Portfolio Optimization using Linear, Non-linear and Integer Programming and Black-schole Theory - Term Paper Example

PORTFOLIO OPTIMIZATION USING LINEAR, NON-LINEAR AND INTEGER PROGRAMMING AND BLACK-SCHOLE THEORY By University Name City, State Date of Submission Portfolio Optimization Using Linear, Non-Linear and Integer Programming and Black-Schole Theory In the stock market, a stock portfolio refers to the component of an investor in relation to the stocks they hold and the value based on the prices. A portfolio holds risks that may affect its price. These may lead to the overvaluation undervaluation or correct valuation of the stock. In developing a stock portfolio, one covers the market prices and the changes that the stock is exposed to overtime that will allow it to either grow or fail in the market. A portfolio growth requires proper analysis that will aid in its optimization. Portfolio optimization refers to the selection of various proportions of assets to add to the portfolio in a criterion that promotes them for it to develop further. Portfolio optimization dwells on the improvement of the portfolio and improving the value and level of the stock portfolio to that which proves attractive to the market. The criterion allows for the combination of the various aspects either directly or indirectly. The optimization aspect allows for the analysis of the expected value and the rate of return and their dispersion. The dispersion looks at the distribution of the rate of return to the stock adding value to it and through these measures, the financial risks also undergo analysis to provide evidence and confidence in the investors. It’s through portfolio optimization that investors understand the various risks existing in the market that allows them. Portfolio optimization will require identification of a portfolio that is rich in diversification to ensure it provides good support to the portfolio. Following the yahoo finance, the following portfolio was considered. Considering the modern portfolio theory as developed by Harry Markowitz, standard deviation of the various portfolios and its rate of return as means of maximizing the expected returns are considered for the risks that may engulf the portfolio. These provide an efficient portfolio. Considering the trade-off between the risks involved with the expected returns that the portfolio covers. Considering the historical factors of the stocks provides the investors with better decisions on the direction of the stock overtime. These based on the curves on the value of the stocks, its prices and volumes traded. Considering the standard deviation, the volatility of the stock, and the stock prices and volumes provides a better position on the stock. Considering these enables, the investors provide a better response to their portfolio needs. Considering the initial portfolio selection, the following portfolio is selected. Some of the rapidly growing industries include technology, real estate sector, and the banking industry. The consideration of technology, the investor would consider having apple in the portfolio. This originates from the NASDAQ platform while the other product from the real estate sector that is GSPS from the S&P 500. Considering these two stock products on the portfolio that will aid the investors make decisions on further diversification of their portfolio. Considering the stock GSPC, the following table indicates the volume variance over the period covering Jan 2014 to date. According to the stocks, the fluctuation of volumes sold provides a picture on the nature of stock that GSPC is. These provide a position that investors consider with the price fluctuation and the volatility to make decisions on the stock. Considering the prices that the stock closed with during the week, the investor looks at this stock as one they will keep on their portfolio. The valuation provides a volatility represented by their standard deviation of 0.016. The limited variation of the prices indicates the stock movement. The prices fluctuate in a manner that indicates stability in the stock. The second stock on the portfolio is Apple Inc. The stock whose current price is 127.50 and provides a stable fluctuation in volumes as indicated in the graphs below: The chart indicates a volume variation with the highest volumes over the period traded in the previous week. The prices of this stock indicate as below: Considering the price fluctuation for the stock, the stock indicates a high surge in price from below 100 per week to over 600 per week. The surge explains the increases sale that indicated in the volumes traded. The stock indicates a high volatility of 0.78. The volatility indicates that the stock had major price volatility in which the high volatility and change in volatility indicated a change in price. These provide the stocks that the investor has on the portfolio having bought the stocks earlier in 2014. The current price of the stocks provide a favorable sale price for the stocks while considering the volumes bought which indicates as per the table below: Stock initial week cost 1000 shares current position price & Value GSPC 1833 1833000 2089.27 2089270 Apple 540.67 540670 127.5 127500   Total 2373670   2216770 According to the table, the stock Apple is performing poorly on the portfolio. Part of stock optimization includes the identification of non-performing stocks to indicate to the investors the stocks affecting their portfolio and those that promote it. Considering these, the stock GSPC is performing better compared to the Apple stock. The volatility of the stocks too indicates that Apple is not performing well with a high volatility compared to GSPC. It indicates that the stock is not stable and may affect the general stock as indicated in the table above with the total portfolio indicating a fall from a portfolio value of 2373670 to 2216770 indicating a 156900 fall in value. Portfolio Optimization In optimizing a portfolio, a number of methods may apply. These include the linear, non-linear, integer programming, and the use of the Black-Schole Theory. These play a role in directing the growth of a portfolio. Linear programming method of portfolio optimization Linear programming method seeks to establish the best outcome or approach. The model is used in mathematical problems and the finding of solutions. It is also applicable in the development of details through which decisions in financial investments that involve stocks (Michaud, & Michaud, 2008). The model may help in portfolio optimization through easing the analysis of various stocks and making decisions in the bets stocks to invest in and those to sell or hold. Application of linear programming looks at the linear relationships between the aspects under consideration. It also referred to as mathematical optimization. It is subject to various equality aspects and constraints through which the decisions on stocks stem. The study reveals the relationship looking at the variables applicable and their effect on the stocks to facilitate investment decisions. These problems normally appear in a canonical form that looks at CTX that expresses maximization, Ax≤b reflecting the aspects considered the problem to be subject to and x≥0. X represents the vector that carries the variables of the study or factors to be determined for this case are the future price of the stock. C and b indicate the known coefficients which indicate the know price and the expected volatility aspect. ()T indicates the matrix transpose. The objective function is then easily solved using CTX and the Ax≤b and x≥0 that indicate to the convex polytope through which the relationship is identified. Through this, the objective function becomes optimized. Application of linear programming in business investing embraces shadow pricing. Shadow pricing assumes the nominal value of any money market fund as $1 per share despite the fact that it may not represent a true value of the stock. The graph below indicates the graph that reflects shadow pricing Considering these, an investor faces prices P1, P2 and has the cash M to invest in the stocks. Therefore, the investors formula is max{u(X1,X2) : P1X1+P2X2=m}. From this, the point U that describes the future prices of the investment comes to life. The graph above indicates the same relationship applied to foreign exchange. It works for stock optimization too (Ramaswamy, 2004, p.180). Non-linear programming method of portfolio optimization The application of non-linear programming method deals with the minimizing or the application of maximization options through the creation of a nonlinear objectives function that provides for the development of constraints, those that prove linear and nonlinear. These may prove of equality nature and non-equality nature. The simplicity of the model in detailing the problems allows for the development of optimization options that lead to an easy identification of portfolio optimization. The problem is solved through the application of mathematical aspects in solving vectors x those counts as a local minimum to the appointed nonlinear scalar function f (x). These unconstrained functions the constraints may be solved. X1≥0, X2≥0, X12+X22≥1, X12+X22≤2 Following these, the objective function is maximized with the formula F(x) = X1+X2 Where X= (X1, X2) Based on these, the calculation of the necessary prices for the stocks in the future will provide the investors with models through which the study reflects a successful analysis of the market. The model aids in the determination of the future prices of the stock that the investors will rely on to make decisions with regard to their portfolio. Various portfolios may reflect various results and indications to the investors. Through the application of non-linear programming, the portfolio will indicate future performance through which the investors will decide if to sell, buy or hold a stock. Considering the stocks under consideration that is GSPC and Apple, one discovers that Apple has a more worrying trend that affects the portfolio. This prompts the decision to apply averaging techniques to it and having it disposed. The stock has a high volatility, which affects its market price fluctuations. Integer programming Integer programming refers to the linear programming aspect that considers all variables in the study as integers. The objectives function in these reflects the integer constraints. The integer programming may prove linear and non-linear programming. Special case considerations include 0-1 that reflects the integer linear programming normally considered as a binary aspect. The graph below indicates an integer programming that reflects the formula that holds the following max y, -x+y≤1, 3x+2y≤12, 2x+3y≤12, x,y≤0 and x,y€Z The application of these the feasible integer points appear in red and the convex hull appears as per the red dashed line. These points indicate the polyhedron that reflects the area covered by the integers that reflect the relationship between the integers under consideration. The blue lines on the other line indicate the coordinated axes that provide a definition to the polyhedron that relaxes the linear program. The blue line originates from the inequalities that the factors considered carry that provide the integer programming with the linear aspects. During optimization, the investors aim at moving the black line as furthest up as possible through which the polyhedron is still touched by it. Points (1,2) indicate the optimal points (2,2) indicate an objective value of 2 that indicates the options available. The integers provide a unique optimum value that introduces the relaxation aspect (1,8,2,8) that provides an objective value of 2.8. Through this method, investors identify the peak of their stocks and furthest they may go on the stock market in relation to the prices (Klatte, Luthi, & Schmedders, 2012, p.196). The stocks have a higher price and when they reach there, the investor disposes the stock. It is at this point that the stocks on their portfolio reaches a peak and needs disposal while when buying investors make decisions based on their prediction of the furthest that the stock may raise on the market. Black-Schole Theory of portfolio optimization The theory applies the formula for the calculation of the portfolio optimization that leads to the obtaining of the option price at which the investors’ base on to make portfolio related decisions. The theory applies a mathematical model that relates to the financial markets with specific derivative instruments applicable in investing. The model deduced provides the theoretical price estimate that provides the confidence in investors on the future prices of their portfolio and making decisions on whether to hold the shares, sell them or buy. The various adjustments made on the stock prices provide the method with a close proximity to the actual prices of the stock. The corrections normally provided by the options participants in the market. The Black-Scholes price provides a “fairly close” price to that which the market observes. The theory involves an “option smile” that covers the various discrepancies in the market. Through this formula, the price option provides a price that overtime the stock will hold. The stock aids in the elimination of risk and providing the right way of buying and selling of the various assets in a manner that the investors find more secure and more researched. The theory makes assumptions on the market assets. The major assumption is that the market will always have at least a single risky asset or stock and another riskless aspect called the money market, bond or the cash. Other risks on the asset include; The risk free rate of interest exists on the riskless rate. The random walk has a constant volatility and drift on the market. The stocks do not pay dividends. The assumptions the theory makes on the market include: The market has no arbitrage options or opportunities. At a riskless rate, it is deemed possible to borrow on amount. The stock provides the ability to buy and sell at any amounts. The market is frictionless with no fees or application costs that may affect the transitions above. Through the theory, one estimates the possibility of a payoff in the future. These depend on the values taken on that future date by the stock with the prices determined at the current time. The theory also added in the creation of a hedge position for the asset or stock. The consideration of the initial formula for the theory follows the following: OP= In these, D1= D2= In these, S stands for stock price, X strike price, t time remaining to the expiration in percent of the year, r the current risk free rate that is continuous and compounded, v the annual volatility of the stock price. The In represents the natural logarithm of the stock N(x) represents the standard normal cumulative distribution function and e is for the exponential function (Hoadley Trading & Investment Tools, n.d). Based on the stocks above, starting with GSPC, the theory yields the following: D1= S=2,089.27, X= 2500, r= 0.50% (Based on UK’s base lending rate), v2=0.830, t= 50% D1= D1= D1= D1= D1= D1= D1= D2= D2= D2= Inserting the values in the OP formula, the following is obtained: OP= OP= OP= OP= OP=17.08 Considering this, the price of the stock will have increased by 17.08 six months from now. Using the same formula, it’s possible to determine for apple and identify the portfolios that the investor may decide to retain and those they will decide to sell or buy. Conclusion The consideration of these methods provides the investors with the decision abilities that have informed direction of the stocks. All the methods provide the investors with better approaches to decision-making and stock price fluctuations. The volatility of stocks allows for the improvement of the ease in identifying the stocks that allows for the investors to profit from their portfolio more and easing the approach to stock markets. Reference List Bank of England, n.d, Central Bank Rates, Viewed on March 20, 205 from http://www.cbrates.com/england/index.htm Hoadley Trading & Investment Tools, n.d, Option Pricing Model and the “Greeks”, Viewed on March 20, 2015 from http://www.hoadley.net/options/bs.htm http://finance.yahoo.com/q/hp?s=AAPL&a=00&b=12&c=2014&d=02&e=20&f=2015&g=w http://finance.yahoo.com/q?s=^GSPC Klatte, D Luthi, H & Schmedders, K 2012, Operations Research Proceedings 2011: Selected Papers of the International Conference on Operations Research (OR 2011), August 30-September 2, 2011, Zurich, Switzerland, Springer Science & Business Media. Michaud, O R & Michaud, R O 2008, Efficient Asset Management: A Practical Guide to Stock Portfolio Optimization and Asset Allocation, Oxford University Press. Ramaswamy, S 2004, Managing Credit Risk in Corporate Bond Portfolios: A Practitioner’s Guide. John Wiley & Sons. Read More