Mathematical Aspects of Financial Markets with Frictions

Frictions are factors that hinder trading of securities in financial markets. Typical frictions include limited market depth, transaction costs, lack of infinite divisibility of securities, and taxes. Conventional models used in mathematical finance often gloss over these issues, which affect almost all financial markets, by arguing that the impact of frictions is negligible and, consequently, the frictionless models are valid approximations. This dissertation consists of three research papers, which are related to the study of the validity of such approximations in two distinct modeling problems. Models of price dynamics that are based on diffusion processes, i.e., continuous strong Markov processes, are widely used in the frictionless scenario. The first paper establishes that diffusion models can indeed be understood as approximations of price dynamics in markets with frictions. This is achieved by introducing an agent-based model of a financial market where finitely many agents trade a financial security, the price of which evolves according to price impacts generated by trades. It is shown that, if the number of agents is large, then under certain assumptions the price process of security, which is a pure-jump process, can be approximated by a one-dimensional diffusion process. In a slightly extended model, in which agents may exhibit herd behavior, the approximating diffusion model turns out to be a stochastic volatility model. Finally, it is shown that when agents' tendency to herd is strong, logarithmic returns in the approximating stochastic volatility model are heavy-tailed. The remaining papers are related to no-arbitrage criteria and superhedging in continuous-time option pricing models under small-transaction-cost asymptotics. Guasoni, Rásonyi, and Schachermayer have recently shown that, in such a setting, any financial security admits no arbitrage opportunities and there exist no feasible superhedging strategies for European call and put options written on it, as long as its price process is continuous and has the so-called conditional full support (CFS) property. Motivated by this result, CFS is established for certain stochastic integrals and a subclass of Brownian semistationary processes in the two papers. As a consequence, a wide range of possibly non-Markovian local and stochastic volatility models have the CFS property.

Examining and Modeling the Dynamics of the Volatility Surface

The objective of this paper is to investigate and model the characteristics of the prevailing volatility smiles and surfaces on the DAX- and ESX-index options markets. Continuing on the trend of Implied Volatility Functions, the Standardized Log-Moneyness model is introduced and fitted to historical data. The model replaces the constant volatility parameter of the Black & Scholes pricing model with a matrix of volatilities with respect to moneyness and maturity and is tested out-of-sample. Considering the dynamics, the results show support for the hypotheses put forward in this study, implying that the smile increases in magnitude when maturity and ATM volatility decreases and that there is a negative/positive correlation between a change in the underlying asset/time to maturity and implied ATM volatility. Further, the Standardized Log-Moneyness model indicates an improvement to pricing accuracy compared to previous Implied Volatility Function models, however indicating that the parameters of the models are to be re-estimated continuously for the models to fully capture the changing dynamics of the volatility smiles.

Pricing Index Options with Stochastic Volatility

The objective of this paper is to investigate the pricing accuracy under stochastic volatility where the volatility follows a square root process. The theoretical prices are compared with market price data (the German DAX index options market) by using two different techniques of parameter estimation, the method of moments and implicit estimation by inversion. Standard Black & Scholes pricing is used as a benchmark. The results indicate that the stochastic volatility model with parameters estimated by inversion using the available prices on the preceding day, is the most accurate pricing method of the three in this study and can be considered satisfactory. However, as the same model with parameters estimated using a rolling window (the method of moments) proved to be inferior to the benchmark, the importance of stable and correct estimation of the parameters is evident.

Essays on Labor Market Frictions, Technological Change and Macroeconomic Fluctuations

The dissertation consists of an introductory chapter and three essays that apply search-matching theory to study the interaction of labor market frictions, technological change and macroeconomic fluctuations. The first essay studies the impact of capital-embodied growth on equilibrium unemployment by extending a vintage capital/search model to incorporate vintage human capital. In addition to the capital obsolescence (or creative destruction) effect that tends to raise unemployment, vintage human capital introduces a skill obsolescence effect of faster growth that has the opposite sign. Faster skill obsolescence reduces the value of unemployment, hence wages and leads to more job creation and less job destruction, unambiguously reducing unemployment. The second essay studies the effect of skill biased technological change on skill mismatch and the allocation of workers and firms in the labor market. By allowing workers to invest in education, we extend a matching model with two-sided heterogeneity to incorporate an endogenous distribution of high and low skill workers. We consider various possibilities for the cost of acquiring skills and show that while unemployment increases in most scenarios, the effect on the distribution of vacancy and worker types varies according to the structure of skill costs. When the model is extended to incorporate endogenous labor market participation, we show that the unemployment rate becomes less informative of the state of the labor market as the participation margin absorbs employment effects. The third essay studies the effects of labor taxes on equilibrium labor market outcomes and macroeconomic dynamics in a New Keynesian model with matching frictions. Three policy instruments are considered: a marginal tax and a tax subsidy to produce tax progression schemes, and a replacement ratio to account for variability in outside options. In equilibrium, the marginal tax rate and replacement ratio dampen economic activity whereas tax subsidies boost the economy. The marginal tax rate and replacement ratio amplify shock responses whereas employment subsidies weaken them. The tax instruments affect the degree to which the wage absorbs shocks. We show that increasing tax progression when taxation is initially progressive is harmful for steady state employment and output, and amplifies the sensitivity of macroeconomic variables to shocks. When taxation is initially proportional, increasing progression is beneficial for output and employment and dampens shock responses.

A Cointegration Approach to Topics in Empirical Macroeconomics (summary section only)

Mikael Juselius’ doctoral dissertation covers a range of significant issues in modern macroeconomics by empirically testing a number of important theoretical hypotheses. The first essay presents indirect evidence within the framework of the cointegrated VAR model on the elasticity of substitution between capital and labor by using Finnish manufacturing data. Instead of estimating the elasticity of substitution by using the first order conditions, he develops a new approach that utilizes a CES production function in a model with a 3-stage decision process: investment in the long run, wage bargaining in the medium run and price and employment decisions in the short run. He estimates the elasticity of substitution to be below one. The second essay tests the restrictions implied by the core equations of the New Keynesian Model (NKM) in a vector autoregressive model (VAR) by using both Euro area and U.S. data. Both the new Keynesian Phillips curve and the aggregate demand curve are estimated and tested. The restrictions implied by the core equations of the NKM are rejected on both U.S. and Euro area data. These results are important for further research. The third essay is methodologically similar to essay 2, but it concentrates on Finnish macro data by adopting a theoretical framework of an open economy. Juselius’ results suggests that the open economy NKM framework is too stylized to provide an adequate explanation for Finnish inflation. The final essay provides a macroeconometric model of Finnish inflation and associated explanatory variables and it estimates the relative importance of different inflation theories. His main finding is that Finnish inflation is primarily determined by excess demand in the product market and by changes in the long-term interest rate. This study is part of the research agenda carried out by the Research Unit of Economic Structure and Growth (RUESG). The aim of RUESG it to conduct theoretical and empirical research with respect to important issues in industrial economics, real option theory, game theory, organization theory, theory of financial systems as well as to study problems in labor markets, macroeconomics, natural resources, taxation and time series econometrics. RUESG was established at the beginning of 1995 and is one of the National Centers of Excellence in research selected by the Academy of Finland. It is financed jointly by the Academy of Finland, the University of Helsinki, the Yrjö Jahnsson Foundation, Bank of Finland and the Nokia Group. This support is gratefully acknowledged.

Production process of Scots pine.

The accompanying collective research report is the result of the research project in 1986­90 between The Finnish Academy and the former Soviet Academy of Sciences. The project was organized around common field work in Finland and in the former Soviet Union and theoretical analyses of tree growth determining processes. Based on theoretical analyses, dynamic stand growth models were made and their parameters were determined utilizing the field results. Annual cycle affects the tree growth. Our theoretical approach was based on adaptation to local climate conditions from Lapland to South Russia. The initiation of growth was described as a simple low and high temperature accumulation driven model. Linking the theoretical model with long term temperature data allowed us to analyze what type of temperature response produced favorable outcome in different climates. Initiation of growth consumes the carbohydrate reserves in plants. We measured the dynamics of insoluble and soluble sugars in the very northern and Karelian conditions. Clear cyclical pattern was observed but the differences between locations were surprisingly small. Analysis of field measurements of CO2 exchange showed that irradiance is the dominating factor causing variation in photosynthetic rate in natural conditions during summer. The effect of other factors is so small that they can be omitted without any considerable loss of accuracy. A special experiment carried out in Hyytiälä showed that the needle living space, defined as the ratio between the shoot cylindric volume and needle surface area, correlates with the shoot photosynthesis. The penetration of irradiance into Scots pine canopy is a complicated phenomenon because of the movement of the sun on the sky and the complicated structure of branches and needles. A moderately simple but balanced forest radiation regime submodel was constructed. It consists of the tree crown and forest structure, the gap probability calculation and the consideration of spatial and temporal variation of radiation inside the forest. The common field excursions in different geographical regions resulted in a lot of experimental data of regularities of woody structures. The water transport seems to be a good common factor to analyse these properties of tree structure. There are evident regressions between cross-sectional areas measured at different locations along the water pathway from fine roots to needles. The observed regressions have clear geographical trends. For example, the same cross-sectional area can support three times higher needle mass in South Russia than in Lapland. Geographical trends can also be seen in shoot and needle structure. Analysis of data published by several Russian authors show, that one ton of needles transpire 42 ton of water a year. This annual amount of transpiration seems to be independent of geographical location, year and site conditions. The produced theoretical and experimental material is utilised in the development of stand growth model that describes the growth and development of Scots pine stands in Finland and the former Soviet Union. The core of the model is carbon and nutrient balances. This means that carbon obtained in photosynthesis is consumed for growth and maintenance and nutrients are taken according to the metabolic needs. The annual photosynthetic production by trees in the stand is determined as a function of irradiance and shading during the active period. The utilisation of the annual photosynthetic production to the growth of different components of trees is based on structural regularities. Since the fundamental metabolic processes are the same in all locations the same growth model structure can be applied in the large range of Scots pine. The annual photosynthetic production and structural regularities determining the allocation of resources have geographical features. The common field measurements enable the application of the model to the analysis of growth and development of stands growing on the five locations of experiments. The model enables the analysis of geographical differences in the growth of Scots pine. For example, the annual photosynthetic production of a 100-year-old stand at Voronez is 3.5 times higher than in Lapland. The share consumed to needle growth (30 %) and to growth of branches (5 %) seems to be the same in all locations. In contrast, the share of fine roots is decreasing when moving from north to south. It is 20 % in Lapland, 15 % in Hyytiälä Central Finland and Kentjärvi Karelia and 15 % in Voronez South Russia. The stem masses (115­113 ton/ha) are rather similar in Hyytiälä, Kentjärvi and Voronez, but rather low (50 ton/ha) in Lapland. In Voronez the height of the trees reach 29 m being in Hyytiälä and Kentjärvi 22 m and in Lapland only 14 m. The present approach enables utilization of structural and functional knowledge, gained in places of intensive research, in the analysis of growth and development of any stand. This opens new possibilities for growth research and also for applications in forestry practice.

Kirjallisuusosa: Alzheimerin taudin rottamallien vastaavuus tautiin ihmisessä ja mallien käyttökelpoisuus

Kirjallisuusosa: Alzheimerin taudin hoitoon olisi tarvetta uusille taudinkulkuun vaikuttaville lääkeaineille. Niiden kehittämiseksi tarvitaan eläinmalleja, joissa esiintyy taudin patofysiologisia piirteitä. Rottamalleista vanhemmat skopolamiini- tai MK-801-häirintä sekä ikääntyneiden rottien käyttö eivät kovin hyvin vastaa taudin patofysiologiaa, vaikka niissä eläimen muisti käyttäytymiskokeissa onkin heikentynyt. Uudemmat transgeeniset rottamallit ja mallit, joissa annetaan Aβ:a aivoihin, ilmentävät huomattavasti paremmin Alzheimerin taudin kaltaista tilaa aivoissa ainakin Aβ:n osalta. Taupatofysiologiaa ei silti kummassakaan näistä malleista juuri esiinny. Toisaalta Aβ:lla näyttäisi olevan huomattavasti tau:ta suurempi rooli taudissa, joten sen ilmeneminen mallissa onkin keskeisempi tekijä. Nämä mallit ilmentävät melko suurelti osin yhtä hyvin Alzheimerin taudin patofysiologiaa. Aβ:n antaminen on hieman yksinkertaisempi suorittaa käytännössä, sillä siinä ei tarvitse luoda transgeenista kantaa. Toisaalta transgeenisessa mallissa Aβ-patofysiologia syntyy enemmän Alzheimerin taudin kaltaisesti solujen sisällä eikä valmiita aggregoituvia Aβ-peptidejä anneta ulkopuolelta aivoihin. Molemmat mallit ovat kuitenkin käyttökelpoisia, ja soveltuvat erityisesti Aβ:an vaikuttavien lääkeaineiden kehittämiseen. Kokeellinen osa: Kokeen tarkoituksena oli validoida kohotettu ristikko-sokkelo (elevated plus-maze, EPM) hiirillä kognitiomallina. Kokeessa käytettiin kahden koekerran (trial, T) menetelmää, jossa koekertojen pituus oli viisi minuuttia. Näin saatiin useita oppimista kuvaavia parametreja. Hiirille yritettiin saada muistihäiriö aikaviiveen avulla (koekertojen väli 1-18 vrk) tai antamalla muskariinireseptoriantagonistia skopolamiinia (0,1-0,8 mg/kg i.p.) 30 minuuttia ennen T1:tä. Nämä kokeet suoritettiin sekä C57BL/6J- että ICR:(CD-1)-hiirillä. Aikaviivekokeissa ainut ryhmä, jolla oli viitettä unohtamisesta, oli ICR:(CD-1)-hiirien 18 vrk:n ryhmä. Tämän perusteella tutkittiin vielä 21 vuorokauden aikaväli, mutta selvää muistihäiriötä ei esiintynyt. Skopolamiini ei häirinnyt muistia ICR:(CD-1)-hiirillä, mutta C57BL/6J-hiirillä 0,2 mg/kg:n annoksesta ylöspäin merkitsevä muistihäiriö esiintyi. Näin ollen jatkokokeissa käytettäväksi valittiin skopolamiinin annos 0,2 mg/kg C57BL/6J-hiirillä, ja siinä tutkittiin donepetsiilin (0,3, 0,8 ja 1,5 mg/kg s.c), memantiinin (5,0 ja 10,0 mg/kg s.c) ja kokeellisen 5-HT6-antagonistin SB742457:n (1,5 ja 6,0 mg/kg s.c) muistia parantavia vaikutuksia. Tutkittavat lääkeaineet annettiin 40 minuuttia ennen T1:tä ja skopolamiini 30 minuuttia ennen. Memantiinilla (5,0 mg/kg) oli selkeä skopolamiinin heikentämää kognitiota parantava vaikutus ja donepetsiilillakin (1,5 mg/kg) suuntaus tähän. Tulosten perusteella malli näyttäisi soveltuvan muisti- ja oppimisvaikutusten tutkimiseen käytettäväksi malliksi.