Application of Quasi Monte Carlo methods and Global Sensitivity Analysis in finance

Transcription

1 Applcaton of Quas Monte Carlo methods and Global Senstvty Analyss n fnance Serge Kucherenko, Nlay Shah Imperal College London, UK skucherenko@mperalacuk Daro Czraky Barclays Captal DaroCzraky@barclayscaptalcom 1

2 Outlne Applcaton of MC methods to path dependent ntegrals MC and QMC smulaton of opton prcng Is the Brownan brdge dscretzaton always more effcent than the Standard scheme? Global Senstvty Analyss and Sobol Senstvty Indces MC/QMC smulaton of the Cox, Ingersoll and Ross nterest rate model Comparson of dfferent Sobol sequence generators 2

3 Applcaton of MC methods to path dependent ntegrals I = F[ x( t)] d x, (1) C W xt ( ) contnuous n 0 t T, x(0) = x I = E( F[ W( t)]), W( t) random Wener processes (a Brownan moton) Monte Carlo approach: to construct many random paths W( t), evaluate functonal and average results 0 (1) can be reduced to r r I[ f] = n f( x) dx H 3

4 Monte Carlo ntegraton methods r I[ f] = E[ f( x)] N 1 r Monte Carlo : IN[ f] = f( z) N = 1 r { z } s a sequence of random ponts n H Error: ε = I[ f] I [ f] N 2 1/2 σ ( f ) εn = ( E( ε )) = 1/2 N Convergence does not depent on dmensonalty but t s slow n 4

5 Quas random sequences (Low( dscrepancy sequences) Dscrepancy s a measure of devaton from unformty: r r Q y H Q y = y y y n Defntons:: ( ), ( ) [0, 1) [0, 2) [0, n ), mq ( ) volume of Q D N N r Q( y) = sup m( Q) r n Q( y) H N D N N N 1/2 1/2 Random sequences: N (ln ln ) / ~ 1/ n (ln N) DN c( d) Low dscrepancy sequences (LDS) N Convergence: ε = I[ f] I [ f] V( f) D, n O(ln N) εqmc = N Assymptotcally ε ~ O(1/ N) much hgher than ε MC ~ O(1/ N) QMC N N QMC 5

6 Approxmatons of path dependent ntegrals wth Standard and Brownan brdge schems SDE: dw = z dt, z ~ N (0,1) Standard algorthm: Brownan brdge algorthm: W( t ) = W( t ) + tz, t = T / n, 0 n t 0 T/2 T WT ( ) = W+ Tz, W( T /2) = ( W( T) + W0) + Tz2, W( T /4) = ( W( T /2) + W0) + T /2 z3, W(3 T/ 4) = ( WT ( / 2) + WT ( )) + T/ 2 z4, 2 2 M 1 1 W(( n 1) T / n) = ( W(( n 2) T / n) + W( T)) + 2 T / nzn 2 2 6

7 Opton prcng Dscretzaton of the Wener process Share prce follows geometrcal Brownan moton: 1/2 ds =µ Sdt +σ SdW, dw = z( dt), z ~ N(0,1) Usng Ito's lemma 1 2 St ( ) = S0 exp[( µ σ ) t+σwt ( ))], Wt ( ) Wener path 2 For tme step t 1 2 St ( + t) = St ( )exp[( r σ ) t+σ ( Wt ( + t) W( t))] 2 For the standard dscretzaton algorthm a termnal asset value: 1 2 ST ( ) = S0exp[( r σ ) T+σ t( z1+ z2 + + zn)] 2 7

8 MC smulaton of opton prcng The value of European style optons rt ( ) C( KT, ) = e E P St ( ), K The payoff functon for an Asan call opton P A =max( S- K,0), For a geometrc average Asan call: S=( S ) There s a closed form soluton Q n 2 rt σ T 1 C( KT, ) = e max[0,( S0 exp[( r ) t +σ Φ ( uj)] n 1 2 n H = j= 1 K)] du du 1 n n =1 1/ n 1/ n 8

9 MC smulaton of opton prcng Dscretzaton In a general case N rt 1 ( () () C (, ) ) N KT = e P S0, S1, L, ST, K N = 1 For the case of European-style call N N 1 () rt 1 () CN ( K, T) = C = e max( ST K,0) N = 1 N = 1 9

10 MC and QMC methods wth standard and Brownan Brdge dscretzatons Asan Call (32 observatons) S=100, K=105, r=005, s=02, T=05, C=384 (analytcal) 45 4 Opton Value 35 QMC, Brownan Brdge QMC, Standard Approxmaton MC, Brownan Brdge 3 Analytcal value N_path Call prce vrs the number of paths MC - slow convergences, convergence curve s hghly oscllatng QMC convergence monotonc Convergence s much faster for Brownan brdge 10

11 Asan call Convergence curves Asan Call wth geometrc averagng 252 observatons S=100, K=105, r=005, s=02, T=10, C=556 (analytcal) 10 K 1 k ε = ( I IN ) K k = 1 2 1/2 1 Log(RMSE) 01 ε ~ N α, 0< α < QMC, Brownan Brdge QMC, Standard Approxmaton MC, Brownan Brdge Trendlne -QMC, BB, 1/N^082 Trendlne - QMC, Stand, 1/N^056 Trendlne - MC, 1/N^ Log(N_path) Log-log plot of the root mean square error versus the number of paths Brownan brdge much faster convergence wth QMC methods: ~1/N 08 11

12 Y = f ( x) x ( x, x,, x k ) 0 x 1 Consder a model x s a vector of nput varables = Y s the model output s 1 s k k ( ) ( ) ( ) Y = f ( x) = f + f x + f x, x + + f x, x,, x, ANOVA decomposton and Senstvty Indces x Ω ANOVA decomposton: 0 j j 1,2,, k 1 2 k = 1 j> f ( x,,, x ) dx = 0, k, 1 k s Varance decomposton: Model, f(x) , j 1,2,, n Y K σ = σ + σ + σ j Sobol SI: k = 1 + = 1 S + S j + S jl + S 1, 2,, < j < j < l k 12

13 Sobol Senstvty Indces (SI) Defnton: σ S σ σ 2 2 = / 1 s 1 s ( ) 1 = 1,,,, s f x s x 1 s dx x 1 s σ = ( f ( x) f0 ) dx 0 - partal varances - varance Senstvty ndces for subsets of varables: σ m 2 2 y = σ 1, K, s= 1 Κ ( ) Introducton of the total varance: 1 s s x ( tot ) σ = σ σ y z = ( y, z) Correspondng global senstvty ndces: 2 2 tot S = / σ, ( ) 2 S = σ tot / σ y σ y y y 2 13

14 How to use Sobol Senstvty Indces? tot y y tot 0 S S 1 = 1 y S S accounts for all nteractons between y and z, x=(y,z) The mportant ndces n practce are and ( ) ( ) tot S = 0 f x does not depend on ; only depends on ; If y S tot S corresponds to the absence of nteractons between and other varables 1, then functon has addtve structure: f ( x) f f ( x ) Fxng unessental varables If S f x tot S = S n s= 1 S = ( ) S << 1 f x z tot z ( ) (, ) does not depend on so t can be fxed f x f y z n z complexty reducton, from to n n varables 0 x x x =

15 Applcatons of Global Senstvty Analyss Global Senstvty Analyss can be used to dentfy key parameters whose uncertanty most strongly affects the output; rank varables, fx unessental varables and reduce model complexty; select a model structure from a set of known competng models; dentfy functonal dependences; analyze effcences of numercal schemes 15

16 Effectve dmensons Let u be a cardnalty of a set of varables u The effectve dmenson of f( x) n the superposton sense s the smallest nteger d such that 0< u< d S S u (1 ε), ε << 1 S It means that f( x) s almost a sum of -dmensonal functons d S The functon f( x) has effectve dmenson n the truncaton sense d f u {1,2,, d } T Example: d d S T S (1 ε), ε << 1 u ( ) f x n = = 1 x d = 1, d = n S T does not depend on the order n whch the nput varables are sampled, - depends on the order by reoderng varables d can be reduced T T 16

17 Classfcaton of functons 17

18 Global Senstvty Analyss of standard dscretzaton and Brownan Brdge Apply global SA to payoff functon PA ({Z })=max( S({Z })- K,0), {Z }, = 1, n 1 Standard Approxmaton Brownan Brdge 01 S_total tme step number Log of total senstvty ndces versus tme step number Standard dscretzaton - S_total slowly decrease wth Brownan brdge - S_total of the frst few varables are much larger than those of the subsequent varables 18

19 Global Senstvty Analyss of two algorthms at dfferent n Standard approxmaton: the effectve dmenson d T n the effectve dmenson d S > 2 Brownan Brdge approxmaton: the effectve dmenson d T 2 the effectve dmenson d S 2 19

20 Opton prcng: Why Brownan Brdge s more effcent than standard dscretzaton n the case of QMC? The ntal coordnates of LDS are much better dstrbuted than the later hgh dmensonal coordnates Global SA: for the Brownan brdge dscretzaton the low ndex varables are much more mportant than hgher ndex varables For the Brownan brdge dscretzaton well dstrbuted coordnates are used for mportant varables and hgher not so well dstrbuted coordnates are used for far less mportant varables The standard constructon does not account for the specfcs of LDSs dstrbuton propertes Applcaton of QMC wth the Brownan brdge dscretzaton results n the tme reducton of CPU tme compared wth MC! 20

21 Cox, Ingersoll and Ross nterest rate model ( ) dy = α+β y dt +σ y dw t t t t where α > 0, β > 0, σ > 0 The generalsed method of moments (GMM) estmaton ) ) ) s used to obtan α, β and σ estmates Data: the 9-month Eurbor nterest rate daly tme seres usng 250 daly observatons, startng at 30 Dec 1998 ) ) ) α= 1e-005, β= 01109, σ= Euler dscretsaton: ( ) y y = α+βy t+σ y tε t+ t t t t t 21

22 5-days ahead backtest for the 9-month Eurbor nterest rate seres Forecast s accurate but can be prohbtvely expensve f ran over a large tme span or f a large number of MC runs s needed 22

23 CIR model: 50 day forecasts of the 9-month Eurbor rate QMC Euler QMC Mlsten 50-perod forecast perod forecast Number of smulaton runs (N) MC Euler Number of smulaton runs (N) MC Mlsten 50-perod forecast perod forecast Number of smulaton runs (N) Number of smulaton runs (N) MC and QMC estmators wth usng standard Euler and Mlsten schemes QMC produces much smother and much faster convergence than MC 23

24 RMSE convergence for 250-days forecastng horzon K 1 k ε = ( I IN ) K k = 1 2 1/ MC Euler (H = 250) QMC Euler (H = 250) Trendlne MC (-053) Trendlne QMC (-079) log2(rm SE) ε ~ N α, 0< α < log2(n) QMC estmator dsplays much faster convergence compared to MC regardless of the forecastng horzon (dmenson) 24

25 RMSE convergence for 250-days forecastng horzon K 1 k ε = ( I IN ) K k = 1 2 1/2 ε ~ N α, 0< α < 1 There s no notable advantage of Brownan brdge over the standard Euler scheme 25

26 Global Senstvty Analyss of StandardS and Brownan Brdge dscretzatons (CIR model) Apply global SA to a functon yt (Z, Z,, Z ), n s a number of days 1 2 n 1 08 Standard Approxmaton Brownan Brdge S_total tme step number Log of total senstvty ndces versus tme step number Standard dscretzaton - S_total are constant Brownan brdge - S_total of the frst varable s much larger than those of the subsequent varables 26

27 Global Senstvty Analyss of StandardS and Brownan Brdge dscretzatons (CIR model) n S Standard S BB Standard approxmaton: the effectve dmenson d T n the effectve dmenson d S =1 Brownan Brdge approxmaton: the effectve dmenson d T 2 the effectve dmenson d S =1 The low effectve dmenson d S for both schemes =1, hence QMC effcency can not be further mproved by changng samplng strategy 27

28 What s the optmal way to arrange N ponts n two dmensons? Regular Grd Sobol Sequence Low dmensonal projectons of low dscrepancy sequences are better dstrbuted than hgher dmensonal projectons 28

29 Why Sobol sequences are so effcent? n O(ln N) Convergence: ε = for all LDS N n 1 O(ln N) k For Sobol' LDS: ε =,f N = 2, k nteger N Sobol' LDS: 1 Best unformty of dstrbuton as N goes to nfnty 2 Good dstrbuton for farly small ntal sets 3 A very fast computatonal algorthm 29

30 Sobol LDS Property A and Property A A Property A Consder n-dmensonal hypercube whch s cut by plans x j =1/2 nto 2 n subcubes Sequence of Sobol ponts satsfes Property A, f after dvdng the sequence nto blocks of 2 n ponts, each one of the ponts n any one block belong to a dfferent subcube Property A Consder n-dmensonal hypercube whch s cut by plans x j =k/4, j=1,,n, k=1,2,3 nto 4 n subcubes Sequence of Sobol ponts satsfes Property A, f after dvdng the sequence nto blocks of 4 n ponts, each one of the ponts n any one block belong to a dfferent subcube Property A Property A 30

31 Comparson of Sobol sequence generators SobolSeq generator: Sobol' sequences satsfy two addtonal unformty propertes: Property A for all dmensons and Property A' for adjacent dmensons I = [,1] s n 0 = 1 ( 1+ c ( x 05) ) dx 31

32 Summary Global Senstvty Analyss s a general approach for uncertanty, complexty reducton and structure analyss of non-lnear models It can be wdely appled n fnance Quas MC methods outperform MC regardless of nomnal dmensonalty for problems wth low effectve dmensons n ether truncaton or superposton sense The effcency of the Brownan Brdge or any other constructon wthn the framework of the Quas MC method depends on effectve dmensons of the ntegrand The Sobol Sequence generator satsfyng unformty propertes A and A' has superor performance over other generators 32

33 Acknowledgments Prof Sobol EPSRC grant EP/D506743/1 33