Short bio
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.
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 .
Mean–variance portfolio allocation without a riskfree asset
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]
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., sigmafinitevariation purejump
process), one that is closed under stochastic integration, composition, and smooth
transformations.
Sigmafinitevariation 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 "purejump" 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 sigmafinite variation 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 sigmafinitevariation purejump processes.
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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 nailbiting 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.
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Simplified stochastic calculus
December 7th, 2021. The second paper of the series has just been accepted for publication 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 to get a feel for the
new calculus (many thanks to the referees and associate editor for suggesting them). Preprint is available here.
December 18th, 2020. The first paper of the series has just been accepted for publication in the
European Journal of Operational Research . We are very grateful to two anonymous referees for their expert suggestions. In particular, they 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).
[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 