Short intro
The theory of asset pricing
and risk measurement in incomplete markets is concerned with the methodology
and practical implementation of optimal hedging and pricing of derivative
securities in the presence of hedging errors. The standard asset pricing
theory assumes that all sources of risk are priced in the market; this
assumption is most famously embedded in the BlackScholes option pricing
formula. In reality, even extremely frequent hedging leaves a significant
amount of risk. In most models this risk is unaccounted for, as LTCM found
to its own detriment. My work proposes standardized measurement of risk
across different utility functions, allows for attribution of performance
among different assets (for example stocks and options) in a dynamic framework,
and provides extremely fast implementation of optimal dynamic hedge ratios
and risk measurements using Fourier transform.
Simplified stochastic calculus
April 28th, 2023. Slides from the talk at LMU workshop .
April 12th, 2023. The final paper of the simplified calculus trilogy has just
been accepted for publication in Stochastic Processes and Their Applications. This work
illustrates how powerful and parsimonious the calculus can be, whether in integral
transforms or in measure changes. The preprint is available here. More applications to come soon …
January 12th, 2022. The second paper of the series has appeared in the
Electronic Journal of Probability.
The paper provides a theoretical underpinning of the ‘calculus of predictable
variations,’ which substantially simplifies and streamlines manipulation of general
stochastic processes. Check out the very helpful tables in the introduction (thanks to the referees and
AE) to get a feel for the new calculus. The preprint is available here.
April 7th, 2021. The first paper of the series has
appeared in the
European Journal of Operational Research . The referees made us think how to derive the Riccati equations in affine models à la Duffie, Pan, and Singleton (2000). The outcome of that exercise is available
here.
June 22nd, 2020. Just completed a suite of four papers dealing with the
practical applications and the theoretical underpinning of simplified stochastic calculus. If you have
never heard of this topic, you may wish to start here. There is more to come on the application front. Mathematicians will find
something of interest
here,
here, and
here. Joint work with
Johannes Ruf (LSE Mathematics).
Mean–variance portfolio allocation without a riskfree asset
April 7th, 2023.
This paper has now been accepted in the Mathematics of Operations Research.
October 19th, 2021.
A piece of research that has taken more than a decade to complete is now available on SSRN and arXiv. Have you ever wondered why the mean–variance theory looks more complicated when there is no riskfree asset? Somewhat surprisingly, it is about three times more demanding to compute the efficient frontier in the absence of a riskfree asset than in its presence.
We provide, for the first time in the literature, unified formulae for dynamically mean–variance efficient portfolios that work the same way in discrete and continuoustime models and with or without a riskfree asset. Joint work with Christoph Czichowsky (LSE Mathematics) and Jan Kallsen (CAU Kiel, Mathematisches Seminar).
[read more]
The law of one price in mean–variance hedging
November 2nd, 2022.
The latest piece of research, available on SSRN and arXiv, disentangles the difference between no arbitrage and the lesser requirement of the law of one price (LOP). The latter broadly asserts that that identical financial flows should command the same price. We uncover a new mechanism through which LOP can fail in a continuoustime L^{2}(P) setting without frictions, namely trading from just before a predictable stopping time,
which surprisingly identifies LOP violations even for continuous price processes. Closing this loophole allows to give a version of the Fundamental Theorem of Asset Pricing'' appropriate in the quadratic context, establishing the equivalence of the economic concept of LOP with the probabilistic property of the existence of a local Emartingale state price density. The latter provides unique prices for all squareintegrable contingent claims in an extended market and subsequently plays an important role in meanvariance hedging. Joint work with Christoph Czichowsky (LSE Mathematics)
[read more]
A talk in memory of Peter Carr at the 11th Bachelier Congress
June 16th, 2022. Gvariations from Carr and Lee (2013, Finasto)
are linked to the simplified stochastic calculus .
Purejump processes
January 12th, 2021. A piece of research that originated in the development
of the simplified stochastic calculus has just been accepted for publication in
Bernoulli. The paper gives a better notion of a purejump process (i.e., sigmalocally finite variation purejump process), one that
is closed under stochastic integration, composition, and smooth transformations.
Sigmalocally finite variation purejump processes require the ability to sum jumps that are not absolutely summable, a seemingly impossible task. Naturally, sigmafinitevariation purejump process is
uniquely determined by its jumps. Observe, however, that the literature operates with multiple
classes of purejump processes without always distinguishing them as such. For example, a "pure
jump" Lévy process is not necessarily determined by its jumps. All this is tidied up in the paper
written jointly with Johannes Ruf
(LSE Mathematics) .
The reward for all the hard work is a very natural measureinvariant
decomposition of every semimartingale into a discretetime sigmafinitevariation purejump
component and a process that has no jumps at predictable times (the latter is typically exemplified by a
jumpdiffusion in concrete models). Such decomposition is not possible without the notion of sigma
finitevariation purejump processes.
[read more]
The Hansen ratio in mean–variance portfolio theory
August 17th, 2020. The Sharpe ratio is well known in the meanvariance
portfolio theory. Most people do not know that the Hansen ratio is even better suited to the description
of the efficient frontier. Simple concepts can be surprisingly difficult to spot; see
here.
Latest research on the fundamentals of expected utility maximization
July 3rd, 2020. The January 2020 issue of Mathematical Finance
features the paper Convex duality and Orlicz
spaces in expected utility maximization coauthored with
Sara Biagini (LUISS,
Rome). One reviewer called it a quest. It certainly stands as the longest, most openended and nail
biting piece of research I have been involved in. I will try to give a flavour of the intricacies
involved.
The utility function, say U, more specifically its left tail, generates a
normed space of random variables L^{U} (the Orlicz space from the title). Think of
L^{U} as a repository of all hypothetical net trading positions whose smallenough multiple
gives finite expected utility, both long and short. L^{U} contains in its centre a heart
H^{U}; this is the set of all hypothetical net positions that give finite expected
utility after arbitrary scaling.
Utility U has a conjugate function V that generates another Orlicz space L^{V}. Roughly speaking, L^{V} contains all pricing
rules such that trading on those rules in a complete market yields finite expected utility. We show that
the results are driven by the link between L^{U} and L^{V}. This is
hard to see because the link is a real shapeshifter. To begin with, when the probability space is
finite, the duality (L^{U},L^{V}) is compatible with norms on L^{U}
and L^{V}, regardless of the shape of U. If the probability space is
diffuse (has no atoms), then the normcompatibility is dictated purely by the shape of U and
occurs if and only if L^{U} = H^{U}, resp. L^{V} = H^{
V}. When the probability space is made of countably many atoms and U is the wrong
shape, the link may still be normcompatible but only for specific choices of probability weights.
Why is the normcompatibility so important? For one thing, if we maximize expected
utility over a convex subset of L^{U} and then maximize over its normclosure, the
expected utility cannot increase, which is what one wants and expects. It was a shock to discover that in
the incompatible case the utility can actually jump upwards on the (L^{U},L^{V})
closure of a convex set. Such jump must be assumed away, otherwise it generates a duality gap and
destroys the theory as we know it. Interestingly, in the seminal work of Kramkov and Schachermayer (1999,
2003), the (L^{U},L^{V})–closedness of the set of terminal wealths follows
from the absence of arbitrage via the celebrated Fundamental Theorem of Asset Pricing (FTAP). We will get
back to this shortly.
Our paper explains the mechanism causing the utility increase on the closure. In the
incompatible case, there are trading positions in L^{U} that cannot be scaled
arbitrarily. Furthermore, it is possible to create a sequence of such trades that converges to zero in
the (L^{U},L^{V})–sense but not in the L^{U} norm. When
these nuisance trades are added to a fixed genuine trading opportunity, they prevent the latter from
being exploited fully because the nuisance trades, being outside of the heart H^{U}, are
allowed only in small multiples. However, in the closure the genuine trading opportunity appears
unencumbered (because the limit of the nuisance trades is zero). Hence, in the closure the genuine
trading opportunity can be traded fully and this causes the jump in expected utility.
Let us now return to the smallmarket (finite number of assets) FTAP. From the construction in the
previous paragraph it is very clear that the FTAP link between the absence of arbitrage and the
(L^{U},L^{V})–closedness of terminal wealths is an artefact of small
financial markets. In a large financial market (even with just one time period) one can easily
construct an example that is arbitragefree on the closure and yet displays a duality gap.
[read more]
[23] 
with Christoph Czichowsky and Jan Kallsen, Numeraireinvariant quadratic hedging and mean–variance portfolio
allocation, to appear in Mathematics of Operations Research 
[22] 
with Johannes Ruf,
Simplified calculus for semimartingales: Multiplicative compensators and
changes of measure, Stochastic Processes and Their Applications,
161, 572–602, 2023 
[21] 
with Johannes Ruf, Simplified stochastic calculus via
semimartingale representations, Electronic Journal of Probability, 27, article no.3,
pp. 1–32, 2022 
[20] 
with Johannes Ruf, Purejump semimartingales,
Bernoulli, 27(4), pp. 2624–2648, 2021 
[19] 
with Johannes Ruf, Simplified stochastic calculus
with applications in Economics and Finance, European Journal of Operational Research,
293(2), pp. 547–560, 2021 
[18] 
Semimartingale theory of monotone mean–variance
portfolio allocation, Mathematical Finance, 30(3), pp. 1168–1178, 2020 
[17] 
with Igor Melicherčík, Simple explicit
formula for nearoptimal stochastic lifestyling, European Journal of Operational Research
284(2), 769–778, 2020 
[16] 
with Sara Biagini, Convex duality and Orlicz spaces
in expected utility maximization, Mathematical Finance 30(1), 85–127, 2020 
[15] 
with Pavol Brunovský and Ján Komadel,
Optimal trade execution under endogenous pressure to liquidate: Theory and numerical solutions,
European Journal of Operational Research 264(3), 1159–1171, 2018 
[14] 
with Fabio Maccheroni, Massimo Marinacci and Aldo
Rustichini, On the computation of optimal monotone mean–variance portfolios via truncated quadratic
utility, Journal of Mathematical Economics 48(6), 386–395, 2012 
[13] 
with Chris Brooks and Joelle Miffre, Optimal hedging with
higher moments, Journal of Futures Markets 32(10), 909–944, 2012 
[12] 
with Ioannis Kyriakou, An improved convolution algorithm
for discretely sampled Asian options, Quantitative Finance 11(3), 381–389, 2011 
[11] 
with Sara Biagini, Admissible strategies in semimartingale portfolio
optimization, SIAM Journal on Control and Optimization, 49(1), 42–72, 2011 
[10] 
Mathematical Techniques in Finance: Tools for Incomplete Markets, Princeton University
Press, 2nd edition, July 2009, pp. 416 

 handson introduction to asset
pricing, optimal portfolio selection and evaluation of investment performance
 simple EXCEL spreadsheets and MATLAB codes integrated in the text
 large number of examples and solved exercises
 more advanced topics include
 fast Fourier transform
 finite difference methods
 multinomial lattices and Levy processes

[9] 
with Jan Kallsen, Hedging by sequential regressions
revisited, Mathematical Finance 19(4), 591–617, 2009 
[8] 
with Jan Kallsen, Mean–variance hedging and optimal
investment in Heston's model with correlation, Mathematical Finance 18(3), 473–492, 2008

[7] 
with Jan Kallsen, A counterexample concerning the
varianceoptimal martingale measure, Mathematical Finance 18(2), 305–316, 2008 
[6] 
with Jan Kallsen, On the structure of general
mean–variance hedging strategies, The Annals of Probability 35(4), 1479–1531, 2007

[5] 
Optimal continuoustime hedging with leptokurtic returns,
Mathematical Finance 17(2), 175–203, 2007. 
[4] 
with David K. Miles, Risk, return, and portfolio
allocation under alternative pension systems with incomplete and imperfect financial markets, The
Economic Journal 116(2), 529–557, 2006. 
[3] 
Introduction to fast Fourier transform in finance,
Journal of Derivatives 12(1), 73–88, 2004 
[2] 
Generalized Sharpe ratios and asset pricing in incomplete
markets, Review of Finance 7(2), 191–233, 2003. Presented at AFA Annual Meeting 2001, New
Orleans. 
[1] 
with Stewart D. Hodges, The theory of gooddeal pricing in
financial markets, in Geman, Madan, Pliska, Vorst (eds.): Mathematical Finance — Bachelier
Congress 2000, 175–202, Springer Verlag 2002. 
Research Projects
 with Prof. David Miles, 2000–2004, Economics of Social Security in Japan, £200,000+
 with Prof. James Sefton, 2002–2004, Design of Behavioural Tax Model, £80,000
Selected refereed conferences and *invited talks
(full list here)
[22] 
16/06 2022 
11th Bachelier Congress, Hong Kong Simplified stochastic calculus: A
talk in memory of Peter Carr 

[21] 
12/05 2022 
*Talks in Financial and Insurance
Mathematics, ETH Zurich Simplified stochastic calculus via semimartingale
representations 
[20] 
17/07 2018 
10th Bachelier Congress, Dublin
Convex duality and Orlicz spaces in expected utility maximization 

[19] 
08/06
2017 
*Convex Stochastic Optimization Workshop, Kings College London
Convex duality and Orlicz spaces in expected utility maximization 
[18] 
03/11 2016 
*London Mathematical Finance Seminar, UCL
Optimal trade execution under endogenous pressure to liquidate 
[17] 
28/08 2015 
*George Boole Mathematical Sciences Conference, Cork
Quadratic hedging with and without numeraire change 
[16] 
25/09 2014 
*LondonParis Bachelier Workshop, Paris
Gooddeal prices for log contract 
[15] 
04/06 2014 
8th Bachelier Congress, Brussels
Asymptotics of quadratic hedging in Lévy models 
[14] 
26/08 2013 
6th Summer School of Mathematical Finance, Vienna
Computation of optimal monotone mean–variance portfolios 
[13] 
05/09 2012 
*Finance and Actuarial Science Talks, ETH Zurich
Optimal hedging with higher moments 
[12] 
12/07
2010 
AnStaP10, Conference in Honour of W. Schachermayer, Vienna
Admissible strategies for semimartingale portfolio optimization 
[11] 
24/06
2010 
6th Bachelier Congress, Toronto
Admissible strategies for semimartingale portfolio optimization 
[10] 
18/07
2008 
5th Bachelier Congress, London
Mean–variance hedging and optimal investment in Heston's model 
[9] 
24/08
2007 
EFA 2007 Annual Meeting, Ljubljana
Optimal Hedging with Higher Moments 
[8] 
25/05
2007 
*Stanford Unversity
MeanVariance Hedging and Optimal Investment in Heston's Model 
[7] 
29/09
2005 
*Courant Institute
for Mathematical Sciences, NYU
On the structure of general mean–variance hedging strategies 
[6] 
28/09
2005 
*Columbia University,
New York
On the structure of general mean–variance hedging strategies 
[5] 
14/09
2005 
*Summer
School Bologna, Frontiers of Financial Mathematics, Bologna,
Oneday workshop
on the theory and applications of gooddeal pricing 
[4] 
19/04
2005 
*Developments
in Quantitative Finance, Isaac Newton Institute, Cambridge
On the structure of general mean–variance hedging strategies 
[3] 
24/09
2004 
ESF Exploratory Workshop, London Business School
The risk of optimal, continuously rebalanced hedging strategies 
[2] 
23/05
2002 
*Workshop on
Incomplete Markets, Carnegie Mellon
University, Pittsburgh
Derivatives without differentiation 
[1] 
05/01
2001 
AFA 2001 Annual Meeting, New Orleans
Generalized Sharpe ratios 
Refereeing Activity
Applied Mathematical Finance, Annals of Operations Research, Automatica, Bernoulli, Economic Journal, European Financial Management, European Journal of Finance, European Journal of Operational Research, Finance and Stochastics, IEEE Transactions on Automatic Control, International Journal of Computer Mathematics, International Journal of Theoretical and Applied Finance, Journal of Computational and Applied Mathematics, Journal of Computational Finance, Journal of Finance, Journal of Financial Econometrics, Journal of Futures Markets, Mathematical Finance, Mathematics and Financial Economics, Mathematical Reviews, Mathematics of Operations Research, Operations Research, Princeton University Press, Quantitative Finance, Review of Derivatives Research, Risk, SIAM Journal on Financial Mathematics, Statistics and Decisions
Editorial Appointments
06/2007 Review of Derivatives Research
PhD Supervision
[9] 
09/2014 08/2018

Ján Komadel, Comenius University Bratislava
Optimization in financial mathematics

[8] 
10/2011 10/2017 
Juraj Špilda, Cass Business School
On Sources of Risk in Quadratic Hedging and Incomplete Markets

[7] 
10/2013 09/2017 
Xuecan Cui, Luxembourg School of Finance
Asset Pricing Models with underlying Lévy Processes

[6] 
10/2008
10/2013 
Nicolaos Karouzakis, Cass Business School
Three Essays on the Dynamic Evolution of Market Interest Rates…

[5] 
10/2007
06/2012 
Ka Kei Chan, Cass Business School
Theoretical essays on bank risk taking and financial stability

[4] 
10/2006
11/2010 
Ioannis Kyriakou, Cass Business School
Efficient valuation of exotic derivatives with pathdependence and earlyexercise feature

[3] 
10/2002
10/2006 
Lubomír Schmidt, Imperial College Business School
Optimal lifecycle consumption and asset allocation with applications to pension finance and public economics

[2] 
10/2001
09/2006 
Mariam HarfushPardo, Imperial College Business School
An investigation on portfolio choice and wealth accumulation in fully funded pension systems with a guaranteed minimum benefit

[1] 
10/2001
10/2004 
YungChih Wang, Imperial College Business School
Topics in investment appraisal and real options

Media Coverage
Other
 Erdős number: 4 (AČ → J. Ruf → V. Prokaj → L. Gerencsér → PE)
 Kolmogorov number: 3 (AČ
→ J. Kallsen → A.N. Shiryaev → ANK)
Last revised June/23/2020 