Portfolio theory and derivative pricing

27.03.2019 - 29.03.2019 | Warsaw



Elisa Alos

On smile properties of volatility derivatives and exotic products: understanding the VIX skew

We develop a method to study the implied volatility for exotic options and volatility derivatives with European payoffs such as VIX options. Our approach, based on Malliavin calculus techniques, allows us to describe the properties of the at-the-money implied volatility (ATMI) in terms of the Malliavin derivatives of the underlying process. More precisely, we study the short-time behaviour of the ATMI level and skew. As an application, we describe the short-term behavior of the ATMI of VIX and realized variance options in terms of the Hurst parameter of the model, and most importantly we describe the class of volatility processes that generate a positive skew for the VIX implied volatility. In addition, we find that our ATMI asymptotic formulae perform very well even for large maturities. Several numerical examples are provided to support our theoretical results. 


Przemysław Juszczuk, Ignacy Kaliszewski, Janusz Miroforidis, Dmitry Podkopaev

Deriving a new dataset for the Markowitz portfolio model

In this work, we present the procedure to acquire and initial preprocess the financial data, which can be used for the different portfolio-related problems. The mean-variance model originally proposed by Harry Markowitz in 1952 was since then subject of extensive research in the financial field. Multiple extensions of the original bi-criteria model are still on the pick of modern research. However, a serious drawback is the number of reliable datasets, easy to acquire and use in tests is still very small. Despite the well-known Beasley OR Library including the 5 different datasets, there is no common testing environment including a more robust number of data, at the same time unified with the original format.

We investigate the possibilities of acquiring such data and present details of our procedure. We focus on initial preprocessing of such data, especially on the aspects of outliers and incomplete data. We also target the multi-period investments, for which every dataset covers more than one investing period. Such data can be further used to investigate the properties of the Pareto front.


Przemysław Juszczuk, Ignacy Kaliszewski, Janusz Miroforidis, Dmitry Podkopaev

Mean return - standard deviation Pareto front with low-cardinality portfolios in the presence of the risk-free asset

Derivation of the whole Pareto front for the standard mean return - standard deviation problem with all-risky assets for the number of assets well over thousand, is a fairly easy task for contemporary portfolio optimization methods and software. However, the classic mean return - standard deviation model is a simplification of the investing reality in many aspects. One important aspect is that it admits portfolios with numerous assets whereas in practice investors tend to manage portfolios with limited number of them. Although limiting the number of assets in portfolios can be modeled with binary variables and the resulting models can be solved with mixed-integer solvers, the specific  optimization methods devoted to solving the standard mean return - standard deviation problem are no longer applicable. As the result, solving large-scale instances of the extended model can be problematic.

We assume that the investor, next to risky assets, can also invest in a risk-free asset. If so, the Pareto front in the presence of risk-free asset is composed of two parts: a fragment of the Pareto problem for all-risky assets and a line segment.

In view of this, in this work we analyze the viability of approximating the relevant segment of the all-risky asset Pareto front by pairwise convex combinations of assets and then by pairwise convex combinations of assets and the resulting portfolios. Proceeding along this line, we get a series of approximation accuracy - portfolio cardinality trade-offs, with growing Pareto front approximation accuracy and growing portfolio cardinality.


Agnieszka Rygiel

Semi-static hedging with frictions

A finite discrete-time financial market model with transaction costs is considered. We give dual representations of the superhedging costs of path dependent European options. This is obtained without making any probabilistic assumptions on the behavior of the risky asset. Instead, we assume that in addition to trading the stock, the investor is allowed to take static positions in a finite number of options (written on this risky asset) with initially known prices.


Andrzej M.J. Skulimowski

Applying real options in the strategic planning of a knowledge repository exploitation

This paper presents an application of the roadmapping methodology to detect, value and apply real options related to the exploitation strategy building for an intelligent knowledge repository. Such repositories, endowed with AI tools, will ful­fill increasingly relevant information provision tasks for ever-growing communities of users. They can provide dynamically updated economic and financial data, business profiles, online courses, scholarly publications and other information. The diversified scope of planned actions of the repository contributes to the complexity of strategy planning and its multicriteria selection problem.

Real options constitute an essential component of the strategic plan of a universal digital knowledge repository. They are a natural and useful tool to describe the relations between different variants of the roadmapping objects and their deployment plans. The variants correspond to the external random factors and planning scenarios, and to the associa­ted financial yields. Rights gained during the knowledge repository operation as well as the liabilities are modeled by long or short [1] real-option positions, respectively. The application of real options in AI-based innovation planning may be threefold. First, the real option valuations occur in the financial criteria where they augment discounted cash flow estimations. The iterative dependence of future investment opportunities on previous outcomes will be modeled by nested real options, and embedded into an anticipatory network [2] that allows to model the expected consequen­ces of implementing an operation strategy. Second, we define social impact real options (SIRO), which are valued in social return units such as the number of satisfied users or their competence growth. They can be embedded in social impact criteria in a similar way as the financial options occur in the ENPV. When solving the ID-MP multicriteria optimization problem, techniques based on social return on investment (SROI) may be used to aggregate financial and social impact option valuations. Observe that in non-profit and academic institution exploitation planning, the financial criteria may reduce to the real option valuation as no financial investment efficiency calculation is under­taken. The real options detected most often are project expansion, abandonment, and switch options. Third, when optimizing the risk-related criteria, the real as well as financial options may be used to hedging. The latter may also be regarded as a specific way to aggregate the risk measures based on volatility and the criterion expressed as the volatile factor. We will also show that the detection and financial valuation of real options may be easily combined with the SWOTC (SWOT with Challenges) assessment of roadmapping objects. Specifically, licensing (chances), mergers (challenges), and acquisitions (all) can be valued as real options. Thus, the real options contribute to the efficiency of multicriteria analysis in a roadmapping-based strategic planning problem solving.

During the planning process, repository’s owners or stakeholders take into account financial, social and specific qualitative strategic criteria. The provision of new content and services on the platform will be modelled as an innovation development and market placement problem (ID-MP). The latter is a dynamic four-criteria problem with real-options-enhanced NPV (ENPV) – aggregating subordinated momentary financial performan­ce criteria,



where OVli and OVsj are long and short real option positions, respectively, after matching. The other criteria are options-affected risk measure (ER), Social Impact index (SII), and the Strategic Position Index (SPI). IÎJ is the variable to be optimized, where J is the set of action plans modelled as paths in the roadmapping hypergraph. T is the planning horizon, and d is the forecasted or simulated discount rate which may vary during the planning period. The strategy choice is an adaptive process, where the strategy results from a roadmapping algorithm transforming the external economic, technological and research scenarios into a dynamic ranking of strategic goals. They, in turn, determine the current investment and other activity priorities, termed action plans, according to a preference-based multicriteria optimization algorithms. The multicriteria optimization problem so arisen will be solved during an interactive group decision procedure with the roadmapping methodology extended to detect and value real options. As a final result, we will provide an example of applying the methodology presented to building an exploitation strategy for the digital knowledge repository established within a recent Horizon 2020 project [3].

Keywords: Intelligent knowledge repositories, real options, technological roadmapping, strategic planning, multicriteria analysis


[1] Skulimowski, A.M.J.: Methods of technological roadmapping and foresight. Chemik: Nauka-Technika-Rynek 42(5), 197–204 (2009)

[2] Skulimowski, A.M.J.: Anticipatory Network Models of Multicriteria Decision-Making Processes. Int. J. Systems Sci. 45(1), 39-59, http://www.tandfonline.com/doi/full/10.1080/00207721.2012.670308  (2014)

[3] Web site of the MOVING project: www.moving-project.eu



Martin Schweitzer

Dynamic mean-variance optimisation problems without information

We want to solve the problems of mean-variance hedging (MVH) and mean-variance portfolio selection (MVPS) under restricted information. We work in a setting where the underlying price process S is a semimartingale, but not adapted to the filtration G which models the information available for constructing trading strategies. We choose as G = F^det the zero-information filtration and assume that S is a time-dependent affine transformation of a square-integrable martingale. This class of processes includes in particular arithmetic and exponential Levy models with suitable integrability. We give explicit solutions to the MVH and MVPS problems in this setting, and we show for the Levy case how they can be expressed in terms of the Levy triplet. Explicit formulas are obtained for hedging European call options in the Bachelier and Black--Scholes models.

This is based on joint work with Danijel Zivoi and Mario Sikic.


Lukasz Stettner

Shadow price - inductive direct approach

In the talk we consider the problem of market with proportional transaction costs i.e. with bid and ask prices. We are looking for a price, called shadow price which is between bid and ask price and is such that optimal utility from terminal wealth is the same as on the market with bid and ask prices. The approach we present is direct, we are not using duality theory and is using induction with a sequence of recursive static models, which can be solved.

The talk is based on the paper: T. Rogala, L. Stettner, Optimal strategies for utility from terminal wealth with general bid and ask prices, published online 2018, Applied Mathematics & Optimization doi: 10.1007/s00245-018-9550-5


Ralph E. Steuer, Sebastian Utz

Domain of Expertise that Needs to Be Passed for Transiting from Conventional Bi-Criterion to Theme Tri-Criterion Investing

In mutual fund investing, there is the area of theme investing. Theme investing includes ethical-impact, socially responsible, sustainable, green energy, and Shariah-compliant investing. Admittedly, not everyone is interested in theme investing. However, there is enough interest in the area that the area cannot be ignored.  Whereas the goal in conventional investing is to build portfolios exhibiting optimal risk/return tradeoffs, the goal in theme investing is to build portfolios that exhibit optimal risk/return/theme tradeoffs. While the mutual fund industry is able to build portfolios that are able to pass for theme portfolios in the current market, it is our position that these portfolios are customarily far from being optimal theme portfolios.

This is because theme investing uses the same basic two-stage approach used in conventional investing, that is, where the purpose of Stage I is to produce an approved list and the purpose of Stage II is to construct out of securities on the approved list the portfolio exhibiting the most preferred risk/return tradeoff attainable at this point. The only difference is that in the theme case, the Stage I screening is more rigorous. Other than for an almost certainly different approved list, the process of Stage II remains the same. While it is hard to find fault with the two-stage approach in conventional investing, when applied to theme investing it causes the theme and financial objectives of a problem to be treated sequentially, which almost certainly results in serious amounts of theme achievement being left on the table at insignificant cost. How this is our conclusion and the new knowledge that must be incorporated into the mutual fund industry for it to do a better job in endeavoring to produce optimal portfolios in theme investing are discussed in the paper.


Maximilian Wimmer

Sustainable Tracking of Different Indices

We introduce a new methodology for modeling optimal institutional portfolios, and using this methodology, we report on results that combine the benefits of passive investing with the needs of socially responsible (SR) investors. Our methodology is based on the hypothesis that in SR investing, social responsibility is a third criterion, and this causes the classical bi-criterion efficient frontier to become a tri-criterion efficient surface. Using this surface, in an empirical study, we estimate the costs resulting from adding a social responsibility threshold to several passive index investments. Our findings are that by our implementation, the costs are marginal.



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