Brown Ian
Content
A DISSERTATION REPORT ON
MODELING SCENARIO PLANNING FOR STOCK PRICE FOR INVESTMENT DECISIONS BY HNI CUSTOMERS A WIENER AND BROWNIAN MOTION APPROACH.
(submitted as partly fulfillment of PGDM programme)
Under aegies of
Submitted by:
B.NAVEEN SWAROOP
ROLL NO:GBS/02/10-12/01
Under The Esteemed Guidence Of Prof.K.C.MISHRA GURUNANAK BUSINESS SCHOOL,HYDERABAD
GURUNANAK BUSINESS SCHOOL (Approved by AICTE ministry of HRD,NEW DELHI) ADITYA COURT ROAD NO:3 BANJARAHILLS,HYDERABAD - 522001
1
DECLARATION
I hereby declare that the project work entitled “Modelling scenario planning for stock prices for investment decisions by HNI Customers -A wiener And Brownian motion approach.” submitted to the GURUNANAK BUSINESS SCHOOL, Hyderabad, is a record of an original work done by me under the guidance of Mr.K.C.MISHRA, PROFESSOR,GURUNANAK
BUSINESS SCHOOL,HYDERABAD and this project work is submitted in the partial fulfilment of the requirements for the award of the degree of PGDM. The results embodied in this thesis have not been submitted to any other University or Institute for the award of any degree or diploma.
Signature of student
2
CERTIFICATE
This is to certify that the project report entitled “Modelling scenario planning for stock
prices for investment decisions by HNI Customers -A wiener And Brownian motion approach.” is a bonafide record of work done by Mr. B.NAVEEN SWAROOP, and
submitted in partial fulfillment of the requirements of PGDM program of GURUNANAK BUSINESS SCHOOL, Hyderabad.
Prof.K.C.Mishra Gurunanak business school Hyderabad
3
Abstract
This project helps all the stock market investors mainly for thee HNI’s to know where to buy and where to exit the stock with all different probabilities which we simulate in excel using a Brownian motion/Wiener process the best pricing technique. As we all know there were various methods to price a stock , but in all the cases we need to calculate volatility, again for which there are various methods. This study will help in finding the best pricing for the stock.
4
INDEX
Topic Introduction Research study Literature review Modeling Stochastic HNI clients Understanding Stock Price Model Valuation Findings and Interpretations
References
Page No 6 10 11 15 18 24 31 38 39
5
Introduction and History of Brownian motion
Brownian motion refers to either the physical phenomenon that minute particles immersed in a fluid move around randomly or the mathematical models used to describe those random movements which will be explored in this project.
History: Brownian motion was discovered by the biologist Robert Brown [2] in 1827. While Brown was studying pollen particles floating in water in the microscope, he observed minute particles in the pollen grains executing the jittery motion. After repeating the experiment with particles of dust, he was able to conclude that the motion was due to pollen being “alive” but the origin of the motion remained unexplained. The first one to give a theory of Brownian motion was Louis Bachelier in 1900 in his PhD thesis “The theory of speculation”. However, it was only in 1905 that Albert Einstein, using a probabilistic model, could sufficiently explain Brownian motion. He observed that if the kinetic energy of fluids was right, the molecules of water moved at random. Thus, a small particle would receive a random number of impacts of random strength and from random directions in any short period of time. This random bombardment by the molecules of the fluid would cause a sufficiently small particle to move exactly just how Brown described it While the primary domain of Brownian Motion is science, it has other real world applications and in this project the stock market is mentioned as early as the second paragraph. If the stock market is truly random, as described by Brownian Motion, then it follows that the market is unpredictable. If the market is unpredictable, is it efficient, at least for most asset classes. One definition of an efficient market is that it is impossible to 'beat the market' since all relevant information is already reflected in the market price. In other words, the random price movement is impacted by so many variables the average investor is unable to sniff out the inefficiencies. When the subject of inefficient markets and active managers emerges, it is difficult to pass up Rex Sinquefield's quip, "So who still believes markets don't work? Apparently it is only the North Koreans, the Cubans and the active managers." If one were to classify investors into two groups, efficient or inefficient, approximately 20% of all investors fall into the efficient market camp while 80% argue the stock market is inefficient.
6
Why such a wide gap of opinion? Even the most ardent supporters of either camp recognize there are times when there is slippage within these rigid definitions, but investing practitioners, by their behavior, primarily fall into the inefficient camp. Are they Brownian Motion nay-sayers? It is useful to scan a historical review of the Efficient Market Hypothesis (EMH) in condensed form as shown at this project. EMH has a longer history than many might predict. Even its formalized history from the early 1950s is moving into advanced age. Will the efficient vs. inefficient market debate never end? It appears as if there will always be mathematical algorithms where super computers can skim small amounts per trade, but do it thousands of times each day. There are ways to slow this non-productive activity, but that can wait for a later explanation. For the vast number of small unskilled investors, pitching your tent in the efficient market or passive management camp is preferred. For non-professional, and many professionals, who advocate the inefficient market hypothesis, it is difficult to knock the claim that there are a few "300 hitters" of investing. We certainly hear more about the Warren Buffett's of investing than we do the Charlie Steadman's, yet there are far more Steadman's planted in the inefficient market graveyard. Within the actively managed mutual fund industry, we know the "beat the market" average is dismal. What we now need is a 20 to 30 year study of actively managed accounts by hundreds of small investors. Or should we be satisfied with William Sharpe's article, "The Arithmetic of Active Management," and be done with the argument as to which investing style is best suited for the average investor. If we use Harold Evensky's narrow definition of active management, few investors fall into the efficient market camp where passive management is king. Here is Evensky's definition from Wealth Management. "Active Management is the art and science of security selection based on a belief in a manager's ability to consistently and accurately evaluate current and/or future events better than other investors. The core philosophical basis is that by brains, hard work, and/or technology the active manager can, over time and net of costs, beat the system. As selecting one asset class in lieu of another is an 'active' decision, market timing and active asset allocation are subsets of active management." While it is difficult to be a pure efficient market investor, it is well worth the time exploring the advantages of tilting toward a Brownian Motion investing style. One such example is presented
7
in the Mebane Faber and Eric Richardson book, "The Ivy Portfolio: How to Invest Like the Top Endowments and Avoid Bear Markets." A sample portfolio based on averages when the book was written includes five asset classes. The are: Domestic Stocks , Foreign Stocks , Bonds , Real Estate , and Commodities . Readers will note the projected risk or uncertainty is 16.6% or a little higher than idea, based on the projected return. In addition, the Diversification Metric is one 23%, largely due to the portfolio holding only five ETFs. Faber and Richardson provide several examples of portfolios that are constructed for the passively oriented investor. I track a number of portfolios built around ETFs and the performance results are available at this project. Instead of focusing on five asset classes, as is the case in the portfolio presented above, most portfolios are built around 12 to 17 asset classes. The reason for including more asset classes is so one can tilt the portfolio toward the value end of the investing spectrum. This is a form of active management or another way of saying the market is not pure "Brownian Motion."
8
9
OBJECTIVES: The objective of the study is to address the following problem, that wrong decision in investing in stock markets to avoid this main objective to provide a powerful tool to take decision by HNI clients. As they invest in huge amounts to avoid losses or to avoid risk all possible parameters are taken and powerful tool is modeled to have a good investment decision.
SCOPE OF THE STUDY The scope of the study is to price the stock by using different parameters it will give a scope to know about all parameters involved in pricing stock, how those parameters vary in pricing a stock. This gives about all knowledge about the stock pricing by simulating the price with different probabilities.
10
LITERATURE REVIEW
Testing Geometric Brownian motion: An Analytical Look at the Black Scholes Option Pricing Model
This project tests the geometric Brownian motion and constant elasticity of variance diffusions for S&P 500 Index prices from April 1998 to February 1999 and from March 1994 to January 1995. The results show that over time periods of strong transitions of volatility, the CEV model predicts price returns more accurately than the GBM. During times of relatively constant volatility, both the CEV and GBM models predict returns equally well. Although the CEV model did not drastically outperform the GBM model, the results imply that it is necessary to incorporate a stochastic element to volatility when predicting returns. This could perhaps also be done using other models 30 such as GARCH models or perhaps even Jump Diffusion models which would allow for the volatility to jump. Further research is necessary to explore these different avenues to search for an ideal method for modeling price returns and pricing options.
DOUBLE LOOKBACKS
It presented a technique for pricing lookback options on two assets following correlated geometric Brownian motions. The essential part of this technique is to derive the probability density/distribution functions of the extreme values of two correlated Brownian motions. With this technique, the prices of many kinds of lookback and barrier options can be calculated efficiently. We hope that our pricing technology will be useful for any research in the future that involves the extreme values of two correlated geometric Brownian motions.
11
Simulation of fractional Brownian motion
In this report, several methods to simulate fractional Brownian motion were described and discussed. The three (theoretically) most promising `families' of methods were evaluated: _ The RMDl;r method proposed by Norros et al. _ The Paxson method proposed by Paxson [49] and a generalization of the Paxson method, the approximate circulant method; _ A wavelet-based simulation method by Abry and Sellan [2]. A fair comparison between different methods is a difficult task. Almost any method can be favored above the others by changing the comparison criterion. In this report, the comparison was made by an error analysis and by checking theoretically or statistically a number of properties that fully characterize fractional Brownian motion: besides that the sample should be centered and Gaussian, its increments should be stationary. The property that the variance at time t should be t2H was checked by estimating the Hurst parameter with various estimators. We arrive at the following conclusions: _ So-called spectral simulation provides the theoretical foundation for the Paxson method, which clarifiees many empirical observations and leads to an important improvement of the method. Moreover, a natural extension of the Paxson method, the approximate circulant method, has a direct connection with the exact Davies and Harte method. The obtained insights show that the accuracy of both methods increases as the sample size grows, and that the methods are even asymptotically exact in the sense that the finite-dimensional distributions of the approximate sample converge in probability to the corresponding distributions of the exact sample. _ These insights make it also possible to perform an error analysis for the approximate circulant method, in which the theoretical results are confirmed. The same errors can be analyzed for the RMD1;2 method, which shows better performance. _ The covariance structure of approximate Paxson samples, approximate circulate samples and RMDl;r samples can numerically be calculated. Since all approximate methods under study produce zero mean Gaussian samples, these covariances fully characterize the approximate
12
method. While the errors in the covariance’s are quite big for the Paxson method (although they vanish asymptotically), The auto covariance function of the approximate circulant method is visually almost indistinguishable from the exact auto covariance function. Still, this auto covariance function does not show the desired hyperbolical tail behavior. Since the RMDl;r produces in general non-stationary samples, it takes more report to study the tail behavior of the covariance’s. We showed how the covariance’s of an RMDl;r sample can numerically be computed for general l _ 0 and r _ 1. An analysis of these covariance’s of the RMD1;2 and the RMD3;3 method showed that these methods also do not have the desired decay in the covariance’s This can be improved by removing some begin and end points in the approximate RMDl;r sample. 71 The overall impression from the accuracy analysis is that only the RMD1;2, the RMD3;3, and the approximate circulant method are reasonable alternatives for the exact Davies and Harte method. The wavelet method studied in this report is disappointing, in terms of both speed and accuracy. In fact, it produces (approximate) samples of a process that does not have stationary increments, but is also called fractional Brownian motion in the old literature. The RMDl;r method is the only studied method of which the time complexity grows linearly in the trace length. Since the extra time needed by the RMD3;3 compared to the RMD1;2 method is almost negligible, the RMD3;3 method is the most interesting. The Paxson method is faster than the exact Davies and Harte method, while the approximate circulant method is only faster for relatively large sample sizes. As mentioned before, their accuracy also improves as the trace length grows. Since they are based on the FFT, these methods have time complexity N log(N). _ The wavelet method is very slow, although the time complexity is approximately N log(N). From these observations, a _nal choice is readily made: the RMD3;3 method. Moreover, it is possible to generate RMD3;3 samples on-the-y, which is an advantage that is not shared by the Davies and Harte method (nor by the two spectral methods). In both methods it is possible to `recycle' some computations when more samples of the same size are needed, but the RMD3;3 method stays an order N method, and the Davies and Harte an order N log(N) method.
13
However, in a simulation study, for instance, it may be far more desirable to know that the samples are exact. Therefore, the Davies and Harte method should be used in principle. When long traces or many runs are needed in a simulation study, it may computationally be impossible to generate exact samples. The RMD3;3 method is then a good alternative. Before celebrating the RMDl;r method too much, we have also seen in a study of the relative error in a network model that the approximate circulant method performs in this model better than the RMD1;2 Method for reasonable sample sizes. This indicates that our conclusions are not as robust as they may seem. If the approximate method is not _rst evaluated in the framework in which it is used, it is quite risky to just use an approximate method. This is particularly the case when rare events are simulated, e.g., loss probabilities in queues. The use of approximate methods should thus be avoided, especially since a quite fast exact algorithm exists.
14
On Stochastic and Worst-case Models for Investing:
In practice, most investing is done assuming a probabilistic model of stock price returns known as the Geometric Brownian Motion (GBM). While often an acceptable approximation, the GBM model is not always valid empirically. This motivates a worst-case approach to investing, called universal portfolio management, where the objective is to maximize wealth relative to the wealth earned by the best fixed portfolio in hindsight. In this paper we tie the two approaches, and design an investment strategy which is universal in the worst-case, and yet capable of exploiting the mostly valid GBM model. Our method is based on new and improved regret bounds for online convex optimization with exp-concave loss functions.
“Average-case” Investing:
Much of mathematical finance theory is devoted to the modeling of Stock prices and devising investment strategies that maximize wealth gain, minimize risk while doing so, and so on. Typically, this is done by estimating the parameters in a probabilistic model of stock prices. Investment strategies are thus geared to such average case models (in the formal computer science sense), and are naturally susceptible to drastic deviations from the model, as witnessed in the recent stock market crash.
Even so, empirically the Geometric Brownian Motion (GBM) ([Osb59, Bac00]) has enjoyed great predictive success and every year trillions of dollars are traded assuming this model. Black and Scholes [BS73] used this same model in their Nobel prize winning work on pricing options on stocks. “Worst-case” Investing:
The fragility of average-case models in the face of rare but dramatic deviations led Cover [Cov91] to take a worst-case approach to investing in stocks. The performance of an online investment algorithm for arbitrary sequences of stock price returns is measured with respect to
15
the best CRP (constant rebalanced portfolio, see [Cov91]) in hindsight. A universal portfolio selection algorithm is one that obtains sublinear (in the number of trading periods T ) regret,which is the difference in the logarithms of the final wealths obtained by the two. Cover [Cov91] gave the first universal portfolio selection algorithm with regret bounded by O(log T ). There has been much follow-up work after Cover’s seminal work, such as which focused on either obtaining alternate universal algorithms or improving the efficiency of Cover’s algorithm. However, the best regret bound is still O(log T ).
This dependence of the regret on the number of trading periods is not entirely satisfactory for two main reasons. First, a priori it is not clear why the online algorithm should have high regret (growing with the number of iterations) in an unchanging environment. As an extreme example, consider a setting with two stocks where one has an “upward drift” of 1% daily, whereas the second stock remains at the same price. One would expect to “figure out” this pattern quickly and focus on the first stock, thus attaining a constant fraction of the wealth of the best CRP in the long run, i.e. constant regret, unlike the worst-case bound of O(log T ).
The second problem arises from trading frequency. Suppose we need to invest over a fixed period of time, say a year. Trading more frequently potentially leads to higher wealth gain, by capitalizing on short term stock movements. However, increasing trading frequency increases T , and thus one may expect more regret. The problem is actually even worse: since we measure regret as a difference of logarithms of the final wealths, a regret bound of O(log T ) implies a poly(T ) factor ratio between the final wealths. In reality, however, experiments show that some known online algorithms actually improve with increasing trading frequency.
Bridging Worst-case and Average-case Investing: Both these issues are resolved if one can show that the regret of a “good” online algorithm depends on total variation in the sequence of stock returns, rather than purely on the number of iterations. If the stock return sequence has low variation, we expect our algorithm to
16
be able to perform better. If we trade more frequently, then the periteration variation should go down correspondingly, so the total variation stays the same.
We analyze a portfolio selection algorithm and prove that its regret is bounded by O(log Q), where Q (formally defined in Section 1.2) is the sum of squared deviations of the returns from their mean.Since Q ≤ T (after appropriate normalization), we improve over previous regret bounds and retain the worst-case robustness. Furthermore, in an average-case model such as GBM, the variation can be tied very nicely to the volatility parameter, which explains the experimental observation the regret doesn’t increase with increasing trading frequency. Our algorithm is efficient, and its implementation requires constant time per iteration (independent of the number of game iterations).
New Techniques and Comparison to Related Work
Cesa-Bianchi, Mansour and Stoltz [CBMS07] initiated work on relating worst case regret to the variation in the data for the related learning problem of prediction from expert advice, and conjectured that the optimal regret bounds should depend on the observed variation of the cost sequence.Recently, this conjectured was proved and regret bounds of O( Q) were obtained in the full information and bandit linear optimization settings where Q is the variation in the cost sequence. In this paper we give an exponential improvement in regret, viz. O(log Q), for the case of online exp-concave optimization, which includes portfolio selection as a special case.
Another approach to connecting worst-case to average-case investing was taken by Jamshidian [Jam92] and Cross and Barron. They considered a model of “continuous trading”, where there are T “trading intervals”, and in each the online investor chooses a fixed portfolio which is rebalanced k times with k → ∞. They prove familiar regret bounds of O(log T ) (independent of k) in this model w.r.t. the best fixed portfolio which is rebalanced T × k times. In this model our algorithm attains the tighter regret bounds of O(log Q), although our algorithm has more flexibility. Furthermore their algorithms, being extensions of Cover’s algorithm, may require exponential timein general
17
1 .Our bounds of O(log Q) regret require completely different techniques compared to the O( Q) regret bounds of. These previous bounds are based on first-order gradient descent methods which are too weak to obtain O(log Q) regret. Instead we have to use the second-order Newton step ideas based on (in particular, the Hessian of the cost functions).
The second-order techniques of are, however, not sensitive enough to obtain O(log Q) bounds. This is because progress was measured in terms of the distance between successive portfolios in the usual Euclidean norm, which is insensitive to variation in the cost sequence. In this paper, we introduce a different analysis technique, based on analyzing the distance between successive predictions using norms that keep changing from iteration to iteration and are actually sensitive to the variation.
A key technical step in the analysis is a lemma (Lemma 6) which bounds the sum of differences of successive Cesaro means of a sequence of vectors by the logarithm of its variation. This lemma, which may be useful in other contexts when variation bounds on the regret are desired, is proved using the Kahn-Karush-Tucker conditions, and also improves the regret bounds in previous projects
18
HNI clients:
A high-net-worth individual (HNWI) is a person with a high net worth. In the private banking business, these individuals typically are defined as having investable finance (financial assets not including primary residence) in excess of US$1 million.
19
First, HNIs suffer in being targeted by the wealth management professionals. HNIs offer size, which makes it worthwhile for advisory businesses to hire highly paid resources to source business from them. Anyone with net worth exceeding $1 million (Rs 5 cr) is searched and acquired as a client. There are two options for the client. First, they can stay loyal to the brand that acquires them, with the hope that the proposition of the bank or broking house is good. This option means that the client ends up speaking to a new private banker or wealth manager ever too often. Second, they can stay loyal to the wealth manager and move with him or her. This means that the client has traded off the core wealth management proposition of the bank or broking business for the relationship with the individual adviser. Both the options are harmful for the HNI's wealth and the latter is downright unethical. However, it is not uncommon for a wealth manager to be hired on the basis of the number of HNI clients who will move with him. Nor is a bank or broking house able to retain talent and provide its HNI client the benefit of dealing with the same person for a longer time. The strategic investment that an institution makes to create a solid wealth management proposition for its HNI clients is shortchanged by the job-hopping wealth managers. Independent advisers offer stability, but not all of them have the capability to reach and service this demanding segment of clients. Second, HNIs are easy prey when it comes to meeting the revenue targets of wealth managers and advisers. The product teams of banks and broking houses create a 'white list' of products to be offered to their clients. This exercise, at its best, is expected to hold those products that are the most suitable for its clients to choose from. In its worst form, this list may have the products that earn the maximum revenue for the advisory business. Individual wealth managers have to achieve certain revenue targets. The strategy of pushing the chosen products to the clients is very similar to a sales exercise. Each product
20
category has its weightage for the wealth manager and the overall revenue has to be achieved within a defined time frame. This process makes most wealth managers mere sellers of products, eager to achieve their numbers. The same holds true for independent advisers, who need the highest revenue at the lowest cost. HNIs are easy victims. We may theoretically believe that there is a large range of products being offered to clients after completing the necessary procedures for profiling. On the ground, though, the hunt is for large cheques for the highest revenue products, from as few customer calls as possible. While businesses like to differentiate themselves as wealth managers offering good quality advice to clients, the focus on selling and pushing products shortchanges this aspiration. Until the business grows to a size, where asset gathering is not the core focus, sharp practices are likely to continue, hurting the core customers they all chase. HNIs comprise this core.
Recent problems for HNI clients
The recent crash in the stock market has not shaken the confidence of high net worth individuals (HNIs), according to bankers managing the funds for these preferred clients. Though their clients may have seen a slight diminishing of their wealth, it is unlikely that they suffered losses, bank officials said. Typically, this segment has a higher exposure to equities as against smaller depositors, due to the larger amounts of surplus cash available with them. While some banks advised their HNI clients to use the opportunity to book profits, others advised clients to use the slump as an opportunity to buy value stocks. According to Mr C. Jayaram, Executive Director, Head-Wealth Management, Kotak Mahindra Bank, the bank's advice to clients was to invest across various asset classes and maintain discipline.
21
Unfortunately, investors in India are faced with limited asset classes such as equity, fixed income, real estate, and commodity to some extent. Therefore, by default, equities form a significant part of the HNIs' investment portfolio, Mr Jayaram said. "In the last 1-2 years, the proportion of our HNI clients who have increased their exposure to equity has gone up. The stock market has gone up so much that people did not feel the need to get out of equities during the crash." However, he said, if the stock markets continue to be in decline mode, then clients might want to change their asset classes. Kotak Mahindra has about 3,600 HNI clients with a cut-off over Rs 5 crore investible assets. HSBC, which manages about 1,000 clients with assets under management of about Rs 2,500 crore, advised its clients to book profits on a systematic basis. "In the falling market, we advised clients to take profit on a systematic basis and not to exit completely. Because, then, they may miss out on the upside," said Mr Subir Mittra, Head, Private Banking, India, HSBC. In an extreme volatile market, it is also important to pick the right stock and buy good stocks on dips, he added. Most banks offer extensive research reports and technical advice about individual companies to help their clients make the right choice regarding stocks. ICICI Bank advised its clients to reinvest when the Sensex crashed from 12,000 to 9,000, said Mr Anup Bagchi, General Manager. "Clients who followed this strategy have overall made higher returns."
22
Another reason that many HNIs chose to stay invested during the fall is many of the HNI clients are SME entrepreneurs, who are more comfortable with risk. "HNIs prefer high risk and high reward. Therefore, most of them have not moved away from equities as an asset class as the fundamentals are good
23
Understanding Stock Price Model
Black-Scholes Model Assumptions
There are several assumptions underlying the Black-Scholes model of calculating options pricing. The most significant is that volatility, a measure of how much a stock can be expected to move in the near-term, is a constant over time. The Black-Scholes model also assumes stocks move in a manner referred to as a random walk; at any given moment, they are as likely to move up as they are to move down. These assumptions are combined with the principle that options pricing should provide no immediate gain to either seller or buyer. The exact 6 assumptions of the Black-Scholes Model are : 1. Stock pays no dividends 2. Option can only be exercised upon expiration 3. Market direction cannot be predicted, hence "Random Walk" 4. No commissions are charged in the transaction 5. Interest rates remain constant 6. Stock returns are normally distributed, thus volatility is constant over time
The Black-Scholes Formula Is:
C0 = S0N(d1) - Xe-rTN(d2) Where: d1 = [ln(S0/X) + (r + σ2/2)T]/ σ √T And: d2 = d1 - σ √T And where:
24
C0 = current option value S0 = current stock price N(d) = the probability that a random draw from a standard normal distribution will be less than (d). X = exercise price e = 2.71828, the base of the natural log function r = risk-free interest rate (annualized continuously compounded rate on a safe asset with the same maturity as the expiration of the option; usually the money market rate for a maturity equal to the option's maturity.) T = time to option's maturity, in years ln = natural logarithm function σ = standard deviation of the annualized continuously compounded rate of return on the stock C0 = Se-dTN(d1) - Xe-rTN(d2) Where: d1 = [ln(S0/X) + (r - d + σ2/2)T]/ σ √T And: d2 = d1 - σ √T
25
MONTE CARLO SIMULATION In general terms, the Monte Carlo method (or Monte Carlo simulation) can be used to describe any technique that approximates solutions to quantitative problems through statistical sampling. As used here, 'Monte Carlo simulation' is more specifically used to describe a method for propagating (translating) uncertainties in model inputs into uncertainties in model outputs (results). Hence, it is a type of simulation that explicitly and quantitatively represents uncertainties. Monte Carlo simulation relies on the process of explicitly representing uncertainties by specifying inputs as probability distributions. If the inputs describing a system are uncertain, the prediction of future performance is necessarily uncertain. That is, the result of any analysis based on inputs represented by probability distributions is itself a probability distribution. Whereas the result of a single simulation of an uncertain system is a qualified statement ("if we build the dam, the salmon population could go extinct"), the result of a probabilistic (Monte Carlo) simulation is a quantified probability ("if we build the dam, there is a 20% chance that the salmon population will go extinct"). Such a result (in this case, quantifying the risk of extinction) is typically much more useful to decision-makers who utilize the simulation results. In order to compute the probability distribution of predicted performance, it is necessary to propagate (translate) the input uncertainties into uncertainties in the results. A variety of methods exist for propagating uncertainty. Monte Carlo simulation is perhaps the most common technique for propagating the uncertainty in the various aspects of a system to the predicted performance.
In Monte Carlo simulation, the entire system is simulated a large number (e.g., 1000) of times. Each simulation is equally likely, referred to as a realization of the system. For each realization, all of the uncertain parameters are sampled (i.e., a single random value is selected from the specified distribution describing each parameter). The system is then simulated through time (given the particular set of input parameters) such that the performance of the system can be computed. This results is a large number of separate and independent results, each representing a possible “future” for the system (i.e., one possible path the system may follow through time).
26
The results of the independent system realizations are assembled into probability distributions of possible outcomes. As a result, the outputs are not single values, but probability distributions.
•
Stochastic Process • • • • Any variable (like stock price) whose value changes with time in uncertain way is said to follow stochastic process Process can be classified as Discrete time and Continuous time process Process can also be classified as Discrete variable and Continuous variable Stock price model is continuous time, continuous variable process
Weiner Process • • When two independent normal distributions are added, the resulting distribution has mean and variance equal to the sum of the mean and variance of respective distributions Consider two normal distributions with mean of 0 and variance of 1; the resulting distribution will have mean of 0 and variance of 2 and standard deviation of √ Weiner Process/Brownian Motion •
Assume these two distributions represent price for year 1 and year 2. Then it can be said that total variance in two years is 2 and standard deviation is √
• •
On the same lines, for the variance in period
is
with standard deviation of √
Weiner/Brownian process is Markov process with mean of 0 and variance of 1.
Formal Definition
A variable z follows Weiner process if – If small change • – And in two different periods are independent of each other in small time is given by
27
Generalized Weiner Process •
Generalized Weiner process for • – The first term is drift -
is defined as
increases by
in a small period is the standard Weiner process given by
– Second term is the noise where
–
√ and Variance is
– Mean of the process is – Stock Price Model • • • • • • • • • • • Writing VBA Code • – and √ √ The discrete version is √
is the stock price with expected drift of stock
where
is the expected rate of return on
In a short period of time, the increase in stock price is given by If stock volatility is assumed to be zero then
But stock price has volatility and in short period of time it is proportional to stock price Hence modifying the equation yields
and initial stock price
is given
28
–
is defined
– Generate random number – Compute and add this to and add it to previous – For the next period, generate random number, compute price
Simulate Geometric Brownian motion with Excel:
Learn about Geometric Brownian motion Stock prices are often modeled as the sum of
the deterministic drift, or growth, rate and a random number with a mean of 0 and a variance that is proportional to dt
This is known as Geometric Brownian Motion, and is commonly model to define stock price paths. It is defined by the following stochastic differential equation.
St is the stock price at time t, dt is the time step, μ is the drift, σ is the volatility, Wt is a Weiner process, and ε is a normal distribution with a mean of zero and standard deviation of one .
Substituting Equation 2 into Equation 1 gives
Hence dSt is the sum of a general trend, and a term that represents uncertainty.
29
Converting Equation 3 into finite difference form gives
Bear in mind that ε is a normal distribution with a mean of zero and standard deviation of one. This can be represented in Excel by NORM.INV(RAND(),0,1).
The spreadsheet linked to at the bottom of this post implements Geometric Brownian Motion in Excel using Equation 4.
30
Valuation:
Stock Price Simulation Inputs Stock Price Expected Return Per year (mu) Annulized Volatality (sigma) Expiry Period (year) Time Steps to follow (N) dt (It is one day) 50 5% 10% 1.0 250 0.00
Price Evolution Day1 Day2 Day3 Day4 Day5 Day6 Day7 Day8 Day9 Day10 Day11 Day12 Day13 Day14 Day15 Day16 Day17 Day18 Day19 Day20
Formula 50 49.82 49.02 48.76 48.80 48.60 48.96 48.85 48.74 48.71 48.79 48.72 48.18 48.16 48.32 48.24 48.05 47.84 47.86 47.81
VBA Code 50.00 50.18 50.22 50.29 50.12 49.97 50.22 49.53 49.76 50.05 50.24 49.71 49.65 50.01 50.27 50.18 50.75 51.13 50.62 51.16 31
Day21 Day22 Day23 Day24 Day25 Day26 Day27 Day28 Day29 Day30 Day31 Day32 Day33 Day34 Day35 Day36 Day37 Day38 Day39 Day40 Day41 Day42 Day43 Day44 Day45 Day46 Day47 Day48 Day49 Day50 Day51 Day52 Day53 Day54 Day55 Day56 Day57 Day58 Day59 Day60 Day61
47.68 47.88 47.88 47.49 47.10 46.62 46.71 46.77 46.49 46.22 45.65 45.54 45.75 45.14 45.15 45.62 46.01 46.06 46.11 46.36 46.34 46.28 46.22 46.06 46.19 46.04 45.77 45.56 45.64 45.83 45.60 45.12 44.87 44.49 44.60 44.56 44.79 44.47 44.75 44.58 44.73
51.06 51.09 51.33 50.82 50.90 50.89 50.73 50.84 50.97 50.78 50.60 50.92 51.23 51.31 52.03 52.49 52.25 52.43 53.12 52.90 52.94 52.53 53.62 53.78 53.06 53.14 52.72 52.31 52.59 52.42 51.87 51.70 51.61 51.45 51.99 52.68 52.61 52.42 52.10 51.79 51.92 32
Day62 Day63 Day64 Day65 Day66 Day67 Day68 Day69 Day70 Day71 Day72 Day73 Day74 Day75 Day76 Day77 Day78 Day79 Day80 Day81 Day82 Day83 Day84 Day85 Day86 Day87 Day88 Day89 Day90 Day91 Day92 Day93 Day94 Day95 Day96 Day97 Day98 Day99 Day100 Day101 Day102
44.43 44.72 44.49 43.78 44.04 43.49 43.69 43.65 43.68 43.55 43.11 43.48 43.34 42.63 42.86 43.10 43.29 42.94 43.43 43.23 42.86 42.71 43.09 43.21 43.17 43.55 43.55 43.66 43.86 43.95 44.46 44.78 45.36 45.38 44.97 45.49 46.03 46.08 46.10 45.55 45.62
51.86 51.80 51.99 51.85 51.97 51.72 51.44 51.51 51.07 51.04 51.48 51.28 51.55 51.46 51.29 51.75 51.87 51.99 51.94 51.53 51.59 51.76 52.22 52.55 51.90 51.94 52.41 52.36 52.52 52.53 52.56 52.54 52.42 52.35 52.16 51.64 51.43 52.10 51.60 51.52 51.42 33
Day103 Day104 Day105 Day106 Day107 Day108 Day109 Day110 Day111 Day112 Day113 Day114 Day115 Day116 Day117 Day118 Day119 Day120 Day121 Day122 Day123 Day124 Day125 Day126 Day127 Day128 Day129 Day130 Day131 Day132 Day133 Day134 Day135 Day136 Day137 Day138 Day139 Day140 Day141 Day142 Day143
45.59 45.44 45.69 45.70 45.79 45.19 45.30 45.53 45.66 45.75 46.11 45.79 46.02 45.77 45.73 45.79 45.79 46.03 45.90 45.69 46.01 46.37 46.16 46.08 46.19 46.24 46.21 45.57 46.32 46.03 46.43 46.47 46.67 46.28 46.36 46.07 45.98 46.15 46.55 46.53 46.55
51.42 51.10 51.09 50.89 51.01 51.05 50.74 51.24 51.38 51.39 51.31 50.92 51.19 51.16 51.39 51.48 51.81 51.14 50.89 50.43 50.04 49.92 49.57 48.48 48.52 48.65 48.70 49.00 48.58 48.32 48.47 48.45 48.34 48.04 48.21 48.67 48.70 48.29 48.52 48.45 48.43 34
Day144 Day145 Day146 Day147 Day148 Day149 Day150 Day151 Day152 Day153 Day154 Day155 Day156 Day157 Day158 Day159 Day160 Day161 Day162 Day163 Day164 Day165 Day166 Day167 Day168 Day169 Day170 Day171 Day172 Day173 Day174 Day175 Day176 Day177 Day178 Day179 Day180 Day181 Day182 Day183 Day184
46.44 46.70 46.82 46.95 46.81 46.38 46.30 46.18 46.07 46.00 46.16 45.71 45.37 45.59 45.99 45.41 45.54 45.59 44.88 44.82 44.90 44.68 44.69 44.58 44.61 44.35 44.79 44.75 44.78 44.66 44.73 44.64 44.90 44.84 44.80 44.87 44.74 44.88 45.44 45.60 45.50
48.43 48.20 48.07 47.68 47.76 47.48 47.93 47.55 47.51 47.34 47.69 47.91 47.73 47.88 47.69 47.29 46.75 46.62 46.87 46.72 46.51 46.51 46.32 46.21 45.73 45.72 45.49 45.82 45.89 46.10 46.54 46.42 46.46 46.06 46.17 46.11 46.63 46.29 46.71 46.81 46.71 35
Day185 Day186 Day187 Day188 Day189 Day190 Day191 Day192 Day193 Day194 Day195 Day196 Day197 Day198 Day199 Day200 Day201 Day202 Day203 Day204 Day205 Day206 Day207 Day208 Day209 Day210 Day211 Day212 Day213 Day214 Day215 Day216 Day217 Day218 Day219 Day220 Day221 Day222 Day223 Day224 Day225
45.32 45.29 45.32 45.35 45.15 45.26 45.09 45.39 45.67 45.82 45.65 45.64 45.58 45.65 45.68 45.49 45.62 45.72 45.41 45.03 45.02 45.00 45.07 44.93 45.02 45.21 45.37 45.19 45.13 45.40 45.42 44.73 44.57 44.73 43.92 44.25 44.57 44.93 45.19 45.08 45.39
46.41 46.40 46.19 46.92 46.60 46.05 45.94 45.99 46.41 46.45 46.39 46.70 46.98 47.13 47.31 48.14 48.02 48.03 47.97 48.13 47.86 47.81 47.86 48.14 48.18 48.13 48.15 47.93 48.03 48.03 48.19 48.56 48.47 48.32 48.17 47.86 47.89 47.67 47.75 47.65 48.01 36
Day226 Day227 Day228 Day229 Day230 Day231 Day232 Day233 Day234 Day235 Day236 Day237 Day238 Day239 Day240 Day241 Day242 Day243 Day244 Day245 Day246 Day247 Day248 Day249 Day250
45.52 45.27 45.33 44.87 45.25 44.95 44.85 44.90 45.39 45.47 45.95 45.79 45.55 45.48 45.21 45.08 45.14 44.74 44.65 44.49 44.91 45.43 44.94 44.66 44.39
48.00 47.74 47.90 48.11 48.21 48.46 48.16 48.44 48.19 48.72 48.27 47.81 48.06 47.98 47.96 47.62 47.26 46.99 46.51 46.68 46.72 46.77 46.55 46.54 46.74
37
VBA CODE:
Sub stock_price() 'Read the Inputs Sheets("Stock Price Model").Select S = Range("d4") mu = Range("d5") sigma = Range("d6") dt = 1 / 250 N = 250
Range("f12") = S 'First Day Price is input For xx = 2 To N ' Range("f" & xx + 11) = Range("f" & xx + 10) + (mu * Range("f" & xx + 10) * dt + sigma
* S * dt ^ 1 / 2) eps = Application.WorksheetFunction.NormSInv(Rnd()) Range("f" & xx + 11) = Range("f" & xx + 10) + (mu * Range("f" & xx + 10) * dt + sigma * Range("f" & xx + 10) * eps * dt ^ (1 / 2)) Next
End Sub
38
GRAPH:
70 60 50 40 Price 30 20
10
0 1 51 101 Days--> 151 201 251
39
Findings and Interpretation:
The stock pricing is done by simulating the stock price for so many times and noting down the values or we can observe the graph which we get from the values calculated by using formula manually or we assigned program to automatically simulate the stock. By observing these the decision will be made.
Here our main motive is to take right decision by the HNI investors that is to provide a good tool to take decision by them in investing stock markets.
As there will be lot of parameters in taking decision for investing in stock markets like volatility, time and some other variables we assume that as the drift .Here we will not consider the fundamental parameter in calculating the price of the stock. So as we are considering all small and minute parameters to get effective efficient output, so it is a suitable tool for the HNI’s to use the tool in investing into stock markets.
40
APPENDICES:
www.nseindia.com www.yahoofinance.com www.investopedia.com www.wikipedia.com
41
MODELING SCENARIO PLANNING FOR STOCK PRICE FOR INVESTMENT DECISIONS BY HNI CUSTOMERS A WIENER AND BROWNIAN MOTION APPROACH.
(submitted as partly fulfillment of PGDM programme)
Under aegies of
Submitted by:
B.NAVEEN SWAROOP
ROLL NO:GBS/02/10-12/01
Under The Esteemed Guidence Of Prof.K.C.MISHRA GURUNANAK BUSINESS SCHOOL,HYDERABAD
GURUNANAK BUSINESS SCHOOL (Approved by AICTE ministry of HRD,NEW DELHI) ADITYA COURT ROAD NO:3 BANJARAHILLS,HYDERABAD - 522001
1
DECLARATION
I hereby declare that the project work entitled “Modelling scenario planning for stock prices for investment decisions by HNI Customers -A wiener And Brownian motion approach.” submitted to the GURUNANAK BUSINESS SCHOOL, Hyderabad, is a record of an original work done by me under the guidance of Mr.K.C.MISHRA, PROFESSOR,GURUNANAK
BUSINESS SCHOOL,HYDERABAD and this project work is submitted in the partial fulfilment of the requirements for the award of the degree of PGDM. The results embodied in this thesis have not been submitted to any other University or Institute for the award of any degree or diploma.
Signature of student
2
CERTIFICATE
This is to certify that the project report entitled “Modelling scenario planning for stock
prices for investment decisions by HNI Customers -A wiener And Brownian motion approach.” is a bonafide record of work done by Mr. B.NAVEEN SWAROOP, and
submitted in partial fulfillment of the requirements of PGDM program of GURUNANAK BUSINESS SCHOOL, Hyderabad.
Prof.K.C.Mishra Gurunanak business school Hyderabad
3
Abstract
This project helps all the stock market investors mainly for thee HNI’s to know where to buy and where to exit the stock with all different probabilities which we simulate in excel using a Brownian motion/Wiener process the best pricing technique. As we all know there were various methods to price a stock , but in all the cases we need to calculate volatility, again for which there are various methods. This study will help in finding the best pricing for the stock.
4
INDEX
Topic Introduction Research study Literature review Modeling Stochastic HNI clients Understanding Stock Price Model Valuation Findings and Interpretations
References
Page No 6 10 11 15 18 24 31 38 39
5
Introduction and History of Brownian motion
Brownian motion refers to either the physical phenomenon that minute particles immersed in a fluid move around randomly or the mathematical models used to describe those random movements which will be explored in this project.
History: Brownian motion was discovered by the biologist Robert Brown [2] in 1827. While Brown was studying pollen particles floating in water in the microscope, he observed minute particles in the pollen grains executing the jittery motion. After repeating the experiment with particles of dust, he was able to conclude that the motion was due to pollen being “alive” but the origin of the motion remained unexplained. The first one to give a theory of Brownian motion was Louis Bachelier in 1900 in his PhD thesis “The theory of speculation”. However, it was only in 1905 that Albert Einstein, using a probabilistic model, could sufficiently explain Brownian motion. He observed that if the kinetic energy of fluids was right, the molecules of water moved at random. Thus, a small particle would receive a random number of impacts of random strength and from random directions in any short period of time. This random bombardment by the molecules of the fluid would cause a sufficiently small particle to move exactly just how Brown described it While the primary domain of Brownian Motion is science, it has other real world applications and in this project the stock market is mentioned as early as the second paragraph. If the stock market is truly random, as described by Brownian Motion, then it follows that the market is unpredictable. If the market is unpredictable, is it efficient, at least for most asset classes. One definition of an efficient market is that it is impossible to 'beat the market' since all relevant information is already reflected in the market price. In other words, the random price movement is impacted by so many variables the average investor is unable to sniff out the inefficiencies. When the subject of inefficient markets and active managers emerges, it is difficult to pass up Rex Sinquefield's quip, "So who still believes markets don't work? Apparently it is only the North Koreans, the Cubans and the active managers." If one were to classify investors into two groups, efficient or inefficient, approximately 20% of all investors fall into the efficient market camp while 80% argue the stock market is inefficient.
6
Why such a wide gap of opinion? Even the most ardent supporters of either camp recognize there are times when there is slippage within these rigid definitions, but investing practitioners, by their behavior, primarily fall into the inefficient camp. Are they Brownian Motion nay-sayers? It is useful to scan a historical review of the Efficient Market Hypothesis (EMH) in condensed form as shown at this project. EMH has a longer history than many might predict. Even its formalized history from the early 1950s is moving into advanced age. Will the efficient vs. inefficient market debate never end? It appears as if there will always be mathematical algorithms where super computers can skim small amounts per trade, but do it thousands of times each day. There are ways to slow this non-productive activity, but that can wait for a later explanation. For the vast number of small unskilled investors, pitching your tent in the efficient market or passive management camp is preferred. For non-professional, and many professionals, who advocate the inefficient market hypothesis, it is difficult to knock the claim that there are a few "300 hitters" of investing. We certainly hear more about the Warren Buffett's of investing than we do the Charlie Steadman's, yet there are far more Steadman's planted in the inefficient market graveyard. Within the actively managed mutual fund industry, we know the "beat the market" average is dismal. What we now need is a 20 to 30 year study of actively managed accounts by hundreds of small investors. Or should we be satisfied with William Sharpe's article, "The Arithmetic of Active Management," and be done with the argument as to which investing style is best suited for the average investor. If we use Harold Evensky's narrow definition of active management, few investors fall into the efficient market camp where passive management is king. Here is Evensky's definition from Wealth Management. "Active Management is the art and science of security selection based on a belief in a manager's ability to consistently and accurately evaluate current and/or future events better than other investors. The core philosophical basis is that by brains, hard work, and/or technology the active manager can, over time and net of costs, beat the system. As selecting one asset class in lieu of another is an 'active' decision, market timing and active asset allocation are subsets of active management." While it is difficult to be a pure efficient market investor, it is well worth the time exploring the advantages of tilting toward a Brownian Motion investing style. One such example is presented
7
in the Mebane Faber and Eric Richardson book, "The Ivy Portfolio: How to Invest Like the Top Endowments and Avoid Bear Markets." A sample portfolio based on averages when the book was written includes five asset classes. The are: Domestic Stocks , Foreign Stocks , Bonds , Real Estate , and Commodities . Readers will note the projected risk or uncertainty is 16.6% or a little higher than idea, based on the projected return. In addition, the Diversification Metric is one 23%, largely due to the portfolio holding only five ETFs. Faber and Richardson provide several examples of portfolios that are constructed for the passively oriented investor. I track a number of portfolios built around ETFs and the performance results are available at this project. Instead of focusing on five asset classes, as is the case in the portfolio presented above, most portfolios are built around 12 to 17 asset classes. The reason for including more asset classes is so one can tilt the portfolio toward the value end of the investing spectrum. This is a form of active management or another way of saying the market is not pure "Brownian Motion."
8
9
OBJECTIVES: The objective of the study is to address the following problem, that wrong decision in investing in stock markets to avoid this main objective to provide a powerful tool to take decision by HNI clients. As they invest in huge amounts to avoid losses or to avoid risk all possible parameters are taken and powerful tool is modeled to have a good investment decision.
SCOPE OF THE STUDY The scope of the study is to price the stock by using different parameters it will give a scope to know about all parameters involved in pricing stock, how those parameters vary in pricing a stock. This gives about all knowledge about the stock pricing by simulating the price with different probabilities.
10
LITERATURE REVIEW
Testing Geometric Brownian motion: An Analytical Look at the Black Scholes Option Pricing Model
This project tests the geometric Brownian motion and constant elasticity of variance diffusions for S&P 500 Index prices from April 1998 to February 1999 and from March 1994 to January 1995. The results show that over time periods of strong transitions of volatility, the CEV model predicts price returns more accurately than the GBM. During times of relatively constant volatility, both the CEV and GBM models predict returns equally well. Although the CEV model did not drastically outperform the GBM model, the results imply that it is necessary to incorporate a stochastic element to volatility when predicting returns. This could perhaps also be done using other models 30 such as GARCH models or perhaps even Jump Diffusion models which would allow for the volatility to jump. Further research is necessary to explore these different avenues to search for an ideal method for modeling price returns and pricing options.
DOUBLE LOOKBACKS
It presented a technique for pricing lookback options on two assets following correlated geometric Brownian motions. The essential part of this technique is to derive the probability density/distribution functions of the extreme values of two correlated Brownian motions. With this technique, the prices of many kinds of lookback and barrier options can be calculated efficiently. We hope that our pricing technology will be useful for any research in the future that involves the extreme values of two correlated geometric Brownian motions.
11
Simulation of fractional Brownian motion
In this report, several methods to simulate fractional Brownian motion were described and discussed. The three (theoretically) most promising `families' of methods were evaluated: _ The RMDl;r method proposed by Norros et al. _ The Paxson method proposed by Paxson [49] and a generalization of the Paxson method, the approximate circulant method; _ A wavelet-based simulation method by Abry and Sellan [2]. A fair comparison between different methods is a difficult task. Almost any method can be favored above the others by changing the comparison criterion. In this report, the comparison was made by an error analysis and by checking theoretically or statistically a number of properties that fully characterize fractional Brownian motion: besides that the sample should be centered and Gaussian, its increments should be stationary. The property that the variance at time t should be t2H was checked by estimating the Hurst parameter with various estimators. We arrive at the following conclusions: _ So-called spectral simulation provides the theoretical foundation for the Paxson method, which clarifiees many empirical observations and leads to an important improvement of the method. Moreover, a natural extension of the Paxson method, the approximate circulant method, has a direct connection with the exact Davies and Harte method. The obtained insights show that the accuracy of both methods increases as the sample size grows, and that the methods are even asymptotically exact in the sense that the finite-dimensional distributions of the approximate sample converge in probability to the corresponding distributions of the exact sample. _ These insights make it also possible to perform an error analysis for the approximate circulant method, in which the theoretical results are confirmed. The same errors can be analyzed for the RMD1;2 method, which shows better performance. _ The covariance structure of approximate Paxson samples, approximate circulate samples and RMDl;r samples can numerically be calculated. Since all approximate methods under study produce zero mean Gaussian samples, these covariances fully characterize the approximate
12
method. While the errors in the covariance’s are quite big for the Paxson method (although they vanish asymptotically), The auto covariance function of the approximate circulant method is visually almost indistinguishable from the exact auto covariance function. Still, this auto covariance function does not show the desired hyperbolical tail behavior. Since the RMDl;r produces in general non-stationary samples, it takes more report to study the tail behavior of the covariance’s. We showed how the covariance’s of an RMDl;r sample can numerically be computed for general l _ 0 and r _ 1. An analysis of these covariance’s of the RMD1;2 and the RMD3;3 method showed that these methods also do not have the desired decay in the covariance’s This can be improved by removing some begin and end points in the approximate RMDl;r sample. 71 The overall impression from the accuracy analysis is that only the RMD1;2, the RMD3;3, and the approximate circulant method are reasonable alternatives for the exact Davies and Harte method. The wavelet method studied in this report is disappointing, in terms of both speed and accuracy. In fact, it produces (approximate) samples of a process that does not have stationary increments, but is also called fractional Brownian motion in the old literature. The RMDl;r method is the only studied method of which the time complexity grows linearly in the trace length. Since the extra time needed by the RMD3;3 compared to the RMD1;2 method is almost negligible, the RMD3;3 method is the most interesting. The Paxson method is faster than the exact Davies and Harte method, while the approximate circulant method is only faster for relatively large sample sizes. As mentioned before, their accuracy also improves as the trace length grows. Since they are based on the FFT, these methods have time complexity N log(N). _ The wavelet method is very slow, although the time complexity is approximately N log(N). From these observations, a _nal choice is readily made: the RMD3;3 method. Moreover, it is possible to generate RMD3;3 samples on-the-y, which is an advantage that is not shared by the Davies and Harte method (nor by the two spectral methods). In both methods it is possible to `recycle' some computations when more samples of the same size are needed, but the RMD3;3 method stays an order N method, and the Davies and Harte an order N log(N) method.
13
However, in a simulation study, for instance, it may be far more desirable to know that the samples are exact. Therefore, the Davies and Harte method should be used in principle. When long traces or many runs are needed in a simulation study, it may computationally be impossible to generate exact samples. The RMD3;3 method is then a good alternative. Before celebrating the RMDl;r method too much, we have also seen in a study of the relative error in a network model that the approximate circulant method performs in this model better than the RMD1;2 Method for reasonable sample sizes. This indicates that our conclusions are not as robust as they may seem. If the approximate method is not _rst evaluated in the framework in which it is used, it is quite risky to just use an approximate method. This is particularly the case when rare events are simulated, e.g., loss probabilities in queues. The use of approximate methods should thus be avoided, especially since a quite fast exact algorithm exists.
14
On Stochastic and Worst-case Models for Investing:
In practice, most investing is done assuming a probabilistic model of stock price returns known as the Geometric Brownian Motion (GBM). While often an acceptable approximation, the GBM model is not always valid empirically. This motivates a worst-case approach to investing, called universal portfolio management, where the objective is to maximize wealth relative to the wealth earned by the best fixed portfolio in hindsight. In this paper we tie the two approaches, and design an investment strategy which is universal in the worst-case, and yet capable of exploiting the mostly valid GBM model. Our method is based on new and improved regret bounds for online convex optimization with exp-concave loss functions.
“Average-case” Investing:
Much of mathematical finance theory is devoted to the modeling of Stock prices and devising investment strategies that maximize wealth gain, minimize risk while doing so, and so on. Typically, this is done by estimating the parameters in a probabilistic model of stock prices. Investment strategies are thus geared to such average case models (in the formal computer science sense), and are naturally susceptible to drastic deviations from the model, as witnessed in the recent stock market crash.
Even so, empirically the Geometric Brownian Motion (GBM) ([Osb59, Bac00]) has enjoyed great predictive success and every year trillions of dollars are traded assuming this model. Black and Scholes [BS73] used this same model in their Nobel prize winning work on pricing options on stocks. “Worst-case” Investing:
The fragility of average-case models in the face of rare but dramatic deviations led Cover [Cov91] to take a worst-case approach to investing in stocks. The performance of an online investment algorithm for arbitrary sequences of stock price returns is measured with respect to
15
the best CRP (constant rebalanced portfolio, see [Cov91]) in hindsight. A universal portfolio selection algorithm is one that obtains sublinear (in the number of trading periods T ) regret,which is the difference in the logarithms of the final wealths obtained by the two. Cover [Cov91] gave the first universal portfolio selection algorithm with regret bounded by O(log T ). There has been much follow-up work after Cover’s seminal work, such as which focused on either obtaining alternate universal algorithms or improving the efficiency of Cover’s algorithm. However, the best regret bound is still O(log T ).
This dependence of the regret on the number of trading periods is not entirely satisfactory for two main reasons. First, a priori it is not clear why the online algorithm should have high regret (growing with the number of iterations) in an unchanging environment. As an extreme example, consider a setting with two stocks where one has an “upward drift” of 1% daily, whereas the second stock remains at the same price. One would expect to “figure out” this pattern quickly and focus on the first stock, thus attaining a constant fraction of the wealth of the best CRP in the long run, i.e. constant regret, unlike the worst-case bound of O(log T ).
The second problem arises from trading frequency. Suppose we need to invest over a fixed period of time, say a year. Trading more frequently potentially leads to higher wealth gain, by capitalizing on short term stock movements. However, increasing trading frequency increases T , and thus one may expect more regret. The problem is actually even worse: since we measure regret as a difference of logarithms of the final wealths, a regret bound of O(log T ) implies a poly(T ) factor ratio between the final wealths. In reality, however, experiments show that some known online algorithms actually improve with increasing trading frequency.
Bridging Worst-case and Average-case Investing: Both these issues are resolved if one can show that the regret of a “good” online algorithm depends on total variation in the sequence of stock returns, rather than purely on the number of iterations. If the stock return sequence has low variation, we expect our algorithm to
16
be able to perform better. If we trade more frequently, then the periteration variation should go down correspondingly, so the total variation stays the same.
We analyze a portfolio selection algorithm and prove that its regret is bounded by O(log Q), where Q (formally defined in Section 1.2) is the sum of squared deviations of the returns from their mean.Since Q ≤ T (after appropriate normalization), we improve over previous regret bounds and retain the worst-case robustness. Furthermore, in an average-case model such as GBM, the variation can be tied very nicely to the volatility parameter, which explains the experimental observation the regret doesn’t increase with increasing trading frequency. Our algorithm is efficient, and its implementation requires constant time per iteration (independent of the number of game iterations).
New Techniques and Comparison to Related Work
Cesa-Bianchi, Mansour and Stoltz [CBMS07] initiated work on relating worst case regret to the variation in the data for the related learning problem of prediction from expert advice, and conjectured that the optimal regret bounds should depend on the observed variation of the cost sequence.Recently, this conjectured was proved and regret bounds of O( Q) were obtained in the full information and bandit linear optimization settings where Q is the variation in the cost sequence. In this paper we give an exponential improvement in regret, viz. O(log Q), for the case of online exp-concave optimization, which includes portfolio selection as a special case.
Another approach to connecting worst-case to average-case investing was taken by Jamshidian [Jam92] and Cross and Barron. They considered a model of “continuous trading”, where there are T “trading intervals”, and in each the online investor chooses a fixed portfolio which is rebalanced k times with k → ∞. They prove familiar regret bounds of O(log T ) (independent of k) in this model w.r.t. the best fixed portfolio which is rebalanced T × k times. In this model our algorithm attains the tighter regret bounds of O(log Q), although our algorithm has more flexibility. Furthermore their algorithms, being extensions of Cover’s algorithm, may require exponential timein general
17
1 .Our bounds of O(log Q) regret require completely different techniques compared to the O( Q) regret bounds of. These previous bounds are based on first-order gradient descent methods which are too weak to obtain O(log Q) regret. Instead we have to use the second-order Newton step ideas based on (in particular, the Hessian of the cost functions).
The second-order techniques of are, however, not sensitive enough to obtain O(log Q) bounds. This is because progress was measured in terms of the distance between successive portfolios in the usual Euclidean norm, which is insensitive to variation in the cost sequence. In this paper, we introduce a different analysis technique, based on analyzing the distance between successive predictions using norms that keep changing from iteration to iteration and are actually sensitive to the variation.
A key technical step in the analysis is a lemma (Lemma 6) which bounds the sum of differences of successive Cesaro means of a sequence of vectors by the logarithm of its variation. This lemma, which may be useful in other contexts when variation bounds on the regret are desired, is proved using the Kahn-Karush-Tucker conditions, and also improves the regret bounds in previous projects
18
HNI clients:
A high-net-worth individual (HNWI) is a person with a high net worth. In the private banking business, these individuals typically are defined as having investable finance (financial assets not including primary residence) in excess of US$1 million.
19
First, HNIs suffer in being targeted by the wealth management professionals. HNIs offer size, which makes it worthwhile for advisory businesses to hire highly paid resources to source business from them. Anyone with net worth exceeding $1 million (Rs 5 cr) is searched and acquired as a client. There are two options for the client. First, they can stay loyal to the brand that acquires them, with the hope that the proposition of the bank or broking house is good. This option means that the client ends up speaking to a new private banker or wealth manager ever too often. Second, they can stay loyal to the wealth manager and move with him or her. This means that the client has traded off the core wealth management proposition of the bank or broking business for the relationship with the individual adviser. Both the options are harmful for the HNI's wealth and the latter is downright unethical. However, it is not uncommon for a wealth manager to be hired on the basis of the number of HNI clients who will move with him. Nor is a bank or broking house able to retain talent and provide its HNI client the benefit of dealing with the same person for a longer time. The strategic investment that an institution makes to create a solid wealth management proposition for its HNI clients is shortchanged by the job-hopping wealth managers. Independent advisers offer stability, but not all of them have the capability to reach and service this demanding segment of clients. Second, HNIs are easy prey when it comes to meeting the revenue targets of wealth managers and advisers. The product teams of banks and broking houses create a 'white list' of products to be offered to their clients. This exercise, at its best, is expected to hold those products that are the most suitable for its clients to choose from. In its worst form, this list may have the products that earn the maximum revenue for the advisory business. Individual wealth managers have to achieve certain revenue targets. The strategy of pushing the chosen products to the clients is very similar to a sales exercise. Each product
20
category has its weightage for the wealth manager and the overall revenue has to be achieved within a defined time frame. This process makes most wealth managers mere sellers of products, eager to achieve their numbers. The same holds true for independent advisers, who need the highest revenue at the lowest cost. HNIs are easy victims. We may theoretically believe that there is a large range of products being offered to clients after completing the necessary procedures for profiling. On the ground, though, the hunt is for large cheques for the highest revenue products, from as few customer calls as possible. While businesses like to differentiate themselves as wealth managers offering good quality advice to clients, the focus on selling and pushing products shortchanges this aspiration. Until the business grows to a size, where asset gathering is not the core focus, sharp practices are likely to continue, hurting the core customers they all chase. HNIs comprise this core.
Recent problems for HNI clients
The recent crash in the stock market has not shaken the confidence of high net worth individuals (HNIs), according to bankers managing the funds for these preferred clients. Though their clients may have seen a slight diminishing of their wealth, it is unlikely that they suffered losses, bank officials said. Typically, this segment has a higher exposure to equities as against smaller depositors, due to the larger amounts of surplus cash available with them. While some banks advised their HNI clients to use the opportunity to book profits, others advised clients to use the slump as an opportunity to buy value stocks. According to Mr C. Jayaram, Executive Director, Head-Wealth Management, Kotak Mahindra Bank, the bank's advice to clients was to invest across various asset classes and maintain discipline.
21
Unfortunately, investors in India are faced with limited asset classes such as equity, fixed income, real estate, and commodity to some extent. Therefore, by default, equities form a significant part of the HNIs' investment portfolio, Mr Jayaram said. "In the last 1-2 years, the proportion of our HNI clients who have increased their exposure to equity has gone up. The stock market has gone up so much that people did not feel the need to get out of equities during the crash." However, he said, if the stock markets continue to be in decline mode, then clients might want to change their asset classes. Kotak Mahindra has about 3,600 HNI clients with a cut-off over Rs 5 crore investible assets. HSBC, which manages about 1,000 clients with assets under management of about Rs 2,500 crore, advised its clients to book profits on a systematic basis. "In the falling market, we advised clients to take profit on a systematic basis and not to exit completely. Because, then, they may miss out on the upside," said Mr Subir Mittra, Head, Private Banking, India, HSBC. In an extreme volatile market, it is also important to pick the right stock and buy good stocks on dips, he added. Most banks offer extensive research reports and technical advice about individual companies to help their clients make the right choice regarding stocks. ICICI Bank advised its clients to reinvest when the Sensex crashed from 12,000 to 9,000, said Mr Anup Bagchi, General Manager. "Clients who followed this strategy have overall made higher returns."
22
Another reason that many HNIs chose to stay invested during the fall is many of the HNI clients are SME entrepreneurs, who are more comfortable with risk. "HNIs prefer high risk and high reward. Therefore, most of them have not moved away from equities as an asset class as the fundamentals are good
23
Understanding Stock Price Model
Black-Scholes Model Assumptions
There are several assumptions underlying the Black-Scholes model of calculating options pricing. The most significant is that volatility, a measure of how much a stock can be expected to move in the near-term, is a constant over time. The Black-Scholes model also assumes stocks move in a manner referred to as a random walk; at any given moment, they are as likely to move up as they are to move down. These assumptions are combined with the principle that options pricing should provide no immediate gain to either seller or buyer. The exact 6 assumptions of the Black-Scholes Model are : 1. Stock pays no dividends 2. Option can only be exercised upon expiration 3. Market direction cannot be predicted, hence "Random Walk" 4. No commissions are charged in the transaction 5. Interest rates remain constant 6. Stock returns are normally distributed, thus volatility is constant over time
The Black-Scholes Formula Is:
C0 = S0N(d1) - Xe-rTN(d2) Where: d1 = [ln(S0/X) + (r + σ2/2)T]/ σ √T And: d2 = d1 - σ √T And where:
24
C0 = current option value S0 = current stock price N(d) = the probability that a random draw from a standard normal distribution will be less than (d). X = exercise price e = 2.71828, the base of the natural log function r = risk-free interest rate (annualized continuously compounded rate on a safe asset with the same maturity as the expiration of the option; usually the money market rate for a maturity equal to the option's maturity.) T = time to option's maturity, in years ln = natural logarithm function σ = standard deviation of the annualized continuously compounded rate of return on the stock C0 = Se-dTN(d1) - Xe-rTN(d2) Where: d1 = [ln(S0/X) + (r - d + σ2/2)T]/ σ √T And: d2 = d1 - σ √T
25
MONTE CARLO SIMULATION In general terms, the Monte Carlo method (or Monte Carlo simulation) can be used to describe any technique that approximates solutions to quantitative problems through statistical sampling. As used here, 'Monte Carlo simulation' is more specifically used to describe a method for propagating (translating) uncertainties in model inputs into uncertainties in model outputs (results). Hence, it is a type of simulation that explicitly and quantitatively represents uncertainties. Monte Carlo simulation relies on the process of explicitly representing uncertainties by specifying inputs as probability distributions. If the inputs describing a system are uncertain, the prediction of future performance is necessarily uncertain. That is, the result of any analysis based on inputs represented by probability distributions is itself a probability distribution. Whereas the result of a single simulation of an uncertain system is a qualified statement ("if we build the dam, the salmon population could go extinct"), the result of a probabilistic (Monte Carlo) simulation is a quantified probability ("if we build the dam, there is a 20% chance that the salmon population will go extinct"). Such a result (in this case, quantifying the risk of extinction) is typically much more useful to decision-makers who utilize the simulation results. In order to compute the probability distribution of predicted performance, it is necessary to propagate (translate) the input uncertainties into uncertainties in the results. A variety of methods exist for propagating uncertainty. Monte Carlo simulation is perhaps the most common technique for propagating the uncertainty in the various aspects of a system to the predicted performance.
In Monte Carlo simulation, the entire system is simulated a large number (e.g., 1000) of times. Each simulation is equally likely, referred to as a realization of the system. For each realization, all of the uncertain parameters are sampled (i.e., a single random value is selected from the specified distribution describing each parameter). The system is then simulated through time (given the particular set of input parameters) such that the performance of the system can be computed. This results is a large number of separate and independent results, each representing a possible “future” for the system (i.e., one possible path the system may follow through time).
26
The results of the independent system realizations are assembled into probability distributions of possible outcomes. As a result, the outputs are not single values, but probability distributions.
•
Stochastic Process • • • • Any variable (like stock price) whose value changes with time in uncertain way is said to follow stochastic process Process can be classified as Discrete time and Continuous time process Process can also be classified as Discrete variable and Continuous variable Stock price model is continuous time, continuous variable process
Weiner Process • • When two independent normal distributions are added, the resulting distribution has mean and variance equal to the sum of the mean and variance of respective distributions Consider two normal distributions with mean of 0 and variance of 1; the resulting distribution will have mean of 0 and variance of 2 and standard deviation of √ Weiner Process/Brownian Motion •
Assume these two distributions represent price for year 1 and year 2. Then it can be said that total variance in two years is 2 and standard deviation is √
• •
On the same lines, for the variance in period
is
with standard deviation of √
Weiner/Brownian process is Markov process with mean of 0 and variance of 1.
Formal Definition
A variable z follows Weiner process if – If small change • – And in two different periods are independent of each other in small time is given by
27
Generalized Weiner Process •
Generalized Weiner process for • – The first term is drift -
is defined as
increases by
in a small period is the standard Weiner process given by
– Second term is the noise where
–
√ and Variance is
– Mean of the process is – Stock Price Model • • • • • • • • • • • Writing VBA Code • – and √ √ The discrete version is √
is the stock price with expected drift of stock
where
is the expected rate of return on
In a short period of time, the increase in stock price is given by If stock volatility is assumed to be zero then
But stock price has volatility and in short period of time it is proportional to stock price Hence modifying the equation yields
and initial stock price
is given
28
–
is defined
– Generate random number – Compute and add this to and add it to previous – For the next period, generate random number, compute price
Simulate Geometric Brownian motion with Excel:
Learn about Geometric Brownian motion Stock prices are often modeled as the sum of
the deterministic drift, or growth, rate and a random number with a mean of 0 and a variance that is proportional to dt
This is known as Geometric Brownian Motion, and is commonly model to define stock price paths. It is defined by the following stochastic differential equation.
St is the stock price at time t, dt is the time step, μ is the drift, σ is the volatility, Wt is a Weiner process, and ε is a normal distribution with a mean of zero and standard deviation of one .
Substituting Equation 2 into Equation 1 gives
Hence dSt is the sum of a general trend, and a term that represents uncertainty.
29
Converting Equation 3 into finite difference form gives
Bear in mind that ε is a normal distribution with a mean of zero and standard deviation of one. This can be represented in Excel by NORM.INV(RAND(),0,1).
The spreadsheet linked to at the bottom of this post implements Geometric Brownian Motion in Excel using Equation 4.
30
Valuation:
Stock Price Simulation Inputs Stock Price Expected Return Per year (mu) Annulized Volatality (sigma) Expiry Period (year) Time Steps to follow (N) dt (It is one day) 50 5% 10% 1.0 250 0.00
Price Evolution Day1 Day2 Day3 Day4 Day5 Day6 Day7 Day8 Day9 Day10 Day11 Day12 Day13 Day14 Day15 Day16 Day17 Day18 Day19 Day20
Formula 50 49.82 49.02 48.76 48.80 48.60 48.96 48.85 48.74 48.71 48.79 48.72 48.18 48.16 48.32 48.24 48.05 47.84 47.86 47.81
VBA Code 50.00 50.18 50.22 50.29 50.12 49.97 50.22 49.53 49.76 50.05 50.24 49.71 49.65 50.01 50.27 50.18 50.75 51.13 50.62 51.16 31
Day21 Day22 Day23 Day24 Day25 Day26 Day27 Day28 Day29 Day30 Day31 Day32 Day33 Day34 Day35 Day36 Day37 Day38 Day39 Day40 Day41 Day42 Day43 Day44 Day45 Day46 Day47 Day48 Day49 Day50 Day51 Day52 Day53 Day54 Day55 Day56 Day57 Day58 Day59 Day60 Day61
47.68 47.88 47.88 47.49 47.10 46.62 46.71 46.77 46.49 46.22 45.65 45.54 45.75 45.14 45.15 45.62 46.01 46.06 46.11 46.36 46.34 46.28 46.22 46.06 46.19 46.04 45.77 45.56 45.64 45.83 45.60 45.12 44.87 44.49 44.60 44.56 44.79 44.47 44.75 44.58 44.73
51.06 51.09 51.33 50.82 50.90 50.89 50.73 50.84 50.97 50.78 50.60 50.92 51.23 51.31 52.03 52.49 52.25 52.43 53.12 52.90 52.94 52.53 53.62 53.78 53.06 53.14 52.72 52.31 52.59 52.42 51.87 51.70 51.61 51.45 51.99 52.68 52.61 52.42 52.10 51.79 51.92 32
Day62 Day63 Day64 Day65 Day66 Day67 Day68 Day69 Day70 Day71 Day72 Day73 Day74 Day75 Day76 Day77 Day78 Day79 Day80 Day81 Day82 Day83 Day84 Day85 Day86 Day87 Day88 Day89 Day90 Day91 Day92 Day93 Day94 Day95 Day96 Day97 Day98 Day99 Day100 Day101 Day102
44.43 44.72 44.49 43.78 44.04 43.49 43.69 43.65 43.68 43.55 43.11 43.48 43.34 42.63 42.86 43.10 43.29 42.94 43.43 43.23 42.86 42.71 43.09 43.21 43.17 43.55 43.55 43.66 43.86 43.95 44.46 44.78 45.36 45.38 44.97 45.49 46.03 46.08 46.10 45.55 45.62
51.86 51.80 51.99 51.85 51.97 51.72 51.44 51.51 51.07 51.04 51.48 51.28 51.55 51.46 51.29 51.75 51.87 51.99 51.94 51.53 51.59 51.76 52.22 52.55 51.90 51.94 52.41 52.36 52.52 52.53 52.56 52.54 52.42 52.35 52.16 51.64 51.43 52.10 51.60 51.52 51.42 33
Day103 Day104 Day105 Day106 Day107 Day108 Day109 Day110 Day111 Day112 Day113 Day114 Day115 Day116 Day117 Day118 Day119 Day120 Day121 Day122 Day123 Day124 Day125 Day126 Day127 Day128 Day129 Day130 Day131 Day132 Day133 Day134 Day135 Day136 Day137 Day138 Day139 Day140 Day141 Day142 Day143
45.59 45.44 45.69 45.70 45.79 45.19 45.30 45.53 45.66 45.75 46.11 45.79 46.02 45.77 45.73 45.79 45.79 46.03 45.90 45.69 46.01 46.37 46.16 46.08 46.19 46.24 46.21 45.57 46.32 46.03 46.43 46.47 46.67 46.28 46.36 46.07 45.98 46.15 46.55 46.53 46.55
51.42 51.10 51.09 50.89 51.01 51.05 50.74 51.24 51.38 51.39 51.31 50.92 51.19 51.16 51.39 51.48 51.81 51.14 50.89 50.43 50.04 49.92 49.57 48.48 48.52 48.65 48.70 49.00 48.58 48.32 48.47 48.45 48.34 48.04 48.21 48.67 48.70 48.29 48.52 48.45 48.43 34
Day144 Day145 Day146 Day147 Day148 Day149 Day150 Day151 Day152 Day153 Day154 Day155 Day156 Day157 Day158 Day159 Day160 Day161 Day162 Day163 Day164 Day165 Day166 Day167 Day168 Day169 Day170 Day171 Day172 Day173 Day174 Day175 Day176 Day177 Day178 Day179 Day180 Day181 Day182 Day183 Day184
46.44 46.70 46.82 46.95 46.81 46.38 46.30 46.18 46.07 46.00 46.16 45.71 45.37 45.59 45.99 45.41 45.54 45.59 44.88 44.82 44.90 44.68 44.69 44.58 44.61 44.35 44.79 44.75 44.78 44.66 44.73 44.64 44.90 44.84 44.80 44.87 44.74 44.88 45.44 45.60 45.50
48.43 48.20 48.07 47.68 47.76 47.48 47.93 47.55 47.51 47.34 47.69 47.91 47.73 47.88 47.69 47.29 46.75 46.62 46.87 46.72 46.51 46.51 46.32 46.21 45.73 45.72 45.49 45.82 45.89 46.10 46.54 46.42 46.46 46.06 46.17 46.11 46.63 46.29 46.71 46.81 46.71 35
Day185 Day186 Day187 Day188 Day189 Day190 Day191 Day192 Day193 Day194 Day195 Day196 Day197 Day198 Day199 Day200 Day201 Day202 Day203 Day204 Day205 Day206 Day207 Day208 Day209 Day210 Day211 Day212 Day213 Day214 Day215 Day216 Day217 Day218 Day219 Day220 Day221 Day222 Day223 Day224 Day225
45.32 45.29 45.32 45.35 45.15 45.26 45.09 45.39 45.67 45.82 45.65 45.64 45.58 45.65 45.68 45.49 45.62 45.72 45.41 45.03 45.02 45.00 45.07 44.93 45.02 45.21 45.37 45.19 45.13 45.40 45.42 44.73 44.57 44.73 43.92 44.25 44.57 44.93 45.19 45.08 45.39
46.41 46.40 46.19 46.92 46.60 46.05 45.94 45.99 46.41 46.45 46.39 46.70 46.98 47.13 47.31 48.14 48.02 48.03 47.97 48.13 47.86 47.81 47.86 48.14 48.18 48.13 48.15 47.93 48.03 48.03 48.19 48.56 48.47 48.32 48.17 47.86 47.89 47.67 47.75 47.65 48.01 36
Day226 Day227 Day228 Day229 Day230 Day231 Day232 Day233 Day234 Day235 Day236 Day237 Day238 Day239 Day240 Day241 Day242 Day243 Day244 Day245 Day246 Day247 Day248 Day249 Day250
45.52 45.27 45.33 44.87 45.25 44.95 44.85 44.90 45.39 45.47 45.95 45.79 45.55 45.48 45.21 45.08 45.14 44.74 44.65 44.49 44.91 45.43 44.94 44.66 44.39
48.00 47.74 47.90 48.11 48.21 48.46 48.16 48.44 48.19 48.72 48.27 47.81 48.06 47.98 47.96 47.62 47.26 46.99 46.51 46.68 46.72 46.77 46.55 46.54 46.74
37
VBA CODE:
Sub stock_price() 'Read the Inputs Sheets("Stock Price Model").Select S = Range("d4") mu = Range("d5") sigma = Range("d6") dt = 1 / 250 N = 250
Range("f12") = S 'First Day Price is input For xx = 2 To N ' Range("f" & xx + 11) = Range("f" & xx + 10) + (mu * Range("f" & xx + 10) * dt + sigma
* S * dt ^ 1 / 2) eps = Application.WorksheetFunction.NormSInv(Rnd()) Range("f" & xx + 11) = Range("f" & xx + 10) + (mu * Range("f" & xx + 10) * dt + sigma * Range("f" & xx + 10) * eps * dt ^ (1 / 2)) Next
End Sub
38
GRAPH:
70 60 50 40 Price 30 20
10
0 1 51 101 Days--> 151 201 251
39
Findings and Interpretation:
The stock pricing is done by simulating the stock price for so many times and noting down the values or we can observe the graph which we get from the values calculated by using formula manually or we assigned program to automatically simulate the stock. By observing these the decision will be made.
Here our main motive is to take right decision by the HNI investors that is to provide a good tool to take decision by them in investing stock markets.
As there will be lot of parameters in taking decision for investing in stock markets like volatility, time and some other variables we assume that as the drift .Here we will not consider the fundamental parameter in calculating the price of the stock. So as we are considering all small and minute parameters to get effective efficient output, so it is a suitable tool for the HNI’s to use the tool in investing into stock markets.
40
APPENDICES:
www.nseindia.com www.yahoofinance.com www.investopedia.com www.wikipedia.com
41
