Unbiased simulation of distributions with explicitly known integral transforms

Transcription

1 Unbiased simulation of distributions with explicitly known integral transforms Denis Belomestny, Nan Chen, and Yiwei Wang Abstract In this paper, we propose an importance-sampling based method to obtain an unbiased simulator to evaluate expectations involving random variables whose probability density functions are unknown while their Fourier transforms have an explicit form. We give a general principle about how to choose appropriate importance samplers under different models. Compared with the existing methods, our method avoids time-consuming numerical Fourier inversion and can be applied effectively to high dimensional financial applications such as option pricing and sensitivity estimation under Heston stochastic volatility model, high dimensional affine jumpdiffusion model, and various Levy processes. Introduction Nowadays there are three main techniques for option pricing in finance: Monte Carlo, PDE and Fourier transform methods. The use of Monte Carlo technique in option pricing has a long history going back to Boyle [2] and we refer to Glasserman [7] for an overview. In the case of Lévy-driven models, a basic building block of any Monte- Carlo method is the simulation of the increments of the underlying Lévy process. In some situations - for instance, for the Variance Gamma model - the process can be expressed in terms of subordinated Brownian motion, and hence its increments can be simulated (almost) exactly. However, in other cases no exact simulation algorithm is known. If our aim is to compute some expectation with respect to the distribution Denis Belomestny Duisburg-Essen University, Thea-Leymann-Str. 9, Essen, Germany denis.belomestny@ uni-due.de and National Research University Higher School of Economics, Moscow, Russia Nan Chen Yiwei Wang The Chinese University of Hong Kong, China nchen@se.cuhk.edu.hk, ywwang@se. cuhk.edu.hk Corresponding author

2 2 Denis Belomestny, Nan Chen, and Yiwei Wang of the Lévy increments, we can use the so-called Fourier transform approach. Since the seminal work of Carr and Madan [3] Fourier techniques have become wellestablished in computational finance to efficiently price financial instruments, like European, Asian, multi-asset or barrier options. Recently, Biagini et al. [] and Hurd and Zhou [8] extend the Fourier method to price options on several assets, considering basket options, spread options and catastrophe insurance derivatives. The Fourier methods belong to the class of numerical integration methods and such they are limited to low dimensional distributions. In this paper we propose a novel approach for computing high-dimensional integrals with respect to distributions with explicitly known Fourier transforms based on the ge! nuine combination of Fourier and Monte Carlo techniques. In order to illustrate the main idea of our approach, let us first consider a simple problem of computing expectations with respect to one-dimensional stable distributions. Let p α (x) be the density of a random variable X having a symmetric stable law with the stability parameter α (0,2), i.e., F [p α ](u) = exp( u α ). Suppose we want to compute the expectation Q = E[g(X)] for some nonnegative function g. Since there are several algorithms of sampling from stable distribution (see, e.g. [4]), we could use Monte Carlo to construct the estimate Q n = n n i= g(x i ), X,...,X n is an i.i.d. sample from the corresponding α-stable distribution. Take, for example, g(x) = (max{x,0}) β with some β (0,α), then we have by the Parseval s identity E[g(X)] = = α π g(x) x [x p α(x)]dx 0 u α exp( u α )Im[F [g( )/ ](u)] du, = α Γ (β)sin(βπ/2) π 0 u α β exp( u α )du provided α >. This means that E[g(X)] = E[g (X )], X has a power exponential distribution with the density f α (x) = Γ ( + /α) exp( xα ) Γ (/α)γ (β)sin(βπ/2) and g (x) = C(α,β) x α β with C(α,β) = π. In particular, if β = α, then Var[g(X)] > B(2 α) for some B > 0 not depending on α, while Var[g (X )] = 0. This shows that even in the above very simple situation, moving to the Fourier domain can significantly reduce the variance of Monte Carlo estimates. More importantly, by using our approach, we replace the problem of sampling from Page: 2 job: author_9.09_final macro: svmult.cls date/time: 9-Sep-204/5:03

3 Unbiased simulation of distributions with explicitly known integral transforms 3 the stable distribution p α by a much simpler problem of drawing from the exponential power distribution f α. Of course, the main power of Monte Carlo methods can be observed in high-dimensional integration problems, which will be considered in the next section. 2 General framework Let g be a real-valued function on and let p be a probability density on. Our aim is to compute the integral of g with respect to p : V = g(x)p(x)dx. Suppose that there is a vector R, such that g(x)e x,r L ( ), f (x)e x,r L ( ), then we have by the Parseval s formula V = (2π) d F [g](ir u)f [p](u ir)du. () Rd Let q be a probability density function with the property that q(x) = 0 whenever F [p](u ir) = 0. That is, q has the same support as F [p](u ir). Then we can write V = [p](u ir) (2π) d F [g](ir u)f q(u)du = E q [h(x)], (2) Rd q(u) h(x) = [p](x ir) F [g](ir x)f. (2π) d q(x) and X is a random variable distributed according to q. The variance of the corresponding Monte Carlo estimator is given by ( ) 2d F [p](u ir) 2 Var q [h(x)] = F [g](ir u) 2 du V 2. 2π Rd q(u) Note that the function F [p](ir u) is, up to a constant, a probability density and in order to minimize the variance, we need to find a density q, that minimizes the ratio F [p](u ir) q(u) and that we are able to simulate from. In the next section, we discuss how to get a tight upper bound for F [p](ir u) in the case of an infinitely divisible distribution p, corresponding to the marginal distributions of Lévy processes. Such Page: 3 job: author_9.09_final macro: svmult.cls date/time: 9-Sep-204/5:03

4 4 Denis Belomestny, Nan Chen, and Yiwei Wang a bound can be then used to find a density q leading to small values of variance Var q [h(x)]. 3 Lévy processes Let (Z t ) be a pure jump d-dimensional Levy process with the characteristic exponent ψ, that is [ ] E e i u,z t = e tψ(u), u. Consider the process X t = ΛZ t, Λ is a real m d matrix. Let a vector R R m be such that ν R (dz). = e Λ R,z ν(dz) is again a Lévy measure, i.e. ( z 2 ) ν R (dz) <. Suppose that there exist a constant C ν > 0 and a real number α (0,2), such that, for sufficiently small ρ > 0, the following estimate holds {z R: z,h ρ} z,h 2 ν R (dz) C ν ρ 2 α, h, h =. (3) Lemma. Suppose that (3) holds, then there exists constant A R > 0 such that, for any u R m and sufficiently large Λ u, ( φ t (u ir) A R exp 2tC ) ν π 2 Λ u α, φ t (z). = E [ e i z,x t ]. Proof. For any u R m, we have ( Rd ] φ t (u ir) = exp t [ e Λ R,z cos( Λ u,z ) + Λ R,z { z } ( Rd ] ) = exp t [ e Λ R,z + Λ R,z { z } ν(dz) ( Rd [ ] ) exp t e Λ R,z { cos( Λ u,z )} ν(dz) ( ) = A R exp t { cos( Λ u,z )} ν R (dz), ) ν(dz) ( Rd ) ) A R = exp t (e Λ R,z Λ R,z { z } ν(dz) <, Page: 4 job: author_9.09_final macro: svmult.cls date/time: 9-Sep-204/5:03

5 Unbiased simulation of distributions with explicitly known integral transforms 5 since e Λ R,z Λ R,z { z } C (Λ R) z 2 { z } +C 2 (Λ R)e Λ R,z { z >}. First, note that the condition (3) is equivalent to the following one z,k 2 ν R (dz) C ν k α, {z R: z,k } for sufficiently large k, say k c 0. To see this, it is enough to change in (3) the vector h to the vector ρk. Fix u R m with u and Λ u c 0, then using the inequality cos(x) 2 x 2, x π, we find π 2 ( cos( Λ u,z )) ν R (dz) 2 π 2 Λ u,z 2 ν R (dz) {z R: Λ u,z } 2C ν π 2 Λ u α. Discussion Let us comment on the condition (3) (for simplicity we take R = 0). Clearly, if (Z t ) is a d-dimensional α-stable process which is rotation invariant (ψ(h) = c α h α, for h ), then (3) holds. Consider now general α-stable processes. It is known that Z is α-stable if and only if its components Z,...,Z d are α-stable and if the Levy C copula of Z is homogeneous of order, i.e. C (r ξ,...,r ξ d ) = r C (ξ,...,ξ d ) for all ξ = (ξ,...,ξ d ) and r > 0. As an example of such homogeneous Levy copula one can consider C (ξ,...,ξ d ) = 2 2 d ( d j= ) /θ ξ j θ ) (ηξ... ξd 0 ( η) ξ... ξd<0, θ > 0 and η [0,]. If the marginal tail integrals given by with Π j (x j ) = ν (R,...,I (x j ),...R)sgn(x j ) I (x) = { (x, ), x 0, (,x], x < 0, are absolutely continuous, we can compute the Lévy measure ν for the Lévy copula C by differentiation as follows: Page: 5 job: author_9.09_final macro: svmult.cls date/time: 9-Sep-204/5:03

6 6 Denis Belomestny, Nan Chen, and Yiwei Wang ν(dx,...,dx d ) =... d C ξ =Π (x ),...,ξ d =Π d (x d ) ν (dx )... ν d (dx d ), ν (dx ),...,ν d (dx d ) are the marginal Lévy measures. Suppose that the marginal Lévy measures are absolutely continuous with a stable-like behaviour: ν j (dx j ) = k j (x j )dx j = l j( x j ) x j +α dx j, j =,...,d, l,...,l d are some nonnegative bounded nonincreasing functions on [0, ) with l j (0) > 0 and α [0,2]. Then ν(dx,...,dx d ) = G(Π (x ),...,Π d (x d ))k (x )... k d (x d )dx...dx d with G(ξ,...,ξ d ) =... d C ξ...,ξ d. Note that for any r > 0, k j (rx j ) = r α k j (x j,r), Π j (rx j ) = r α Π j (x j,r), j =,...,d, k j (x j,r) = l j(rx j ) x x j +α, Π j j(x j,r) = {x j 0} k j (s,r) ds + {x j <0} k j (s,r) ds. x j Since the function G is homogeneous with order d, we get for ρ (0,), {z R: z,h ρ} z,h 2 ν(dz) = ρ 2 α {z R: y,h } y,h 2 G ( Π (y,ρ),...,π d (y d,ρ) ) k (y,ρ)... k d (y d,ρ)dy...dy d ρ 2 α y,h 2 G ( Π (y,),...,π d (y d,) ) {z R: y,h } k (y,)... k d (y d,)dy...dy d and the condition (3) holds, provided inf z,h 2 ν(dz) > 0. h: h = {z R: z,h } If for some R = (R,..., ) the fucntions e xr il i (x), i =,...,d, are bounded, the condition (3) holds for ν R (dz). = e R,z ν(dz). 4 Positive definite densities Let p be a probability density on, which is positive definite. For example, all symmetric infinite divisible absolute continuous distributions have positive definite densities. Let furthermore g be a nonnegative integrable function on. Suppose Page: 6 job: author_9.09_final macro: svmult.cls date/time: 9-Sep-204/5:03

7 Unbiased simulation of distributions with explicitly known integral transforms 7 that we want to compute the expectation V = E p [g(x)] = g(x)p(x)dx. We have by the Parseval s identity V = (2π) d F [g](x)f [p](x)dx. Rd Note that p (x) = F [p](x)/((2π) d p(0)) is a probability density and therefore we have another dual representation for V : (Remark: Dividing by (2π) d ) makes p a density) V = E p [g (X)] with g (x) = p(0)f [g](x). Let us compare the variances of the random variables g(x) under X p and g (X) under X p. It holds and Var p [g(x)] = g2 (x)p(x)dx V 2 Var p [g (X)] = p(0) (2π) d F [g](x) 2 F [p](x)dx V 2 R d = p(0) (g g)(x)p(x)dx V 2, (g g)(x) = g(x + y)g(y)dy. As a result, Var p [g(x)] Var p [g [ (X)] = g 2 (x) p(0)(g g)(x) ] p(x)dx. Note that if p(0) > 0 is small, then it is likely that Var p [g(x)] > Var p [g (X)]. This means that estimating V under p with Monte Carlo can be viewed as a variance reduction method in this case. Apart from the variance reduction effect, the density p may have in many cases (for example, for infinitely divisible distributions) much simpler form than p and therefore is easy to simulate from. Page: 7 job: author_9.09_final macro: svmult.cls date/time: 9-Sep-204/5:03

8 8 Denis Belomestny, Nan Chen, and Yiwei Wang 5 Numerical examples European Call option under CGMY model The CGMY process (X t ) with drift µ is a pure jump Lévy with the Lévy measure [ exp(gx) ν CGMY (x) = C x +Y x<0 + exp( Mx) ] x +Y x>0, C,G,M > 0, 0 < Y < 2. As can be easily seen, the Lévy measure ν CGMY satisfies the condition (3) with α = Y. The characteristic function of X T is given by φ(u) = E[e iux T ] = exp { iµut + TCΓ ( Y )[(M iu) Y M Y + (G + iu) Y G Y ] }, µ = r CΓ ( Y )[(M ) Y M Y + (G + ) Y G Y ensures that (e rt e X T ) is a martingale. Suppose the stock price follows the model S t = S 0 e X t, then due to (), for any R >, the price of the European call option is given by F [g](ir u) = e rt E[(S T K) + ] = e rt 2π F [g](ir u)f [p](u ir)du, (4) K R e iulnk (iu + R )(iu + R), F [p](u ir) = ei(u ir)lns0 E[e i(u ir)x T ]. (Remark: Sorry the original expression is its conjugate.) Lemma implies F [p](u ir) Ae u α θ for α Y, some A > 0, θ > 0 and large enough u. So we can use the exponential power density u α q(u) = 2θ α Γ ( + α θ )e as the important sampler in (2), the parameter θ can be chosen by minimizing the simulated second moment. Feng, [6] used the parameters C =, G = 5, M = 5, Y = 0.5 to calculate the price of the European call option via numerical inversion of the corresponding characteristic function. The obtained option price was Our numerical results are shown in Table. Page: 8 job: author_9.09_final macro: svmult.cls date/time: 9-Sep-204/5:03

9 Unbiased simulation of distributions with explicitly known integral transforms 9 Table Call option in CGMY model, exponential power IS (R = 4.5,α = 0.49,θ = 0.4) No. of simulation Price 95%-interval RMSE Time(s) 00, [9.7793,9.8520] , [9.7927,9.8292] ,600, [9.8036,9.828] European Put option under NIG model The Normal Inverse Gaussian Lévy process can be constructed by subordinating BM with an Inverse Gaussian process: X t (α,β,δ) = βt t (ν,δ) +W(T t (ν,δ)), α = β 2 + ν 2 and T t (ν,δ) is the Inverse Gaussian Lévy process defined by T t (ν,δ) = inf{s > 0 : νs + B s = δt}. We have ( ]) E[e iux t ] = exp δt[ α 2 β 2 α 2 (β + iu) 2 and the corresponding Lévy measure ν NIG fulfils the condition (3) with α =. Suppose the stock price is modelled by S t = S 0 e at+x t, the choice a = r q δ( α 2 β 2 α 2 (β + ) 2 ) ensures the martingale condition. Then for any R < 0, the price of European put option is given by e rt E[(K S T ) + ] = e rt 2π F [g](ir u)f [p](u ir)du, F [g](ir u) = K R e iulnk (iu + R )(iu + R), F [p](u ir) = ei(u ir)(lns 0+aT ) E[e i(u ir)x T ]. Lemma implies that one can use the Laplace density q(u) = u e θ 2θ as the important sampler, the parameter θ can be chosen by minimizing the simulated second moment. Chen, Feng and Lin [5] used parameters α = 5, β = 5, δ = 0.5, r = 0.03, S 0 = K = 00, T = 0.5 to calculate the price of the European put option and obtained the value Table 2 shows numerical results with the same parameters and compares the RMSE and the computational time of our method with that of the method direct simulating the subordinator. Page: 9 job: author_9.09_final macro: svmult.cls date/time: 9-Sep-204/5:03

10 0 Denis Belomestny, Nan Chen, and Yiwei Wang Table 2 Put option in NIG model, Exponential power IS (R = 9.3, θ = 2.4) No. of simulation Price 95%-interval RMSE Time RMSE (Direct) Time 00, [4.5879,4.5922] , [4.5886,4.5907] ,600, [4.589,4.5902] Barrier option under CGMY model The payoff function of barrier option with m monitoring time points is (S T K) + {L St...S tm U}, S t = S 0 e X t and 0 < t <... < t m < T. According to the Parseval s Theorem, the option price is equal to e rt (2π) m+ F [g](ir u)f [p](u ir)du, Rm+ u = (u,...,u m+ ), R = (0,...,0,R), e ( iu m+ R+)lnK e (iu m+ iu m +R)lnU e (iu m+ iu m +R)lnL F [g](ir u) = (iu m+ + R )(iu m+ + R) iu m+ iu m + R e (iu m iu m )lnu e (iu m iu m )lnl iu m iu m e(iu2 iu)lnu e (iu2 iu)lnl iu 2 iu 2 and F [p](u ir) = e iu [ m lns 0 φ (u j ) ] φ (u m+ ir). j= We use parameters C =, G = 5, M = 5, Y =.5, r = 0., s = 00, K = 00, T = 2, U = 05, L = 95 and calculate the price of the barrier option when there is only one monitoring time at t =. The benchmark price calculated by numerical integration is We use the method described in Section 2 with the importance sampler of the form h(u,u 2 ) = u α u 2 α 2 2θ /α Γ ( + α ) e θ 2θ /α 2 2 Γ ( + α 2 ) e θ 2 α =.5, θ = 0.9, α 2 =.5, θ 2 = 0.25 and the damping factor R =.06. The results are presented in Table 3. Acknowledgements The second and third authors are grateful for the financial support of a GRF grant from HK SAR government (Grant ID: CUHK43). Page: 0 job: author_9.09_final macro: svmult.cls date/time: 9-Sep-204/5:03

11 Unbiased simulation of distributions with explicitly known integral transforms Table 3 Barrier option in CGMY model, exponential power IS (α =.5, θ = 0.9, α 2 =.5, θ 2 = 0.25, R =.06) No. of simulation Price 95%-interval RMSE Time(s) 6,400, [2.02,2.524] ,600, [2.2349,2.4450] References. Biagini, F., Bregman, Y., Meyer-Brandis, T.: Pricing of catastrophe insurance options written on a loss index with reestimation. Insurance: Mathematics and Economics 43(2), (2008) 2. Boyle, P.P.: Options: A Monte Carlo approach. Journal of financial economics 4(3), (977) 3. Carr, P., Madan, D.: Option valuation using the fast Fourier transform. Journal of computational finance 2(4), 6 73 (999) 4. Chambers, J.M., Mallows, C.L., Stuck, B.: A method for simulating stable random variables. Journal of the American Statistical Association 7(354), (976) 5. Chen, Z., Feng, L., Lin, X.: Simulating Lévy processes from their characteristic functions and financial applications. ACM Transactions on Modeling and Computer Simulation (TOMACS) 22(3), 4 (202) 6. Feng, L., Lin, X.: Inverting analytic characteristic functions and financial applications. SIAM Journal on Financial Mathematics 4(), (203) 7. Glasserman, P.: Monte Carlo methods in financial engineering, vol. 53. Springer (2004) 8. Hurd, T.R., Zhou, Z.: A Fourier transform method for spread option pricing. SIAM Journal on Financial Mathematics (), (200) Page: job: author_9.09_final macro: svmult.cls date/time: 9-Sep-204/5:03