t n,t n+1 ] of width h f as a simpe Brownian motion with constant drift an f and voatiity bn f conditiona on the simuated vaues Ŝn f and Ŝ f n+1. As shown in [8] we can then simuate the oca minimum Ŝ f n,min = 1 Ŝ Ŝn f + Ŝ f n+1 2 f n+1 Ŝ n f 2 2b f n 2 h f ogu n 26 n with U n a uniform random variabe on [0,1]. We define the fine eve s payoff this way choosing b f n = bŝ f n,t n and considering the minimum over a timesteps to get the goba minimum of the path. At the coarse eve we sti consider a simpe Brownian motion on each timestep of width h c = 2h f. To get high strong convergence rates, we reuse the fine increments by defining a midpoint vaue for each step Ŝn+1/2 c = 1 Ŝc 2 n + Ŝn+1 c bc n W n+1/2 W n, 27 where W n+1/2 W n is the difference of the corresponding fine Brownian increments on [t n+1/2,t n+1 ] and [t n,t n+1/2 ]. Conditiona on this vaue, we then define the minimum over the whoe step as the minimum of the minimum over each haf step, that is

13 Computing Greeks using mutieve path simuation 13 [ Ŝn,min c 1 2 = min Ŝn c + Ŝ Ŝc n+1/2 c 2 n+1/2 Ŝn c b c n 2 h c ogu 1,n, ] 1 2 Ŝ Ŝc n+1/2 c 2 + Ŝc n+1 n+1 Ŝn+1/2 c b c n 2 h c ogu 2,n 28 where U 1,n and U 2,n are the vaues we samped to compute the minima of the corresponding timesteps at the fine eve. Once again we use the tower property to check that condition 5 is verified and that this coarse-eve estimator is adequate. Using the treatment described above, we can then appy straighforward pathwise sensitivities to compute the mutieve estimator. This gives the foowing resuts: Estimator β MLMC Compexity Vaue 1.9 Oε 2 Deta 1.9 Oε 2 Vega 1.3 Oε 2 For the vaue s estimator, Gies, Debrabant and Rösser [7] have proved that VŶ = Oh 2 δ for a δ > 0, thus we expected β 2. In the Back & Schoes mode, we can prove that Deta = V /S 0. We therefore expected β 2 for Deta too. The strong convergence speed of Vega s estimator cannot be derived that easiy and wi be anaysed in our future research. Unike the reguar ca option, the payoff of the ookback ca is perfecty smooth and so therefore there is no benefit from using conditiona expectations and associated methods. 5 European barrier ca Barrier options are contracts which are activated or deactivated when the underying asset S reaches a certain barrier vaue B. We consider here the down-and-out ca for which the payoff can be written as P = S T K + 1 min t [0,T ] S t > K 29 Both the naive estimators and the approach used with the ookback ca are unsatisfactory here: the discontinuity induced by the barrier resuts in a higher variance than before. Therefore we use the approach deveoped in [4] where we compute the probabiity p n that the minimum of the interpoant crosses the barrier within each timestep. This gives the conditiona expectation of the payoff conditiona on the Brownian increments of the fine path: N f 1 P f = Ŝ f N f K + 1 p f n n=0 30

14 14 Syvestre Burgos, Michae B. Gies with p f n = exp 2Ŝ n f B + Ŝ f n+1 B+ b f n 2 h f At the coarse eve we define the payoff simiary: we first simuate a midpoint vaue Ŝ c n+1/2 as before and then define pc n the probabiity of not hitting B in [t n,t n+1 ], that is the probabiity of not hitting B in [t n,t n+1/2 ] and [t n+1/2,t n+1 ]. Thus P c = Ŝ c N c K + N c 1 n=0 1 p c n = ŜN c c K + N c 1 1 p n,1 1 p n,2 31 n=0 with 2Ŝ n c B + Ŝn+1/2 c p n,1 = exp B+ b c n 2 h f 2Ŝ n+1/2 c B+ Ŝn+1 c B+ p n,2 = exp b c n 2 h f 5.1 Pathwise sensitivities The mutieve estimators Ŷ = P f P c 1 are Lipschitz with respect to a Ŝ f n n=1...nf and Ŝ c n n=1...nc, so we can use pathwise sensitivities to compute the Greeks. Our numerica simuations give Estimator β MLMC Compexity Vaue 1.6 Oε 2 Deta 0.6 Oε 2.4 Vega 0.6 Oε 2.4 Gies proved β = 3 2 δ δ > 0 for the vaue s estimator. We are currenty working on a numerica anaysis supporting the observed convergence rates for the Greeks. 5.2 Conditiona Expectations The ow convergence rates observed in the previous section come from both the discontinuity at the barrier and from the ack of smoothness of the ca around K. To address the atter, we can use the techniques described in section 1. Since path spitting and Vibrato Monte Caro offer rates that are at best equa to those of conditiona expectations, we have therefore impemented conditiona expectations and obtained the foowing resuts:

15 Computing Greeks using mutieve path simuation 15 Estimator β MLMC Compexity Vaue 1.7 Oε 2 Deta 0.7 Oε 2.3 Vega 0.7 Oε 2.3 We see that the maximum benefits of these techniques are ony margina. The barrier appears to be responsibe for most of the variance of the mutieve estimators. Concusion and future work In this paper we have shown for a range of cases how mutieve techniques can be used to reduce the computationa compexity of Monte Caro Greeks. Smoothing a Lipschitz payoff with conditiona expectations reduces the compexity to Oε 2. From this technique we derive the Path spitting and Vibrato methods: they offer the same efficiency and avoid intricate integra computations. Payoff smoothing and Vibrato aso enabe us to extend the computation of Greeks to discontinuous payoffs where the pathwise sensitivity approach is not appicabe. Numerica evidence shows that with we-constructed estimators these techniques provide computationa savings even with exotic payoffs. So far we have mosty reied on numerica estimates of β to estimate the compexity of the agorithms. Our current anaysis is somewhat crude ; this is why our current research now focuses on a rigorous numerica anaysis of the agorithms compexity. References [1] S. Asmussen and P. Gynn. Stochastic Simuation. Springer, New York, [2] M. Broadie and P. Gasserman. Estimating security price derivatives using simuation. Management Science, 42, , [3] P. L Ecuyer. A unified view of the IPA, SF and LR gradient estimation techniques. Management Science, 36, , [4] M.B. Gies. Improved mutieve Monte Caro convergence using the Mistein scheme. In A. Keer, S. Heinrich, and H. Niederreiter, editors, Monte Caro and Quasi-Monte Caro Methods 2006, Springer-Verag, [5] M.B. Gies. Mutieve Monte Caro path simuation. Operations Research, 56, , [6] M.B. Gies. Vibrato Monte Caro sensitivities. In P. L Ecuyer and A. Owen, editors, Monte Caro and Quasi-Monte Caro Methods 2008, Springer, [7] M.B. Gies, K. Debrabant and A. Rösser. Numerica anaysis of mutieve Monte Caro path simuation using the Mistein discretisation. In preparation. [8] P. Gasserman. Monte Caro Methods in Financia Engineering. Springer, New York, [9] P.E. Koeden and E. Paten. Numerica Soution of Stochastic Differentia Equations. Springer, Berin, 1992.