Advanced Strategies of Portfolio Management in the Heston Market Model
AAdvanced Strategies of Portfolio Management in theHeston Market Model
Jaros law Gruszka a, ∗ , Janusz Szwabi´nski a a Hugo Steinhaus Center, Faculty of Pure and Applied Mathematics, Wroc law University ofScience and Technology
Abstract
There is a great number of factors to take into account when building andmanaging an investment portfolio. It is widely believed that a proper set-up ofthe portfolio combined with a good, robust management strategy is the key tosuccessful investment. In this paper, we aim at an analysis of two aspects thatmay have an impact on investment performance: diversity of assets and inclusionof cash in the portfolio. We also propose two new management strategies basedon the MACD and RSI factors known from technical analysis. Monte Carlosimulations within the Heston model of a market are used to perform numericalexperiments.
Keywords: portfolio management, diversification, MACD, RSI
1. Introduction
Managing an investment portfolio is a vital research topic, which can beapproached by means of multiple branches of science. Beginning from thepioneering work of Harry Markovitz in 1952 [1], for which he was awarded aNobel Prize, a plethora of works have been written concerning this subject.Scientists use tools developed by various disciplines in order to find the mosteffective methods of setting up and managing a financial portfolio. It shouldnot come as a surprise, that a huge part of the world’s portfolio managementresearch is related to the realm of mathematics. Among the well-established waysof approaching the topic one could definitely mention theoretical mathematics,specifically — the theory of stochastic differential equations [2] and the theoryof martingales [3]. Advanced statistical methods are also extremely popular,especially when it comes to the accurate estimation of risk associated with eachinvestment opportunity [4, 5, 6]. However, there also exist some less renownedbranches of mathematics which are used in order to study portfolio dynamics and ∗ Corresponding author
Email addresses: [email protected] (Jaros law Gruszka), [email protected] (Janusz Szwabi´nski)
Preprint submitted to Elsevier July 29, 2020 a r X i v : . [ q -f i n . P M ] J u l ne of them is certainly the theory of fuzzy sets [7, 8]. Recently, with the raise ofpopularity of machine learning techniques, some advances in the field of portfoliomanagement are also done, which utilise this new and powerful toolbox [9, 10].Finally, one of the branches of science which has recently contributed a lot tothe research on investment portfolio dynamics is econophysics [11]. Looking atthe vibrant financial environment from the perspective of natural sciences shednew light on the issue in general and allowed researchers to attempt seekingcorrelations between well-studied physical systems and the ones associated withthe world of economy.Recently, the impact of some specific factors (e.g. transaction fees) on variousportfolio management strategies was studied within the Geometric BrownianMotion (GBM) model [12]. The authors looked at strategies based mainly on acyclic investor’s activity called portfolio rebalancing. We broadened their analysisin our previous paper [13] as well as applied it to some real data from the Polishstock market. In this paper, we are going to extend the strategies presentedtherein by additional factors that may be crucial while making investmentdecisions: portfolio diversification and inclusion of cash in an investment portfolio.Moreover, we are going to study the results of performing buy and sell transactionsbased on signals triggered by well-known technical analysis’ indicators — MACDand RSI. To this end, we introduce our own portfolio management strategiesthat incorporate and utilise those signals. In contrast to both of previous worksmentioned [12, 13], here we will use the Heston model [14] of a market to generatesynthetic asset data as it offers more flexibility than GBM and is more realisticas well.The paper is structured as follows: in section 2 we describe and justify generalassumptions for all our numerical experiments, including the Heston model. Insection 3 we outline all the factors of portfolio management we decided to study,we also describe there how we approached modelling them. Results of MonteCarlo experiments are presented and discussed in section 4. Finally, we drawsome conclusions in section 5.
2. Modelling framework
In most part of this paper, we will be operating on terms and notions whichhave been introduced in our previous article [13]. However, we will repeat someof the most crucial concepts here, along with their definitions, to make this papereasier to understand for a reader.The most basic term which we will be using throughout this entire work isa portfolio — a collection of investment assets held by a given party e.g. anindividual or a company. In more technical terms, to have a full information of aninvestor’s portfolio consisting of n assets, at any given moment of time t , one needsto posses information about prices of each of those n assets — S , S , . . . , S n ,as well as their amounts — q , q , . . . , q n . Having this information, the mostbasic measure which can be used to evaluate portfolio performance is its value,changing over time. We call it portfolio wealth ( t ) = n X i =1 S i ( t ) q i ( t ) . (1)In order to make it easier to compare various portfolios with each other,different measures are often introduced, which aim to make the wealth of portfoliomore relative — both in terms of the growing value of assets and the passingtime. After Alper et al. [12] we use a measure called a growth of portfolio , g ( t ) = log W ( t ) W (0) t . (2)The most important assumption that we make within all our experiments,which will be described later, is that the prices of assets do not depend ondecisions a single investor makes. That means the character of trajectories S i for any i ∈ { , . . . , n } is fully random and modelled by a dedicated stochasticprocess. Therefore, the only way an investor can influence performance of theirportfolio is by altering quantities q i of the assets held. Another, very naturalassumption is that at any point of time t the investor only knows prices of eachof the n assets up to this moment and not further, i.e. for every i ∈ { , . . . , n } values of S i ( u ) are known only for u < t . Finally, we assume that an investor isnot able to alter the price of an asset itself by performing any market transaction.Admittedly, this is not always true in case of real markets, especially if weconsider transactions performed by big market players like banks, mutual fundsor hedge funds. However, for a single, individual investor, this does not need tobe treated as a limitation as most of market participants operate within a rangeof financial means way too small to be able to influence a typical stock market.In our previous paper [13], multiple portfolio management strategies werestudied. Among them, a passive portfolio was the simplest one. Here, we recallits definition, as it is often used as a benchmark. In a portfolio of such kindwe only decide which stocks to buy at the beginning as well as how much ofthem we want, but later on we do not change those quantities. Hence, for every i ∈ { , , . . . , n } and for any t ∈ [0 , T ] we have q i ( t ) = q i (0) = const. (3) A big part of a modern research related to the methods of portfolio manage-ment focuses on testing obtained results based on real market data. Although itmay seem justified and legitimate, studying financial markets only that way hassome drawbacks as well. First of all, research which focuses on one particularmarket, stock, or even basket of stocks cannot be treated as fully universal. Thisis because any conclusions of such research are only fully applicable to this oneparticular class of assets which have been used to prove authors’ claims. In orderto avoid that, we will use synthetically generated data to make experimentsand hence — we will draw more general conclusions which can be applied to3ny market and any stocks or sets of stocks. Using the Monte Carlo frameworkenabled us to generate arbitrary number of trajectories with help of stochasticprocesses that we considered to model our assets. Only then we performed all ournumerical experiments and average the results on. Thanks to that we eradicatedthe bias related to picking some particular assets from existing markets. Weconsider our results to be more generic since they represent an average scenarioof what may happen at any possible market in the world.
Like we mentioned in section 2.1, we assume that a single investor can onlywatch the prices of assets randomly changing on the market, not being able toinfluence them. As a model describing the behaviour of the assets, we chose the
Heston model [14]. This model is often used as a description of the movement ofan underlying assets while pricing derivative instruments, particularly options.Heston model as well as some of its variants, are heavily studied in physics —and are often referred to as diffusive diffusivity models [15]. Within this model,the price of a financial asset is considered to be a stochastic process which solvesthe following stochastic differential equation : dS ( t ) = µS ( t ) dt + p v ( t ) S ( t ) dB S ( t ) , (4) dv ( t ) = κ ( θ − v ( t )) dt + σ p v ( t ) S ( t ) dB v ( t ) . (5)As one can see, the process S ( t ) is built upon the value of µ which is constantand the value of v ( t ) which is a stochastic process on its own (hence — thediffusive diffusivity term, often used by econophisicists). µ is called a drift andit represents a general tendency of an asset to grow (if µ >
0) or fall (if µ < v ( t ) represents the volatility of the asset and is actuallymodelled by a process known as CIR, originally introduced by Cox, Ingersolland Ross to model the movement of interest rates [16]. One of the definingfeatures of the CIR model is the so called mean-reversion . The value of theprocess generally oscillates around a long-term average θ , randomly convergingto and diverging from it with the rate of κ . A random factor of severity of thoseoscillations is reflected in the value of σ , which hence can be called ”a volatilityof a volatility”. Thus, the Heston model, featuring volatility of an asset changingin time, appears to reflect the behaviour of the real-life markets well. In reality,we indeed observe periods of time when prices of assets do not move significantly(practitioners often refer to such behaviour as staying in a consolidation ) butanother times prices of those very same assets fluctuate strongly, achieving e.g.daily return rates, which can be orders of magnitude greater than during thepeaceful times [17]. Geometric Brownian Motion, which is very often used for in the equations related to the Heston model, indices designating the i -th asset have beendropped for the clarity of the record B S ( t ) and B v ( t ) respectively. One should allow for the possibilitythat those two are correlated with instantaneous correlation ρdB S ( t ) dB v ( t ) = ρdt. (6)This can also be explained from the practical point of view since there seemto be an actual correlation between prices of assets and their volatility. It isusually observed to be negative — i.e. an increased volatility of a market usuallyoccurs when prices drop, especially as a consequence of some kind of a marketevent, often related to some critical political or economical news. On the otherhand, when the prices casually grow, lower market volatility can be observed[19].To complete the set-up of the Heston model — initial conditions for both S ( t ) and v ( t ) are required S (0) = S > , (7) v (0) = v > . (8)Here, s represents the initial price of an asset at time t = 0, and v is the valueof the market volatility at that point of time.
3. Elements of portfolio construction and management strategy
Most investors argue that one of the most important factors of creating asuccessful investment portfolio is diversification [20], which is a rule behind areal-life advice not to put all one’s eggs into one basket . This essentially meansnot to use all available money to only buy one kind of a financial asset. Investorswho do not diverse their portfolios are usually beginners [21], either unconsciousof the risk they undertake or hoping for a one-off ”golden-shot”, which wouldallow them to quickly earn a lot of money — a behaviour scheme arguablymore similar to gambling than to responsible investing. It is worth to notethat diversification is much more than just selecting more than one asset to beincluded in our portfolio. Stocks that we choose should belong to companiesoperating in different industries, be of a various size and should differ in theirbusiness model. This allows the entire portfolio to be more resilient to somebigger movements which may take place in one branch of economy, as it oftenhappens that while some companies lose money because of some market events,others take advantage of it.In terms of simulations themselves, there are a few ways to create someportfolios as more diversified than others. The first method simply uses the5imulation parameters. As mentioned in the previous paragraph, stocks in awell-diversified portfolio should differ from one another. In simulations, we canachieve it by varying parameters of the model generating trajectories of stockprices. For example, in the Heston model, described in section 2.3, one can usedifferent values of µ, κ, θ or σ parameters to differentiate behaviour of each ofthe stocks. Another important concept, when it comes to the mathematicalmodelling of diversification, is correlation. Since companies operating in thesame branches of economy usually have similar client base, not differ significantlyin an internal structure and generally face almost the same external risk factors— one can assume they will react in a similar way for all market events relevant tothe given industry and hence — their stock price plots will look quite similarly.Correlation allows us to introduce this similarity to the trajectories of stochasticprocesses representing those prices. Since in our case we have n assets, we canspeak about an entire correlation matrix % = [ ρ i,j ] n × n where ρ i,j = corr( S i ( t ) , S j ( t )) for any t > . (9)In a well diversified portfolio, an investor should aim to maximise the numberof assets which are not correlated to each other or they are correlated negatively,i.e. ρ i,j (cid:54) i, j ∈ { , , . . . , n } . (10) Another important question investors often ask themselves is whether theyshould leave some money prepared for investment in form of cash and — ifso — how much should they leave. Not investing all possessed money intostocks at the very first moment has some immediate advantages. It reducesthe risk, as the more money remains inside a portfolio, the more stable it is,since cash, unlike stocks, does not change its value in time at all . Moreover,leaving some cash aside allows an investor to react when an opportunity on amarket appears, without the need to make any changes in the existing portfoliostock arrangement. On the other hand however, holding cash in a portfoliodiminishes the amount of money earned form an actual investment, which seemsto be especially dissatisfying in case of a very successful arrangement of the riskyassets.From the modelling perspective, including a possibility of having cash inportfolio is trivial. It is only needed to introduce a new kind of an asset, say S and state that S ( t ) = 1 for all t ∈ [0 , T ] . (11) this statement is not entirely true due to existence of inflation and a possibility of placingcash into a risk-free interest bearing deposit, although we consider neither of those in this work
6n such a set-up, q ( t ) can represent the amount of cash in a portfolio andthe definition of wealth (in formula (1)) needs to be adjusted to also take intoaccount the ”zeroth” asset W ( t ) = n X i =0 S i ( t ) q i ( t ) = q ( t ) + n X i =1 S i ( t ) q i ( t ) . (12) The last factor discussed in this paper is trading indicators. Throughout theyears, investors have been attempting to find some mathematical tools whichwould allow them to ”predict the future” or, at least, to help them find thecorrect moment to buy or sell a stock. This resulted in creation of a wide set ofindicators and markers for this purpose. A branch of trading activities whichstudies usefulness of those markers is called technical analysis [22]. One of themost widespread indicators, well known among all investors using technicalanalysis (often referred to as traders, due to usually very short time horizons oftheir investments) is MACD — Moving Average Convergence Divergence [23].MACD was invented by Gerard Appel in 1979. It is based on three time serieswhich are derived from the asset price process by means of a transformation verycommonly used in technical analysis — EMA — exponential moving average.EMA, as the name itself suggests, is a kind of a moving average, but withexponentially decreasing weights of factors more distant in time from the currentone. It can easily be described by a recursive formulaEMA
X,p ( t ) = ( X ( t ) , for t = 0 αX ( t ) + (1 − α ) EMA X,p ( t − ∆ t ) , for t > α = p +1 . EMA is time dependent and has two parameters. One is thebase process X . In case of MACD — this process is simply the stock priceof an i -th asset S i . The second parameter p is called the lag and it is largelyresponsible for the weight of the past values taken to the average. The bigger p , the bigger is the weight of older values of the underlying process in the finalresult, which has an effect in a smoother EMA curve.EMA as a discrete operator, requires time discretisation. In order to useeq. (13), one needs to fix a small time interval of a length ∆ t and EMA willonly be available at the time points t = k ∆ t, k ∈ { , , . . . , K − } such that( K − t (cid:54) T and K ∆ t > T .MACD indicator makes its predictions based on the difference between twotime series. The first is the MACD line, i.e. a line obtained by subtracting twoEMAs of different lags.MACD i,p,q ( t ) = EMA S i ,p ( t ) − EMA S i ,q ( t ) (14)for p < q . EMA related to the lag parameter p is called the fast line whereasthe one related to the parameter q is called the slow line. Buy and sell signalsare generated by places where the MACD line crosses what is called, the signal7ine — another EMA, with a new lag parameter s < p . In other words, we canintroduce the final indicator line as F i,p,q,s ( t ) = EMA S i ,s ( t ) − MACD i,p,q ( t ) . (15)Whenever this line changes its value from negative to positive — MACDgives a ”buy” signal. Contrarily, when this line drops form positive values to thenegative ones — we obtain a ”sell” signal. Hence we, can introduce the buyingand selling indicators: + i,p,q,s ( t ) = ( , if F i,p,q,s ( t − ∆ t ) < ∧ F i,p,q,s ( t ) > , otherwise. (16) − i,p,q,s ( t ) = ( , if F i,p,q,s ( t − ∆ t ) > ∧ F i,p,q,s ( t ) < , otherwise. (17)Since the MACD indicator only tells when to buy or sell a particular asset,but not how much , we created our own strategy for that purpose. Let us thereforeintroduce factors ψ and φ , such that ψ, φ ∈ [0 , ψ is called the sell factor. Whenever MACD generates a sell signal for agiven stock, a ψ part of the amount of this asset is sold and the money fromselling is converted into portfolio cash, q i ( t ) = q i ( t − ∆ t )(1 − ψ − i,p,q,s ( t )) , (18) q ( t ) = q ( t − ∆ t ) + n X i =1 S i ( t ) q i ( t − ∆ t ) ψ − i,p,q,s ( t ) . (19)Similarly, φ is called the buy factor. This time however, it is used to decide,what part of available cash (including new portion obtained from selling somestocks) will be used to buy new stocks c ( t ) = φq ( t ) . (20)The fraction of portfolio cash c ( t ) is then used to buy assets indicated by abuy signal q i ( t ) = q i ( t ) + c ( t ) S i ( t ) n X i =1 + i,p,q,s ( t ) ! − + i,p,q,s ( t ) , (21) q ( t ) = q ( t ) − c ( t ) . (22)It can easily be shown that this strategy is self-financing and the amount ofcash in portfolio will never drop below zero (see Appendix A for a proof).8 .4. RSI trading indicator An other, commonly used trading indicator is called RSI — the relativeStrength Index. It was discovered by J. Welles Wilder Jr. in 1978 [24]. Valuesof the index can only be in the interval [0 , t and thendifferences between subsequent prices need to be calculated D i ( t ) = ( , for t = 0 ,S i ( t ) − S i ( t − ∆ t ) , for t > . (23)Next, positive and negative differences are sorted out from each other D + i ( t ) = ( D ( t ) , where D ( t ) > , where D ( t ) (cid:54) . (24) D − i ( t ) = ( , where D ( t ) (cid:62) D ( t ) , where D ( t ) < . (25)Those two time series are transformed through EMA and additionally av-eraged by a classical, arithmetic mean. This gives two crucial coefficients forcalculating the value of RSI — a i and b i a i ( t ) = 1 { k : k ∈ N , k ∆ t (cid:54) t } X { k : k ∈ N ,k ∆ t (cid:54) t } EMA D + i ,s ( k ∆ t ) , (26) b i ( t ) = 1 { k : k ∈ N , k ∆ t (cid:54) t } X { k : k ∈ N ,k ∆ t (cid:54) t } EMA D − i ,s ( k ∆ t ) . (27)The ratio of a i to b i is called relative strength RS i ( t ) = a i b i . (28)Having the relative strength, to calculate RSI one should only rescale RS i ,so that all values are between 0 and 100 using a simple formulaRSI i ( t ) = 100 − RS i ( t ) . (29)In order to construct the actual strategy out of the values of RSI, we canconstruct dedicated indicators, similar to the ones that have been proposedfor the MACD in eqs. (16) and (17). This time however arbitrary levels ofoverbuying and overselling need to be fixed additionally. Let us denote themby d + and d − respectively. As mentioned above, practitioners usually stick to d + = 30 and d − = 70. Having that fixed, we can define the following indicatorsfor RSI strategy 9 + i,d + ( t ) = ( , if RSI i ( t ) < d + , , otherwise, (30) − i,d − ( t ) = ( , if RSI i ( t ) > d − , , otherwise. (31)With those indicators, we can construct the exact same strategy as in caseof MACD (reacting for signals, as described by equations (18) – (22)) simply byreplacing MACD-related buy and sell indicators by the newly defined RSI-basedones.
4. Results
In this section, we present the results of our Monte Carlo experimentsillustrating the impact of the factors introduced in the previous sections onthe portfolio performance. In each experiment, a number of portfolios weregenerated independently by sweeping the possible values of one particular factorand keeping the others fixed. Results were averaged over a certain number ofindependent runs, varying depending on an experiment . In all figures presentedbelow, the growth of portfolio (2) is used as a measure of portfolio performance.In Fig. 1 the importance of diversification is shown. We compared 3 typesof passively run portfolios. The well diversified portfolio consisted of 3 uncor-related assets. Their trajectories were simulated with slightly different valuesof parameters κ and θ . That made the variance process of each asset have adifferent stochastic character. The poorly diversified portfolio also consisted ofthree assets, although this time they were all positively correlated. Moreover,this time simulation parameters for all of the assets where exactly the same.The third portfolio was not diversified and it only had one asset in it, for whichprices were generated with identical parameters as in case of a poorly diversifiedone. As one can see, a not diversified portfolio performs definitely the worst formost of the time, whereas the well diversified and poorly diversified portfoliosseem to initially compete, but the well diversified one eventually turns out to bethe best in case of a 2-year time horizon.Figures 2–4 illustrate the impact of storing cash in an investment portfolio.The intuition here would be that a portfolio with left-away money will performworse compared to an analogous portfolio with all money resources invested instocks. Rather interestingly however, it turns out it might not always be thecase. We found out that stocks’ drift is the decisive parameter here. In Fig.2 two passive portfolios are compared — one with some share of cash, an onewithout it. We used a relatively big value of the µ parameter, meaning stock each plot, along with other simulation parameters, has an MCt indicated in its caption —this is exactly the number of M onte C arlo independent t rials which were run to obtain givenresult igure 1: (Colour online) Portfolio growth in time for various types of portfolios with differentlevel of diversification. Simulation parameters: T = 2 , ∆ t = 2 − , s = 100 , µ = 0 . , v =0 . , σ = 0 . , ρ = − . , MCt = 1000. prices had a strong tendency to grow over time. In this case our expectationseems to be true — the non-cash portfolio performs much worse. If we howeverchange µ to a much smaller value and hence — purge the stocks of their growthpotential — we observe quite the opposite result, pictured in Fig. 3. In suchcase results are better for a conservative portfolio in which cash was not entirelyspent. Those two results are special cases of the third analysis focusing on theaspect of cash in the investment portfolio more generally, presented in Fig. 4.The plot shows the difference between the final (measured at t = T ) growth of aclassical passive portfolio, with all money invested and an analogous portfolio,having some initial share of cash — varying for each of the few lines on theplot. Positive values of the curves mean that a portfolio with cash is better,negative ones — quite the opposite — an all-in portfolio gives a better outcome.Entire analysis has been performed against the µ parameter. We can clearly seeit confirms that for small values of the drift a cash-featuring portfolios lead tobetter results. What we can also read from the plot is that the share of cashmatters a lot. A portfolio consisting mostly of cash will lead to much worseresults in case if stocks have a strong tendency to grow, but it will also give bestresults in case assets have lower growth potential.It turns out that the drift is a critical parameter not only when it comes tothe cash inclusion, but it also has a huge impact on the effectiveness of strategiesbased on using technical analysis indicators like MACD or RSI. In Fig. 5, acomparison between a passive and MACD-driven portfolio has been presented,for a relatively small value of µ . As one can see, it is not obvious to assesswhich of those two perform better in this case. If we however increase µ — itbecomes clear that portfolios managed by a strategy using MACD as an indicatoroutperforms a simple buy-and-hold strategy.One could expect that it would be the case for strategies utilising various11 igure 2: (Colour online) Portfolio growth in time for a passive portfolio and a portfolio witha cash contribution (approx. 28%) for µ = 0 .
1. Other simulation parameters: T = 2 , ∆ t =2 − , n = 5 , s = 100 , v = 0 . , κ = 1 . , θ = 0 . , σ = 0 . , ρ = − . , MCt = 1000. indicators, that the bigger the drift, the better the results. It turns out thatit is very much dependent on the actual indicator being used. To demonstratethis, one can compare Fig.7 and 8 which both present the comparison between apassive and a RSI-driven portfolio — the former for a small value of µ and thelatter — for a bigger one. We can see that the tendency is exactly the oppositein this case. When the stocks behave more indecisively, the RSI-driven portfoliocompetes with the passive one on a level playing field and although the passiveone wins eventually — a shorter (e.g. a − to a -year) portfolio would turnout to perform better if driven by RSI. However, when assets grow fast — anRSI-based portfolio performs much worse than the passive one.In order to obtain even more general results, similarly to what we have donefor the cash inclusion experiment, also here we have drawn the plot of final (at t = T ) differences between actively managed and passive portfolios in dependenceof the drift parameter µ shown in Fig. 9. As we can see, for the range of µ thathas been used, MACD performs best for big values of the drift, whereas RSIturns out to be useful around the value of µ = 0 and even then it does not reallyoutperform a passive portfolio.
5. Conclusions
In the article we focused on a few important aspects of a portfolio that canbe considered a well-managed one and we studied their actual importance, basedon the synthetic data, simulated according to the Heston model, in a MonteCarlo-type experiments. First of those aspects is portfolio diversification. Wemanaged to numerically show that a simple passive portfolio which consistsof independent assets, all having different inner characteristics (modelled by12 igure 3: (Colour online) Portfolio growth in time for a passive portfolio and a portfolio with acash contribution (approx. 28%) for µ = 0 . T = 2 , ∆ t = 2 − , n =5 , s = 100 , v = 0 . , κ = 1 . , θ = 0 . , σ = 0 . , ρ = − . , MCt = 1000. different simulation parameters) lead to better results compared to portfolioswith correlated assets of similar character. An utterly not-diversified portfolio,meaning the one with only one asset, performed definitely worst than the twodiversified ones, which makes the importance of diversification even more evident.The other factor that we considered in our research was the share of cash ina portfolio. Rather surprisingly, we found out that keeping cash in a portfoliomay be beneficial and that the main factor which needs to be considered hereis the assets’ drift. Our results reveal that if the stocks have a big tendencyto grow — leaving cash in one’s portfolio deteriorates portfolio’s performance.However, if this is not the case and the stocks do not represent a very significantdrift — it is exactly the opposite. The more money one keeps uninvested, thebetter results the portfolio attains. Finally, for our last experiment, we wantedto examine the effectiveness of popular trading indicators, commonly used byinvestors utilising technical analysis. We chose two of them — MACD and RSI.For the purpose of testing we designed special portfolio management strategies,in which transactions were performed when an indicator sent a buy or sell signal.The comparison between those strategies and a simple passive one was very muchdependent on the drift parameter again, similarly as in case of cash inclusion.Moreover, results obtained for MACD were very different from those for RSIwhich can serve as a confirmation that different trading indicators should beused for different market conditions.There are really wide perspectives of continuing the above research. Includingprices jumps, which can be seen in the real markets, is certainly one of them.The other one can be comparing various portfolio performance measures, as thegrowth of portfolio, which we used is not universally acclaimed to be the bestone. Finally, it would be very interesting to find new management strategies,13 igure 4: (Colour online) Impact of the amount of cash stored in the portfolio for variousvalues of the drift parameter µ . Simulation parameters: T = 2 , ∆ t = 2 − , n = 5 , s = 100 , v =0 . , κ = 1 . , θ = 0 . , σ = 0 . , ρ = − . , MCt = 5000. which would be more successful than the ones based on MACD and RSI andpossibly less dependent on the assets drift. Those and multiple other questionswhich arose during writing this work definitely constitute a great incentive forcontinuing the research in the field. Acknowledgement
This work was supported by the Polish Ministry of Science and HigherEducation (MNiSW) core funding for statutory R&D activities.
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Appendix A.
Below we present the proof of the fact that the strategy described by equations(18) – (22) is self financing, i.e. at any point of time (except t = 0 and we setup our portfolio) only resources available in portfolio are used and no value isadded to or withdrawn from portfolio.We start by fixing the initial amount of cash in the portfolio q (0) > q i (0) > , i ∈ { , , . . . , n } .Let us now look into the movements which happen at an arbitrary point oftime t . On one hand, we know that if we consider having cash in our portfolio,the wealth of it should be expressed by formula (12). This is what we call the actual wealth , W ( t ) = q ( t ) + n X i =1 S i ( t ) q i ( t ) . (A.1)On the other hand however, we know that before we do any buy or selltransaction at time t , the value of our portfolio arises from the amount of assetswe had in the previous step. This is what we call a temporary wealth temp ( t ) = q ( t − ∆ t ) + n X i =1 S i ( t ) q i ( t − ∆ t ) . (A.2)Proving that the strategy is self financing means proving that those twowealths are equal, i.e. we only use financial resources that we have availablebecause of our previous investment decisions, W temp ( t ) = W ( t ) . (A.3)Using equations (21) and (22) we have W ( t ) = q ( t ) + n X i =1 S i ( t ) q i ( t ) == q ( t ) − c ( t ) + n X i =1 S i ( t ) q i ( t ) + c ( t ) S i ( t ) n X j =1 + j,p,q,s ( t ) − + i,p,q,s ( t ) == q ( t ) − c ( t ) + n X i =1 S i ( t ) q i ( t ) + n X i =1 S i ( t ) c ( t ) S i ( t ) n X j =1 + j,p,q,s ( t ) − + i,p,q,s ( t ) == q ( t ) − c ( t ) + n X i =1 S i ( t ) q i ( t ) + n X i =1 c ( t ) + i,p,q,s ( t ) n X j =1 + j,p,q,s ( t ) − == q ( t ) − c ( t ) + n X i =1 S i ( t ) q i ( t ) + c ( t ) n X j =1 + j,p,q,s ( t ) − n X i =1 + i,p,q,s ( t ) == q ( t ) − c ( t ) + n X i =1 S i ( t ) q i ( t ) + c ( t ) = q ( t ) + n X i =1 S i ( t ) q i ( t )Now, using equations (18) and (19) we obtain20 ( t ) = q ( t ) + n X i =1 S i ( t ) q i ( t ) == q ( t − ∆ t ) + n X i =1 S i ( t ) q i ( t − ∆ t ) ψ − i,p,q,s ( t )++ n X i =1 S i ( t ) q i ( t − ∆ t )(1 − ψ − i,p,q,s ( t )) == q ( t − ∆ t ) + n X i =1 S i ( t ) q i ( t − ∆ t ) ψ − i,p,q,s ( t )++ n X i =1 S i ( t ) q i ( t − ∆ t ) − n X i =1 S i ( t ) q i ( t − ∆ t ) ψ − i,p,q,s ( t ) == q ( t − ∆ t ) + n X i =1 S i ( t ) q i ( t − ∆ t ) = W temp ( t )Thus, since W ( t ) = W temp ( t ), the strategy is indeed self-financing.It can also be proven that the amount of cash after each step will never benegative. More precisely, we will prove that if an amount of cash brought fromthe previous time step is non-negative, it will remain such in the next step. That,together with an assumption that the initial amount of cash at time t = 0 isnon-negative proves it will always be that way.The ultimate amount of cash at time t is given by equation (22) q ( t ) = q ( t ) − c ( t ) . Plugging equation (20) into the above one we get q ( t ) = q ( t ) − φq ( t ) = q ( t )(1 − φ ) . Using equation (19) what we obtain is q ( t ) = q ( t − ∆ t ) + n X i =1 S i ( t ) q i ( t − ∆ t ) ψ − i,p,q,s ( t ) ! (1 − φ ) == q ( t − ∆ t ) + ψ n X i =1 S i ( t ) q i ( t − ∆ t ) − i,p,q,s ( t ) ! (1 − φ ) . q ( t − ∆ t ) (cid:62) ψ ∈ [0 , S i ( t ) > − i,p,q,s ( t ) (cid:62) q i ( t − ∆ t ) (cid:62) q ( t − ∆ t ) + ψ P ni =1 S i ( t ) q i ( t − ∆ t ) − i,p,q,s ( t ) must be non-negative.Also, since φ ∈ [0 , − φ ) (cid:62)
0. Hence, multiplication of thosetwo expressions must also be non-negative and thus q ( t ) ≥ . for any t ∈ [0 , T, T