Download as PDF, PPTX
![Solving
Systems
of
Linear
Equa%ons
Quick
Review
q)A:3 3#1 3 -2 3 5 6 2 4
3f;
q)B:3 1#5 7 8f;
q)X:.qml.mls[A;B];
q)X
-15
8
2
MATHEMATICS
FUNDAMENTALS](https://image.slidesharecdn.com/quantitativefinanceinq-161013115245/75/Quantitative-finance-in-q-4-2048.jpg)
![Mul%ple
Regression
Least-‐Squares
(LSQ)
Approxima%on
Quick
Review
q)ingredients:(“FFFF”;”,”)
0:`ingredients.csv;
q)heat:”F” $ read0`heat.csv;
q)ingredients:flip ingredients;
q)heat:13 1#heat;
q).qml.mm[.qml.mm[.qml.minv[.qml.
mm[flip
ingredients;ingredients]];flip
ingredients];heat]
2.193046
1.153326
0.7585091
0.4863193
MATHEMATICS
FUNDAMENTALS](https://image.slidesharecdn.com/quantitativefinanceinq-161013115245/75/Quantitative-finance-in-q-5-2048.jpg)
![Demonstra%on
Results
// Problematic Multiple Regression example
// Based on R Tutorial Data
// http://www.cyclismo.org/tutorial/R/linearLieastSquares.html
//
// Car Loans regressed on year from U.S. Federal Reserve
// http://www.federalreserve.gov/releases/g19/20050805
//
l qml.q
year:1f cross 2000f+til 5
rate:9.34 8.50 7.62 6.93 6.60f;
// Reorient the rate matrix
rate:5 1#rate;
// Run least squares approximation
a:(flip year) $ year;
b:inv a;
y:(b $ (flip year)) $ rate;
x:.qml.mm[.qml.mm[.qml.minv[.qml.mm[flip year;year]];flip
year];rate];
show "Year Matrix";
show year;
show "Intermediate Result a";
show a;
show "Matrix Inversion of a";
show "Ill-conditioned Matrix!";
show "a's 2-norm condition number is 8.0321e+12"
show b;
show "Results Using q"
show y;
show "Results Using QML"
show x;
Welcome
to
kdb+
32bit
edi%on
For
support
please
see
hbp://groups.google.com/d/forum/personal-‐kdbplus
Tutorials
can
be
found
at
hbp://code.kx.com/wiki/Tutorials
To
exit,
type
To
remove
this
startup
msg,
edit
q.q
"Year
Matrix"
1
2000
1
2001
1
2002
1
2003
1
2004
"Intermediate
Result
a"
5
10010
10010
2.004003e+07
"Matrix
Inversion
of
a"
"Ill-‐condi%oned
Matrix!"
"a's
2-‐norm
condi%on
number
is
8.0321e+12"
"Results
Using
q"
"Results
Using
QML"
1419.208
-‐0.705
MATHEMATICS
FUNDAMENTALS](https://image.slidesharecdn.com/quantitativefinanceinq-161013115245/75/Quantitative-finance-in-q-9-2048.jpg)
![Problem
Avoidance
by
Input
Transforma%on
// Problematic Multiple Regression example
// Based on R Tutorial Data
// http://www.cyclismo.org/tutorial/R/linearLieastSquares.html
//
// Car Loans regressed on year from U.S. Federal Reserve
// http://www.federalreserve.gov/releases/g19/20050805
//
l qml.q
year:1f cross 0f+til 5
rate:9.34 8.50 7.62 6.93 6.60f;
// Reorient the rate matrix
rate:5 1#rate;
// Run least squares approximation
a:(flip year) $ year;
b:inv a;
y:(b $ (flip year)) $ rate;
x:.qml.mm[.qml.mm[.qml.minv[.qml.mm[flip year;year]];flip
year];rate];
show "Year Matrix";
show year;
show "Intermediate Result a";
show a;
show "Matrix Inversion of a";
show "a's 2-norm condition number is 22.4555"
show b;
show "Results Using q"
show y;
show "Results Using QML"
show x;
Welcome
to
kdb+
32bit
edi%on
For
support
please
see
hbp://groups.google.com/d/forum/personal-‐kdbplus
Tutorials
can
be
found
at
hbp://code.kx.com/wiki/Tutorials
To
exit,
type
To
remove
this
startup
msg,
edit
q.q
"Year
Matrix"
1
0
1
1
1
2
1
3
1
4
"Intermediate
Result
a"
5
10
10
30
"Matrix
Inversion
of
a"
"a's
2-‐norm
condi%on
number
is
22.4555"
0.6
-‐0.2
-‐0.2
0.1
"Results
Using
q"
9.208
-‐0.705
"Results
Using
QML"
9.208
-‐0.705
MATHEMATICS
FUNDAMENTALS](https://image.slidesharecdn.com/quantitativefinanceinq-161013115245/75/Quantitative-finance-in-q-10-2048.jpg)
![PCA
in
q/kdb+
with
qml
//
Load
q
Math
Library
(qml)
l
qml.q
//
Read
in
historical
JPY
LIBOR
data
histLIBOR:
("SFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF";enlist
",")
0:`QProject.csv;
//
Calculate
daily
differences
histDeltas:
deltas
delete
Tenor
from
histLIBOR;
delete
from
`histDeltas
where
i
=
0;
//
Define
a
func%on
to
calculate
variance-‐covariance
matrix
varCov:{a
cov/::
a:flip
x};
//
Calculate
an
annualize
variance-‐covariance
matrix
histCov:varCov[histDeltas]*252%10000;
//
Extract
eigenvalues
and
eigenvectors
sorted
by
decreasing
modulus
eigen:.qml.mev[value
each
value
histCov];
//
Write
out
data
for
graphing
elsewhere
graphData1:(1
51)#raze
eigen[0];
save
`graphData1.csv
graphData2:(1
51)#sums
eigen[0]%sum
eigen[0];
save
`graphData2.csv
graphData3:(3
51)#eigen[1][%l
3];
save
`graphData3.csv
ADVANCED
APPLICATION](https://image.slidesharecdn.com/quantitativefinanceinq-161013115245/75/Quantitative-finance-in-q-17-2048.jpg)
![Monte
Carlo
Simula%on
Random
Number
Genera%on
• For
each
%mestep,
we
need
3
normally
distributed
random
values
• Use
the
Box_Muller
transform
as
implemented
in
Nick
Psaris’
Q
Tips
• Computa%onally
more
intense,
however
beber
accuracy
/ box-muller
bm:{
if[count[x] mod
2;'`length];
x:2 0N#x;
r:sqrt -2f*log first x;
theta:2f*acos[-1f]*last
x;
x: r*cos theta;
x,:r*sin theta;
x
ADVANCED
APPLICATION](https://image.slidesharecdn.com/quantitativefinanceinq-161013115245/75/Quantitative-finance-in-q-20-2048.jpg)
More Related Content
Quantitative finance in q
- 1. Quan%ta%ve Finance in q/kdb+ Pricing Fixed Income Deriva%ves Mark Lefevre Quan%ta%ve Analyst
- 2. Introduc%on • q/kdb+ can be used for much more than lightning fast table manipula%on • Excellent analy%cal capabili%es • Combining these strengths with 3rd-‐party soIware and/or linking with addi%onal libraries provides excep%onal opportuni%es to analyze, model and predict
- 3. Mo%va%on • q/kdb+ is oIen used in combina%on with MatLab and R for heavy quan%ta%ve finance work • q/kdb+ integrates well with many foreign languages and applica%ons – C, C#, Java, Python, Perl, Fortran, Scala – ODBC, GPUs (CUDA) – Matlab, R • Moving algorithmic code into q/kdb+ is not always straighWorward – Out-‐of-‐the-‐box q/kdb+ lacks high-‐level sta%s%cs, op%miza%on, linear algebra – Algorithmic code from other languages/environments usually was developed with a different mindset • Let’s explore this topic by breaking in a dyadic manner 1. Mathema%cs Fundamentals/Numerical Analysis 2. Advanced, Prac%cal Applica%on: Pricing a Fixed Income Deriva%ve
- 4. Solving Systems of Linear Equa%ons Quick Review q)A:3 3#1 3 -2 3 5 6 2 4 3f; q)B:3 1#5 7 8f; q)X:.qml.mls[A;B]; q)X -15 8 2 MATHEMATICS FUNDAMENTALS
- 5. Mul%ple Regression Least-‐Squares (LSQ) Approxima%on Quick Review q)ingredients:(“FFFF”;”,”) 0:`ingredients.csv; q)heat:”F” $ read0`heat.csv; q)ingredients:flip ingredients; q)heat:13 1#heat; q).qml.mm[.qml.mm[.qml.minv[.qml. mm[flip ingredients;ingredients]];flip ingredients];heat] 2.193046 1.153326 0.7585091 0.4863193 MATHEMATICS FUNDAMENTALS
- 6. Problema%c Linear Regression • One area of focus in Numerical Analysis is numerical stability • We can demonstrate a numerical stability problem using a simple linear regression as an example • Simple prac%cal example – Car loans regressed on year from U.S Fed Reserve • References • R tutorial hbp://www.cyclismo.org/tutorial/R/ linearLeastSquares.html MATHEMATICS FUNDAMENTALS
- 7. Reference Matlab Implementa%on MATHEMATICS FUNDAMENTALS
- 8. q/kdb+ Implementa%on • Live Demonstra%on MATHEMATICS FUNDAMENTALS
- 9. Demonstra%on Results //Problematic Multiple Regression example // Based on R Tutorial Data // http://www.cyclismo.org/tutorial/R/linearLieastSquares.html // // Car Loans regressed on year from U.S. Federal Reserve // http://www.federalreserve.gov/releases/g19/20050805 // l qml.q year:1f cross 2000f+til 5 rate:9.34 8.50 7.62 6.93 6.60f; // Reorient the rate matrix rate:5 1#rate; // Run least squares approximation a:(flip year) $ year; b:inv a; y:(b $ (flip year)) $ rate; x:.qml.mm[.qml.mm[.qml.minv[.qml.mm[flip year;year]];flip year];rate]; show "Year Matrix"; show year; show "Intermediate Result a"; show a; show "Matrix Inversion of a"; show "Ill-conditioned Matrix!"; show "a's 2-norm condition number is 8.0321e+12" show b; show "Results Using q" show y; show "Results Using QML" show x; Welcome to kdb+ 32bit edi%on For support please see hbp://groups.google.com/d/forum/personal-‐kdbplus Tutorials can be found at hbp://code.kx.com/wiki/Tutorials To exit, type To remove this startup msg, edit q.q "Year Matrix" 1 2000 1 2001 1 2002 1 2003 1 2004 "Intermediate Result a" 5 10010 10010 2.004003e+07 "Matrix Inversion of a" "Ill-‐condi%oned Matrix!" "a's 2-‐norm condi%on number is 8.0321e+12" "Results Using q" "Results Using QML" 1419.208 -‐0.705 MATHEMATICS FUNDAMENTALS
- 10. Problem Avoidance by Input Transforma%on // Problematic Multiple Regression example // Based on R Tutorial Data // http://www.cyclismo.org/tutorial/R/linearLieastSquares.html // // Car Loans regressed on year from U.S. Federal Reserve // http://www.federalreserve.gov/releases/g19/20050805 // l qml.q year:1f cross 0f+til 5 rate:9.34 8.50 7.62 6.93 6.60f; // Reorient the rate matrix rate:5 1#rate; // Run least squares approximation a:(flip year) $ year; b:inv a; y:(b $ (flip year)) $ rate; x:.qml.mm[.qml.mm[.qml.minv[.qml.mm[flip year;year]];flip year];rate]; show "Year Matrix"; show year; show "Intermediate Result a"; show a; show "Matrix Inversion of a"; show "a's 2-norm condition number is 22.4555" show b; show "Results Using q" show y; show "Results Using QML" show x; Welcome to kdb+ 32bit edi%on For support please see hbp://groups.google.com/d/forum/personal-‐kdbplus Tutorials can be found at hbp://code.kx.com/wiki/Tutorials To exit, type To remove this startup msg, edit q.q "Year Matrix" 1 0 1 1 1 2 1 3 1 4 "Intermediate Result a" 5 10 10 30 "Matrix Inversion of a" "a's 2-‐norm condi%on number is 22.4555" 0.6 -‐0.2 -‐0.2 0.1 "Results Using q" 9.208 -‐0.705 "Results Using QML" 9.208 -‐0.705 MATHEMATICS FUNDAMENTALS
- 11. Q Math Library • qml is a free library for q/kdb+ that links into various numerical libraries including LAPACK – Linear algebra – Sta%s%cs – Op%miza%on • Compiling qml with 32-‐bit q/kdb+ on a 64-‐bit machine requires pa%ence and the correct compilers, but works – LAPACK is wriben in Fortran 90, so you will need a Fortran compiler, as well as the usual suspects References LAPACK http://www.netlib.org/lapack/ qml http://althenia.net/qml
- 12. Fixed Income Deriva%ves Source: Bank of Interna%onal Seblements (BIS) Quarterly Review, June 2015 • Most common fixed income deriva%ves – Op5ons • Caps, floors, swaps – Futures/Forwards – Credit Default Swaps (CDS) – Interest Rate Swaps – Cross Currency Swaps – Total Return Swaps 0 5000 10000 15000 20000 25000 30000 35000 40000 Jun.05 Jun.06 Jun.07 Jun.08 Jun.09 Jun.10 Jun.11 Jun.12 Jun.13 Jun.14 USD Billions OTC Deriva5ves (No5onal Outstanding) Unallocated Credit default swaps Commodity contracts Equity-‐linked contracts Foreign exchange contracts ADVANCED APPLICATION
- 13. Pricing Caps and Caplets • A cap is an op%on on an underlying floa%ng interest rate, such as LIBOR, that provides investors with a hedge against the interest rate rising above the cap rate • The cap is composed of individual caplets • If the interest rate is above the strike, the caplet is in-‐the-‐money • Rate set at %me t • Payoff at %me t+1 • Present Value max( 𝑟↓𝑡 − 𝑘,0)× 𝜏× 𝑁 𝐷 𝐹↓𝑂𝐼𝑆 ↓𝑡+1 ×max( 𝑟↓𝑡 − 𝑘,0)× 𝜏× 𝑁 ADVANCED APPLICATION
- 14. Heath-‐Jarrow-‐Morton Model • HJM models the dynamics of the en%re instantaneous forward rate curve • Purple is instantaneous rate agreed at t for %me s • Blue is a ZCB at that future %me over the future %me period • Green is differen%al and integral forms of the dynamics. Wt is GBM • Red is key rela%onship between driI (α) and vola%lity (σ)! • Widely used in prac%ce • Uses Monte-‐Carlo methods and discrete %me finance • This presenta%on glosses over many important implementa%on details in the interest of %me References hbp//www.columbia.edu/~mh2078/ HJM_models.pdf 𝑓(𝑡, 𝑠)=− 𝜕/𝜕𝑠 log 𝑍↓𝑡↑𝑠 𝑍↓𝑡↑𝑠 =−∫𝑡↑𝑠▒𝑓(𝑡, 𝑢)𝑑𝑢 𝑑𝑓(𝑡, 𝑠)= 𝛼(𝑡, 𝑠)𝑑𝑡+ 𝜎( 𝑡, 𝑠)↑𝑇 𝑑 𝑊↓𝑡 𝛼(𝑡, 𝑠)= 𝜎( 𝑡, 𝑠)↑𝑇 ∫𝑡↑𝑠▒𝜎(𝑡, 𝑢)𝑑𝑢 𝑓(𝑡, 𝑠)= 𝑓(0, 𝑠)+∫0↑𝑡▒𝛼(𝑢, 𝑠)𝑑𝑢 +∫0↑𝑡▒𝜎( 𝑢, 𝑠)↑𝑇 𝑑 𝑊↓𝑢 ADVANCED APPLICATION
- 15. Historical Data Analysis ADVANCED APPLICATION
- 16. Principal Component Analysis (PCA) • PCA is a sta%s%cal procedure that u%lizes orthogonal transforma%ons to convert a set of observa%ons of possibly correlated variables into a set of values of linearly uncorrelated variables called principal components. • In a word, decorrela(on • The principal components are the eigenvectors of a symmetric variance-‐ covariance matrix • Eigenvectors are ordered by their corresponding eigenvalues, which is also the amount of variance explained by the component • If we take just a few of the first principal components, we can achieve – dimensionality reduc5on – work in linear parameter space • This is very useful for simula%ng high dimensionality problems, such as instantaneous forward curve evolu%ons ADVANCED APPLICATION
- 17. PCA in q/kdb+ with qml // Load q Math Library (qml) l qml.q // Read in historical JPY LIBOR data histLIBOR: ("SFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF";enlist ",") 0:`QProject.csv; // Calculate daily differences histDeltas: deltas delete Tenor from histLIBOR; delete from `histDeltas where i = 0; // Define a func%on to calculate variance-‐covariance matrix varCov:{a cov/:: a:flip x}; // Calculate an annualize variance-‐covariance matrix histCov:varCov[histDeltas]*252%10000; // Extract eigenvalues and eigenvectors sorted by decreasing modulus eigen:.qml.mev[value each value histCov]; // Write out data for graphing elsewhere graphData1:(1 51)#raze eigen[0]; save `graphData1.csv graphData2:(1 51)#sums eigen[0]%sum eigen[0]; save `graphData2.csv graphData3:(3 51)#eigen[1][%l 3]; save `graphData3.csv ADVANCED APPLICATION
- 18. PCA Results ADVANCED APPLICATION
- 19. Vola%lity Fi•ng • In a mul%-‐factor HJM framework, we need to integrate over vola%lity func%ons of a form like • What this means in English, is we have several eigenvalue / eigenvector func%ons driving the dynamics at par%cular tenors • Consistent with Musiela parameteriza%on • This permits a rich evolu%on of forward rates exhibi%ng – Level shiIs – Twists – Buberfly-‐like inflec%on about specific tenors • Subject to regimes ADVANCED APPLICATION
- 20. Monte Carlo Simula%on Random Number Genera%on • For each %mestep, we need 3 normally distributed random values • Use the Box_Muller transform as implemented in Nick Psaris’ Q Tips • Computa%onally more intense, however beber accuracy / box-muller bm:{ if[count[x] mod 2;'`length]; x:2 0N#x; r:sqrt -2f*log first x; theta:2f*acos[-1f]*last x; x: r*cos theta; x,:r*sin theta; x ADVANCED APPLICATION
- 21. Pricing • Live Demonstra%on ADVANCED APPLICATION
- 22. Conclusion • q/kdb+ can be used for much more than table manipula%on • Demonstrated some basic mathema%cs and an advanced, prac%cal applica%on to pricing caplets • Used 3rd-‐party soIware for visualiza%on, however the new HTML5 capabili%es hold lots of promise • Linked to and leveraged industrial-‐strength mathema%cs libraries • Hopefully, this has provide a glimpse at the excep%onal opportuni%es to analyze, model and predict that q/kdb + can provide
- 23. About Me • Mark is currently consul%ng at one of the largest banks in Tokyo as a quan%ta%ve analyst developing high-‐performance algorithmic trading systems on the e-‐FX desk. Prior to moving to Japan, he worked in London for Unicredit on the Equity-‐ Linked Origina%on desk crea%ng conver%ble bonds for European corporates, consulted in the US on e-‐commerce analy%cs and worked for several high-‐tech soIware companies. • Earlier in his career, he worked for Mitsubishi Semiconductor America designing semiconductors and a startup developing a DSP. He then moved into applica%ons engineering for an Electronic Design Automa%on (EDA) company and, subsequently, internet soIware companies in CA and Europe. • Mark has a bachelors degree in Electrical Engineering and Computer Science from Duke University, a masters degree in Computer Engineering from North Carolina State University and an MBA in Quan%ta%ve Finance from the Wharton School of Business. He recently completed a Cer%ficate in Quan%ta%ve Finance (CQF). • He dreams of the day when he can create soIware without encountering a single type error