Stochastic Finance: Nyse, Nasqad and Amex Markets - Assignment Example

Table of Contents Question 3 Question 2 6 Question 3 10 APPENDIX 15 Appendix 15 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). Using Yahoo finance I settled on three companies; AAC Holdings, Inc., 1347 Capital Corp. and Alexco Resource Corp traded at NYSE, NASQAD and AMEX respectively and obtained daily closing stock prices for each company for six consecutive weeks commencing 13th April 2015 to 22nd May 2015. (see excel calculations) (b). The table in the appendix 1 also shows the results of the calculation for the logarithmic daily changes in the closing stock price for each of the three companies for six consecutive weeks. The formula; (Refer to excel calculations) (Kennedy, 2010), (c). 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; Table 1.   AAC 1347 Capital corp Alexco RG Min -0.066782 -0.004154 -0.070618 Q1 -0.007734 0.000000 -0.023024 Median 0.001664 0.000000 0.000000 Q3 0.023888 0.001025 0.043259685 Max 0.101649 0.009322 0.112477983       Box 1 - Hidden -0.007734 0.000000 0.000000 Box 2 - Lower 0.009398 0.000000 0.023024 Box 3 - Upper 0.022224 0.001025 0.043260       Whiskier Top 0.077761 0.008297 0.069218299 Wiskier Bottom 0.059048 0.004154 0.047594 Figure1. (d) The graphs below are obtained from plotting the z value against the proportion for the three companies in different stock markets; the plots show a normal probability trend. Figure 2 The curve shown in the graph above clearly indicates the performance in the stock markets. The daily closing stock prices for AAC Holdings, Inc, a company in the NYSE markets tends to be stable by showing a regular trend in growth over the first few weeks. The growth rate however slows down in the last weeks. Figure 3 As compared to the scenario in AAC Holdings, Inc, the trend here is a steady upward growth till the end. This pattern in growth however cannot be reliably adopted to predict the future state of the stock price for 1347 Capital corp. Investors without proper analysis can easily be lured to invest in such stocks since they promise high level of returns in a short time but can also drop in value drastically. Figure 4. Alexco Resource Corp though in a different stock market (AMEX) has the lowest stock prices with an average of$ 0.41. At this price Investors are attracted to invest but only when the trend in growth is consistent. This case proves the same as the stock prices keep on going up for the entire six week period. For all the three companies the curve plotted shows a general trend in growth. Being an ideal market situation, there are a number of factors that might affect this general trend; Inflation Inflation weakens the value of the local currency against the foreign currency in other words foreign exchange (Ross, S. 2003). A general rise in prices for most of the products means that the input prices for production of various goods and services are rising. In this case market analysts and fund managers will always consider the net impact on the margin of the entity that they are tracking. Interest rates Sometimes the changing interest rates in the market cause the value of the investment to go down. In evaluating the best investment the trend of changes in interest rates should be keenly observed (Ross, S. 2003). Changes in economic policy Changes in some factors within the market environment sometimes cause a change in the value of the investment. In most occasions these changes affects the company negatively by causing the value of the investment to decrease there by lowering prices of the share in the stock market. Financial risk diversification The main goal and objective of a companies is to maximize profit but with the assurance of its future existence for long term continuality. In achieve this goal, companies tend to seek diversification in their operations which most of the time lands them to investment portfolio in bide of diversifying the business risk. In spite that every investment of any kind has a risk factor a touched to it, this which in most of the cases cannot be avoided but by forming the portfolio, the Financial risk manager shall have mitigated against risk premium. Consequently, this allows the company to evaluate risk levels before the company goes ahead to venture into business of any kind (Ross, S. 2003). The manager of a company therefore needs to know the value of perfect information in order to make these critical decisions on investment to be ventured into. However, any slight error or misinformation could see the company suffer huge losses both at present and in the long run loss. It is therefore prudent to conduct a risk assessment on the investment in question. A team of individuals with relevant knowledge on the area of investment is picked to conduct the risk assessment. This group then tables their findings before the management which intern makes the best decision based on the team’s findings. The following are the risks that are common to the financial market. 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 For easy understanding of the model, we divide it into two 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 is equal to zero it indicates that security C1has no sensitivity to change in the market portfolio i.e RM=RJ. 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 Lamberton, D., Bernard, L. (2007), Introduction to Stochastic Calculus Applied to Finance, 2nd , Chapman and Hall/CRC Shiryaev, A.N., Grossinho, M.d.R., Oliveira, P.E., Esquível, M.L., (2006), Stochastic Finance, Springer. Ross, S. (2003) An elementary introduction to mathematical finance: options and other topics, 2nd edition, Cambridge University Press APPENDIX Appendix 1       NYSE     NASQAD       AMEX           AAC Holdings, Inc.     1347 Capital Corp.       Alexco Resource Corp       DAYS DATE CLOSING Rt PROB. Z VALUE CLOSING Rt PROB. Z VALUE CLOSING Rt PROB. Z VALUE 1 13-Apr-15 31.16 0 0.00 -3.078 9.60 0 0.0323 -1.8486 0.35 0 0.0323 -1.849 2 14-Apr-15 31.11 2.37E-05 0.00 -2.866 9.60 0 0.0645 -1.5179 0.37 0.05557 0.0645 -1.518 3 15-Apr-15 32.02 2.37E-05 0.00 -2.735 9.60 0 0.0968 -1.3002 0.38 0.026668 0.0968 -1.300 4 16-Apr-15 31.93 2.37E-05 0.00 -2.639 9.60 0 0.1290 -1.1310 0.37 -0.02667 0.1290 -1.131 5 17-Apr-15 31.62 2.37E-05 0.01 -2.562 9.60 0 0.1613 -0.9892 0.37 0 0.1613 -0.989 6 20-Apr-15 33.00 7.12E-05 0.01 -2.498 9.60 0 0.1935 -0.8649 0.37 0 0.1935 -0.865 7 21-Apr-15 33.11 2.37E-05 0.01 -2.443 9.63 0.0031 0.2258 -0.7527 0.37 0 0.2258 -0.753 8 22-Apr-15 33.25 2.37E-05 0.01 -2.394 9.62 -0.0010 0.2581 -0.6493 0.37 0 0.2581 -0.649 9 23-Apr-15 33.04 2.37E-05 0.01 -2.351 9.65 0.0031 0.2903 -0.5524 0.39 0.052644 0.2903 -0.552 10 24-Apr-15 32.36 2.37E-05 0.01 -2.311 9.65 0 0.3226 -0.4605 0.37 -0.05264 0.3226 -0.460 11 27-Apr-15 31.41 7.12E-05 0.01 -2.275 9.65 0 0.3548 -0.3723 0.40 0.077962 0.3548 -0.372 12 28-Apr-15 31.75 2.37E-05 0.01 -2.242 9.65 0 0.3871 -0.2869 0.44 0.09531 0.3871 -0.287 13 29-Apr-15 31.31 2.37E-05 0.01 -2.211 9.61 -0.0042 0.4194 -0.2035 0.41 -0.07062 0.4194 -0.204 14 30-Apr-15 34.66 2.37E-05 0.01 -2.182 9.61 0 0.4516 -0.1216 0.41 0 0.4516 -0.122 15 01-May-15 34.43 2.37E-05 0.02 -2.154 9.61 0 0.4839 -0.0404 0.42 0.024098 0.4839 -0.040 16 04-May-15 34.21 7.12E-05 0.02 -2.128 9.61 0 0.5161 0.0404 0.40 -0.04879 0.5161 0.040 17 05-May-15 32.00 2.37E-05 0.02 -2.104 9.61 0 0.5484 0.1216 0.42 0.04879 0.5484 0.122 18 06-May-15 31.02 2.37E-05 0.02 -2.081 9.70 0.00932 0.5806 0.2035 0.41 -0.0241 0.5806 0.204 19 07-May-15 30.98 2.37E-05 0.02 -2.059 9.70 0 0.6129 0.2869 0.40 -0.02469 0.6129 0.287 20 08-May-15 31.68 2.37E-05 0.02 -2.037 9.67 -0.0031 0.6452 0.3723 0.40 0 0.6452 0.372 21 11-May-15 32.79 7.12E-05 0.02 -2.017 9.67 0 0.6774 0.4605 0.38 -0.05129 0.6774 0.460 22 12-May-15 33.60 2.37E-05 0.02 -1.997 9.70 0.00310 0.7097 0.5524 0.39 0.025975 0.7097 0.552 23 13-May-15 34.19 2.37E-05 0.02 -1.979 9.74 0.00412 0.7419 0.6493 0.40 0.025318 0.7419 0.649 24 14-May-15 36.38 2.37E-05 0.02 -1.960 9.75 0.00103 0.7742 0.7527 0.42 0.04879 0.7742 0.753 25 15-May-15 37.01 2.37E-05 0.03 -1.943 9.79 0.00409 0.8065 0.8649 0.47 0.112478 0.8065 0.865 26 18-May-15 37.65 7.12E-05 0.03 -1.926 9.80 0.00102 0.8387 0.9892 0.50 0.061875 0.8387 0.989 27 19-May-15 35.98 2.37E-05 0.03 -1.910 9.79 -0.0010 0.8710 1.1310 0.51 0.019803 0.8710 1.131 28 20-May-15 35.69 2.37E-05 0.03 -1.894 9.79 0 0.9032 1.3002 0.50 -0.0198 0.9032 1.300 29 21-May-15 37.95 2.37E-05 0.03 -1.878 9.85 0.0061 0.9355 1.5179 0.47 -0.06188 0.9355 1.518 30 22-May-15 39.35 2.37E-05 0.03 -1.863 9.82 -0.0031 0.9677 1.8486 0.48 0.021053 0.9677 1.849 Read More