European pricing with monte carlo simulation

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  • The Milshtein scheme only differs from the Euler scheme in the second line. Notice that the drift and volatility in the right-hand side of both schemes are evaluated at time ti, this means that these are exlicit schemes.
  • European pricing with monte carlo simulation

    1. 1. 1 European Pricing with Monte Carlo Simulation Giovanni Della Lunga University of Insubria - Varese - Italy SUMMER SCHOOL FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “COPULA METHODS IN FINANCE” Department of Mathematics for Economics and Social Science of the University of Bologna Bologna - September, 13-15, 2004
    2. 2. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 2 The “path” 1. Scenario generation  exact solution advancement  generating scenarios by numerical integration of the stochastic differential equations 2. Correlation and co-movement  joint normals distributions by the Cholesky Decomposition Approach  joint with copulae: an example using gaussian and t-copula 3. Quasi-random sequences 4. European Pricing with simulation  the workflow of pricing with Monte Carlo  Increasing simulation efficiency with variance reduction methods
    3. 3. 3 1 Scenario Generation
    4. 4. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 4 Introduction  Generating scenarios of the underlying processes that determine the derivative’s price is an essential and delicate task from an analytical perspective as well as from a system design viewpoint;  As we know, the main objective of scenarios in pricing is to compute the expectation that gives us the value of the financial instrument;
    5. 5. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 5 Scenario Nomenclature  It’s well known that the value of a derivative security with payoffs at a known time T is given by the expectation of its payoff, normalized with the numeraire asset;  The value of an European derivative whose payoff depends on a single underlying process, S(t), is given by where all stochastic processes in this expectation are consistent with the measure induced by the numeraire asset B(t).       = )( ]),([ )0(]0),0([ TB TTSV EBSV B
    6. 6. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 6 Scenario Nomenclature  We consider an underlying process S(t) described by the sde  A scenario is a set of values that are an approximation to the j-th realization,S j (ti), of the solution of the sde evaluated at times  A scenario is also called a trajectory  A trajectory can be visualized as a line in the state-vs-time plane describing the path followed by a realization of the stochastic process (actually by an approzimation to the stochastic process). dWtSbdttSatdS ),(),()( += IitS i j ,,1,)(ˆ = IiTti ,,1,0 =≤≤
    7. 7. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 7 Scenario Nomenclature
    8. 8. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 8 Scenario Construction  There are several ways to construct scenario for pricing  Constructing a path of the solution to the SDE at times ti by exact advancement of the solution;  This method is only possible if we have an analytical expression for the solution of the stochastic differential equation  Approximate numerical solution of the stochastic differential equation;  This is the method of choice if we cannot use the previous one;  Just as in the case of ODE there are numerical techniques for discretizing and solving SDE.
    9. 9. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 9 Scenario Construction Exact Solution Advancement  Example: Log-normal process with constant drift and volatility dWdt S dS σµ += ( ) [ ]      −+−      −= −−− )()( 2 1 exp)()( 11 2 1 iiiiii tWtWtttStS σσµ       +      −= )( 2 1 exp)0()( 2 tWtStS σσµ Solution Trajectories construction
    10. 10. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 10 Scenario Construction Exact Solution Advancement  How to obtain a sequence of Wiener process?  The simplest way  Defining the outcomes of successive drawings of the random variable Z corresponding to the j-th trajectory by Zj i, we get the following recursive expression for the j-th trajectory of S(t) )1,0()()( 11 NZZtttWtW iiii ≈−+= −− ( ) ( )       −+−      −= −−− j iiiiii j i j ZtttttStS 11 2 1 2 1 exp)()( σσµ Note. If the time spacing is uniform, all the increments we add to construct the Wiener path have the same variance! Note. If the time spacing is uniform, all the increments we add to construct the Wiener path have the same variance!
    11. 11. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 11 Scenario Construction Exact Solution Advancement  Some observations are in order...  The set W(ti) must be viewed as the components of a vector of random variables with a multidimensional distribution. This means that for a fixed j Zj i are realizations of a multidimensional standard normal random variable which happen to be independent;  Wheter we view the Zj i as coming from a multidimensional distribution of independent normals or as drawings from a single one-dimensional distribution does not affect the outcome as long as the Zj i are generated from pseudo-random numbers;  This distinction, however, is conceptually important and it becomes essential if we generate the Zj i not from pseudo- random numbers but from quasi-random sequences.
    12. 12. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 12 Scenario Construction Numerical Integration of SDE  The numerical integration of the SDE by finite difference is another way of generating scenarios for pricing;  In the case of the numerical integration of ordinary differential equations by finite differences the numerical scheme introduces a discretization error that translates into the numerical solution differing from the exact solution by an amount proportional to a power of the time step.  This amount is the truncation error of the numerical scheme.
    13. 13. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 13 Scenario Construction Numerical Integration of SDE  In the case of the numerical integration of SDE by finite differences the interpretation of the numerical error introduced by the discretization scheme is more complicated;  Unlike the case of ODE where the only thing we are interested in computing is the solution itself, when dealing with SDE there are two aspects that interest us:  One aspect is the accuracy with which we compute the trajectories or paths of a realization of the solution  The other aspect is the accuracy with which we compute functions of the process such as expectations and moments.
    14. 14. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 14 Scenario Construction Numerical Integration of SDE  The order of accuracy with which a given scheme can approximate trajectories of the solution is not the same as the accuracy with which the same scheme can approximate expectations and moments of functions of the trajectories;  The convergence of the numerically computed trajectories to the exact trajectories is called strong convergence and the order of the corresponding numerical scheme is called order of strong convergence;  The convergence of numerically computed functions of the stochastic process to the exact values is called weak convergence and the related order is called order of weak convergence. See Kloeden and Platen For an exhaustive treatment of numerical schemes for SDE.
    15. 15. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 15 Scenario Construction Numerical Integration of SDE  The two most popular schemes for integrating a SDE are the Explicit Euler scheme and the Milshtein scheme ))()()(),(ˆ()),(ˆ()(ˆ)(ˆ 11 iiiiiiii tWtWttSbtttSatStS −+∆+= ++ ( )[ ]ttWtW S ttSb ttSb ii ii ii ∆−− ∂ ∂ + + 2 1 )()( )),(ˆ( )),(ˆ( 2 1 dWtSbdttSatdS ),(),()( += EULER MILSHSTEIN
    16. 16. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 16 Scenario Construction Numerical Integration of SDE  Order of Strong Convergence  The scheme has q order of strong convergence if exist a constant α that does not depend on ∆ such that ( ) q TSSE ∆≤− →∆ α)(ˆlim 0
    17. 17. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 17 Scenario Construction Numerical Integration of SDE  Order of Weak Convergence  Let f(S(t)) denote a function of the trajectory of the solution. A finite difference scheme has q order of weak convergence if exist a constant β that does not depend on ∆ such that [ ] [ ] q TSfESfE ∆≤− →∆ β))(()ˆ(lim 0
    18. 18. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 18 Scenario Construction Numerical Integration of SDE  Why the order of strong and weak convergence differ?  Let’s consider a simplified and intuitive analysis of the Euler scheme applied to the standard log-normal stochastic differential equation;  The following analysis is not rigoruous and is meant to show why the same scheme has different accuracy when it is used to compute trajectories than when it is used to compute expectations. For a complete and formal treatment of this subject the reader is referred to the classical work of Kloeden et al.
    19. 19. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 19 Scenario Construction Numerical Integration of SDE  Log-normal process with constant drift and volatility )( )( )( tdWdt tS tdS σµ += ( ))(1)(ˆ)(ˆ 1 iii tWttStS ∆+∆+=+ σµ       +      −= )( 2 1 exp)0()( 2 tWtStS σσµ Solution Euler scheme ( ))(1)(ˆ)(ˆ 1 0 0 ∏ − = ∆+∆+= k i ik tWttStS σµ
    20. 20. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 20 Scenario Construction Numerical Integration of SDE  Assuming I time steps in the interval 0 ≤ t ≤ T, the error for strong convergence is (S0 = 1 for simplicity)  Now, to determine how well the product in the right approximates the exponential, we expand in Taylor series keeping terms just past order ∆t... ( )               +      −−∆+∆+∏ − = )( 2 1 exp)(1 2 1 0 tWttWtE k i i σσµσµ
    21. 21. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 21 Scenario Construction Numerical Integration of SDE ( ) ( ) )()( 6 1 )( 2 1 )( 2 1 )( 2 1 1 6 1 2 1 1)( 2 1 exp 233 2 22 2 322 tOtW tWt tW tWt xxxtWt i i t i i x i ∆+∆ +∆∆      − +∆ +∆+∆      −+= ++++≈           ∆+∆      − ∆= σ σσµ σ σσµ σσµ      ( ) )()( 6 1 )( 2 1 )( 2 1 exp)(1 233 2 2 tOtW tWt tWttWt i i ii ∆−∆− ∆∆      −−       ∆+∆      −≈∆+∆+ σ σσµ σσµσµ
    22. 22. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 22 Scenario Construction Numerical Integration of SDE { ( ) } )()()( )( 2 1 exp )()( 6 1 )( 2 1 )( 2 1 exp)(1 23 2 233 2 2 1 0 1 0 tIOWIOWtIO TWT tOtW tWt tWttWt i i i I i I i i ∆+∆+∆∆ +      +      −≈ ∆−∆− ∆∆      −−       ∆+∆      −≈∆+∆+ ∏∏ − = − = σσµ σ σσµ σσµσµ
    23. 23. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 23 Scenario Construction Numerical Integration of SDE  Strong order [ ] [ ] ( )  ( )tO WO t WtO t TE WOWtO t T E WIOWtIOETSTSE t t t t ∆=             ∆ ∆ +∆∆ ∆ =       ∆+∆∆ ∆ = ∆+∆∆=− ∆≈ ∆≈ ∆≈ ∆≈   )( 1 )( 1 )()( )()()()(ˆ 2/32/3 3 3 3
    24. 24. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 24 Scenario Construction Numerical Integration of SDE  To analyze the weak convergence consider as example the expectation of the process itself... [ ] )()()( )( 2 1 exp )()( 2 1 exp )( 2 1 exp)(1)()(ˆ 22 2 22 2 1 0 tOtO t T tIO TWTE tIOTWTE TWTEtWtETSTSE I i i ∆=∆ ∆ =∆=             +      −−       ∆+      +      − =             +      −−      ∆+∆+=− ∏ − = σσµ σσµ σσµσµ When we replace the expansion of the product, the terms with ∆W do not contribute to the expectation!!!
    25. 25. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 25 Scenario Construction Numerical Integration of SDE  Now we can see clearly why the order of convergence is different when we look at paths and when we look at moments;  When we look at properties such expectations, the odd power of ∆W don’t contribute to the expectation because in weak convergence we take the expectation first and then the norm;  When we look at the individual paths, on the other hand, these terms are the main contributors to the difference between the exact and the numerical solutions, this is because we take the norm first and then the expectation.
    26. 26. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 26 Working with VBAWorking with VBA A Simple Example of a Stochastic Discrete Time Simulation A Simple Example of a Stochastic Discrete Time Simulation
    27. 27. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 27 Scenario Construction Numerical Integration of SDE  Observations  When the volatility is deterministic, the order of strong convergence of the Euler scheme is 1;  In this case there is nothing to be gained by using the Milshstein scheme in place of the Euler scheme to construct scenario trajectories.
    28. 28. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 28 The Brownian Bridge  Assume you have a Wiener process defined by a set of time- indexed random variables {W(t1), W(t2), ... , W(tn)}.  How do you insert a random variable W(tk) where ti ≤ tk ≤ ti+1 into the set in such a manner that the resulting set still consitutes a Wiener process?  The answer is: with a Brownian Bridge!  The Brownian Bridge is a sort of interpolation tat allows you to introduce intermediate points in the trajectory of a Wiener process.
    29. 29. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 29 The Brownian Bridge  Brownian Bridge Construction  Given W(t) and W(t + ∆t1 + ∆t2) we want to find W(t + ∆t1 );  We assume that we can get the middle point by a weighted average of the two end points plus an independent normal random variable: where α, β and λ are constants to be determined and Z is a standard normal random variable. ZtttWtWttW λβα +∆+∆++=∆+ )()()( 211
    30. 30. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 30 The Brownian Bridge  We have to satisfy the following conditions:      ∆+=∆+ ∆+=∆+∆+∆+ =∆+=∆+ 11 1211 11 )](var[ )](),(cov[ ),min()](),(cov[ ttttW tttttWttW tttttWttW      ∆+=+∆+∆+++ ∆+=∆+∆++ =+ 1 2 21 22 121 )(2 )( 1 ttttttt tttttt λβαβα βα βα
    31. 31. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 31 The Brownian Bridge      ∆+=+∆+∆+++ ∆+=∆+∆++ =+ 1 2 21 22 121 )(2 )( 1 ttttttt tttttt λβαβα βα βα         ∆= −= ∆+∆ ∆ = αγ αβ α 1 21 2 1 t tt t
    32. 32. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 32 The Brownian Bridge  We can use the brownian bridge to generate a Wiener path and then use the Wiener path to produce a trajectory of the process we are interested in;  The simplest strategy for generating a Wiener path using the brownian bridge is to divide the time span of the trajectory into two equal parts and apply the brownian bridge construction to the middle point. We then repeat the procedure for the left and right sides of the time interval.
    33. 33. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 33 The Brownian Bridge  Notice that as we fill in the Wiener path, the additional variance of the normal components we add has decreasing value;  Of course the total variance of all the Wiener increments does not depend on how we construct the path, however the fact that in the brownian bridge approach we use random variables that are multiplied by a factor of decreasing magnitude means that the importance of those variables also decreases as we fill in the path;  The dimension of the random variables with larger variance need to be sampled more efficiently than the dimension with smaller variance;
    34. 34. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 34 The Brownian Bridge  In standard Monte Carlo this is not an issue because standard Monte Carlo is equally efficient at sampling from any dimension;  But standard Monte Carlo is very slow!  As we will see, an alternative to standard Monte Carlo is the use of deterministic or quasi-Monte Carlo or low-discrepancy sequences;  Usually low-discrepacy sequences differs in their ability to cover lower dimension as compared with higher dimension (this may be not completely true!);  The brownian bridge method for path construction reduces the burden on the simulation from having to sample efficiently from the higher dimension.
    35. 35. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 35 Working with VBAWorking with VBA A Simple Example of a Brownian Bridge Construction of Wiener Path A Simple Example of a Brownian Bridge Construction of Wiener Path
    36. 36. 36 2 Correlation and Co-movement
    37. 37. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 37 Joint Normals by the Choleski Decomposition Approach  The standard procedure for generating a set of correlated normal random variables is through a linear combination of uncorrelated normal random variables;  Assume we have a set of n independent standard normal random variables Z and we want to build a set of n correlated standard normals Ž with correlation matrix Σ AZZ =  Σ=t AA
    38. 38. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 38 Joint Normals by the Choleski Decomposition Approach  We can find a solution for A in the form of a triangular matrix               = nnnn AAA AA A A     21 2221 11 0 00 ∑ − = −= 1 1 2 i k ikiiii aa σ       −= ∑ − = 1 1 1 i k jkikij ii ji aa a a σ
    39. 39. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 39 Joint Normals by the Choleski Decomposition Approach  For a two-dimension random vector we have simply  Thus we can sample from a bivariate distribution by setting         − = 2 22 1 1 0 ρσρσ σ A 2 2 21222 1111 1 ZZX ZX ρσρσµ σµ −++= +=
    40. 40. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 40 Simulation with Copula  Gaussian Copula  The copula of the n-variate normal distribution with linear correlation matrix R is where ΦR n denotes the joint distribution function of the n-variate standard normal distribution function with linear correlation matrix R, and Φ-1 denotes the inverse of the distribution function of the univariate standard normal distribution.  Copulas of the above form are called Gaussian Copulas. In the bivariate case the copula expression can be written as ( ))(,),()( 1 1 1 n n R Ga R uuC −− ΦΦΦ= u ( ) ( ) dtds R tstRs R vuC u v Ga R ∫ ∫ − − Φ ∞− Φ ∞−       − +− − − = )( )( 2 12 2 12 2 2/12 12 1 1 )1(2 2 exp 12 1 , π
    41. 41. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 41 Simulation with Copula  Generation of random variates from the Gaussian n-copula 1. Find the Cholesky decomposition A of R 2. Simulate n independent random variates z = (z1,..., zn)’ from N(0,1) 3. Set x = Az 4. Set ui = Φ(xi) with i = 1,2,...,n where Φ denotes the univariate standard normal distribution function 5. (y1,...,yn)’ =[F1 -1 (u1),...,Fn -1 (un)] where Fi denotes the i-th marginal distribution.
    42. 42. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 42 Simulation with Copula  Generation of random variates from the Student T n-copula 1. Find the Cholesky decomposition A of R 2. Simulate n independent random variates z = (z1,..., zn)’ from N(0,1) 3. Simulate a random variate s from χ2 ν independent of z 4. Set y = Az 5. Set x = (ν/s)1/2 y 6. Set ui = Tν(xi) with i = 1, 2, ..., n and where Tν denotes the univariate Student t distribution function 7. (g1, ..., gn)’ =[F1 -1 (u1), ..., Fn -1 (un)] where Fi denotes the i-th marginal distribution
    43. 43. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 43 Working with VBAWorking with VBA Generation of random variates from the Gaussian and Student t bivariate copula Generation of random variates from the Gaussian and Student t bivariate copula
    44. 44. 44 3 Quasi-Random Sequences
    45. 45. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 45  Microsoft Knowledge Base #828795  The RAND function in earlier versions of Excel used a pseudo-random number generation algorithm whose performance on standard tests of randomness was not sufficient (...) the pseudo- random number generation algorithm that is described here was implemented for Excel 2003. It passes the same battery of standard tests.  The battery of tests is named Diehard (see note 1). The algorithm that is implemented in Excel 2003 was developed by B.A. Wichman and I.D. Hill (see note 2 and note 3). (...) It has been shown by Rotz et al (see note 4) to pass the DIEHARD tests and additional tests developed by the National Institute of Standards and Technology (NIST, formerly National Bureau of Standards).  Notes  The tests were developed by Professor George Marsaglia, Department of Statistics, Florida State University and are available at the following Web site:  http://www.csis.hku.hk/~diehard  Wichman, B.A. and I.D. Hill, Algorithm AS 183: An Efficient and Portable Pseudo-Random Number Generator, Applied Statistics, 31, 188-190, 1982.  Wichman, B.A. and I.D. Hill, Building a Random-Number Generator, BYTE, pp. 127-128, March 1987.  Rotz, W. and E. Falk, D. Wood, and J. Mulrow, A Comparison of Random Number Generators Used in Business, presented at Joint Statistical Meetings, Atlanta, GA, 2001. Random Number Generator in Microsoft Excel
    46. 46. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 46  Results in Earlier Versions of Excel  The RAND function in earlier versions of Excel was fine in practice for users who did not require a lengthy sequence of random numbers (such as a million). It failed several standard tests of randomness, making its performance an issue when a lengthy sequence of random numbers was needed.  Results in Excel 2003  A simple and effective algorithm has been implemented. The new generator passes all standard tests of randomness (?????). The RAND function returns negative numbers in Excel 2003 SYMPTOMS When you use the RAND function in Microsoft Office Excel 2003, the RAND function may return negative numbers. CAUSE This problem may occur when you try to use a large number of random numbers, and you update the RAND function multiple times. For example, this problem may occur when you update your Excel worksheet by pressing F9 ten times or more. RESOLUTION How to obtain the hotfix This issue is fixed in the Excel 2003 Hotfix Package that is dated January 12, 2004. For additional information, click the following article number to view the article in the Microsoft Knowledge Base: 833618 Excel 2003 hotfix package released January 12, 2004 STATUS Microsoft has confirmed that this is a problem in the Microsoft products that are listed in the "Applies to" section of this article. Random Number Generator in Microsoft Excel
    47. 47. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 47 Low-discrepacy Sequences  The basic problem with standard Monte Carlo is that, unless special techniques are used (which we will briefly discuss forward) it is intrinsecally slow!  The reason why regular MC is so slow is because randomly sampling from a multidimensional distribution does not fill in the space with the regularity that would be desirable;  Random points tend to cluster and this clustering limits the efficiency with which regular MC can capture payoff features.  Quasi-random sequences are a deterministic way of filling in multidimensional unit intervals in a way that garanties a higher uniformity;  The concept of higher uniformity in filling a generic intervall is captured by the concept of discrepancy.
    48. 48. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 48 Discrepancy  A measure for how inhomogeneously a set of d-dimensional vectors {ri} is distributed in the unit hypercube is the so called discrepancy;  A simple geometrical interpretation of this concept is as follows:  Generate a set of N multivariate draws {ri} from a selected uniform number generation method of dimensionality d ;  All of this N vectors descrive the coordinates of points in the d-dimensional unit hypercube [0,1]d ;  Now select a sub-hypercube S(y) by choosing a point y delimiting the upper right corner of the hyper-rectangular domain from 0 to y;  In other words the sub-hypercube S can be written as ),[),[)( d1 y0y0yS ××=  
    49. 49. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 49 Discrepancy  Next let nS(y) denote the number of all those draws that are in S(y)  In the limit N → ∞ we require perfect homogeneity from the sequence generator which means for all y in [0,1]d { } { }∑∏∑ = = ≥ = ∈ == N 1i d 1k ry N 1i ySryS ikki n 11 )()(  ∏= ∞→ = d 1i i yS N y N n )( lim 
    50. 50. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 50 Discrepancy  The previous equation simply results from the fact that for a perfectly omogeneous and uniform distribution on a unit hypercube the probability of being in a subdomain is equal to the volume of that subdomain, and the volume V of S( y ) is given by the right-hand side of the previous equation;  Then we can now compare for all y ))((e )( ySV N n yS 
    51. 51. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 51  Choosing L∞ -norm we have the followind definition for the discrepancy  For example in the 1-dimensional case we have simply ∏=∈ −= d 1k k yS 10y d N y N n D d )( ],[ )( sup   10 21 ≤≤≤≤≤ nxxx  n2 1k2 x n2 1 D k n1k 1 n − −+= = , )( max Discrepancy
    52. 52. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 52  We now arrive at the number-theoretical definition of low- discrepancy sequences; A sequence in [0,1]d is called a low-discrepancy sequence if for all N > 1 the first N points in the sequence satisfy for some constant c(d) that is only a function of d. ( ) N N dcD d d N ln )()( ≤ Discrepancy
    53. 53. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 53 Halton Sequence ( ) 21212020:1 012 /⇒⋅+⋅+⋅ ( ) 2012 21202120:2 /⇒⋅+⋅+⋅ ( ) 2012 2121212120:3 // +⇒⋅+⋅+⋅ ( ) 3012 21202021:4 /⇒⋅+⋅+⋅ ( ) 3012 2121212021:5 // +⇒⋅+⋅+⋅ ( ) 32012 2121202121:6 // +⇒⋅+⋅+⋅ ( ) 32012 212/121212121:7 // ++⇒⋅+⋅+⋅
    54. 54. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 54 Halton Sequence  Halton sequence can be generated by reflecting the expansion in base b about the decimal point  At step j-th write j in base b (with b prime number)  When we “reflect” j about the decimal point we obtain the number of the sequence H(j) 01 10 bdbdbdj n nn +++= −  1 0 2 1 1 −−− − − +++= n nnj bdbdbdH 
    55. 55. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 55 Halton Sequence  The new numbers that are added tend to fill in the gaps in the existing sequence;  Example with b = 2 125.02120202020214 75.0212121213 25.0212020212 5.021211 321 4 012 21 3 01 21 2 01 1 1 0 =×+×+×=⇒×+×+×= =×+×=⇒×+×= =×+×=⇒×+×= =×=⇒×= −−− −− −− − H H H H
    56. 56. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 56 VBA code for Halton Sequence Public Function Halton(n As Integer, _ x As Integer) As Double ' n è il numero da trasformare ' x è la base ' H è il numero generato dall'algoritmo Dim H As Double Dim z As Double Dim m As Integer Dim na As Integer Dim nb As Integer H = 0 na = n z = 1 / x While (na > 0) nb = Int(na / x) m = na - nb * x H = H + m * z na = nb z = z / x Wend Halton = H End Function 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1
    57. 57. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 57 Working with VBAWorking with VBA Generation of Halton Sequence Generation of Halton Sequence
    58. 58. 58 4 Derivatives Pricing Applications
    59. 59. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 59 European Pricing with MC  In this case we are interested in computing (or estimating) an Expectation;  The expectation we are interested in is where B(.) is the numeraire and V(T) the known payoff at maturity.       = )( )( )0()0( TB TV EBV B
    60. 60. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 60 The Workflow of Monte Carlo Pricing  In applying MC to pricing a European derivative, we face two challenges  How do we construct the function V(T)/B(T) from the underlying process?  How do we estimate the expectation efficiently and accurately?  The first item is important because we may not have an analytical formula for V(T)/B(T) as a function of the underlying process;  The second item is important because we must get the answer with known error bounds (if possible!) and within time constraints.
    61. 61. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 61 The Workflow of Monte Carlo Pricing  In computing the distribution of discounted payoff, we must work with processes hat are specified in an appropriate measure;  In its simplest form MC pricing is carried out in the pricing measure used to derive the derivative price;  However since we are interested only in the expectation we do not necessarily have to carry out our simulation in the pricing measure;  We may be able to carry out our simulation in a different measure than the pricing measure, a measure which is more suitable for speed and accuracy;
    62. 62. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 62 The Workflow of Monte Carlo Pricing  The workflow  In its simplest form MC pricing works by evaluating the payoff function repeatedly and taking the average of these evaluation. Each evaluation is called a MC Cicle;  Each evaluation is preceeded by the computation of the underlying asset or process;  The underlying price needed to evaluated the payoff function is captured by the concept of scenario.  Each MC cycle gives us a number which is the realization of a random variable. In some implementation of MC the random variables that get realized at each cycle may not be independent.  The objective of MC in pricing is to infer primarily the mean from the properties of the sample generated by the simulation.
    63. 63. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 63 Why Monte Carlo Pricing?  In numerical analysis there is an informal concept known as the curse of dimensionality. This refers to the fact that the computational load (CPU time, memory requirements, etc...) may increase exponentially with the numer of dimensions of the problem;  The computational work needed to estimate the expectation through MC does not depend explicitly on the dimensionality of the problem, this means that there is no curse of dimensionality in MC computation when we are only interested in a simple expectation (this is the case with european derivatives, thinghs are more complicated with early exercise features).
    64. 64. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 64 Simulation Efficiency  The efficiency of a simulation refers to the computational cost of achieving a given level of confidence in the quantity we are trying to estimate;  Both te uncentainty in the estimation of the expectation as well as the uncertainty in the error of our estimation depend on the variance of the population from which we sample;  However whatever we do to reduce the variance of the population will most likely tend to increase the computational time per MC cycle;  As a result in order to make a fair comparison between different estimators we must take into account not only their variance but also the computational work for each MC cycle.
    65. 65. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 65 Simulation Efficiency  Suppose we want to compute a parameter P, for example the price of a derivative security, and that we have a choice between two types of Monte Carlo estimates which we denote by  Suppose that both are unbiased, so that  but niPi ,...,1,ˆ 1 = niP i ,...,1,ˆ 2 = [ ] PPE =1 ˆ [ ] PPE =2 ˆ 21 σσ <
    66. 66. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 66 Simulation Efficiency  A sample mean of n replications of P1 gives a more precise estimate of P than does a sample mean of n replications of P2;  This oversimplifies the comparison because it fails to capture possible differences in the computational effort required by the two estimators;  Generating n replications of P1 may be more time-consuming than generating n replications of P2; Smaller variance is not sufficient grounds for preferring one estimator over another!
    67. 67. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 67 Simulation Efficiency  To compare estimators with different computational requirements, we argue as follows;  Suppose the work required to generate one replication of Pj is a constant, bj (j= 1,2);  With computing time t , the number of replications of Pj that can be generated is t / bj;  The two estimators available with computing time t are therefore: ∑= 1/ 1 11 ˆ bt i iP t b ∑= 2/ 1 22 ˆ bt i iP t b
    68. 68. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 68 Simulation Efficiency  For large t these are approximately normally distributed with mean P and with standard deviations  Thus for large t the first estimator should be preferred over the second if  The important quantity is the product of variance and work per run; t b t b 2 2 1 1 σσ 2 2 21 2 1 bb σσ <
    69. 69. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 69 Increasing Simulation Efficiency  If we do nothing about efficiency, the number of MC replications we need to achieve acceptable pricing acccuracy may be surprisingly large;  As a result in many cases variance reduction techiques are a practical requirement;  From a general point of view these methods are based on two principal strategies for reducing variance  Taking advantage of tractable features of a model to adjust or correct simulation output  Reducing the variability in simulation input
    70. 70. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 70 Increasing Simulation Efficiency  The most commonly used strategies for variance reduction are the following  Antithetic variates  Control variates  Importance sampling  Stratification  Low-discrepancy sequences
    71. 71. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 71 Antithetic variates  In this case we construc the estimator by using two brownian trajectories that are mirror images of each other;  This causes cancellation of dispersion;  This method tends to reduce the variance modestly but it is extremely easy to implement and as a result very commonly used;
    72. 72. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 72 Antithetic variates  To apply the antithetic variate technique, we generate standard normal random numbers Z and define two set of samples of the underlying price  Similarly we define two sets of discounted payoff samples  At last we construct our mean estimator by averagin these samples ZTTr eSST σσ +−+ = )2/( 0 2 )()2/( 0 2 ZTTr eSST −+−− = σσ [ ]0,)(max KTSVT −= ++ [ ]0,)(max KTSVT −= +− ( )∑= −+ += n j jj VV n V 1 2 11 )0(
    73. 73. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 73 Antithetic variates  For the antithetic method to work we need V+ and V- to be negatively correlated;  This will happen if the payoff function is a monotonic function of Z;  Attemps to combine this method with other methods tipically don’t work well!
    74. 74. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 74 Control Variates  The method of “Control Variate” is among the most effective methods of the first kind;  It exploits information about the errors in estimates of known quantities to reduce the error in an estimate of an unknown quantity.
    75. 75. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 75 Control Variates  To describe the method, let’s suppose that Y1,….,Yn are outputs fron n replications of a simulation;  Suppose that the Yi are iid and that our goal is to estimate E[Yi ] , the usual estimator is the sample mean <Y> = (Y1+…+Yn)/n. This estimator is umbiased and converges with probability 1 as n →∞;  Suppose now that on each replication we calculate another output Xi along with Yi, let te pairs (Xi,Yi) iid and the expectation of Xi be E[X] (known!);  Then we calculate from the i-th replication… [ ]( )XEXbYbY iii −−=)(
    76. 76. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 76 Control Variates  … and then compute the sample mean  This is a control variate estimator;  The observed error <X>-E[X] serves as a control in estimating E[Y];  The control variate estimator has smaller variance than the standard estimator if ( ) ( )( )∑ −−=−−= i ii XEXbY N XEXbYbY ][ 1 ][)( XYYX bb ρσσ 2<
    77. 77. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 77 Control Variates  The effectiveness of a control variate  is mainly determined by the strength of the correlation between the quantity of interest, Y, and the control X.  Can vary widely with the parameters of a problem!  With a single control variable the optimal coefficient b* is given by )var( ),cov( * X YX =β
    78. 78. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 78 Control Variate  As in all the situations when the parameters determining the result are calculated from the same simulation, this can introduce a bias that is difficult to estimate;  In the limit of very large numbers of iterations this bias vanishes but the main goal of variance reduction techniques is to require fewer simulations;  The efficiency of this method is strongly dependent on the correlation between the variable of which we have to compute the estimator and the control variate itself.  Finding efficient control variate is more an art than a science!  In the following slides we give an example of the use of copula function in finding control variate.
    79. 79. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 79 Pricing Rainbow Options with Monte Carlo  Rainbow options are extensively used in structured finance equity linked products, such as Everest, Altiplanos and the like.  Very often Monte Carlo simulation is the only pricing technique available, even though it turns out to be pretty costly and slow, particularly for high dimensional problems.  Copula methods, and particularly Fréchet bounds, provide approximated closed form solutions to the problem.
    80. 80. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 80 Copula functions and the pricing of structured financial products  Digital products structured with digital bivariate options:  Coupon = 10% if Nasdaq > K1 and Nikkey > K2  Coupon = 10% C(Q(Nasdaq > K1),Q(Nikkey > K2))  Call options on the minimum of a basket of assets (Everest) ( ) [ ] ηηηη dTSQTSQTSQCTtP TKSSSCall K NN N ∫ ∞ >>> = ))((),...)((),)((, ),),,...,(min( 2211 21
    81. 81. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 81 Super-replication (lower bound)  With perfect negative dependence we have with K** defined in such a way that Q1(K**) + Q2(K**) = 1.  So, super-replication requires two call spreads and a debt position for an amount K**- K. ( ) ( ) [ ] ( ) ( )[ ] ( ) ( )[ ] [ ]       −− −+− = =       −−>+>= > > ∫∫ KKB KtSCKtSCKtSCKtSC KKBdSQBdSQBCALL KK K K K K KKMax ** ;,**;,;,**;, ** 2211 ** ** 22 ** 11** 1 1 αηηη
    82. 82. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 82 Super-replication (upper bound)  With perfect positive dependence with K* such that Q1(K*) = Q2(K*).  Super-replication requires a call spread on the first asset and a call option on the second one ( ) ( ) ( ) ( )[ ] ( ))*,max(;, ;,*;, 2 11* *],max[ 2 * 11* 2 KKtSC KtSCKtSC dSQBdSQBCMax KK KK K K KK + +−= =>+>= > ∞ > ∫∫ 1 1 ηηηη
    83. 83. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 83 The indipendence case  In the independence case  …where numerical integration is required. ( ) ( )∫ ∞ >>= K Ind dSQSQBC ηηη 211 2
    84. 84. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 84 Fréchet pricing  Using the closed form solutions for the extreme dependence cases, we may come up with an approximated evaluation of a rainbow option with a rank correlation equal to ρ Cρ = (1 – ρ)Cind + ρCmax  Conjecture: Cρ should be a good control variate for any Monte Carlo Simulation of a two colour Rainbow Option with correlation ρ.
    85. 85. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 85 Efficiency of Reduction In the Money Rho = 0.25 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Standard Monte Carlo "Fréchet" Reduction Variate Expiration 0.25 Asset 1 110 Asset 2 105 Volatility Asset 1 0.25 Volatility Asset 2 0.25-0.4 Strike 95 Risk Free Rate 0.05 Option Data
    86. 86. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 86 Efficiency of Reduction In the Money Rho = 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Expiration 0.25 Asset 1 110 Asset 2 105 Volatility Asset 1 0.25 Volatility Asset 2 0.25-0.4 Strike 95 Risk Free Rate 0.05 Option Data
    87. 87. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 87 Efficiency of Reduction In the Money Rho = 0.75 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Expiration 0.25 Asset 1 110 Asset 2 105 Volatility Asset 1 0.25 Volatility Asset 2 0.25-0.4 Strike 95 Risk Free Rate 0.05 Option Data
    88. 88. FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula 88 Bibliography  U. Cherubini, E. Luciano and W. Vecchiato  Copula Methods in Finance, Wiley Finance (2004)  P. Glasserman  Monte Carlo Methods in Financial Engineering, Springer (2004)  P. Jackel  Monte Carlo Methods in Finance, Wiley Finance (2002)  P. Kloeden, E. Platen and H. Schurz  Numerical Solution of SDE through Computer Experiments, Springer (1994)  D. Tavella  Derivatives Pricing, An Introduction to Computational Finance, Wiley Finance (2002)  P. Wilmott  Derivatives, Wiley (1998)

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