1. Using the NAG Library in Excel
Marcin Krzysztofik
NAG Ltd.
Experts in numerical algorithms
and HPC services

2. Agenda
 NAG Introduction
 Why do Quants love NAG?
 NAG from VBA
 Examples
• Least Squares Optimisation
• Local Volatility
• Nearest Correlation Matrix
2

3. The Numerical Algorithms Group
 NAG provides mathematical and statistical algorithm
libraries and compiler widely used in industry and academia
 Founded in 1970 as a co-operative project in UK
 Operates as a commercial, not-for-profit organization
 Worldwide operations
• Offices in Oxford & Manchester, Chicago, Tokyo, Taipei
• Technical Support Engineer in Germany
• Distributors in Brazil, India, Singapore, Mexico…
 Over 3,000 customer sites worldwide
 Staff of ~100, ~70 technical, includes ~25 HPC
3

4. NAG Portfolio
 Mathematical, statistical, data analysis components
• NAG Numerical libraries
• Connectors for Excel, .NET, R, Java
 NAG Fortran Compiler
• Windows GUI - Fortran Builder
 HPC software engineering services
• HECToR support
 Consultancy work for bespoke application
development
4

5. NAG is a HPCFinance partner
http://www.hpcfinance.eu
The network is recruiting for
 Early Stage Researchers (ESRs ~ PhD Students)
 Experienced Researchers (ERs ~ Post Docs) 5

6. The NAG Library Contents
• Dense Linear Algebra
• Root Finding
• Sparse Linear Algebra
• Summation of Series
• Correlation & Regression Analysis
• Quadrature
• Multivariate Methods
• Ordinary Differential Equations
• Analysis of Variance
• Partial Differential Equations
• Random Number Generators
• Numerical Differentiation
• Univariate Estimation
• Integral Equations
• Nonparametric Statistics
• Mesh Generation
• Smoothing in Statistics
• Interpolation
• Contingency Table Analysis
• Curve and Surface Fitting
• Survival Analysis
• Optimisation
• Time Series Analysis
• Approximations of Special Functions
• Operations Research
6

7. The NAG Library is now at Mark 23
Now available as:
 The NAG Fortran Library
 The NAG Library for SMP & Multicore
 The NAG C Library
 The NAG Toolbox for MATLAB
7
7

8. NAG Library : new Mark 23
Mark 23 has new functions in many areas including...
 Wavelet Transforms
• One dimensional continuous transforms  Optimisation
• Two dimensional discrete single level and • Multi-start optimisation
multi-level transforms • Minimization by quadratic approximation
 ODE’s (BOBYQA)
• BVP solution through Chebyshev pseudo- • Stochastic global optimisation using PSO
spectral method  RNG’s
 Matrix Operations • Generators of multivariate copulas
• Matrix exponentials • Skip-ahead for Mersenne Twister
• Functions of real symmetric and Hermitian • L’Ecuyer MRG32K3a generator
matricies  Statistics
• Sparse matrix functions
• Quantiles of streamed data, bivariate
• LAPACK 3.2 Cholesky solvers and Student’s t, and two probability density
factorizations, and many other LAPACK functions
driver functions
• Nearest correlation matrices
 Interpolation • Quantile regression
• Modified Shepard’s method in 4D/5D • Peirce Outlier detection
 New vector functions (in G01 and S) • Anderson–Darling goodness-of-fit
8

9. NAG Library : new Mark 23
Mark 23 has new functions in many areas including...
 Wavelet Transforms
• One dimensional continuous transforms  Optimisation
• Two dimensional discrete single level and • Multi-start optimisation
multi-level transforms • Minimization by quadratic approximation
 ODE’s (BOBYQA)
• BVP solution through Chebyshev pseudo- • Stochastic global optimisation using PSO
spectral method  RNG’s
 Matrix Operations • Generators of multivariate copulas
• Matrix exponentials • Skip-ahead for Mersenne Twister
• Functions of real symmetric and Hermitian • L’Ecuyer MRG32K3a generator
matricies  Statistics
• Sparse matrix functions
• Quantiles of streamed data, bivariate
• LAPACK 3.2 Cholesky solvers and Student’s t, and two probability density
factorizations, and many other LAPACK functions
driver functions
• Nearest correlation matrices
 Interpolation • Quantile regression
• Modified Shepard’s method in 4D/5D • Peirce Outlier detection
 New vector functions (in G01 and S) • Anderson–Darling goodness-of-fit
9

10. Why do Quants love NAG?
Senior quant from Tier 1 Investment Bank
“We deploy production code in C++ embedding NAG C Library functions
wherever we can, but often prototype new models in MATLAB before
writing our C++ code. Having the same NAG algorithms in MATLAB via the
NAG Toolbox for MATLAB is a real win for us”
OK, what more?
10

11. Numerical computation – DIY Vs NAG
 DIY implementations of numerical components have
their place, but NOT in production code.
• Handwritten code might be easy to implement, but will…
 NOT be well tested
 NOT be fast
 NOT be stable
 NOT deliver good error handling
• NAG implementations in contrast are fast and
 Accurate
 Well tested & thoroughly documented
 Give “qualified error” messages e.g. tolerances of answers (which
the user can choose to ignore, but avoids proceeding blindly)
11

12. Problem 1: Finance PDE Solvers
 Major banks PDE solvers tend to be “proprietary” so
there is a reluctance to take “end to end” solutions
 Several different numerical components needed
• Sparse Linear Solvers
• Functions for finding a range of Eigenvalues (stability
checkers)
• Special Linear Systems solvers (e.g. Banded Matrices)
• ..
12

13. Problem 1: Finance PDE Solvers
 Major banks PDE solvers tend to be “proprietary” so
there is a reluctance to take “end to end” solutions
NAG to the rescue
 Several different numerical components needed
• Sparse Linear Solvers √
• Functions for finding a range of Eigenvalues (stability
checkers) √
• Special Linear Systems solvers (e.g. Banded Matrices) √
• .. √ √
13

14. Problem 2: Calibration
 Major banks all need to calibrate their models
 Several different numerical components needed
• Optimisation functions (e.g. constrained non-linear
optimisers)
• Interpolation functions
• Spline functions
• ..
14

15. Problem 2: Calibration
 Major banks all need to calibrate their models
NAG to the rescue
 Several different numerical components needed
• Optimisation functions (e.g. constrained non-linear
optimisers) √
• Interpolation functions (used intelligently*) √
• Spline functions √
• .. √ √
*interpolator must be used carefully –must know the properties to pick appropriate method
15

16. Problem 3: Closed-Form Option Pricing Formulae
 Closed form solutions are useful
• for validating numerical methods
• for some calibration problems as input (objective function)
to optimisers
 Several different numerical components needed
• FFTs, Quadrature
• Root finders (single variable)
• Special functions (Bessel, non-Central Chi, Erf,..)
• Probability distributions
• ..
16

17. Problem 3: Closed-Form Option Pricing Formulae
 Closed form solutions are useful
• for validating numerical methods
• for some calibration problems as input (objective function)
to optimisers
NAG to the rescue
 Several different numerical components needed
• FFTs, Quadrature √
• Root finders (single variable) √
• Special functions (Bessel, non-Central Chi, Erf,..) √
• Probability distributions √
• .. √ √
17

18. Problem 4: Simulation (Monte Carlo)
 Simulation is important for scenario generation
 Several different numerical components needed
• Random Number Generators
• Brownian bridge constructor
• Interpolation/Splines
• Principal Component Analysis
• Cholesky Decomposition
• Distributions (uniform, Normal, exponential gamma,
Poisson, Student’s t, Weibull,..)
• ..
18

19. Problem 4: Simulation (Monte Carlo)
 Simulation is important for scenario generation
NAG to the rescue
 Several different numerical components needed
• Random Number Generators √
• Brownian bridge constructor √
• Interpolation/Splines √
• Principal Component Analysis√
• Cholesky Decomposition √
• Distributions (uniform, Normal, exponential gamma,
Poisson, Student’s t, Weibull,..)√
• .. √ √
19

20. Problem 5: Risk Matrix VaR
 VaR is important for identifying correlation of one
asset to another (eg Yen Vs USD)
 Several different numerical components needed
• Matrix functions
• Correlation and Regressions
•
• ..
20

21. Problem 5: Risk Matrix VaR
 VaR is important for identifying correlation of one
asset to another (eg Yen Vs USD)
NAG to the rescue
 Several different numerical components needed
• Matrix functions √
• Correlation and Regressions √
•
• .. √ √
21

22. Problem 6: Historical VaR
 This is another VaR methodology (again important
for identifying what variables might impact you
most (eg Yen Vs USD))
 Several different numerical components needed
• Time Series
• Correlation and Regressions
• Matrix functions
• Cholesky Decomp
• RNGs ..
22

23. Problem 6: Historical VaR
 This is another VaR methodology (again important
for identifying what variables might impact you
most (eg Yen Vs USD))
NAG to the rescue
 Several different numerical components needed
• Time Series √
• Correlation and Regressions √
• Matrix functions √
• Cholesky Decomp √
• RNGs .. √ √
23

24. NAG provides the atomic bricks
 … for the quants to build the walls, houses and fancy
castles!
 Users know NAG Components are here today,
tomorrow and beyond
• Functions are not removed when new ones added without
sensible notice and advice
• NAG functions are well documented
 Lets take a look at Excel & VBA now…
24

25. NAG from Excel
 Intro
 Why use NAG from Excel
 How to call NAG from Excel
• A simple function
• Callbacks
• Passing arrays
25

26. NAG from Excel
 Use Excel’s underlying VBA to call external library
 NAG provides VBA Declaration Statements
• Shipped together with the DLL version of NAG Library
• Just Ctrl+C Ctrl+V into a VBA module
 Start creating macros and UDFs
• Macros executed for example via a button
• User Defined Functions called via Excel Function Wizard
26

27. Why NAG & Excel?
 Microsoft Office Excel is the industry-standard
spreadsheet application.
 Easy to use, intuitive, practical.
 However it’s not a statistical data analysis package...
 And that’s where NAG being great at serious
numerical calculations comes in!
27

28. How to call a simple function?
 S17AEF – returns value of the Bessel function 𝐽0 (𝑥)
 Note:
• Declaration statements in VB
• Calling straight from the spreadsheet
• Calling through a wrapper function
• Error handling
28

29. The IFAIL parameter
IFAIL serves two purposes:
 On input it determines what action the Library
routine takes when an error is detected:
• -1 (Soft Fail & Noisy Exit) – error message and continue;
• 0 (Hard Fail) – error message and stop;
• 1 (Soft Fail & Quiet Exit) – no error message and continue.
 On output it informs what kind of error has occurred,
depending on the function in use (see
documentation for details).
29

30. How to handle callbacks?
 A callback is a piece of executable code that is
passed as an argument to other code.
 D01BAF – performs numerical integration on a
function of one variable
• uses Gaussian quadrature rules
 Note:
• Visual Basic AddressOf operator to handle callbacks
• Argument Passing by reference ByRef
30

31. How to pass arrays as arguments?
 F07FDF – computes the Cholesky factorization of a
real symmetric positive definite matrix.
 Note:
• Reading the array from the worksheet (Range -> Double)
• Checking if the matrix is a square matrix
• What if the matrix is not symmetric?
32

32. Example
Least Squares Optimisation
33

33. 1st Example: Least Squares Optimisation
 General formulation of an optimisation problem is
min F ( x) subject to constraints on x
xR
n
• Constraints can be bound, linear or fully non-linear
• Example: maximise value of portfolio subject to limits on
capital, individual positions and overall risk exposure
• Have routines in NAG Library to solve these problems
 Least squares optimisation addresses the following:
m
min  f i 2 ( x) subject to constraints on x
xR
n
i 1
• Most commonly used in fitting data
34

34. 1st Example: Least Squares Optimisation
 Important special case:
 x 
m
 Ax 
min   yi  f i ( x)  subject to l   u
2
xR
n
i 1 c( x)
 
• Used in calibration and non-linear regression, typically in
the form
m
min   yi  C ( xi ,  )  subject to l    u
2
 R
n
i 1
• Here 𝐶is a pricing formula (e.g. SABR, Heston, CIR) with
parameters 𝛽, and (𝑥 𝑖 , 𝑦 𝑖 ) are 𝑚 observed market prices
35

35. 1st Example: Least Squares Optimisation
 Another important special case:
 x 
min x A x  b x subject to l  
T T
u
xR
C x 
n
• Called Quadratic Programming
• 𝐴 and 𝐶 and matrices and 𝑏 is a vector
• Objective function is quadratic polynomial, constraints are
linear
• Arises in certain asset allocation problems, interpolation
problems, linear programming problems ...
• Emphasis on speed and scale: can have 𝑛 ≥ 20,000 and
still require a solution in under 0.5s
36

36. 1st Example: Mean-Variance Analysis
 Recall classic Markowitz model
n
P( x)   xi Ri  x R subject to 
n
T
x 1
i 1 i
i 1
• Assume return on each asset 𝑖 is 𝑅 𝑖 ~𝑁1 (𝜇 𝑖 , 𝜎 𝑖2 )
• Assume vector of all 𝑚 returns 𝑅~𝑁 𝑛 (𝜇, A)
• Therefore return on portfolio 𝑃 𝑥 ~𝑁1 (𝑥 𝑇 𝜇, 𝑥 𝑇 𝐴𝑥)
• Define “Risk” as variance of return on portfolio, i.e.
𝑉𝑎𝑟 𝑃(𝑥) = 𝑥 𝑇 𝐴𝑥
• Problem: for given level of expected return 𝑟, find the
portfolio with minimal risk

min Var P( x)   x A x s.t. 
T  xi  1
 
xR n
 EP( x)   i xi   r 
 
37

37. 1st Example: Mean-Variance Analysis
 Let’s look at some market data
• 10 big FTSE100 players
• Market data: 20/05/2003 – 19/06/2009
 Use NAG to compute covariances, returns and
standard deviations of the stock returns
 Set upper & lower investment constraints
 Optimize
• Return portfolio risk for given return, sensitivities, shadow
prices
• Plot the Efficient Frontier
38

38. Example
Local Volatility Model
39

39. 2nd Example: Local Volatility Model
 What’s in a smile?
• Black-Scholes model is not perfect. Main drawbacks?
• So traders pick different vol for each combination of
maturity and strike, in order to get “correct” price
 Local volatility model tries to express this
dSt  r (t )   (t ) St dt   (t , St ) St dWt
• Here 𝜎(𝑡, 𝑥) is a function, to be determined
• Since volatility now depends on time and asset level, we
could hope that it’s possible to capture smile effect
• But what should 𝜎(𝑡, 𝑥) be to achieve this?
• Key is Dupire formula
40

40. 2nd Example: Local Volatility Model
 Let 𝐶(𝑇, 𝐾) denote price of European call option in
Local Volatility model
• By manipulating Kolmogorov forward equation (Fokker-
Plank equation) together with
  0 r ( s ) ds 
T
C (T , K )  E  e
 maxST  K ,0 

 
we can show that
CT   (T )C   (T )C K
 (T , K )  2
K 2C KK
• Now 𝐶(𝑇, 𝐾) we observe in the market – why? So this
gives a formula to translate market call quotes into 𝜎(𝑡, 𝑥)
41

41. 2nd Example: Local Volatility Model
 Let 𝜃(𝑇, 𝐾) be market implied vol for a call option
• Then 𝐶 𝑇, 𝐾 = 𝐵𝑆(𝑇, 𝐾, 𝜃(𝑇, 𝐾)) where 𝐵𝑆(. ) is the
Black-Scholes pricing formula
• Substituting and simplifying, we arrive at the Dupire
formula
2T   / T  2 K r (T )   (T )  K
 2 (T , K ) 
 1 1  
2
K  KK  d  T  K   
2
 d  K  
2
  K T  
 
• So given a smile surface 𝜃(𝑇, 𝐾) we can also recover
𝜎(𝑡, 𝑥). This more common way of doing things.
42

42. 2nd Example: Local Volatility Model
 Sweet – no problems!!
• Well, no. What’s the problem?
43

43. 2nd Example: Local Volatility Model
 Two problems
• 1. Dupire formula assumes infinitely many market quotes.
We only have finitely many. We’ll have to interpolate, and
this is challenging
• 2. Once we’ve obtained 𝜎(𝑡, 𝑥), what then? How do we
price options? Two routes: either Monte Carlo, or PDEs.
44

44. 2nd Example: Interpolation
 Interpolation – but that’s just joining the dots!
• Interpolated surface must be smooth: at least 𝐶 1,2
 Many interpolators can do this, so not such a problem.
• Worse: the interpolated surface must be arbitrage free!
 E.g. suppose had quotes for maturity 𝑇 and 𝐾 = 50,60,70,80,90.
 Delete quote for 𝐾 = 70 and interpolate over 𝐾 = 50,60,80,90.
 Interpolated value at 𝐾 = 70 must be consistent with current
market prices – no arbitrage
 Termed shape preserving interpolation – it’s a very
hard mathematical problem!
• Research is quite young, and not much in financial space
45

45. 2nd Example: Interpolation
 So how is this problem commonly tackled?
• As best you can
• One approach – fit monotonic (Hermite) interpolator in
time, then a cubic spline in space
• Make sure 𝜎 𝑡, 𝑥 ≥ 0
 Backward heat problems are unstable
 NAG Library has all these interpolators (and more!)
• Currently has no routines for shape constrained
interpolation, but watch this space!
46

46. 2nd Example: PDE Solution
 Monte Carlo vs PDEs?
• Local vol model is quite amenable to PDE techniques,
provided interpolation is good enough
• Monte Carlo reserved for heavily path-dependent options
 The local volatility model PDE for option price 𝑉 is
Vt  r (t )   (t ) SVS  1 2  2 (t , S ) S 2VSS  r (t )V
• NAG Library has many PDE solvers, but which kind of
solver should we be using?
• Decision tree
47

47. 2nd Example: NAG PDE Solver
 Try to use most specific solver, rather than most
general solver
 So seems we should use D03PFF
• Provided we can massage our PDE into conservative form
• Conservative form is term used predominantly in
engineering literature
• Related to continuity or conservation laws
 Conservation of mass
 Conservation of momentum
 Conservation of energy
 Etc.
• Many PDEs can be put into conservative form
48

48. 2nd Example: NAG PDE Solver
 Can re-write our PDE as follows
   
Vt  r    SV   V   1 2  2 S 2 VS   rV
 S  S
  2S 2 
1
Vt 

SV   VS   2r   V
r  S 2r    S r 
• Note 𝑟, 𝛾, 𝜎 are still functions. Our boundary conditions
are at 𝑡 = 𝑇, not 𝑡 = 0. So set 𝜏 = 𝑇 − 𝑡 to get
1    2S 2  2r  
V  SV   VS   V
r  S 2r    S r 
49

49. 2nd Example: NAG PDE Solver
 So our PDE can be placed in conservative form
 What about flux? How can we get approximate
solutions?
• Work it out analytically for your approximate Riemann
solver of choice: our flux equation is

Vt  SV   0
S
• Use a NAG routine to approximate it numerically (D03PUF)
• Look at Example 2 in D03PFF doc
 Just so happens to have our exact flux equation, with solution
50

50. 2nd Example: NAG Interpolators
 Requirements for interpolation
• 1D monotonic (Hermite) interpolator in time – E01BEF
• 1D cubic spline interpolator in space – E01BAF
 So far so good. How will we get derivatives?
• Recall need 𝜃 𝑇 , 𝜃 𝐾 , 𝜃 𝐾𝐾
• Simple approach – use finite differences
• Do NOT do this!
• NAG interpolation routines can also return exact analytic
derivatives – use these
51

51. 2nd Example: PDE Solver
 Before we look at the code:
• How will we know whether our model and
implementation are correct ... ?
 Examine the code
52

52. Example
Nearest Correlation Matrix
53

53. 3rd Example: Nearest Correlation Matrix
 Correlation between assets
• We all know what this is ... ?
 Mathematically, a correlation matrix 𝐶 ∈ ℝ 𝑛×𝑛 is ...
1. Square
2. Symmetric
3. Has ones on diagonal
4. Is positive semi-definite: 𝑥 𝑇 𝐶𝑥 ≥ 0 for all 𝑥 ∈ ℝ 𝑛
54

54. 3rd Example: Nearest Correlation Matrix
 How do we estimate correlations?
• Historical data
• Parametric methods such as Gaussian Copulas
• Try to back it out from options markets
 Typically 1, 2 and 3 easy enough to ensure
• Ensuring positive semi-definite can be tricky
55

55. 3rd Example: Nearest Correlation Matrix
 Historical data
• Take time series for several assets and try to estimate
correlation
• Perhaps time series model with exponential weights?
 Gaussian copula
 
G (u )  FZ (u )   C  1 (u1 ), ,  1 (un ) for u  [0,1]n
where 𝐹 𝑍 𝑢 = ℙ(𝑍 ≤ 𝑢) and Z~𝑁 𝑛 (0, C). Then set
FX ( x)  G F1 ( x1 ), , Fn ( xn )  for x  R n
where 𝐹𝑖 is the marginal distribution of asset 𝑋 𝑖
• Now only need 𝐹𝑖 and 𝐶, but still have to estimate 𝐶
56

56. 3rd Example: Nearest Correlation Matrix
 Infer from options markets
• Recent market events yet again highlighted importance of
correlation
• New developments in market-based approaches
• Idea is to combine individual options, options on indexes,
and perhaps best-of-two options, to back out correlation
structure
• Example is Local Correlation Model
• Typically have to supply some “primal” input correlation
matrix which is then evolved over time and space
57

57. 3rd Example: Nearest Correlation Matrix
 Given the importance of estimating correlations, what
happens if estimate not mathematically correct?
 NAG Library can find the “nearest” correlation matrix
to a given square matrix 𝐴
• G02AAF solves the problem min 𝐶 𝐴 − 𝐶 2𝐹 in Frobenius
norm
• G02ABF incorporates weights min 𝐶 𝑊 1/2
𝐴− 𝐶 𝑊 1/2 2
𝐹
• Weights useful when have more confidence in accuracy of
observations for certain assets than for others
58

58. 3rd Example: Nearest Correlation Matrix
 The effect of W:
A =
0.4218 0.6557 0.6787 0.6555
0.9157 0.3571 0.7577 0.1712
0.7922 0.8491 0.7431 0.7060
0.9595 0.9340 0.3922 0.0318
W = diag([10,10,1,1])
W*A*W = Whole rows/cols
42.1761 65.5741 6.7874 6.5548 weighted by 𝑤 𝑖
91.5736 35.7123 7.5774 1.7119
7.9221 8.4913 0.7431 0.7060 Elements weighted by
9.5949 9.3399 0.3922 0.0318 𝑤 𝑖 ∗ 𝑤𝑗
59

59. 3rd Example: Nearest Correlation Matrix
 Can also do dimension reduction (G02AEF)
• So-called factor models
• Very similar to PCA in regression analysis
• Suppose have assets 𝑌1 , ⋯ , 𝑌 𝑛 , an 𝑛 dimensional Brownian
motion 𝑊 and an 𝑛 × 𝑛 “correlation” matrix 𝐴 where
Yt1  Wt1 
   
    Ft  A  
Yt n  Wt n 
   
• For example, a simple multi-asset model
• Assuming one factor (Brownian motion) for each asset, and
𝐴𝐴 𝑇 gives correlation between all factors
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60. 3rd Example: Nearest Correlation Matrix
 Can use NAG Library to reduce the number of factors
• Find a n × 𝑘 matrix D (where 𝑘 < 𝑛) such that
Yt1  Wt1 
   
    Ft  D   
Yt n  Wt k 
   
• Crucially, 𝐷𝐷 𝑇 gives a correlation structure as close as
possible to the original structure implied by 𝐴
• Can be very useful to reduce complexity and
computational cost of some models and applications
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61. 3rd Example: Nearest Correlation Matrix
 Let’s examine the code
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62. Finally….
 Thank you and……
63

63. Finally…
 Contact support@nag.co.uk to get your free 30-day
trial licence key for NAG Library
• NAG C Library, Fortran Library,
• NAG Toolbox for MATLAB
• NAG Library for .NET
 Visit www.nag.co.uk/numeric/nagandexcel.asp
• Many Excel examples available
• Tips & tricks, white papers
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