A Random Walk Through Quantitative Finance 2016... · N.PADAYACHI, Arbitrage-free implied volatility surfaces for options on single stock futures, North American Journal of Economics

A Random Walk Through Quantitative Finance

Coenraad C. A. Labuschagne

Programme in Quantitative FinanceDepartment of Finance and Investment Management

University of JohannesburgSouth Africa

Inaugural Lecture6 April 2016

Coenraad C. A. Labuschagne Inaugural Lecture

Abstract

Quantitative finance has some of its roots in probabilitytheory, which has its roots in functional analysis.Both probability theory and functional analysis are areas ofmathematics.The interaction between these three areas of mathematicswill be discussed.In particular, attention will be given to the basic problems inquantitative finance and the tools from probability theoryand functional analysis that are used to solve them.


Introduction

This presentation is about what I do as a mathematician inquantitative finance – in layman’s terms.I will discuss some of the basic problems in quantitativefinance.Then mention the tools from probability theory that areused to solve them.This will be followed by a discussion on the interactionbetween functional analysis, probability theory andquantitative finance.All this, of course, is from my own perspective andgoverned by my research contributions and interests.


Quantitative Finance


Problem: pricing of financial instruments

Suppose a financial institution enters into a contract with aclient to sell a financial product - called a financialinstrument or derivative - to the client.They agree that at specified time t = T, the financialinstitution will pay the client a specified amount C(t,K)which depends on K, the strike price.The client will make a once off payment at time t = 0.The problem is to determine the selling price of thederivative at time t = 0.

Obtain the price at time t = 0 of a claim with agreed price K attime t = T.


Keep in mind that the value of the claim C(t,K) varies astime t varies.The financial institution does not want to sell the derivativeat time t = 0, be exposed to the claim at time T, and in themean time do nothing with the cash that the product wassold for!That cash has to be invested so as to cover the price of theclaim at time t = T.

This means that the position has to be hedged.

So the question is twofold:How do you price the claim at time t = 0, and hedge theposition.


Problem: management of risk

An important aspect of quantitative finance is themanagement of risk.There are various types of risk that need to managed.These include:

credit risk,market risk,volatility,arbitrage, andinflation.

The source has to be identified, the risk has to bemeasured, and steps have to be formulated to addressthese.An axiomatic approach to Risk Measures was introducedby Artzner, Delbaen, Eber and Heath in the early 1990s.Formulation of the the latter needs the notion of anLp-space (an object from probability theory and functionalanalysis) .


Tools

Solutions to these problems in Quantitative Finance regardingpricing & hedging and risk management

usually require the following fromprobability theory and functional analysis

expectations,conditional expectations,martingales,submartingales and supermartingales,stopping times, andLp-spaces.


Pricing

There are many known approaches to option pricing,which includes heavy-tailed distribution techniques.The continuous time Black-Scholes-Merton (BSM) modelis considered by many financial practitioners to beadequate for option pricing, irrespective of itsover-simplified assumptions.It was, and still is, widely used in practice, as it is wellunderstood and yields a framework in which both pricingand hedging is possible.


Black Scholes Merton (BSM) pricing model

Health warning: if equations make you feel ill, do not look at thenext two slides.

Let T denote the fixed time of maturity of a derivativecontract and σ as the volatility of the underlying security, inthis case a stock price S.The Black-Scholes-Merton partial differential equation(PDE) is given by

∂f∂t

+ rS∂f∂S

+12σ2S2 ∂

2f∂S2 = rf

wheref is the price of a derivative which depends on the stockprice St and time t ∈ [0,T].r is a constant interest rate, known as the risk-free rate.


BSM PDE

For a European call and put option (which are specific financialderivatives) with strike K, the BSM PDE has solution

Vt = α(

S0N (αd1)− Ke−r(T−t)N (αd2))

where

d1 =ln(

S0K

)+(r + 1

2σ2)

(T − t)

σ√

(T − t)

andd2 = d1 − σ

√T − t,

where α = 1 for a call option and α = −1 for a put option andN (x) is the cumulative distribution function of the standardnormal distribution.


2008 Global Financial Crisis - 1

The 2008 Global Financial Crisis (GFC) was a dramaticevent for financial markets.This forced many changes to be made in global markets.The economic meltdown that followed had massive effectson many everyday issues such as house prices, interestrates and inflation.Investment banks were also affected and numerousinvestment banks either defaulted and many banks in theUnited States were taken over by the U.S. Federal Reserveto avoid default.The impact on financial derivative pricing and riskmanagement did not escape the GFC.



Prior to the 2008 credit crisis, pricing the value of aderivative was relatively straightforward. Universally,practitioners and many academics agreed on the pricingmethod used to price a derivative.The method was well-known: discount future expectedcash flows under the risk-neutral measure to the presentdate using the risk-free rate. This method was derived fromthe fundamental theory laid down by Black, Scholes andMerton in the 1970s.



The 2008 credit crisis drove home the fact that what wasused in practice prior to the crisis as an approximation(also called a proxy) for the theoretical notion of a risk-freeinterest rate, as required by the BSM model, is totallyinadequate to yield realistic results.The myth that banks are risk-free was disproved by the2008 credit crisis. The default of what we used to call "toobig to fail“ banks, such as Lehman Brothers and BearStearns, which defaulted in the 2008 credit crisis,disproved the myth that banks are risk-free.



The 2008 credit crisis also exposed the inadequatemanagement of counterparty credit risk.Counterparty credit risk (also known as default risk)between two parties, is the two-sided risk that one of thecounterparties will not pay, as obligated on a trade or atransaction between the two parties.Changes need to be made to the usual ways in which"business was conducted“ prior to the 2008 credit crisisand these changes need to be addressed and incorporatedin the models used prior to the 2008 credit crisis.Improved credit risk management costs money andadjustments have to be made to compensate for the costsinvolved.


Counterparty credit risk

One of the ways to mitigate counterparty credit risk is byposting collateral in a derivative trade. Collateral is aborrower’s pledge of specific assets to a lender, to securerepayment of a liability.Banks required collateral posted from their counterpartieson certain trades prior to the 2008 credit crisis.But as it became apparent that banks are not risk-free,clients require that banks now also post collateral on sometransactions.


What research do I do?

The changes forced on the BSM model by the 2008 GFC areresearched. In particular, the following problems areconsidered:

What is a suitable candidate for the risk-free rate in theSouth African market and how is it estimated?Extensions of the BSM model due to improved riskmanagement brought about by adjustments to costs (XVA).The Piterbarg model for option pricing as a replacement forthe BSM model.

The Piterbarg is an extension of the BSM model which takesthe posting of collateral and multiple interest rates into account.


Research contributions: pricing & hedging andmanagement of risk

Contributions includePricing of convertible bonds.Pricing of exotic options using the Wang transform.Establishing a connection between the Wang andEsscher-Girsanov transforms.Representing set-valued risk measures defined on Banachspace valued Orlicz spaces.Extensions of the BSM model to accommodateshortcomings emphasised by the 2008 GFC.Modelling the stock indices using univariate conditionalvolatility models.


A convertible bond is a corporate debt security that gives theholder the right to exchange future cash payments for aprescribed number of shares of equity.

The Wang transform and the Esscher-Girsanov transformprovide pricing techniques which do not require a change ofmeasure.


Quantitative Finance – Research Papers - UnderReview

C.C.A. LABUSCHAGNE, S.T. VON BOETTICHER, Discretehedging in the Piterbarg option pricing framework.C.C.A. LABUSCHAGNE, S.T. VON BOETTICHER, Dupire’sformula in the Piterbarg pricing formula.C.C.A. LABUSCHAGNE, S.T. VON BOETTICHER,Calculating the Piterbarg price of a derivative using historicreturn distribution methods.C. B. HUNZINGER, A. KOTZÉ, C.C.A. LABUSCHAGNE,Greeks and volatility skews of the Wang transform.C.C.A. LABUSCHAGNE, E. MAJEWSKA, J. OLBRYS, Crisisperiods, contagion and integration effects in the majorAfrican equity markets during the 2007–2009 globalfinancial crisis.A. KOTZÉ, C.C.A. LABUSCHAGNE, A linear algebraapproach to quantify a Basel III compliant default fund.


Quantitative Finance – Selected Research Papers

C.C.A. LABUSCHAGNE, T.M. OFFWOOD-LE ROUX,Representations of set-valued risk measures defined onthe l-tensor product of Banach lattices, Positivity, 18(2014), 619–639.C. B. HUNZINGER, C.C.A. LABUSCHAGNE, The Cox, Rossand Rubinstein tree model which includes bilateral creditrisk and funding costs, North American Journal ofEconomics and Finance, 29 (2014), 200–217.A. KOTZÉ, C.C.A. LABUSCHAGNE, M. NAIR,N.PADAYACHI, Arbitrage-free implied volatility surfaces foroptions on single stock futures, North American Journal ofEconomics and Finance, 26 (2013) 380–399.


C.C.A. LABUSCHAGNE, S.T. VON BOETTICHER,Construction of a BRICS index and option price evaluationrelative to constituent indexes. Proceedings of the WorldFinance Conference, December 2015, Hanoi, Vietnam, inpress.C.C.A. LABUSCHAGNE, S.T. VON BOETTICHER, Anoverview of the Piterbarg option pricing model.Proceedings of the World Finance Conference, December2015, Hanoi, Vietnam, in press.C. B. HUNZINGER, C.C.A. LABUSCHAGNE, The Wangtransform as an option pricing model. Proceedings of theWorld Finance Conference, December 2015, Hanoi,Vietnam, in press.T. JAKARASI, C.C.A. LABUSCHAGNE, O. MAHOMED,Estimating the South African overnight indexed swapcurve. Proceedings of ICOAE 2015, Kazan, Russia:Procedia Ecomomics and Finance, in press.


C.C.A. LABUSCHAGNE, P. VENTER, S.T. VON BOTTICHER,A comparison of Risk Neutral Historic Distribution -,E-GARCH - and GJR-GARCH model generated volatilityskews for BRICS Securities Exchange indexes.Proceedings of ICOAE 2015, Kazan, Russia: ProcediaEcomomics and Finance, in press.C. B. HUNZINGER, C.C.A. LABUSCHAGNE, Pricing acollateralised derivative trade with a funding valueadjustment. In: Proceedings of the Mathematics ofFinance Conference, Kruger National Park, August 2014,Journal of Risk Financial Management 8 (2015), 17-42.doi:10.3390/jrfm8010017.C.C.A. LABUSCHAGNE, W. SURIYAKAT, S.T. VON

BOETTICHER, Tracing the Effect of the Global FinancialCrisis on ASEAN Stock Exchanges via Implied Volatility.Proceedings of the 2nd International Conference onEconomics, Finance and Management Outlooks 20-21December, 2014, Kuala Lumper, Malaysia.C.C.A. LABUSCHAGNE, T.M. OFFWOOD, Pricing exoticoptions using the Wang transform, North American Journalof Economics and Finance, in press.C.C.A. LABUSCHAGNE, T.M. OFFWOOD, PricingConvertible Bonds, International Journal of IntelligentTechnologies and Applied Statistics, 4 (2011), 467 - 488.C.C.A. LABUSCHAGNE, T.M. OFFWOOD, A note on theconnection between the Esscher-Girsanov transform andthe Wang transform, Insurance: Mathematics andEconomics, 47 (2010), 385 - 390.


C.C.A. LABUSCHAGNE, T.M. OFFWOOD, PricingConvertible Bonds using the CGMY model. In:Proceedings of the conference Non-Linear Mathematics forUncertainty and its Applications (NLMUA2011), Beijing,September 2011, Springer-Verlag 2011, Advances inIntelligent and Soft Computing 100, 231 – 238 (2011).C.C.A. LABUSCHAGNE, H.T. NGUYEN, A universalframework for financial and actuarial pricing of risk: mythor reality? In: Proceedings of the third conference of theEconometric Society of Thailand, Chiang Mai University,Thailand, January 2010: The Thai Econometric Society, 2(2010), 59 - 65.C.C.A. LABUSCHAGNE, T.M. OFFWOOD, On thefundamental theorems of asset pricing. In: Proceedings ofthe third conference of the Econometric Society ofThailand, Chiang Mai University, Thailand, January 2010:The Thai Econometric Society, 2 (2010), 49 - 58.


MATHEMATICS


Research in "Pure Mathematics" Related ToQuantitative Finance?

Measure theory is a special case of vector lattice theory.In particular, Lp spaces are special cases of a vector lattice.The natural question that arises is: is it possible to extendthe theory of stochastic processes on Lp spaces to atheory of stochastic processes on vector lattices?Extensive progress and contributions have been made onthis issue.This theory has been developed to such as extent that it ishighly likely that the pricing of options as discussed abovecan be done in the setting of vector lattices.


Stochastic Processes on Riesz spaces

We developed a theory of discrete stochastic processes onRiesz spaces.In this framework, the measure space is replaced by theorder structure of the Riesz space.We were able to give sensible definitions of the basicnotions of probability theory in Riesz spaces.This included the notions of conditional expectation,martingale, submartingale, supermartingale, stopping time,independence, Doob’s decomposition for submartingales,an optional stopping theorem, martingale convergencetheorems, ergodic theorems, various zero one/two laws,and a discrete stochastic integral.


Stochastic Processes on Riesz Spaces – SelectedResearch Papers

J.J. GROBLER, C.C.A. LABUSCHAGNE, The Itô integral formartingales in vector lattices.J.J. GROBLER, C.C.A. LABUSCHAGNE, The Itô integral forBrownian motion in vector lattices, Part II, Journal ofMathematical Analysis and Applications, 423 (2015),820-833.doi:10.1016/j.jmaa.2014.09.063.J.J. GROBLER, C.C.A. LABUSCHAGNE, The Itô integral forBrownian motion in vector lattices, Part I, Journal ofMathematical Analysis and Applications, 423 (2015),797-819.doi:10.1016/j.jmaa.2014.08.013.


C.C.A. LABUSCHAGNE, B.A. WATSON, Discrete timestochastic integrals in Riesz spaces, Positivity 14 (2010),859 - 875.M. KOROSTENSKI, C.C.A. LABUSCHAGNE, A note onregular martingales in Riesz spaces, QuaestionesMathematicae, 31 (2008), 219 - 224.W.-C. KUO, C.C.A. LABUSCHAGNE, B.A. WATSON,Amarts in Riesz spaces, Acta Mathematica Sinica (EnglishSeries), 24 (2008), 329 - 342.W.-C. KUO, C.C.A. LABUSCHAGNE, B.A. WATSON,Ergodic theory and the strong law of large numbers onRiesz spaces, Journal of Mathematical Analysis andApplications, 325 (2007), 422 - 437.


W.-C. KUO, C.C.A. LABUSCHAGNE, B.A. WATSON,Convergence of Riesz space martingales, IndagationesMathematicae (New Series), 17 (2006), 271 - 283.W.-C. KUO, C.C.A. LABUSCHAGNE, B.A. WATSON,Conditional expectations on Riesz spaces, Journal ofMathematical Analysis and Applications, 303 (2005), 509 -521.W.-C. KUO, C.C.A. LABUSCHAGNE, B.A. WATSON,Discrete-time stochastic processes on Riesz spaces,Indagationes Mathematicae (New Series), 15 (2004), 435- 451.


W.-C. KUO, C.C.A. LABUSCHAGNE, B.A. WATSON, Azero-one law for Riesz space and fuzzy processes. In: Y.Liu (ed.) et al., Fuzzy Logic, Soft Computing andComputational Intelligence (Volumes I, II, III). Collection ofpapers presented at the International Fuzzy SystemsAssociation World Congress (IFSA 2005), Beijing, China,July 28 - 31, 2005. Tsinghua University Press, Springer,Beijing, 2005. Fuzzy Logic, Soft Computing andComputational Intelligence, Volume I, 393 - 397 (2005).W.-C. KUO, C.C.A. LABUSCHAGNE, B.A. WATSON, Rieszspace and fuzzy upcrossing theorems. In: M. López-Díaz(ed.) et al., Soft methodology and random informationsystems. Collection of papers presented at the 2ndinternational conference on soft methods in probability andstatistics (SMPS 2004), Oviedo, Spain, September 2 - 4,2004. Berlin: Springer. Advances in Soft Computing, 101 -108 (2004).


Banach Space Valued Stochastic Processes –Contributions

We developed a theory of discrete stochastic processes inBanach spaces and Banach lattices, without the use ofmeasure theory.Guided by known results on stochastic processes onBochner spaces, we replaced the Bochner space by anl-tensor product of a Banach lattice and a Banach space,and derived analogues of these results.The main highlights are

a full description of the convergence of a vector valuedmartingale in terms of its scalar valued component (a newresult, even in the classical case), andcharacterisations of martingale convergence in terms of theRadon-Nikodým property in Banach spaces (an analogueof a known Bochner space result).


We also gave characterisations of submartingale andsupermartingale convergence in terms of theRadon-Nikodým property, anda characterization the Radon-Nikodým property in anl-tensor product in terms of the Radon-Nikodým propertyon each of the component spaces.


Banach Space Valued Stochastic Processes –Selected Research Papers

C.C.A. LABUSCHAGNE, V. MARRAFFA, Operatormartingale decompositions and the Radon-Nikodýmproperty in Banach spaces, Journal of MathematicalAnalysis and Applications, 363 (2010), 357 - 365.S.F. CULLENDER, C.C.A. LABUSCHAGNE, Convergentmartingales of operators and the Radon-Nikodým propertyin Banach spaces, Proceedings of the AmericanMathematical Society, 136 (2008), 3883 - 3893.S.F. CULLENDER, C.C.A. LABUSCHAGNE, Corrigendum to“Unconditional martingale difference sequences in Banachspaces" [J. Math. Anal. Appl. 326 (2007) 1291 - 1309],Journal of Mathematical Analysis and Applications, 338(2008), 751 - 752.


S.F. CULLENDER, C.C.A. LABUSCHAGNE, UnconditionalSchauder decompositions and stopping times in theLebesgue-Bochner spaces, Journal of MathematicalAnalysis and Applications, 336 (2007), 849 - 864.S.F. CULLENDER, W.-C. KUO, C.C.A. LABUSCHAGNE,B.A. WATSON, Measure-free martingales with applicationto classical martingales. In: J. Lawry (ed.) et al., Softmethods for integrated uncertainty modelling. Proceedingsof the 2006 international workshop on soft methods inprobability and statistics (SMPS 2006), Bristol, UnitedKingdom, September 5 - 7, 2006. Berlin: Springer.Advances in Soft Computing, 121 - 128 (2006).


Set Valued Stochastic Processes – Contributions

Robert Aumann received the Nobel Memorial Prize inEconomics in 2005 for his work on conflict and cooperationthrough game-theory analysis.He used set-valued stochastic processes in his work.Hiai and Umegaki were instrumental in developingset-valued stochastic processes on the space of integrablybounded functions (the latter is a complete metric space ofset valued functions).We showed that many of their results can be obtained fromour research on Banach space valued stochasticprocesses.The main tool required is a lattice version of Rådström’sembedding theorem noted by Andrew Pinchuck, Clint vanAlten and I.


Set Valued Stochastic Processes – Research Papers

C.C.A. LABUSCHAGNE, V. MARRAFFA, On Spaces ofBochner and Pettis Integrable Functions and theirSet-valued Counterparts. In: Proceedings of theconference Nonlinear Mathematics for Uncertainty and itsApplications NLMUA2011), Beijing, September 2011,Springer-Verlag 2011, Advances in Intelligent and SoftComputing 100, 51 - 59 (2011).C.C.A. LABUSCHAGNE, A.L. PINCHUCK, Doob’sdecomposition of set-valued submartingales via orderednear vector spaces, Quaestiones Mathematicae, 32(2009), 247 - 264.


C.C.A. LABUSCHAGNE, A Banach lattice approach toconvergent integrably bounded set-valued martingales andtheir positive parts, Journal of Mathematical Analysis andApplications, 342 (2008), 780 - 797.C.C.A. LABUSCHAGNE, Join-semilattices of integrableset-valued martingales, Thai Journal of Mathematics (aspecial conference issue), 5 (2007), 53 - 69. The 8thInternational Conference on Fixed Point Theory and itsApplications, Chiang Mai, Thailand, July 16 - 22, 2007.


Fuzzy Stochastic Processes

What is Fuzzy Mathematics?In Mathematics a set A is associated with its indicator χA

function.χA takes the value 1 on A and 0 outside A.In Fuzzy Mathematics a set A is associated with a functionfA.The function fA can take on any value from 0 to 1.

Fuzzy mathematics has many applications! Check the inside ofthe door or cover of your appliances when you get home.

We made the following contributions to fuzzy stochasticprocesses:


Fuzzy Stochastic Processes – Research Papers

C.C.A. LABUSCHAGNE, A.L. PUNCHUCK, Doob’sdecomposition of fuzzy submartingales via ordered nearvector spaces. In: Proceedings of the Second Asianconference on Nonlinear Analysis and OptimizationPhuket, Thailand, September 2010: Journal of NonlinearAnalysis and Optimization: Theory and Applications (aspecial conference issue), 2, 39 - 52 (2011).C.C.A. LABUSCHAGNE, Positive parts of convergent fuzzymartingales, In: Q.P. Ha and N.M. Kwok (ed.), Proceedingsof the 8th International Conference on IntelligentTechnologies, InTech’07, Sydney, Australia, December 12 -14, 2007. University of Technology Sydney, Australia.Intelligent Technologies in Robotics and Automation, 293 -299 (2007).


Functional Analysis - Contributions

My contributions to functional analysis includethe study of geometric and order properties of tensorproducts of normed spaces and normed Riesz spaces,the geometric and order properties of various spaces ofmaps acting between normed spaces and normed Rieszspaces.


Functional Analysis – Selected Research Papers

C.C.A. LABUSCHAGNE, V. MARRAFFA, A note on theBanach space of preregular maps, QuaestionesMathematicae, 34 (2011), 113 - 117.C.C.A. LABUSCHAGNE, T.M. OFFWOOD, A description ofBanach space-valued Orlicz hearts, Central EuropeanJournal of Mathematics, 8 (2010), 1109 - 1119.C.C.A. LABUSCHAGNE, V. MARRAFFA, On set-valued coneabsolutely summing maps, Central European Journal ofMathematics, 8 (2010), 148 - 157.P.D. ALLENBY, C.C.A. LABUSCHAGNE, On the uniformdensity of C(X)⊗ C(Y) in C(X × Y), IndagationesMathematicae (New Series), 20 (2009), 19 - 22.


S.F. CULLENDER, C.C.A. LABUSCHAGNE, A description ofthe space of Banach space-valued absolutely p-summablesequences, Quaestiones Mathematicae, 31 (2008), 101 -105.S.F. CULLENDER, C.C.A. LABUSCHAGNE, A note on theM-norm of Chaney-Schaefer, Quaestiones Mathematicae,30 (2007), 151 - 158.C.C.A. LABUSCHAGNE, Preduals and nuclear operatorsassociated with bounded, p-convex, p-concave andpositive p-summing operators, Canadian Journal ofMathematics, 59 (2007), 614 - 637.C.C.A. LABUSCHAGNE, A Dodds-Fremlin property forAsplund and Radon-Nikodým operators, Positivity, 10(2006), 391 - 407.


C.C.A. LABUSCHAGNE, Equivalence of the Bochner normand its transpose characterizes Lp-spaces, Journal ofMathematical Analysis and Applications, 308 (2005), 746 -758.C.C.A. LABUSCHAGNE, Characterizing the one-sidedtensor norms ∆p and t∆p, Quaestiones Mathematicae, 27(2004), 339 - 363.C.C.A. LABUSCHAGNE, Riesz reasonable cross norms ontensor products of Banach lattices, QuaestionesMathematicae, 27 (2004), 243–266.C.C.A. LABUSCHAGNE, A note on the order continuity ofthe norm of E⊗̃mF, Archiv der Mathematik (Basel), 62(1994), 335 - 337.


J.J. GROBLER, C.C.A. LABUSCHAGNE, An f-algebraapproach to the Riesz tensor product of ArchimedeanRiesz spaces, Quaestiones Mathematicae, 12 (1989), 425- 438.J.J. GROBLER, C.C.A. LABUSCHAGNE, A note on theapproximation of elements in the Riesz tensor product,Mathematische Zeitschrift, 201 (1989), 273 - 277.J.J. GROBLER, C.C.A. LABUSCHAGNE, The tensor productof Archimedean ordered vector spaces, MathematicalProceedings of the Cambridge Philosophical Society, 104(1988), 331 - 345.J.J. GROBLER, C.C.A. LABUSCHAGNE, An existence prooffor the Riesz tensor product by operator methods, SouthAfrican Journal of Science, 84 (1988), 330 - 331.


Ordered Spaces – Research Papers

C.C.A. LABUSCHAGNE, C.J. VAN ALTEN, On the MacNeillecompletion of MTL-chains, Intech 2008 - Proceedings ofthe 9th International Conference on IntelligentTechnologies (2008), 71 - 75.C.C.A. LABUSCHAGNE, A.L. PINCHUCK, C.J. VAN ALTEN,A vector lattice version of Rådström’s embedding theorem,Quaestiones Mathematicae, 30 (2007), 285 - 308.C.C.A. LABUSCHAGNE, C.J. VAN ALTEN, On the variety ofRiesz spaces, Indagationes Mathematicae (New Series),18 (2007), 61 - 68.


Other – Research Papers

M. KOROSTENSKI, C.C.A. LABUSCHAGNE, Lax propermaps of locales, Journal of Pure and Applied Algebra, 208(2007), 655 - 664.


Acknowledgements

My sincere thanks are due toProfessor Koos Grobler, North-West University(Potchefstroom), who taught me almost all themathematics that I know, who supervised my doctoralthesis, and who is my collaborator from whom I still learnlots on a daily basis.The organisers of the Positivity 7 Conference in honour ofProfessor Adrian Zaanen’s 100th birthday, for inviting me togive a plenary talk at the conference. Professor Zaanenwas my mathematical grandfather (the doctoral supervisorof Koos Grobler).


My research collaborators - it was such a pleasure to workwith you.All my PhD and Masters students - it was a privilege andan honour to work with you, and I learnt more from you thatyou could ever have hoped to learn from me.The University of Johannesburg for the opportunity to givethis lecture.

A special word of thanks is due to Professor Eben Mare for hiswillingness to act as respondent - thank you Eben!


Enlightenment

Before enlightenmentchopping woodcarrying water.

After enlightenmentchopping woodcarrying water.

ZEN PROVERB


THANK YOU FOR YOUR ATTENTION


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A Random Walk Through Quantitative Finance 2016... · N.PADAYACHI, Arbitrage-free implied volatility surfaces for options on single stock futures, North American Journal of Economics