Pricing and capital requirements for with profit contracts: modelling considerations

The aim of this paper is to provide an assessment of alternative frameworks for the fair valuation of life insurance contracts with a predominant financial component, in terms of impact on the market consistent price of the contracts, the embedded options, and the capital requirements for the insurer. In particular, we model the dynamics of the log-returns of the reference fund using the so-called Merton (1976 Merton, RC. 1976. Option pricing when underlying stock returns are discontinuous. J. Finan. Econ., : 125–144.  [Google Scholar]) process, which is given by the sum of an arithmetic Brownian motion and a compound Poisson process, and the Variance Gamma (VG) process introduced by Madan and Seneta (1990 Madan, DB and Seneta, E. 1990. The variance gamma (VG) model for share market returns. J. Bus., 63: 511–524. [Crossref], [Web of Science ®] , [Google Scholar]), and further refined by Madan and Milne (1991 Madan, DB and Milne, F. 1991. Option pricing with VG martingale components. Math. Finan., 1: 39–45. [Crossref] , [Google Scholar]) and Madan et al. (1998 Madan, DB, Carr, P and Chang, E. 1998. The variance gamma process and option pricing. Eur. Finan. Rev., 2: 79–105. [Crossref] , [Google Scholar]). We conclude that, although the choice of the market model does not affect significantly the market consistent price of the overall benefit due at maturity, the consequences of a model misspecification on the capital requirements are noticeable.     

There's more to volatility than volume

It is widely believed that fluctuations in transaction volume, as reflected in the number of transactions and to a lesser extent their size, are the main cause of clustered volatility. Under this view bursts of rapid or slow price diffusion reflect bursts of frequent or less frequent trading, which cause both clustered volatility and heavy tails in price returns. We investigate this hypothesis using tick by tick data from the New York and London Stock Exchanges and show that only a small fraction of volatility fluctuations are explained in this manner. Clustered volatility is still very strong even if price changes are recorded on intervals in which the total transaction volume or number of transactions is held constant. In addition the distribution of price returns conditioned on volume or transaction frequency being held constant is similar to that in real time, making it clear that neither of these are the principal cause of heavy tails in price returns. We analyse recent results of Ane and Geman (2000 Ane, T and Geman, H. 2000. Order flow, transaction clock, and normality of asset returns. J. Finance, 55(5): 2259–2284. [Crossref], [Web of Science ®] , [Google Scholar]: J. Finance, 55, 2259–2284) and Gabaix et al. (2003 Gabaix, X, Gopikrishnan, P, Plerou, V and Stanley, H.E. 2003. A theory of power-law distributions in financial market fluctuations. Nature, 423: 267–270. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]: Nature, 423, 267–270), and discuss the reasons why their conclusions differ from ours. Based on a cross-sectional analysis we show that the long-memory of volatility is dominated by factors other than transaction frequency or total trading volume.     

Optimal time to sell a stock in the Black–Scholes model: comment on ‘Thou shalt buy and hold’, by A. Shiryaev,Z. Xu and X.Y. Zhou

We reconsider the problem of the optimal time to sell a stock studied by Shiryaev et al. (2008 Shiryaev, A, Xu, Z and Zhou, XY. 2008. Thou shalt buy and hold. Quant. Finan., 8: 765–776. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) (following in this issue of Quantitative Finance) using path integral methods. These methods allow us to confirm the results obtained by these authors and extend them to the entire parameter region. We also obtain the full distribution of the time t_m at which the maximum of the price is reached for arbitrary values of the drift.     

A modified Corrado test for assessing abnormal security returns

Event studies typically use the methodology developed by Fama et al. [1969 Fama, E., Fisher, L., Jensen, M. and Roll, R. 1969. The adjustment of stock prices to new information. International Economic Review, 10(1): 1–21. [Crossref] , [Google Scholar]. The adjustment of stock prices to new information. International Economic Review 10, no. 1: 1–21] to segregate a stock's return into expected and unexpected components. Moreover, conventional practice assumes that abnormal returns evolve in terms of a normal distribution. There is, however, an increasing tendency for event studies to employ non-parametric testing procedures due to the mounting empirical evidence which shows that stock returns are incompatible with the normal distribution. This paper focuses on the widely used non-parametric ranking procedure developed by Corrado [1989 Corrado, C. 1989. A nonparametric test for abnormal security price performance in event studies. Journal of Financial Economics, 23(2): 385–95. [Crossref], [Web of Science ®] , [Google Scholar]. A nonparametric test for abnormal security price performance in event studies. Journal of Financial Economics 23, no. 2: 385–95] for assessing the significance of abnormal security returns. In particular, we develop a consistent estimator for the variance of the sum of ranks of the abnormal returns, and show how this leads to a more efficient test statistic (as well as to less cumbersome computational procedures) than the test originally proposed by Corrado (1989 Corrado, C. 1989. A nonparametric test for abnormal security price performance in event studies. Journal of Financial Economics, 23(2): 385–95. [Crossref], [Web of Science ®] , [Google Scholar]). We also use the theorem of Berry [1941 Berry, A. 1941. The accuracy of the Gaussian approximation to the sum of independent variates. Transactions of the American Mathematical Society, 49(1): 122–36. [Crossref] , [Google Scholar]. The accuracy of the Gaussian approximation to the sum of independent variates. Transactions of the American Mathematical Society 49, no. 1: 122–36] and Esseen [1945 Esseen, C. 1945. Fourier analysis of distribution functions: A mathematical study of the Laplace–Gaussian law. Acta Mathematica, 77(1): 1–125. [Crossref], [Web of Science ®] , [Google Scholar]. Fourier analysis of distribution functions: A mathematical study of the Laplace–Gaussian law. Acta Mathematica 77, no. 1: 1–125] to demonstrate how the distribution of the modified Corrado test statistic developed here asymptotically converges towards the normal distribution. This shows that describing the distributional properties of the sum of the ranks in terms of the normal distribution is highly problematic for small sample sizes and small event windows. In these circumstances, we show that a second-order Edgeworth expansion provides a good approximation to the actual probability distribution of the modified Corrado test statistic. The application of the modified Corrado test developed here is illustrated using data for the purchase and sale by UK directors of shares in their own companies.     

Optimal execution strategies in limit order books with general shape functions

We consider optimal execution strategies for block market orders placed in a limit order book (LOB). We build on the resilience model proposed by Obizhaeva and Wang (2005 Obizhaeva, A and Wang, J. 2005. Optimal trading strategy and supply/demand dynamics, Preprint Available online at: http://www.rhsmith.umd.edu/faculty/obizhaeva/OW060408.pdf (accessed 16 February 2009)[Crossref] , [Google Scholar]) but allow for a general shape of the LOB defined via a given density function. Thus, we can allow for empirically observed LOB shapes and obtain a nonlinear price impact of market orders. We distinguish two possibilities for modelling the resilience of the LOB after a large market order: the exponential recovery of the number of limit orders, i.e. of the volume of the LOB, or the exponential recovery of the bid–ask spread. We consider both of these resilience modes and, in each case, derive explicit optimal execution strategies in discrete time. Applying our results to a block-shaped LOB, we obtain a new closed-form representation for the optimal strategy of a risk-neutral investor, which explicitly solves the recursive scheme given in Obizhaeva and Wang (2005 Obizhaeva, A and Wang, J. 2005. Optimal trading strategy and supply/demand dynamics, Preprint Available online at: http://www.rhsmith.umd.edu/faculty/obizhaeva/OW060408.pdf (accessed 16 February 2009)[Crossref] , [Google Scholar]). We also provide some evidence for the robustness of optimal strategies with respect to the choice of the shape function and the resilience-type.     

On a risk model with dependence between interclaim arrivals and claim sizes

We consider an extension to the classical compound Poisson risk model for which the increments of the aggregate claim amount process are independent. In Albrecher and Teugels (2006 Albrecher, H. and Teugels, J. 2006. Exponential behavior in the presence of dependence in risk theory. Journal of Applied Probability, 43(1): 257–273. [Crossref], [Web of Science ®] , [Google Scholar]), an arbitrary dependence structure among the interclaim time and the subsequent claim size expressed through a copula is considered and they derived asymptotic results for both the finite and infinite-time ruin probabilities. In this paper, we consider a particular dependence structure among the interclaim time and the subsequent claim size and we derive the defective renewal equation satisfied by the expected discounted penalty function. Based on the compound geometric tail representation of the Laplace transform of the time to ruin, we also obtain an explicit expression for this Laplace transform for a large class of claim size distributions. The ruin probability being a special case of the Laplace transform of the time to ruin, explicit expressions are therefore obtained for this particular ruin related quantity. Finally, we measure the impact of the various dependence structures in the risk model on the ruin probability via the comparison of their Lundberg coefficients.     

Do emerging markets with consistent returns have better future performance?

Using monthly data for 25 emerging markets around the world, it is found that emerging markets with recently consistent stock returns tend to have future returns that continue in the same direction. The effects are long-lived for negative consistency, and imply that capital flows are much more sensitive to market downturns than market upturns. Additionally, the longer a market has had consistently negative (positive) stock returns, the more negative (positive) are future returns. These results serve as confirmation that the consistency effects of Grinblatt and Moskowitz [J. Finan. Econ., 2004 Grinblatt, M and Moskowitz, T. 2004. Predicting stock price movements from the pattern of past returns. J. Finan. Econ., 71: 541–579. [Crossref], [Web of Science ®] , [Google Scholar], forthcoming] and Watkins [J. Behav. Finan., 2003 Watkins, B. 2003. Riding the wave of investor sentiment: an analysis of consistency as a predictor of future stock returns. J. Behav. Finan., 4: 1–32.  [Google Scholar], 4, 1–32] exist in emerging markets around the world.     

Political influences on the costs of banking crises in emerging market economies: testing the U-shaped veto player hypothesis

While there has been considerable research on the consequences of financial crises, there has been little empirical research on the possible effects of the role of domestic political institutions that influence a government's ability to implement crisis management policies. This paper investigates the impact of domestic institutions, characterized by a U-shaped veto player framework, on the output costs of banking crises. The analysis extends MacIntyre's qualitative study (2001 MacIntyre, Andrew. 2001. Institutions and investors: The politics of the economic crisis in Southeast Asia. International Organization, 55(1): 81–122. [Crossref], [Web of Science ®] , [Google Scholar]) of the relationship between veto players and policy risks in the Asian financial crises. For a large sample of emerging market economies, we find support for McIntyre's hypotheses that both too few and too many veto players are associated with greater costs of banking crises.     

Industry cost of equity capital: European evidence for multifactor models

We estimate the costs of equity capital for 117 industries from 16 European countries employing the CAPM and 8 multifactor asset pricing models as well as a variety of different econometric techniques. In doing so, we extend previous research on cost of equity estimation in mainly two ways. First, our study involves European instead of US or UK industries, which are investigated in previous research, and we find that cost of equity estimates obtained from the CAPM or multifactor asset pricing models are as imprecise for European industries as for US and UK industries. Second, in addition to the CAPM, the Fama and French [1993 Fama, Eugene F., and Kenneth R. French. 1993. “Common Risk Factors in the Returns on Stocks and Bonds.” Journal of Financial Economics 33: 3–56. doi: 10.1016/0304-405X(93)90023-5[Crossref], [Web of Science ®] , [Google Scholar]. “Common Risk Factors in the Returns on Stocks and Bonds.” Journal of Financial Economics 33: 3–56] three-factor model, and the Carhart [1997 Carhart, Mark M. 1997. “On Persistence in Mutual Fund Performance.” The Journal of Finance 52 (1): 57–82. doi: 10.1111/j.1540-6261.1997.tb03808.x[Crossref], [Web of Science ®] , [Google Scholar]. “On Persistence in Mutual Fund Performance.” The Journal of Finance 52 (1): 57–82] four-factor model, which are usually employed, our study includes six multifactor models that have not yet been examined on their ability to provide precise estimates of the costs of equity: the five-factor model of Fama and French [1993 Fama, Eugene F., and Kenneth R. French. 1993. “Common Risk Factors in the Returns on Stocks and Bonds.” Journal of Financial Economics 33: 3–56. doi: 10.1016/0304-405X(93)90023-5[Crossref], [Web of Science ®] , [Google Scholar]. “Common Risk Factors in the Returns on Stocks and Bonds.” Journal of Financial Economics 33: 3–56] as well as the multifactor models of Pástor and Stambaugh [2003 Pástor, Lubos, and Robert F. Stambaugh. 2003. “Liquidity Risk and Expected Stock Returns.” Journal of Political Economy 111 (3): 642–685. doi: 10.1086/374184[Crossref], [Web of Science ®] , [Google Scholar]. “Liquidity Risk and Expected Stock Returns.” Journal of Political Economy 111 (3): 642–685]; Campbell and Vuolteenaho [2004 Campbell, John Y., and Tuomo Vuolteenaho. 2004. “Bad Beta, Good Beta.” American Economic Review 94 (5): 1249–1275. doi: 10.1257/0002828043052240[Crossref], [Web of Science ®] , [Google Scholar]. “Bad Beta, Good Beta.” American Economic Review 94 (5): 1249–1275]; Hahn and Lee [2006 Hahn, Jaehoon, and Hangyong Lee. 2006. “Yield Spreads as Alternative Risk Factors for Size and Book-To-Market.” Journal of Financial &; Quantitative Analysis 41 (2): 245–269. doi: 10.1017/S0022109000002052[Crossref], [Web of Science ®] , [Google Scholar]. “Yield Spreads as Alternative Risk Factors for Size and Book-To-Market.” Journal of Financial &; Quantitative Analysis 41 (2): 245–269]; Petkova [2006 Petkova, Ralitsa. 2006. “Do the Fama–French Factors Proxy for Innovations in Predictive Variables?” The Journal of Finance 61 (2): 581–612. doi: 10.1111/j.1540-6261.2006.00849.x[Crossref], [Web of Science ®] , [Google Scholar]. “Do the Fama–French Factors Proxy for Innovations in Predictive Variables?” The Journal of Finance 61 (2): 581–612]; and Koijen, Lustig, and van Nieuwerburgh [2010 Koijen, Ralph S., Hanno N. Lustig, and Stijn G. van Nieuwerburgh. 2010. “The Cross-Section and Time-Series of Stock and Bond Returns.” Working Paper, University of Chicago, University of California at Los Angeles, New York University. [Google Scholar]. “The Cross-Section and Time-Series of Stock and Bond Returns.” Working Paper, University of Chicago, University of California at Los Angeles, New York University]. Our results suggest that these models provide even more imprecise cost of equity estimates. One main reason for these inaccurate estimates is the large temporal variation of the risk loadings on the non-traded factors in these models.     

The effect of pseudo‐immediacy on intertemporal choices

An interesting phenomenon, which we dub the ‘pseudo‐immediacy effect’, was detected in intertemporal choices. The majority of our participants preferred the smaller but sooner (SS) outcome to the larger but later (LL) outcome when a pseudo‐immediacy reward was framed, but a higher proportion of participants preferred the LL outcome to the SS outcome when the pseudo‐immediate format was removed. Such a shift violated the invariance principle which requires that the preference order between options does not depend on the manner in which they are described. With reference to the pseudo‐certainty effect reported by Kahneman and Tversky in 1984 Kahneman, D. and Tversky, A. 1984. Choice, values, and frames. American Psychologist, 39: 341–50. [Crossref], [Web of Science ®] , [Google Scholar], our findings typically support the notion that risk and delay are psychologically equivalent and that the same psychological process underlies risk and intertemporal choice.     

Stochastic integrals driven by fractional Brownian motion and arbitrage: a tale of two integrals

Recent research suggests that fractional Brownian motion can be used to model the long-range dependence structure of the stock market. Fractional Brownian motion is not a semi-martingale and arbitrage opportunities do exist, however. Hu and Øksendal [Infin. Dimens. Anal., Quant. Probab. Relat. Top., 2003, 6, 1–32] and Elliott and van der Hoek [Math. Finan., 2003 Elliott, RJ and van de Hoek, J. 2003. A general fractional white noise theory applications to finance. Math. Finan., 13: 301–330. [Crossref], [Web of Science ®] , [Google Scholar], 13, 301–330] propose the use of the white noise calculus approach to circumvent this difficulty. Under such a setting, they argue that arbitrage does not exist in the fractional market. To unravel this discrepancy, we examine the definition of self-financing strategies used by these authors. By refining their definitions, a new notion of continuously rebalanced self-financing strategies, which is compatible with simple buy and hold strategies, is given. Under this definition, arbitrage opportunities do exist in fractional markets.     

Nonlinear dynamics in economics and finance and unit root testing

The recent financial crisis exposed the inability of traditional theoretical and empirical models to parsimoniously capture the rich dynamics of the economic environment. This has stimulated the interest of both academics and practitioners in the development and application of more sophisticated models. By allowing for the presence of nonlinearities, complex dynamics, multiple equilibria, structural breaks and spurious trends, these latter models resemble more closely the properties of economic and financial time series. In this article, we illustrate the flexibility of a family of econometric models, namely the exponential smooth transition autoregressive (ESTAR), to encompass several of the above characteristics. We then re-assess the power of the ESTAR unit root test developed by Kapetanios, Shin and Snell ((2003) Kapetanios, G., Shin, Y. and Snell, A. 2003. Testing for a unit root in the nonlinear STAR framework. Journal of Econometrics, 112(2): 359–79. (doi:10.1016/S0304-4076(02)00202-6)[Crossref], [Web of Science ®] , [Google Scholar]) in the presence of nuisance parameters typically encountered in the literature and compare its performance with that of the augmented Dickey-Fuller and the Enders and Granger ((1998) Enders, W. and Granger, C. W.J. 1998. Unit-root tests and asymmetric adjustment with an example using the term structure of interest rates. Journal of Business & Economic Statistics, 16(3): 304–11. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) tests. Our results show the lack of dominance of any particular test and that the power is not independent to priors about the nuisance parameters. Finally, we examine several asset price deviations from fundamentals and one hyper-inflation series and find contradictory results between the nonlinear fitted models and unit root tests. The findings highlight that new testing procedures with higher power are desirable in order to shed light on the behavior of financial and economic series.     

An endogenous volatility approach to pricing and hedging call options with transaction costs

Standard delta hedging fails to exactly replicate a European call option in the presence of transaction costs. We study a pricing and hedging model similar to the delta hedging strategy with an endogenous volatility parameter for the calculation of delta over time. The endogenous volatility depends on both the transaction costs and the option strike prices. The optimal hedging volatility is calculated using the criterion of minimizing the weighted upside and downside replication errors. The endogenous volatility model with equal weights on the up and down replication errors yields an option premium close to the Leland [J. Finance, 1985 Leland, HE. 1985. Option pricing and replication with transaction costs. J. Finance, 40: 1283–1301. [Crossref], [Web of Science ®] , [Google Scholar], 40, 1283–1301] heuristic approach. The model with weights being the probabilities of the option's moneyness provides option prices closest to the actual prices. Option prices from the model are identical to the Black–Scholes option prices when transaction costs are zero. Data on S&P 500 index cash options from January to June 2008 illustrate the model.     

Latin American stock markets’ volatility spillovers during the financial crises: a multivariate FIAPARCH-DCC framework

The main purpose of this paper is to analyse the volatility spillovers in Latin American emerging stock markets. A multivariate Fractionally Integrated Asymmetric Power ARCH model with dynamic conditional correlations of Engle (1982 Engle, R. F. 1982. Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica, 50: 987–1007. [Crossref], [Web of Science ®] , [Google Scholar]) with a Student-t distribution is employed. We examine whether considering for long memory and asymmetry in emerging stock markets behaviour may provide more insights into the volatility spillovers phenomenon. In this paper we select daily frequency stock indexes covering four emerging countries in Latin America for the period (January 1995–September 2009). Our results point out the importance of volatility spillovers in these countries. Moreover, long memory and asymmetry in emerging stock market dynamics seem to provide more insights into the transmission of volatility shocks. More interestingly, the analysis of the DCCEs behaviour over time via multivariate cointegration, vector error correction model and the Cholesky variance decomposition shows shifts behaviour around major Latin American financial crisis and recent subprime crisis. On the practical side, these results may be useful for international portfolio managers and Latin American stock market authorities.     

Accounting for Variation in the Explanatory Power of the Psychometric Paradigm: The Effects of Aggregation and Focus

Most psychometric studies of risk perception have used data that have been averaged over participants prior to analysis. Such aggregation obscures variation among participants and inflates the magnitude of relationships between psychometric dimensions and dependent variables such as overall riskiness. However, most studies that have not averaged data over participants have also shifted the focus of analysis from differences among hazards to differences among participants. Hence, it is unclear whether observed reductions in the explanatory power of psychometric dimensions result from the change in the level of analysis or from the change in the focus of analysis. Following Willis et al.'s (2005 Willis, H. H., DeKay, M. L., Fischhoff, B. and Morgan, M. G. 2005. Aggregate, disaggregate, and hybrid analyses of ecological risk perceptions. Risk Analysis, 25(2): 405–428. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) analysis of ecological risk perceptions, we unconfound these two variables in a study of risk perceptions in Santiago, Chile, although we use more traditional hazards, attributes, and statistical procedures. Results confirm that psychometric dimensions explain less variation in judgments of riskiness and acceptability at the disaggregate level than at the aggregate level. However, they also explain less variation when the focus of analysis is differences among participants rather than differences among hazards. These two effects appear to be similar in magnitude. A simple hybrid analysis economically represents variation among participants' judgments of hazards' riskiness by relating those judgments to a common set of psychometric dimensions from a traditional aggregate‐level analysis.     

Quango reform: the next steps?

The coalition government elected in 2010 in the UK pursued a programme of quango reform focused on reducing the number and expenditure of arm’s-length bodies, increasing transparency, improving accountability and maximizing efficiency and effectiveness. This paper revisits Flinders and Skelcher’s 2012 Flinders, M. and Skelcher, C. (2012), Shrinking the quango state: five challenges in reforming quangos. Public Money &; Management, 32, 5, pp. 327–334.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar] PMM paper ‘Shrinking the quango state: five challenges in reforming quangos’ to assess progress to date and consider future challenges. Drawing insights from the UK programme of quango reform, as well as similar developments in Ireland, the authors identify five new challenges for governments: regulating, managing, reconciling, co-ordinating and reflecting.     

Comment on ‘earnings management around UK open offers’

This note discusses the result of Iqbal, A., S. Espenlaub, and N. Strong. 2008 Iqbal, A., Espenlaub, S. and Strong, N. 2008. Earnings management around UK open offers. European Journal of Finance, this issue [Google Scholar]. Earnings management around UK open offers. European Journal of Finance, this issue, regarding long-run abnormal returns following open offers and announcement abnormal returns, compared with differing results in two previous studies based on similar samples. A survivorship bias explains some of the differences in the reported long-run abnormal returns. The difference in the announcement abnormal returns could be due to use of different data sources.     

The augmented Black–Litterman model: a ranking-free approach to factor-based portfolio construction and beyond

The Fama and French factor-ranking approach (1992, 1993, etc.) has been extensively applied in quantitative fund management. However, this approach suffers from hidden factor view, information inefficiency, etc. issues. Based on the Black–Litterman model (1992; as explained in Cheung 2010b Cheung, W. 2010b. The Black–Litterman model explained. J. Asset Mgmt., 11: 229–243. [Crossref] , [Google Scholar]), we develop a technique that endogenizes the ranking process and elegantly resolves these issues. This model explicitly seeks forward-looking factor views and smoothly blends them to deliver robust allocation to securities. Our numerical experiments show this is an intuitive and practical framework for factor-based portfolio construction, and beyond. This article features: (1) a new and unified framework for strategy combination, factor mimicking and security-specific bets; (2) an elegant and ranking-free approach to factor style construction; (3) worked examples based on the FTSE EUROTOP 100 universe; (4) insight into the classic issue of confidence parameter setting; and (5) implementation guidance in an appendix.     

Dynamic liquidation under market impact

The optimal liquidation problem with transaction costs, which includes a positive fixed cost, and market impact costs, is studied in this paper as a constrained stochastic optimal control problem. We assume that trading is instantaneous and the dynamics of the stock to be liquidated follows a geometric Brownian motion. The solution to the impulse control problem is computed at each time step by solving a linear partial differential equation and a maximization problem. In contrast to results obtained from the static formulation of Almgren and Chriss [J. Risk, 2000 Almgren, R and Chriss, N. 2000. Optimal execution of portfolio transactions. J. Risk, 3: 5–39. [Crossref] , [Google Scholar], 3, 5–39], when risk is not considered, the optimal liquidation strategy from our stochastic control formulation depends on temporary market impact cost and permanent market impact cost parameters. In addition, our computational results indicate the following properties of the optimal execution strategy from the stochastic control formulation. Due to the existence of a no-transaction region, it may not be optimal for some individuals to sell their assets on some trading dates. As the value of the permanent market impact parameter increases, the expected optimal amount liquidated at the terminal time increases. As the value of the quadratic temporary impact cost parameter increases, the expected optimal amount liquidated at trading times tends to be uniform, and the no-transaction region shrinks. In the presence of quadratic temporary market impact costs, in contrast to optimal strategies that result from fixed and/or proportional transaction costs alone, portfolios in the selling region are neither re-balanced into the no-transaction region nor into the sell and no-transaction interface.     

Do financial returns have finite or infinite variance? A paradox and an explanation

One of the major points of contention in studying and modelling financial returns is whether or not the variance of the returns is finite or infinite (sometimes referred to as the Bachelier–Samuelson Gaussian world versus the Mandelbrot stable world). A different formulation of the question asks how heavy the tails of the financial returns are. The available empirical evidence can be, and has been, interpreted in more than one way. The apparent paradox, which has puzzled many a researcher, is that the tails appear to become less heavy for less frequent (e.g. monthly) returns than for more frequent (e.g. daily) returns, a phenomenon not easily explainable by the standard models. Inspired by the prelimit theorems of Klebanov, Rachev and Szekely (1999 Klebanov, L, Rachev, S and Szekely, G. 1999. Pre-limit theorems and their applications. Acta Applicandae Mathematicae, 58: 159–174.  [Google Scholar]) and Klebanov, Rachev and Safarian (2000 Klebanov, L, Rachev, S and Safarian, M. 2000. Local prelimit theorems and their applications to finance. Appl. Math. Lett., 13: 73–78.  [Google Scholar]), we provide an explanation of this paradox. We show that, for financial returns, a natural family of models are those with tempered heavy tails. These models can generate observations that appear heavy tailed for a wide range of aggregation levels before becoming clearly light tailed at even larger aggregation scales. Important examples demonstrate the existence of a natural scale associated with the model at which such an apparent shift in the tails occurs.