Mathematical Problems in Engineering

Volume 2011, Article ID 347604, 15 pages

http://dx.doi.org/10.1155/2011/347604

## Financial Applications of Bivariate Markov Processes

^{1}Department MSIA, University of Bergamo, Via dei Caniana 2, 24127 Bergamo, Italy^{2}Department of Quantitative Methods, University of Brescia, Contrada Santa Chiara 50, 25122 Brescia, Italy

Received 14 April 2011; Accepted 4 September 2011

Academic Editor: Jitao Sun

Copyright © 2011 Sergio Ortobelli Lozza et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper describes a methodology to approximate a bivariate Markov process by means of a proper Markov chain and presents possible financial applications in portfolio theory, option pricing and risk management. In particular, we first show how to model the joint distribution between market stochastic bounds and future wealth and propose an application to large-scale portfolio problems. Secondly, we examine an application to VaR estimation. Finally, we propose a methodology to price Asian options using a bivariate Markov process.

#### 1. Introduction

In this paper, we propose an approach to some classical financial problems based on the analysis of bivariate Markov processes. In particular, we use a bivariate Markov process to examine three possible financial applications: portfolio selection, risk management, and option pricing.

Generally, portfolio selection, risk management, and option pricing problems are studied in financial literature assuming that the stock returns are Gaussian or elliptically distributed. As a matter of fact, the first analysis of the portfolio selection problem was given by Markowitz [1–3] and Tobin [4, 5] in terms of the mean and the variance of the portfolio returns. The portfolio selection based on investors' mean-variance preferences can be justified only assuming that the returns are elliptically distributed.

In risk-management theory, the risk measure mostly adopted by financial institutions to manage and evaluate the market risk exposition of the own portfolios is the value-at-risk (VaR). There exist many methodologies to compute VaR. The most used model (proposed from RiskMetrics) hypothesizes Gaussian or conditionally elliptical distributed returns (see [6, 7]).

Finally, the benchmark model used in option pricing theory, the Black and Scholes model [8], is based on the assumption that the log-return evolves as a Brownian motion and thus log returns are Gaussian distributed.

However, it is well known that the stock returns present heavy tails and skewness and there exist a wide literature on the improvements performed on the Gaussian pioneer models (see, among others, [9] and the references therein). Many efforts have been destined to make the distributional hypothesis more realistic on the price process. Moreover, most of the alternative models are based on different Markovian processes. Effectiveness of Markov processes in describing the portfolios returns has been widely discussed in the literature (see, among others, [9, 10]). When the portfolio returns follow a Markov process, the estimation of the future wealth distribution can be a heavy computational task; nevertheless, the computational complexity can be controlled by means of a Markov chain, where the states are chosen in such a way that they induce a recombining effect on the future wealth (see [11] and the references therein).

Among Markovian models, we essentially distinguish two categories: parametric models (see, among others, [10, 12, 13]) and nonparametric models (see, among others, [11, 14]). In the first category, the Markovian hypothesis is used for diffusive models of the log returns. In the second category of models, the historical series are used to estimate the transition among the states. Nonparametric models have the main advantage in their capacity of adapting to the return distributions. In this paper, we propose a nonparametric Markovian model using an homogeneous Markov chain to describe the returns time evolution. The proposed approach extend the univariate approach proposed by Angelelli and Ortobelli Lozza [11], and for this reason, it is essentially different from other nonparametric models discussed in literature (see, among others, [15–18]).

The main contribution of this paper is twofold.(a)it extends the nonparametric univariate Markovian pricing valuation to the bivariate one to account joint behavior of the stock prices.(b)it shows the financial use and the impact of nonparametric bivariate Markov processes. In particular, we discuss the application of bivariate Markov processes in three financial problems: the large-scale portfolio selection problem, the valuation of the portfolio risk at a given future time, and the pricing of average strike Asian options.

In the first part of the paper, we approach a large-scale portfolio selection problem. The problem is attacked by means of a number of different techniques applied in steps. First, the randomness of the problem is reduced by applying *principal component analysis* (PCA) to the Pearson correlation of the forecasted wealth obtained with the approximating Markov process, which allows to approximate the returns using only few components deriving from the PCA. Secondly, we optimize a proper portfolio selection strategy that accounts for the joint behavior of the future portfolio wealth and of the predicted wealth obtained by the market stochastic bounds (see [19, 20]). The effectiveness of the approach is tested by an ex-post empirical analysis in which the results of this approach are compared to those obtained from a classical mean-variance strategy (see [21]).

In the second part of the paper, we propose to use the covariance matrix obtained by the estimated wealth at a given time to value the percentile of the future wealth. Then, we compare our estimates with the classical methodology used by Riskmetrics (see [7]). Finally, we discuss the pricing of contingent claims that require the use of different random variables. In particular, we show how we can estimate the price of average strike Asian options.

The paper is organized as follows. In Section 2, we discuss how modeling bivariate Markov processes. Section 3 analyzes the large-scale portfolio problem and propose an ex-post empirical comparison. Section 4 discusses the use of bivariate Markov chains for the valuation of the portfolio risk and the pricing of average Asian options. The last section briefly summarizes the paper.

#### 2. Approximating Bivariate Markov Processes with a Markov Chain

Assume that an initial wealth is invested at time in two portfolios of weights and of and risky assets respectively. The vectors and represent the percentage of the initial wealths ( and , resp.) invested in each asset. Denote the prices of these assets at time by and . The portfolios returns during the period are given by the vector with components We assume that the portfolios returns and follow two homogeneous Markov processes. In this section, we introduce an approximation of the bivariate process by a bivariate homogeneous Markov chain. We introduce the multi-index and denote by , the states of the Markov chain. First, we discretize the support of the Markov process . Given a set of past observations , we consider the range of the portfolios returns and divide it into bidimensional intervals , where and are two decreasing sequences given by

The idea is to approximate the returns associated to values of the Markov process in by the state of the Markov chain defined by Introducing we may write and . Assuming the Markov chain homogeneous, we denote its transition matrix by , where represents the probability of observing the returns in being in at time . These probabilities are estimated by the maximum likelihood estimates where is the number of observations that transit from to and is the number of observations in . Let us now consider the bivariate wealth process generated by the gross returns. The wealth at time is a bivariate random variable with possible values where is the wealth at time . Denoting the realized state of Markov chain at time , the value of is given by It is clear that the sequence identifies uniquely the path followed by the bivariate wealth process up to time . Thus, using formulas (2.5), the wealth obtained along the path is given by Notice that vector and represent the percentages of the initial wealths. Thus, if we want to evaluate the sample path of the ex-post wealths, we have to recalibrate each portfolio in order to maintain these percentages constant over time.

Moreover, describing the gross returns by a general bivariate Markov chain with possible states implies that the number of possible values for grows exponentially with the time. However, can take only values. In particular, in this way, the final wealth does not depend on the specific path followed by the process, but only on the sums of the indices of the states traversed by the Markov chain in the first steps. This property is called *recombining effect* of the Markov chain on the wealth process .

Let us denote the possible values of at time by where . The possible values of up to time can be stored in matrices of dimension or in a monodimensional vector of size .

The wealth can be represented by a three-dimensional Markovian tree, starting with a single node and presenting at each time instant the nodes given by , .

We are interested in the evolution of such a process , which is clearly connected to the evolution of . Consider the matrix with componants which represents the probability of obtaining the wealth and to be in state at time and the vector with components The probabilities and can be computed recursively by where is the probability that the return at time zero is . We assume these probabilities to be known from past observations.

#### 3. The Portfolio Selection Problem

In this section, we provide two applications of bivariate Markov processes to the portfolio selection problem(1)to account the joint behavior of the portfolio with the market stochastic bounds,(2)to reduce the dimensionality of large scale portfolio problems.

Finally, we compare a classical (static) portfolio selection strategy and a dynamic one based on the forecasted wealth obtained with Markov processes.

The static portfolio selection problem when no short sales are allowed consists of the maximization of a functional (performance measure or utility functional) defined on the space of possible returns with respect to the portfolio , which is assumed to belong to the -dimensional simplex . In other words, the investors compute the portfolio solution of
Among the various static strategies that have been proposed in the literature, in our empirical application, we consider the *Sharpe ratio *(SR)* strategy* (see [21]) which evaluates the expected excess return for unit of risk (standard deviation); that is,
where is a given benchmark and is the standard deviation of the random variable . When the benchmark is the risk-free rate and is the portfolio return, the Sharpe ratio is isotonic with nonsatiable risk averse preferences. For a discussion on the choice of see [11].

Consider now the dynamic framework. Assume the initial wealth and denote by all the admissible wealth processes depending on an initial portfolio . The dynamic portfolio selection problem consists of the maximization over of a functional depending on the wealth process. In our application, we consider a portfolio selection strategy where investors optimize their portfolio every periods maximizing a functional applied to the forecasted wealth at time . Since the weights represent the percentages of wealth invested in each asset and the value of the assets change every day, we should recalibrate daily the wealth maintaining constant the percentage every day during each period , (where is the time in which we compute the new portfolio composition). Thus, investors periodically compute the portfolio solution of
and then they recalibrate their portfolio every in order to maintain constant the percentages invested in each asset. Moreover, we will make use of a nonmyopic functional, that is, a functional depending on the entire stochastic process . Since we want to value the impact of bivariate processes, we propose to optimize a distance between the wealth of the portfolio and wealth obtained from the market stochastic bounds. Upper and lower market stochastic bounds are, respectively, defined by and and satisfy the relation for all vectors of portfolio weights belonging to the simplex (see [19, 20]). Thus, the returns during the period of stochastic bounds are given by and . Generally, investors would like to minimize a distance measure between the portfolio and the upper market bound and to maximize a distance measure with the lower market bound . To account these investors' preferences, we consider the following *OA-Stochastic Bound Ratio *(OA-SBR) performance functional defined by
where denotes the positive part of a function , are the wealth processes at time , deriving, respectively, from the upper and lower market stochastic bounds. Assuming that the returns follow the Markov chain introduced in Section 2, we can compute the previous expectations for the bivariate processes and exploiting the results of the previous section. In general, for a bivariate wealth process , we have
with .

As we show in the next subsection, bivariate Markov processes are useful even to reduce the dimensionality of large scale portfolio selection problems.

##### 3.1. Large-Scale Portfolios

The number of observations necessary in the optimization process increases proportionally with the dimension of the portfolios considered. Since the number of observations available on the market is relatively small compared to the number of assets, it is clear that a procedure to reduce the dimensionality of large-scale problems is needed. To this purpose, we apply a principal component analysis (PCA). The idea of PCA is to reduce the dimensionality of a data set made of a large number of possible correlated variables (assets) while preserving the largest possible variability in the data. This is done by transforming the initial variables (assets) into a new set of variables (called the principal components) which are uncorrelated and ordered in decreasing order of importance. Consider the assets returns at time
Applying the PCA methodology to the Pearson correlation matrix of the historical series, we replace the original correlated time series with uncorrelated time-series . The dimensionality reduction is obtained by choosing only those components (principal components) whose variability is significantly different from zero. We call these principal components *factors* and denote them by , .

Thus, each series can be written as the linear combination of the identified factors plus a small (uncorrelated) noise
We can further reduce the variability of the error by performing a PCA of the Pearson correlation matrix of the *forecasted wealth *obtained by the single returns. Notice that in order to compute the correlation matrix of the forecasted wealth, it is necessary to use bivariate Markov processes to account the joint behavior of the future wealth (as suggested in Section 2).

Once identified, the factors accounting for most of the variability of the returns and the factors accounting for most of the variability of the forecasted wealth, we regress the return of each asset on the factors as follows: Then, we can use the approximated returns for selecting the optimal portfolio.

##### 3.2. An Empirical Comparison between Portfolio Strategies

In order to value the impact of the bivariate Markovian approximation on portfolio selection strategies, we compare the performance of strategies based either on the Sharpe ratio or on the OA-stochastic bound ratio. The comparison consists of the ex-post evaluation of the wealth produced by the strategies. In particular, we assume that the riskless asset is not allowed; that is, the Sharpe ratio is given by . We approximate the Markovianity assuming states for each asset and a temporal horizon working days.

As dataset, we consider 3805 assets from the main US markets (NYSE and NASDAQ) available in DataStream during the period 05-Aug-2009, 17-Oct-2010. For each optimization, we consider a 6-month time window (about 125 market days) of historical data. Thus, we need a strong dimensionality reduction in order to keep statistical significance of historical data.

For any portfolio optimization, we first preselect the “best” 30 assets following the eight preselection criteria suggested by Ortobelli et al. [20]. The preselection is a methodology to reduce the dimensionality of the portfolio problem. It consists of selecting some assets for their appealing characteristics. In particular, with the proposed preselection criteria, we account the consistency with investors' preferences, the timing of the choices, the association with market stochastic bounds, and the Markovian and asymptotic behavior of wealth (see [20]). On these preselected assets, we apply the principal component analysis as suggested in Section 3.1. In particular, we consider 14 factors: 7 obtained with the PCA applied to the forecasted Pearson correlation matrix of the future wealth and the other 7 obtained with the PCA applied to the Pearson correlation matrix of the historical series. For any estimation, every 20 working days starting from 05 August 2009, we compute the optimal portfolio composition that maximize each performance ratio (SR or OA-SBR) considering the following constrains on the weights . Since portfolio selection problems based on the Markovian hypothesis presents more local optima, we solve the optimization problem using the heuristic for global optimization proposed by [11]. Then, we value the ex-post wealth.

For each strategy, we consider an initial wealth , and we use the last 6 months of daily observations. Thus, starting from 05 August 2009 at the -th recalibration (), three main steps are performed to compute the ex-post final wealth.

*Step 1. *Preselect the “best” 30 assets among 3805 assets (as suggested by [11]). On these assets apply the principal component analysis and approximate the returns, as suggested in Section 3.1.

*Step 2. *Determine the market portfolio that maximizes the performance ratio (SR or OA-SBR) associated to the strategy, that is, the solution of the following optimization problem:

*Step 3. *During the period (where ), we have to recalibrate daily the portfolio maintaining the percentages invested in each asset equal to those of the market portfolio . Thus, the ex-post final wealth is given by
where is the vector of observed daily gross returns between and . The optimal portfolio is the new starting point for the optimization problem.

Steps 1, 2 and 3 are repeated until the observations are available.

Figure 1 reports the ex-post sample paths of the wealth obtained maximizing the Sharpe ratio and the OA-stochastic bound ratio. In particular, we observe that the ex-post wealth of the OA-stochastic bound strategy multiplies of about six times in two months and half during the last week of November 2009 and the first week of February 2010. Instead, the strategy based on the maximization of the Sharpe ratio is not able to produce wealth during the same period. While during the European countries crisis (period from May till September, 2010) the loss of each strategy is no more than the 15% of the wealth. Therefore, this first comparison shows a very high impact (more than 900% in one year) on the ex-post final wealth obtained using the bivariate Markov process.

#### 4. Value at Risk at a Given Time and Applications in Option Pricing Theory

In this section, we consider other two possible applications of the proposed approximation of a bivariate Markov process: the valuation of VaR at a given time and the pricing of average strike Asian options.

##### 4.1. VAR at a Given Time

In the classical risk-management problem, a financial institution has to evaluate the market risk exposition of the owned portfolio. The classical tool proposed and used by practitioners is the value at risk (VaR) that synthesizes in a single value the possible losses which could be realized with a given probability, for a fixed temporal horizon. Namely, indicating with the current time, with the investor's temporal horizon, with the profit/loss realized in the interval and with a level of confidence, the value at risk is the possible loss at time implicitly defined by note that is the opposite of the -percentile of the profit/loss distribution in the interval .

The well-known RiskMetrics model, also called *exponential weighted moving average* (EWMA) model, assumes a Gaussian distribution for the conditional distribution of . Such an hypothesis dramatically simplifies the VaR calculation, in particular for portfolios with many assets whose returns are assumed conditional jointly normal distributed. Thus, if we point out with the composition vector of a portfolio, then the portfolio profit/loss at time is given by
where . We use centered returns to simplify the computation, but clearly, these results can be easily extended to real returns at less of an additive shift. When the conditional joint distribution of centered return vector is Gaussian, every linear combination of the primary components is also normally distributed. Since the expected centered return is null, the 1-day VaR of a portfolio with profit/loss is completely determined from the portfolio standard deviation
where is the percentile of a standard normal distribution, and is the covariance matrix whose evolution over time is described by
where is the so called *decay factor *(see [7]).

Moreover, RiskMetrics proposes to approximate the VaR at a given time by using the time rule However, this approximation can produce very big errors (see, among others, [22, 23]). Straightforward extensions of the RiskMetrics model can be obtained by using any other elliptical distribution (see, among others, [6] and the references therein). That is, if the conditional joint distribution of return vector is elliptically distributed, every linear combination of the primary components follows the same elliptical law. For example, if the joint conditional distribution of is a Student with () degrees of freedom, then(1)formula (4.3) is still valid provided that we substitute with the percentile of a Student with degrees of freedom and;(2)formula (4.5) changes by substituting the percentile of a sum of standard normal distributions with the percentile of the sum of random variables distributed as Student with degrees of freedom, (see [6]); that is,

In order to overcome the approximating error of formulas (4.5) and (4.6), we suggest to value the risk at a given time by using the covariance matrix obtained considering the joint distribution of the forecasted wealth at time for each couple of risky assets. Therefore, if we assume that the vector of the centered returns of wealth at time is conditionally elliptical distributed with null mean and covariance matrix , we get where is either or according to the above definitions of and .

Next, we test and compare the performance of the two alternative models. We compute the VaR with by using a Markov model and the classical EWMA model. Both models are implemented with Gaussian and Student assumption. Tests are executed on 440 NASDAQ assets from January 1997 till July 2010. For the elliptical distributions, the average Student degrees of freedom estimated on 01/01/1997 among the 440 assets is 4.732, and we use this value for all the ex-post analyses. For the Markov processes, we consider 9 states and . For the ex-post computations, we use a time window of 500 working days, and we assume that the historical observations present an exponential probability with . We estimated this value as the average (during the period 1997–2000) of optimal decay factor computed as suggested by Lamantia et al. [6]. Moreover, Kondor et al. [24] suggests to use a large value of the decay factor (near to 1) to compute the covariance matrix for large portfolio in contrast to “the rule of thumb” () proposed by RiskMetrics (see [7]). We consider 22500 random portfolios of the NASDAQ assets. The average of the number of portfolio observations that violate the VaR limits under the two distributional assumptions are shown in Table 1.

The percentages of violations should be, respectively, equal to the VaR limits , , . From this first analysis, we observe that the Markov valuations respect well enough the percentage of violations, while EWMA models generally overestimate the losses. In particular, when we assume that the conditional distribution of the returns follows a Student distribution, both models seem to give better performance than the Gaussian model. Moreover, we test how this valuation is accurate using the conditional (LRc) and unconditional (LRu) likelihood ratio tests proposed by Christoffersen [25] with confidence interval. The percentages of acceptably accurate valuation of VaR are given in Table 2.

Thus, Christoffersen's tests show clearly the best performance of the Markovian approximation even if further analysis are probably still necessary to confirm these studies. In particular, we believe that using other different distributional assumptions that consider also the skewness effects, which are generally observed in the portfolio returns, we should get better results.

##### 4.2. Average Strike Asian Options

In this last subsection, we deal with the problem of pricing average strike price options by using a bivariate Markov process. With average strike Asian options, the final payoff at a maturity is given by(i) for a put option,(ii) for a call option,

where is the stock price at a given time and is the average price during the period . It is well know that when the average is the arithmetic mean, we have not a close form solution for option pricing even when we assume that prices evolve as a geometric Brownian motion. Generally, to price continuous arithmetic average strike Asian options analysts calculate the first and second moments and then fits the approximating lognormal distribution—for the average—to the moments. Further approximations are needed if the average is done on daily prices.

Since by means of the bivariate Markov process we can easily valuate the joint distribution of two random variables, we can describe the joint Markovian behavior of the random vector , where , is the stock price at time , and . Considering a joint Markovian evolution of the vector , we get a pyramidal tree that after steps describes the "wealth process" , where and .

Thus, at less of an increasing transformation, we have the joint distribution of . Therefore, we can price Bermudan and European average strike Asian options using the Iaquinta and Ortobelli's algorithm [26] to compute the risk neutral matrix and the prices. An empirical analysis of this option pricing model requires the use of data from the over-the-counter (OTC) market (market where are priced these derivatives), and it should be object of future discussions and studies.

#### 5. Conclusions

This paper proposes a simple way to value bivariate Markov processes in portfolio, risk management, and option pricing problems. In particular, we have observed that the Markovian previsions of the future present a very big impact on the portfolio choices. Moreover, the bivariate Markov process can be used to estimate the covariance matrix at a given future time. Thus, using the forecasted variability, we can value the risk of a given portfolio at a future time . The comparison of the Markovian prevision with the classical EWMA model shows the highest performance of the first. Finally, we have discussed how to deal with average strike options by using the proposed approximation of a bivariate Markov process.

#### Acknowledgments

The authors give thank for grants ex-MURST 60% 2010, for helpful comments, the referee of this paper, and seminar audiences at 47° Workshop EWGFM, October 2010, Prague.

#### References

- H. Markowitz, “Portfolio selection,”
*Journal of Finance*, vol. 7, pp. 77–91, 1952. View at Google Scholar - H. M. Markowitz,
*Portfolio Selection: Efficient Diversification of Investments*, Cowles Foundation for Research in Economics at Yale University, Monograph 16, John Wiley & Sons, New York, NY, USA, 1959. - H. M. Markowitz,
*Mean-Variance Analysis in Portfolio Choice and Capital Markets*, Basil Blackwell, Oxford, UK, 1987. - J. Tobin, “Liquidity preference as behavior toward risk,”
*Review of Economic Studies*, vol. 25, pp. 65–86, 1958. View at Google Scholar - J. Tobin, “The theory of portfolio selection,” in
*The Theory of Interest Rates*, F. H. Hahn and F. P. R. Brechling, Eds., Macmillan, London, UK, 1965. View at Google Scholar - F. Lamantia, S. Ortobelli Lozza, and S. T. Rachev, “VaR, CVaR and time rules with elliptical and asymmetric stable distributed returns,”
*Investment Management and Financial Innovations*, vol. 3, no. 4, pp. 19–39, 2006. View at Google Scholar - J. Longestaey and P. Zangari,
*Riskmetrics-Technical Document*, J.P. Morgan, New York, NY, USA, 4th edition, 1996. - F. Black and M. Scholes, “The pricing of options and corporate liabilities,”
*Journal of Political Economy*, vol. 81, pp. 637–654, 1973. View at Google Scholar - S. Rachev and S. Mittnik,
*Stable Paretian Models in Finance*, John Wiley & Sons, Chichester, UK, 2000. - J. Duan, E. Dudley, G. Gauthier, and J. Simonato, “Pricing discretely monitored barrier options by a Markov chain,”
*Journal of Derivatives*, vol. 10, pp. 9–23, 2003. View at Google Scholar - E. Angelelli and S. Ortobelli Lozza, “American and European Portfolio Selection Strategies: The Markovian Approach,” in
*Financial Hedging*, P. N. Catlere, Ed., chapter 5, pp. 119–152, Nova Science Publishers, 2009. View at Google Scholar - J.-C. Duan and J.-G. Simonato, “American option pricing under GARCH by a Markov chain approximation,”
*Journal of Economic Dynamics & Control*, vol. 25, no. 11, pp. 1689–1718, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. O. Lozza and A. Staino, “Exotic options with Lévy processes: the Markovian approach,”
*Investment Management and Financial Innovations*, vol. 8, no. 1, pp. 140–156, 2011. View at Google Scholar - G. D'Amico, “Statistical inference for Markov chain European option: estimating the price, the bare risk and the theta by historical distributions of Markov chain,”
*Journal of Statistics & Management Systems*, vol. 9, no. 3, pp. 737–754, 2006. View at Google Scholar · View at Zentralblatt MATH - Y. Aït-Sahalia and A. W. Lo, “Nonparametric estimation of state-price densities implicit in financial asset prices,”
*Journal of Finance*, vol. 53, no. 2, pp. 499–547, 1998. View at Google Scholar - Y. Aít-Sahalia, “Nonparametric pricing of interest rate derivative securities,”
*Econometrica*, vol. 64, no. 3, pp. 527–560, 1996. View at Google Scholar - J. M. Hutchinson, A. W. Lo, and T. Poggio, “A non parametric approach to the pricing and hedging of derivative securities via learning networks,”
*Journal of Finance*, vol. 49, pp. 851–889, 1994. View at Google Scholar - M. Stutzer, “A simple nonparametric approach to derivative security valuation,”
*Journal of Finance*, vol. 51, no. 5, pp. 1633–1652, 1996. View at Google Scholar - S. O. Lozza and F. Pellerey, “Market stochastic bounds with elliptical distributions,”
*Journal of Concrete and Applicable Mathematics*, vol. 6, no. 3, pp. 293–314, 2008. View at Google Scholar · View at Zentralblatt MATH - S. Ortobelli Lozza, D. Toninelli, and E. Angelelli, “Set-portfolio selection with the use of market stochastic bounds,” Tech. Rep., University of Bergamo, 2010. View at Google Scholar
- W. F. Sharpe, “The Sharpe ratio,”
*Journal of Portfolio Management*, pp. 45–58, 1994. View at Google Scholar - F. X. Diebold, A. Hickman, A. Inoue, and T. Schuermann, “Converting 1-day volatility to h-day volatility: scaling by root-h is worse than you think,” Working Paper, Wharton Financial Institutions Center, pp. 97-34, 1998.
- F. X. Diebold, A. Hickman, A. Inoue, and T. Schuermann, “Scale models,”
*Risk*, vol. 11, pp. 104–107, 1998. View at Google Scholar - I. Kondor, S. Pafka, and G. Nagy, “Noise sensitivity of portfolio selection under various risk measures,”
*Journal of Banking and Finance*, vol. 31, no. 5, pp. 1545–1573, 2007. View at Publisher · View at Google Scholar - P. F. Christoffersen, “Evaluating interval forecasts,”
*International Economic Review*, vol. 39, no. 4, pp. 841–862, 1998. View at Publisher · View at Google Scholar · View at MathSciNet - G. Iaquinta and S. Ortobelli Lozza, “Markov chain applications to non parametric option pricing theory,”
*International Journal of Computer Science & Network Security*, vol. 8, no. 6, pp. 199–208, 2008. View at Google Scholar