ACTA UNIVERSITATIS APULENSIS No 16/2008 RISK MANAGEMENT USING VAR SIMULATION WITH APPLICATIONS TO BUCHAREST STOCK EXCHANGE. Alin V.

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1 ACTA UNIVERSITATIS APULENSIS No 16/2008 RISK MANAGEMENT USING VAR SIMULATION WITH APPLICATIONS TO BUCHAREST STOCK EXCHANGE Aln V. Roşca Abstract. In a recent paper, we have proposed and analyzed, from a theoretcal pont of vew, a multdmensonal stock market model (see [?]). In ths paper, we construct a portfolo of stocks for a partcular case of ths market model. We ntroduce the Value at Rsk, as a powerful tool for managng rsks, whch follow from holdng such a portfolo. We present a mathematcal calculaton of Value at Rsk for our market model. Usng ths mathematcal framework, we develop Monte Carlo, Quas-Monte Carlo and Mxed Monte Carlo and Quas-Monte Carlo algorthms for the estmaton of Value at Rsk. We apply the developed methods to portfolos from Bucharest Stock Exchange Mathematcs Subject Classfcaton: 11K36, 65C05, 91B28, 91B30, 91B70. Keywords and phrases: Value at Rsk, Stock Market, Monte Carlo method, Quas-Monte Carlo method, Mxed Monte Carlo and Quas-Monte Carlo Method. 1. Introducton In a recent paper (see [9]), we have ntroduced a multdmensonal stock market model and analyzed some mportant features of t. We have consdered n stock prces S (t), = 1, n, drven by a multdmensonal Brownan moton process B(t) = (B 1 (t), B 2 (t),..., B n (t)) 0 t T on some probablty space (Ω, F, P), together wth the fltraton generated by B(t), denoted by {F(t)} 0 t T. The stock prce processes were defned as follows: 23

2 ds 1 (t) = S 1 (t)[µ 1 (t)dt + σ 1 (t)db 1 (t)], (1) ds (t) = S (t)[µ (t)dt + λσ (t)db 1 (t) + 1 λ 2 σ (t)db (t)], = 2, n,(2) where λ s a real parameter, such that 1 λ 1 and σ(t) = (σ 1 (t),..., σ n (t)) > 0. In ths paper, we assume that λ = 0. We therefore get the followng market model: ds (t) = S (t)[µ dt + σ db (t)], = 1, n, (3) where the drfts µ, = 1, n, and the volatltes σ, = 1, n, are assumed to be constant over tme. In what follows, we consder a portfolo consstng of n rsky assets, whch, n our case, are the n stocks defned n (3). If we denote by p, = 1, n, the number of postons we hold on asset, = 1, n, then we can defne the portfolo value at tme t as V (t) = n p S (t). (4) =1 Holdng such a portfolo of stocks s a rsky busness due the market fluctuatons. As we can expect huge losses from havng a portfolo, we need a powerful tool to measure fnancal rsks. Value at Rsk (VaR) s such a tool for managng rsk n fnancal nsttutons. VaR traces hs roots from the great fnancal dsasters from early 1990s. The valuable lesson that we learned was that poor supervson and rsk management can lead to huge losses n tens of mllons of dollars. The VaR hstory s closely connected wth the name of the Investment Bank J.P. Morgan. Its presdent Denns Weatherstone, n ntenton to evaluate the total rsk hs frm s exposed to, asked to hs drectors to present hm daly a brefng on the fnancal rsk of the company. RskMetrcs Department developed such a rsk measure, wdely used today among fnancal nsttutons, whch they called Value at Rsk (VaR). The defnton of VaR s the maxmum loss that wll occur, over a target horzon, n normal market condtons, wth a certan confdence level (see [4]). For example, a daly VaR of $ at 99% confdence level suggests a 24

3 1 n 100 chance for a loss grater than to occur any sngle day. VaR s a very useful number, as t translates all the complcated market rsk factors nto a sngle number, n a currency, whch everybody can understand. As our portfolo s composed only from shares of stocks, t s mportant to make the followng remark on the market model. The process B(t) s the Brownan moton observed for the assets n the market under the measure P, nduced by the market. In [9], we defned a rsk-neutral probablty measure P and an n-dmensonal Brownan moton under ths rsk-neutral probablty measure, denoted by B(t) = ( B 1 (t),..., B n (t)). Usng ths new defned Brownan moton, the stock prce dynamcs can be expressed as ds = S (rdt + σ d B ), = 1, n, (5) where r s the rsk-free rate. It s mportant to note that the rsk-free nterest rate s used only wth opton prcng. The future values of the stocks should be modelled usng µ, = 1, n, and hence, the market model (3). The parameter µ s replaced wth the rsk-free rate r, only n rsk-neutral valuaton of optons. However, we are not tryng to create a martngale, but model the future behavor of our portfolo. Ths s true for Value at Rsk models, where we are nterested n the future state of the portfolo, not n the present value. Hence, n our Monte Carlo, Quas-Monte Carlo and Mxed Monte Carlo and Quas-Monte Carlo smulatons, we are gong to smulate the real prces of stocks, descrbed n relatons (3). The remanng part of the paper s organzed as follows. In Secton 2, we present a detaled mathematcal calculaton of Value at Rsk for our market model. Usng ths mathematcal framework, we develop Monte Carlo (MC), Quas-Monte Carlo (QMC) and Mxed MC and QMC algorthms for estmaton of Value at Rsk. In Secton 3, we apply the developed methods to two portfolos of stocks from Bucharest Stock Exchange. 2. Monte Carlo Smulaton of VaR There are a varety of methods for computng Value at Rsk. Three of them are shortly summarzed bellow: 1. Delta-gamma approxmaton In ths method (see [3]), t s assumed that all assets are lognormal dstrbuted and reles on hstorcal data, n order to estmate the parameters: means, standard devatons, correlatons and portfolos senstvtes to each of the rsk factors. Ths method s computatonally 25

4 effcent and easest to mplement. However, t gves a poor estmaton for portfolos contanng assets wth hghly non-lnear response to rsk factors. 2. Hstorcal smulaton Hstorcal Smulaton (see [3]) takes a portfolo of assets at a partcular pont n tme and revalues the portfolo a number of tmes, usng a hstory of prces for the assets n the portfolo. The portfolo revaluatons produce a dstrbuton of proft and losses, whch can be examned n order to determne the VaR, wth a chosen level of confdence. The man crtcsm of ths approach s the assumpton that the past can predct the future accurately. Ths method also reles heavly on the tme horzon that s used to capture hstorcal data. 3. Monte Carlo smulaton Monte Carlo smulaton (see [1], [3] and [8]) s a good alternatve to the above two methods because t can handle any non-lnear portfolos and can accommodate any type of dstrbuton of rsk factors. Ths approach smulates possble prce paths, for each of the assets, and values the portfolo. After many smulatons, VAR can be calculated drectly from the smulated dstrbuton of portfolo value change. However, ths method s computatonally ntensve. In ths paper, we focus on the last method: MC smulaton. We wll also use the methods of QMC and Mxed MC and QMC to calculate VaR estmatons. The SDE equaton (3) can be solved usng Ito s theorem (see [10]). Ones obtans S (t) = S (0)e (µ 1 2 σ2 )t+σ B (t), = 1, n. (6) Ths process s called a Geometrc Brownan Moton (see [6]). If we want to smulate ths stochastc process, then the stock prce at tme t s gven by S (t) = S (0)e (µ 1 2 σ2 )t+σ tx (), = 1, n, (7) where x () N(0, 1), = 1, n, are standard normal random varables and t s the holdng tme. Let S(t) = (S 1 (t), S 2 (t),..., S n (t)) T be a vector that contans the values of the stocks at tme t and S(0) = (S 1 (0), S 2 (0),..., S n (0)) T be a vector 26

5 that contans the ntal values of the stocks. Let σ = (σ 1, σ n,..., σ n ) T be the volatlty vector and µ = (µ 1, µ 2,..., µ n ) T the drft vector. Wth these notatons, we can rewrte relatons (7) n matrx form, as follows: S(t) = S(0)e (µ 1 2 σ2 )t+σ t. x, (8) where the symbol. s used for element by element multplcaton and x = (x (1), x (2),..., x (n) ) T s a vector of standard normal varables. Clearly, f the stocks S, = 1, n, are all ndependent, the collecton of stocks can be generated drectly, usng formula (8). But nstead of usng n ndependent Wener processes B (t), to represent the returns, the market model requres correlated underlyng processes. Ths s an mportant assumpton, snce n practce, the stock prces from the market are n general correlated. Let us consder the correlated processes Z 1, Z 2,..., Z n, wth the correlaton matrx C = (ρ j ),j=1,n. We also consder the correspondng covarance matrx Σ. Hence, we are gven wth ds = S (µ dt + σ dz ), = 1, n, (9) where Z, = 1, n, are correlated Brownan motons, wth correlaton matrx C. As a whole, the trends wll be apparent, snce the processes Z are correlated accordng to matrx C. We rewrte the SDE from (9) n the followng form: ds = S (µ dt + n σ j db j ), = 1, n. (10) j=1 Our objectve s to determne the matrx A = (σ,j ),j=1,n, such that AA T = Σ. Proposton 1 If V s an n-dmensonal dagonal matrx such that (V ) = σ, = 1, n, and AA T = Σ, then there s a lower trangular matrx L, such that A = V L. Proof. Because C s a correlaton matrx, t follows that t s a symmetrc, postve defnte matrx. Hence, t has a Cholesky decomposton of the form C = LL T, where L s a lower trangular matrx. We have Hence, we obtan that A = V L. AA T = Σ = V CV = V LL T V = V L(V L) T. 27

6 From Proposton 1, we obtan ( ds = S (µ dt + σ dz ) = S µ dt + σ l j db j ), = 1, n, (11) where (l j ),j=1,n are the components of the lower trangular matrx L. Hence, n order to capture the correlatons among the stocks, we wll replace relaton (8) wth j=1 S(t) = S(0)e (µ 1 2 V σ)t+v Lx t. (12) If the value of the portfolo at tme t s V (t), the holdng perod s t, and the value of the portfolo at tme t + t s V (t + t), then the loss n the portfolo value s defned as Loss = V (t) V (t + t). (13) Havng defned the Loss random varable, we present the Value at Rsk defnton. Defnton 2 (Value at Rsk) For a gven probablty α, the VaR, denoted by δ α, s defned by the followng relaton: P (Loss δ α ) = α. (14) Typcally, the nterval t s fxed to one day or two weeks, and the confdence level α s close to zero, often α = 0.01 or α = In the statstcal termnology, VaR s nothng but the (1 α) th quantle of the Loss dstrbuton. In what follows, we present the mathematcal framework for VaR estmaton, based on MC method. The relaton (14) can be wrtten as or 1 P (Loss < δ α ) = α, (15) F (δ α ) = 1 α = β, (16) where F denotes the (unknown) cumulatve dstrbuton functon (cdf) of random varable Loss. The VaR can be expressed n terms of the nverse cdf, as follows: δ α = F 1 (β) = nf{y F (y) β}. (17) 28

7 For a gven y, the cdf F (y) can be expressed as an expectaton F (y) = E[1 {Loss y} ] = E[1 {V (0) V (t) y} ] = E[1 {V (0) n =1 p S (t) y}] = E[1 {V (0) n =1 p S (0)e (µ 1 2 σ2 )t+σ ], t j=1 l j x j y} where 1 { } s an ndcator functon, whch returns 1 when the relaton { } s true and 0 otherwse. We denote by f(x (1),..., x (n) ) the term 1 {V (0) n =1 p S (0)e (µ 2 1 σ2 )t+σ t j=1 l j x j y} n the last equalty. It follows that F (y) = f(x (1),..., x (n) )dφ(x (1),..., x (n) ) = I, (18) R n where Φ(x (1),..., x (n) ) s a dstrbuton functon on R n, whch can be factored Φ(x (1),..., x (n) ) = Ψ 1 (x (1) )... Ψ n (x (n) ), and Ψ (x () ), = 1, n, represents the standard normal cumulatve dstrbuton functon, denoted by Ψ. We have denoted the last ntegral by I. Usng the MC method, I s estmated by sums of the form Î MC K = 1 K K =1 f(x (1),..., x (n) ), (19) where x = (x (1),..., x (n) ), 1, are ndependent dentcally dstrbuted random ponts on R n, wth the common dstrbuton functon Φ(x (1),..., x (n) ). Another representaton of ths estmaton, n terms of the Loss dstrbuton, s Î K = 1 K 1 {Loss y}, (20) K =1 where {Loss, = 1, K} are samples from the Loss dstrbuton. Sortng the samples {Loss, = 1, K} n ncreasng order, we obtan Loss (1) Loss (2)... Loss (K). (21) 29

8 Then the correspondng sample cumulatve dstrbuton functon s 0 f y < Loss (1) F K (y) = f Loss K () y < Loss (+1), = 1,..., K 1. (22) 1 f y Loss (K) From relatons (19), (20) and (22), we mmedately deduce that y = Loss ([Kβ]) gves F K (y) = β, whch satsfes the defnton of VaR. The algorthm whch generates VaR s presented next. Algorthm 3 VaR Generaton by Monte Carlo Smulaton Method Input data: The ntal stock prces vector S(0) = (S 1 (0),..., S n (0)) T, the horzont tme t, the number of smulatons K and the confdence level α. Step 1. for = 1,..., K do 1.1. Generate a random pont x = (x (1),..., x (n) ) T on R n, wth ndependent dentcally dstrbuted components, each component havng the common dstrbuton functon Ψ Generate the stock prces at tme t, usng formula (12) S (t) = S(0)e (µ 1 2 σ2 )t+σ t. x, (23) where S (t) = (S,1 (t),..., S,n (t)) T Determne the portfolo value at tme t, usng formula (4) V (t) = n p l S,l (t). l= Determne the Loss dstrbuton sample as Loss (t) = V (0) V (t), where V (0) s the value of the portfolo at ntal tme. end for Step 2. Sort the vector Loss = (Loss 1 (t),..., Loss K (t)) n ascendng order,.e. Loss (1) (t) Loss (2) (t)... Loss (K) (t). Output data: V ar = Loss [K(1 α)] (t). 30

9 In order to generate a pont x from Step 1.1, we proceed as follows. We frst generate a random pont ω = (ω (1),..., ω (n) ), where ω (l) s unformly dstrbuted on [0, 1], for each l = 1,..., n. Then, for each component ω (l), l = 1,..., n, we apply the nverson method and obtan that Ψ 1 l (ω (l) ) = x (l) s a random pont wth the dstrbuton functon Ψ. A smlar algorthm can be obtaned for the QMC smulaton method. Frst, we have to transform the ntegraton doman to [0, 1] n. For ths, we use the substtuton Ψ 1 (z () ) = x (), = 1, n, and we obtan I = f(x (1),..., x (n) )dψ 1 (x (1) )... dψ n (x (n) ) R n = f(ψ 1 1 (z (1) ),..., Ψ 1 n (z (n) ))dz (1)... dz (n) [0,1] n = g(z (1),..., z (n) )dz (1)... dz (n). [0,1] n In the last equalty, we have denoted f(ψ 1 1 (z (1) ),..., Ψ 1 n (z (n) ))byg(z (1),..., z (n) ). Usng the QMC method, the ntegral I s estmated by sums of the form Î QMC K = 1 K K =1 g(z (1),..., z (n) ), (24) where (z ) 1 = (z (1),..., z (n) ) 1 s a low-dscrepancy sequence on [0, 1] n. If we replace n Step 1.1 of the Algorthm (3) the random ponts x, = 1, K, wth the low-dscrepancy sequence (z ) 1 = (z (1) the ponts x, = 1, K, from formula (23) wth (v ) 1 = (Ψ 1 1 (z (1),..., z (n) ),..., Ψ 1 n (z (n) )) 1, ) 1 on [0, 1] n, and then we obtan a QMC Algorthm. Durng our experments, we employed as low-dscrepancy sequences on [0, 1] n the Halton sequences (see [2] and [5]). The Mxed MC and QMC method gves the followng estmate: Î MIX K = 1 K K =1 g(q (1),..., q (d) 31, z (d+1),..., z (n) ), (25)

10 where (m ) 1 = (q, z ) 1 s an n-dmensonal mxed sequence on [0, 1] n (see [7]). Frst, we generate a low-dscrepancy sequence (q ) 1, on [0, 1] d, then we generate the ndependent and dentcally dstrbuted random ponts z, 1, on [0, 1] n d. Fnally, we concatenate q and z, for each 1, and we get our mxed sequence on [0, 1] n. In our experments, we used as low-dscrepancy sequences on [0, 1] d for the mxed sequences, the Halton sequences (see [2] and [5]). If we replace n Step 1.1 of the Algorthm (3) the random ponts x, = 1, K, wth the mxed sequence (m ) 1 = (q, z ) 1 on [0, 1] n, and the ponts x, = 1, K, from formula (23) wth (v ) 1 = (Ψ 1 1 (q (1) ), Ψ 1 d (q(d) ), Ψ 1 d+1 (z(d+1) then we obtan a Mxed MC and QMC algorthm. )..., Ψ 1 n (z (n) )) 1, 3. Applcaton of VaR to portfolos from Bucharest Stock Exchange In ths secton, we determne Value at Rsk for two portfolos of stocks from Bucharest Stock Exchange. Frst, we estmate the market model parameters vectors: the drft vector µ and the volatlty vector σ. Then, we estmate the correlaton matrx C. All the estmatons are obtaned based on the log-returns seres calculated as follows. For each stock S, = 1, n, the j-th entry of the return sere R, = 1, n, s R j = log ( S (t j+1 ) S (t j ) ) t j+1 t j, j = 1, M 1, (26) where M s the number of observatons of each of the stock prce seres. The data used for our estmatons are the stock prces from untl The closng prces for each stock are on daly base and can be obtaned freely from the nternet ste of the Bucharest Stock Exchange, The true value for VaR s obtaned from a long MC smulaton of paths for the stock processes n our market model. As ntal prce for each stock path smulaton, we consder the closng prce from the day The VaR s calculated over a horzont tme of 1 day, respectvely 10 days, wth α =

11 We denote by V ar MC, V ar QMC and V ar MIX the outputs of our MC, QMC and Mxed MC and QMC algorthms, respectvely. The estmatons V ar MC and V ar MIX are calculated as an average of m smulaton runs V ar MC(MIX) = 1 m m =1 V ar MC(MIX). (27) We also gve the sample standard devaton ( 1 m ( MC(MIX) s = V ar V ar MC(MIX) ) ) 1 2 2, (28) m 1 =1 where V ar MC(MIX) represents the estmate from run, = 1, m. We use the sample standard devaton to analyze the varance reducton effects. We fx the number of ndependent runs to m = VaR estmaton for Portfolo 1 We assume that Portfolo 1, denoted by Π 1, contans the stocks of two companes: BANCA TRANSILVANIA S.A. (Symbol TLV) and BRD - Groupe Socete Generale S.A. (Symbol BRD), two of the most lqud companes of the Bucharest Stock Exchange market. We hold 150 shares of each company. The parameters of the stock market model are estmated usng Matlab bultn functons and are gven bellow: 1(TLV) 2(BRD) S (0) µ σ Table 1: Parameters of Portfolo Π 1. The estmated correlaton matrx s ( ) C = We consder d = 1 for the Mxed estmate (25). The results of our smulatons are compared n the followng two tables, n terms of ther relatve errors and standard devaton. 33

12 Portfolo Π 1 True Value Smulatons K=10000 K=15000 K=20000 Relatve Error V ar MC Std MC Relatve Error V ar MIX Std MIX Relatve Error V ar QMC Table 2: 1-day VaR smulaton results. Portfolo Π 1 True Value Smulatons K=10000 K=15000 K=20000 Relatve Error V ar MC Std MC Relatve Error V ar MIX Std MIX Relatve Error V ar QMC Table 3: 10-day VaR smulaton results. We see that n all methods, the sample standard devaton Std decreases, as K ncreases from to VaR estmaton for Portfolo 2 We assume that Portfolo 2, denoted by Π 2, contans the stocks of 5 companes: BANCA TRANSILVANIA S.A. (Symbol TLV), BRD - GROUPE SOCIETE GENERALE S.A. (Symbol BRD), ROMPETROL RAFINARE S.A. (Symbol RRC), PETROM S.A. (Symbol SNP) and C.N.T.E.E. TRANS- ELECTRICA (Symbol TEL). We hold 100 shares of each company. The parameters of the stock market model are estmated usng Matlab bult-n functons and are as follows: 1(TLV) 2(BRD) 3(RRC) 4(SNP) 5(TEL) S (0) µ σ

13 Table 4: Parameters of Portfolo Π 2. The estmated correlaton matrx s C = We consder d = 3 for the Mxed estmate (25). The results of our smulatons are compared n the followng two tables, n terms of ther relatve errors and standard devaton. Portfolo Π 2 True Value Smulatons K=10000 K=15000 K=20000 Relatve Error V ar MC Std MC Relatve Error V ar MIX Std MIX Relatve Error V ar QMC Table 5: 1-day VaR smulaton results. Portfolo Π 2 True Value Smulatons K=10000 K=15000 K=20000 Relatve Error V ar MC Std MC Relatve Error V ar MIX Std MIX Relatve Error V ar QMC Table 6: 10-day VaR smulaton results. 35

14 References [1] P. Glasserman, Monte Carlo Methods n Fnancal Engneerng, Sprnger- Verlag, New-York, [2] J. H. Halton, On the effcency of certan quas-random sequences of ponts n evaluatng multdmensonal ntegrals, Numer. Math., 2 (1960), [3] P. Joron, Value at Rsk: The New Bemchmark n Controllng Market Rsk, Irwn, Chcago, [4] A. McNel, R. Frey, P. Embrechts, Quanttatve Rsk Management: Concepts Technques and Tools, Prnceton Unversty Press, Prnceton, [5] H. Nederreter, Random number generaton and Quas-Monte Carlo methods, Socety for Industral and Appled Mathematcs, Phladelpha, [6] B. Øksendal, Stochastc Dfferental Equatons, Sprnger, Berln, [7] G. Okten, B. Tuffn, V. Burago, A central lmt theorem and mproved error bounds for a hybrd-monte Carlo sequence wth applcatons n computatonal fnance, Journal of Complexty, Vol. 22, No. 4 (2006), [8] S. Ross, Smulaton, 3rd ed., Academc Press, San Dego, [9] A. V. Roşca, A Multdmensonal Stock Market Model, Proceedngs of the Internatonal Conference on Numercal Analyss and Approxmaton Theory, Cluj-Napoca, Romana, 2006, [10] S. E. Shreve, Stochastc Calculus for Fnance II: Contnuous-Tme Models, Sprnger-Verlag, New York, Author: Aln V. Roşca Babeş-Bolya Unversty Faculty of Economcs and Busness Admnstraton Str. Teodor Mhal, Nr , Cluj-Napoca, Romana E-mal:arosca@econ.ubbcluj.ro 36