Stochastic Finance - American Airlines Group Inc - Assignment Example

Table of Contents CALCULATIONS 3 Question 3 Question 2 6 APPENDIX 17 Appendix 17 CALCULATIONS Question Stock’s terminal value S has a uniform distribution: that is, it is equally likely to assume any value in the range (0-100) and will not assume any value outside of this range. The random variable x on which this stock’s value is based has a density function p(x) =1 for 0 ≤ x ≤ 1 and 0 elsewhere. The stock’s random terminal value is f(x) =100x. 1. a) Find the distribution function P(x) for p(x) Solution: Assume that the density function {p(xi)=P’(xi)} for a randomly distributed variable x is given by the following: p(x) =1.5x2 + x for 0 ≤ x ≤ 1 and 0 elsewhere. b) Find the expected value of the stock’s terminal S value assuming it will fall within the range (i) 50-100; (ii) 0 – 50; (iii) 0 to 100. Solution i) The distribution function for x will be P(x) =), which since f(x) = x equals The expected value of x given that it falls within this range is a conditional distribution determined by: ii) 0 – 50; iii) 0 to 100. c). Find the variance of S in the range 0 – 100 And since the term to the right of the minus sign is the expected value squared c) What would be the expected future cash flow (contingent on its exercise) of a call option written on this stock if its exercise price were $50? That is, what is the expected cash flow of the option conditional on its exercise? Question 2 (a) The data used in this question has been obtained from Yahoo Finance (EODData.com.) and the companies considered are; AMERICAN AIRLINES GROUP INC, RADIAN GROUP INC and PHARMATHENE INC. These three companies trade in stocks and are listed in National Association of Securities Dealers Automated Quotations (NASQAD), New York Stock Exchange (NYSE) and American Stock Exchange (AMEX) respectively. To analyze the changing trend in stock prices, I have considered daily closing stock prices for each of the three companies listed above and extracted a 30 day information ending 12th May 2015 (see excel work book; sheet 1) (b) Sheet 1 of the excel work book shows the calculations for the logarithmic changes in the daily closing stock prices for each of the three listed companies in their respective stock markets. The table in the appendix section has the results to these calculations. A 30 day result is considered. (c) (Refer to excel calculations) (Kennedy, 2010), (d) the data in my six sets of data are approximately normally distributed; the table and the graph shown below best describes the nature of the data; (see excel work book for full information) Table 1 Box and whisker data   AMERICAN AIRLINES GROUP INC RADIAN GROUP INC PHARMATHENE INC Min -0.046176 -0.020505 -0.045722249 Q1 -0.012985 -0.004121 -0.02139922 Median -0.000204 0.000000 0.003021157 Q3 0.009391 0.010074 0.022688679 Max 0.042046 0.045035 0.047628049     Box 1 - Hidden -0.012985 -0.004121 -0.02139922 Box 2 - Lower 0.012781 0.004121 0.024420377 Box 3 - Upper 0.009595 0.010074 0.019667522     Whiskier Top 0.032655 0.034961 0.02493937 Wiskier Bottom 0.033191 0.016384 0.024323029 Figure1. Box and whisker plot (d). Graphs below are obtained from plotting the market stock prices against the z value for the three companies in different stock markets; the plots show a normal probability trend. Figure 2: AMERICAN AIRLINES GROUP INC STOCK PRICE NORMALITY CURVE The curve shown in the graph above clearly indicates the performance in the stock markets for the American company. The daily closing stock prices for American Airlines Group Inc, a company in the NASQAD markets tends to be stable by showing a regular drop in stock prices over the first few weeks. The growth rate however tends to shoot and hitting its maximum of $52.71 after which the drop is drastic and reaches its minimum of $47.01 before picking up gradually towards the end of the six week period. These changes in daily closing stock prices for American Airlines Group Inc are concentrated around its mean stock price of $49.13 and hence a normal distribution. Figure 3: RADIAN GROUP INC STOCK PRICE NORMALITY CURVE Compared to the American Airlines Group Inc, the trend in Radian Group Inc, listed in NYSE, is a steady upward growth for stock prices for the first days within the analysis period. The price then changes the growth pattern suddenly and hits its highest of $18.45 before reducing to $16.7; the least price during the period. Stability in growth around the mean price of $17.695 for this company is an indicator of normality in distribution of the market stock prices. Figure 4: PHARMATHENE INC STOCK PRICE NORMALITY CURVE The average daily closing stock price for this company is at $1.66. Although being in a different stock market (AMEX) its stock prices are generally low as compared to the American Airlines group and Radian Group in NASQAD and NYSE markets respectively. After achieving its highest price of $1.8 the change in stock price tends to be stable around the average price depicting a normal distribution before dropping to its lowest of $1.55. The stock price-change curves plotted for all these three companies’ shows a general upward trend in growth of the market stock prices and subsequently followed by a drop in stock prices. Being an ideal market situation, there are a number of factors that might have contributed to this general trend; Inflation Inflation normally weakens the value of the local currency compared with the foreign currency which becomes the standard measure (Ross, S. 2003). Steady rise in prices of inputs with regard goods and services means that the prices of such particular finished goods and services in turn increases. This makes the company’s cost of production to be very much high. Accordingly the market analysts and fund managers always consider the net impact on the margin of the entity that they are tracking which in turn influences the share prices in stock market. Interest rates Generally change of interest rates in the market cause the value of the investment to drop tremendously. By evaluating the best investment to pursue, the historical trend of change in interest rates should be intensely observed to draw information for risk adverse investor (Hull, J. 2014). Changes in economic policy Changes in some factors like rule and regulations imposed by host country, within the market environment sometimes cause a change in the value of the investment. For instance, quotas and custom policies affects the company either negatively of positively depending on situation which some time become inevitable (Joshi, M. 2008). To justify a point in case whenever the value of the investment decreases there will be a decline in price of the share in the stock market this makes it to drop in its performance. Financial risk diversification The intent goal of a company is to maximize shareholders wealth, but with the assurance of its future existence for long term continuality. In achieving this goal, a company must command huge market share as compared with its rival competitors (Franke, and Hèardle, 2011). It is therefore very prudent for investment committee to seek diversification in their operations which most of the time leads their decision to investment portfolio so as to diversifying the business risk. Given that very investment project of any kind has a risk factor associated to it, which in most of the cases cannot be avoided but can reduced to some levels by forming the portfolio. In evaluating the risk factors; Financial Manager shall therefore, put mitigating devices against risk premium. As a result, it provides the avenue for the companies to evaluate risk levels before going ahead to venture into business kind of investments (Kijima, M. 2013). Prudently, Managers of the companies tend to know the value of perfect information in order to make a critical decision on investment to be ventured into. This eliminates the flawlessness of the economic decisions in investment. Nevertheless, any slight error or misinformation could see the company suffer huge losses both at present and in the long run loss. It is therefore discreet to conduct a risk assessment on the investment before committing. A team of experts with relevant knowledge on the area of investment should be contracted to conduct the risk assessment. Question 3 Black-Scholes formulae The value of a call option is equal to its expected payoff in a risk-neutral world, discounted at the risk-free interest rate, which can be written in a generalized form as: Solution Call Option Where EQ [ ] denotes expectation with respect to the risk-neutral probability measure Q. Using the approach in Nielsen (1992) and starting with payoff expectation formulae above, we substitute to derive the Black-Scholes option pricing formula as follows: ( Capiński, and Zastawniak, 2011) Under a constant interest rate r the time-t price of a European call option on a non-dividend paying stock when its spot price is St and with strike K and time to maturity is Where: With the following denotation: C=Theoretical call premium S=Current stock price t=time until option expiration K=option striking price N=Cumulative standard normal distribution e=exponential term (2.718) s=standard deviation of the stock r=risk free interest rate Put Option Where: With the following denotation: C=Theoretical put premium S=Current stock price t=time until option expiration K=option striking price N=Cumulative standard normal distribution e=exponential term (2.718) s=standard deviation of the stock r=risk free interest rate In understanding this model, we segment whole question into two separate sections. Section 1, SN (d1), derives the expected benefit from acquiring a stock outright (Cuthbertson, and Nitzsche, 2004). We find this by multiplying stock price [S] by the change in the call premium with respect to a change in the underlying stock price [N (d1)]. Section 2, Ke(-rt)N(d2), gives the present value of paying the exercise price on the day of expiry. Fair market value of the call option is then arrived at by taking the difference between section 1 and 2. This model is mostly used in the pricing of the European call options (Shiryaev, etal 2006), b). Further on consider the data in the panel below related to traded options of a Stock S (traded price at time T of £50.11), exercise price X = £50.11, maturing in one year. The volatility is 22.00% and considers the time now to be 0. The LIBOR rate of 1.17%. The mean and variance set has been computed to be 4.72 and 0.08, respectively. Use the derived Black-Scholes call and put formulae derived in a) and the data given to compute the call and put values. Interpret the results Solution Where: The stock is trading below money with theoretical put premium of £31.56 Consider Capital Asset Pricing Model which is also known as Security Market Line described in the question as under: Whenever a beta of security is equal to one then it means its systematic risk is the same of that of the market. However, beta is equal to zero it indicates that security C1has no sensitivity to change in the market portfolio i.e. RM=RJ. c) Suppose that {Xt, t ≥ 0} is a stochastic process that may be represented as dXt = Ytdt + ZtdWt. For (suitably nice) functions f(x, t) and g(x, t) use Itô’s Lemma to establish the stochastic integration-by-parts formula dt where f, g and the partial derivatives are evaluated at (Xt, t). Further on for the standard Brownian motion {Wt, t ≥ 0}, evaluate the stochastic integral Solution dt Determine the rate of change of the with respect to f and g; differentiating with respect to f keeping g constant we have Determine the rate of change of the with respect to f and g; differentiating with respect to g keeping f constant we have REFERENCES Capiński, M. and Zastawniak, T. (2011) Mathematics for finance, 2nd edition, Springer Stochastic Financial Models, Chapman and Hall/CRC Cuthbertson, K. and Nitzsche, D. (2004) Quantitative Financial Economics: Stocks, Bonds and Foreign Exchange, 2nd edition, Wiley Franke, J. and Hèardle, W. (2011) Statistics of financial markets, 3rd edition, Springer Hull, J. (2014) Options, futures, and other derivatives, 9th edition, Pearson Joshi, M. (2008) The Concepts and Practice of Mathematical Finance, 2nd edition, Cambridge University Press Kijima, M. (2013) Stochastic processes with applications to finance, Chapman and Hall Lamberton, D., Bernard, L. (2007), Introduction to Stochastic Calculus Applied to Finance, 2nd , Chapman and Hall/CRC Ross, S. (2003) An elementary introduction to mathematical finance: options and other topics, 2nd edition, Cambridge University Press Shiryaev, A.N., Grossinho, M.d.R., Oliveira, P.E., Esquível, M.L., (2006), Stochastic Finance, Springer. APPENDIX Appendix 1         NASQAD   NYSE   AMEX           AMERICAN AIRLINES GROUP INC RADIAN GROUP INC PHARMATHENE INC RANK DATE PROPORTION Z VALUE CLOSING Rt CLOSING Rt CLOSING Rt 1 01-Apr-15 0.032258065 -1.8485963 50.44 0 16.7 0 1.64 0 2 02-Apr-15 0.064516129 -1.5179292 49.17 -0.02550083 16.88 0.0107208 1.67 0.0181274 3 03-Apr-15 0.096774194 -1.3001534 49.17 0 16.88 0 1.67 0 4 06-Apr-15 0.129032258 -1.1309776 48.08 -0.022417391 17.08 0.0117787 1.61 -0.0365894 5 07-Apr-15 0.161290323 -0.9891686 47.49 -0.012347128 17.08 0 1.65 0.0245411 6 08-Apr-15 0.193548387 -0.8648944 48.74 0.025980886 17.34 0.0151078 1.64 -0.006079 7 09-Apr-15 0.225806452 -0.7527288 47.84 -0.018637939 17.29 -0.0028877 1.72 0.047628 8 10-Apr-15 0.258064516 -0.6493239 47.73 -0.002301979 17.44 0.0086381 1.74 0.0115608 9 13-Apr-15 0.290322581 -0.5524426 47.79 0.001256282 17.38 -0.0034463 1.8 0.0339016 10 14-Apr-15 0.322580645 -0.4604945 47.42 -0.007772332 17.43 0.0028727 1.79 -0.005571 11 15-Apr-15 0.35483871 -0.3722894 47.75 0.006934986 17.24 -0.0109606 1.71 -0.0457222 12 16-Apr-15 0.387096774 -0.2868939 48.24 0.010209485 17.16 -0.0046512 1.71 0 13 17-Apr-15 0.419354839 -0.2035442 48.19 -0.001037022 17.15 -0.0005829 1.64 -0.0417971 14 20-Apr-15 0.451612903 -0.1215874 49.76 0.032059918 17.94 0.0450347 1.68 0.0240976 15 21-Apr-15 0.483870968 -0.0404405 51.36 0.031648211 17.91 -0.0016736 1.62 -0.0363676 16 22-Apr-15 0.516129032 0.0404405 51.4 0.000778513 18.1 0.0105527 1.65 0.0183491 17 23-Apr-15 0.548387097 0.1215874 51.45 0.00097229 18.45 0.0191524 1.66 0.0060423 18 24-Apr-15 0.580645161 0.2035442 52.71 0.024194729 18.37 -0.0043455 1.62 -0.0243915 19 27-Apr-15 0.612903226 0.2868939 51.87 -0.016064603 18.28 -0.0049113 1.6 -0.0124225 20 28-Apr-15 0.64516129 0.3722894 51.19 -0.013196388 18.41 0.0070864 1.66 0.036814 21 29-Apr-15 0.677419355 0.4604945 48.88 -0.046175886 18.23 -0.0098254 1.61 -0.0305834 22 30-Apr-15 0.709677419 0.5524426 48.28 -0.012350919 17.86 -0.020505 1.65 0.0245411 23 01-May-15 0.741935484 0.6493239 49.39 0.022730578 17.73 -0.0073055 1.59 -0.0370413 24 04-May-15 0.774193548 0.7527288 48.95 -0.008948606 17.85 0.0067454 1.6 0.0062696 25 05-May-15 0.806451613 0.8648944 47.01 -0.040439024 17.66 -0.0107013 1.59 -0.0062696 26 06-May-15 0.838709677 0.9891686 47.04 0.000637959 17.6 -0.0034033 1.55 -0.0254791 27 07-May-15 0.870967742 1.1309776 49.06 0.042045735 17.89 0.016343 1.61 0.0379792 28 08-May-15 0.903225806 1.3001534 49.04 -0.000407747 18.1 0.01167 1.64 0.0184621 29 11-May-15 0.935483871 1.5179292 49.65 0.012362099 18.18 0.0044102 1.66 0.0121214 30 12-May-15 0.967741935 1.8485963 48.85 -0.016244012 18.13 -0.0027541 1.72 0.0355067 Read More