Initially the stocks are analyzed using historical returns to derive expected returns and standard deviations, or deviations from the mean or average market return. Standard deviation is used as a measurement of volatility as it describes the degree of fluctuation in stock prices that occurs. Thereafter the stocks are compared by their respective betas, which measure the risk relative to the market and the responsiveness of the security to macroeconomic events.

To accomplish this task a graphical approach Is used as the stocks are plotted on a Security Market Line (SMALL as It will be easier to visually ampere each stock’s risk to return ratio relative to the market. In this way the stocks will be interpreted by means of a comparison relative to the market’s performance. As mentioned later, the risk and returns are heavily based on economic and accompanied factors.

Subsequently these stocks will be studied by means of regression analysis. A variance-covariance matrix will be used to derive the appropriate weights of each stock for the purpose of forming an optimum portfolio using an efficient frontier approach. The efficient frontier of risky assets will present the combination of stocks and their relative weights needed that will result in the highest possible expected rate of return, given a level of portfolio standard deviation.

The last step of the analysis requires an assessment of what kinds of options, whether put or call, should be purchased for the stocks chosen In the prior step based on option market parameters using the Black-Schools option valuation, a comparison of implied volatility and historical volatilities, and the put-call parity relationship. The main factors affecting the options price, including the stock price, exercise price, time to expiration, the risk free rate (based on the one month and three month labor and implied and historical volatility, are manipulated to understand the different measures affecting the options pricing.

In selecting the stocks chosen for the final step in terms of put/call options, three strike prices are used to understand the wide range of possibilities. Through this thorough analysis one can theoretically deduce the appropriate selection of stocks in the correct ratios in order to selectively purchase options that will attain the highest return on an Investment decision. Analysis of Historical Returns and Standard Deviation The SIX stocks chosen, the majority of which are traded on NASDAQ, are Google, Sony, Exxon-Mobil, Apple, Dell, and Microsoft. They are all actively traded and have actively traded options.

This can be attributed to the overall performance of the particular industry which has seen dramatically lower returns. Specifically, Michael Dell managed to rectify this drastic situation before the fourth quarter of this year, while Microsoft has been involved in antitrust lawsuits overseas throughout this period, and this has contributed to its relatively high volatility and below average returns. Lastly, Exxon- Mobil had a volatility of . 206 with steady returns for the two years individually at approximately 27% and 25% for the first and second years respectively, and a 26% return for the two year period.

The relatively high returns reflect record profits for the company and the oil industry as a whole during these two years. In comparison with the market’s volatility, Exxon-Mobiles relatively high volatility can be traced mainly to abnormally high returns, the aftermath of the local natural disaster hurricane Strain, global geopolitical instability in oil producing countries, and the war in Iraq all took place within this two year span. All stocks were more volatile than the market by a pretty healthy margin, and in general had very high returns overall when compared to the market.

As mentioned earlier, this can be attributed to many specific problems, the current state of each stocks respective industry, as well as outside natural and political factors both locally and globally. [See Appendix Beta and the Security Market Line As mentioned earlier, the beta of a stock measures the sensitivity of a security return to systematic or market factors. As beta measures the responsiveness of a security to macroeconomic events, the specific aforementioned situations for each industry are reflected in the betas of the individual stocks.

A security beta is another way to measure the rockiness of a stock. As a fundamental rule, the beta of he market is always 1, hence if a stock has a beta greater/lower than the markets, that particular stock is considered riskier/less risky than the overall market. Looking below at the security market line (SMS) graph for the first year period, rather than using current 30 day Labor rated of . 26, to getting a better understanding of this case I used rates from November 2005 to October 2006, the SMS has a risk-free rate of 5. 29% and a reward to risk ratio, or slope, of 5. 2% that were used respectively for both years. Almost all stocks lie on a vertical line at or around a beta of one, except or Exxon-Mobil which has a slightly higher beta at 1. 2, and Google with a lower beta at approximately . 8. For the first year Apple appeared to be the best investment as its expected return to risk ratio was highest, and thus it’s Sense’s alpha greatest, with a beta of 1 and expected return of 84. 7%. As it provided the highest return compared to its relatively low risk level, it was the most overvalued and underpinned stock.

The second best choice for the first year was Google, with a lower beta and relatively very high return (beta of . 8 and expected return of 37. 8%). | First Year: I Market Risk Premium | 5. 272% | I Risk Free Rate [pica] 15. 29% I The betas for the second year were more dispersed and relatively much higher. Google, Sony, and Dell registered betas of 1. 46, 1. 42, and 1. 4 respectively. Microsoft and Exxon-Mobil were the only stocks to have their beta drop from the previous year. Apple had the highest beta, and therefore the highest systemic risk for the second year at 1. 67.

It is apparent from the second year graph that Exxon-Mobil provided the best return for a risk-averse investor considering its beta was less than Google and Sony, and provided a higher return (beta of 1 and expected return of 25. %). A risky investor on the other hand would still have chosen Apple for the second year considering its significant returns regardless of its higher risk (beta of 1. 67 and expected return of 33. 5%). All stocks were riskier than the market during this two year period except for Microsoft which had a beta of . 9. Apple was the riskiest stock in the portfolio as it had the highest beta (1. 9). Apple also had the highest return for the two year period at 59. 1%. Apple would therefore be the top choice for a risk taking investor. Google on the other hand would be the top choice of a typical risk- verse investor because its two year beta is near the markets at 1. 03, with its return (31. 3%) considerably higher relative to the other stocks in the portfolio, as well as the market (1 1. 9%). In general most of the stocks were overvalued and underpinned, and although Apple had the highest beta it was the best choice for a risk taking market as can be seen from its Sense’s alpha.

A risk-averse investor would have chosen Google as the optimal choice for the first year and the two year period, and Exxon Mobil as the best choice for the second year, in taking into account their low etas or relatively lower levels of risk and higher expected returns. As the returns on these stocks fluctuated more than that of the market, and because their betas exceeded the market, they were considered relatively risky investments in general. Again, these risks can be attributed to the aforementioned economic factors and company specific risks. See Appendix A] Regression, Weights and the Efficient Frontier Using Regression analysis, Specifically a Variance-covariance matrix, the proportional weights required for portfolio optimization differed significantly between the two years, as would be expected. For the first year the composition of stocks required for an efficient frontier for the chosen risky assets to provide the highest possible expected rate of return included these stocks and weights: Google at 10%, Exxon-Mobil at 23. 4%, Apple at 55. 1%, and Microsoft at 1 1. 5%. For the second year, the stocks chosen and their proportional weights were Google at 12. %, Sony at 19%, Exxon-Mobil at 50. 6% and Apple at 18. 2%. The portfolio mean for the first year was an incredible 59. 5%, and as would be expected Apple was chosen for the majority weight for the first year at 55. % because of its amazingly high return and relatively low level of risk (beta of 1. 1 and 85% return). Google, Exxon-Mobil and Microsoft were chosen to offset each other and efficiently diversify the portfolio. For the second year the portfolio mean was 26. 6% and the majority weight went to Exxon- Mobil at 50. 6% as it had the lowest risk to expected return (beta of 1 and 25% return).

Again, Google, Sony, and Apple were chosen to efficiently diversify the portfolio. Dell, Sony and Microsoft were not chosen because their reward to risk ratio was not conducive to efficiently diversify the portfolio. Additionally Dell’s negative return for the second year was an important disqualifying factor. In all, the proportional weights derived should theoretically provide the most efficient combination of stocks to diversify the portfolio in order to achieve the highest possible expected rate of return given an acceptable level of risk. See Appendix B]. Theoretical Option Prices Based on results in the prior step using the efficient frontier to form an optimal portfolio, two types of options can be purchased in a futures contract for the researched stocks chosen, known as calls and puts, to further hedge risk and examine profits. Call and put options were calculated for the second year efficient portfolio of stocks which included Sony, Google, Exxon-Mobil, and Apple.

As a definition, call options allow the purchaser the right, but not the obligation, to buy a security, bond, or other instrument, at a pre-specified exercise or strike price for a predetermined period of time, at a cost known as a call premium. A put option on the other hand gives the purchaser the right, but not the obligation, to sell a stock, bond or commodity, at an agreed upon price over a fixed period of time in return for a remit. One crucial element of options is that they hold value over a limited and fixed period of time. Moreover, this value increases as time to expiration increases.

In the analysis of the stocks chosen, theoretical call and put prices were derived using software for options calculations. Specifically the binomial option pricing or “tree price 3) historical standard deviations, 4) the relevant interest rates and 5) times to expiration. The two time horizons were five and nine weeks, ending the third week of December 2007 and January 2008. For these two different time periods the interest ate used was based on the one month and three month labor, with a rate of 4. 65% for the December calculations and a rate of 4. 87% for January calculations.

These prices were then compared to actual call and put option prices maturing in the third week of December 2007 and January 2008, for three different strike prices; strike price closest to stock price, strike price above, and strike price below. As an example, the current stock price for [email protected] as of November 19th, 2007 was $47. 43. Thus, the three strike prices that were chosen to compute various call and put prices were $45. 00, $50. 0, and $40. 00. For the $45. 00 strike price, one closest to the actual price of $47. 43 and with an expiration or exercise date of December 20th, 2007, the theoretical call price calculated was $3. 4, while the actual historical listed price was $3. 20. The theoretical put price calculated at this specific strike price and expiration date was $0. 62, while the actual put price was $0. 85. In addition to calculating the theoretical call and put option prices, a binomial tree was formulated to approximate the probability of up and down movements in the price of the stock over small “step” intervals of time. A total of five steps were calculated, with step intervals of roughly seven to twelve days for December and January respectively.

Probabilities of roughly 50% for up and down movements were observed for both calls and puts. Hence the up movement for [email protected] at a theoretical call price of $3. 24 was found to be 1. 039 and the down movement was 0. 9617. The probability of upward or downward movement, which translates as the risk neutral probability of movement, was 0. 5017 in this case. An additional important calculation in reference to options is the hedge ratio, also now as delta, was computed for each of the stocks, options, strike prices, and time intervals.

An options hedge ratio can be defined as the change in the price of an option for a $1 increase in the stock price. As a general rule, a call option has a positive hedge ratio while a put option has a negative hedge ratio. Thus when the price of a stock goes up or down, the price of its option fluctuates in tandem whether directly or inversely as mentioned earlier, the change however is not a one to one ratio. The hedge ratio simply calculates the number of options needed for a relatively risk free portfolio for a given change in a stocks price.

Thus for Sony the call option hedge ratio at the $45 strike price is . 7593, and the put option hedge ratio -. 2419. Similar calculations were made for the remainder of the stocks and a cumulative table was generated with the relevant information for each. What can be derived from this relationship between the primary factors affecting an options price. Again, the primary factors influencing the price of an option are the stock price, the exercise price, the standard deviation or volatility of the security, the time to expiration, the interest rate or risk free rate, and dividend payouts.

As seen room this analysis, the price of a call option increases as the price of the stock increases, and conversely the price of a put option decreases as the price of the security increases. As exercise price increases, the value of a call option will move inversely and decrease, while the opposite will hold true for the value of a put option, as it will increase. In the analysis volatility played an important role as the put and volatility increased both call and put option values increased. Similarly, both put and call option values increased as time to expiration increased.

As the interest rate, or sis free rate, increased, the call option value increased, while the put option value decreased. Lastly as the payment of dividends increases, the effect is an increase in put option value, while the effect on call option value is a decrease. Finally it is important to note that the changes in option values are heavily determined by the weights of each variable, as they can often work to offset each other. For example an increase in exercise price may reduce the value of a call option, while an increase in stock price will raise a call option value.

The theoretical prices calculated for each UT and call option were different than actual prices found, the theoretical put and call prices were lower than the actual prices in every case except one. This disparity was more pronounced with certain stocks compared to others. For example the disparity of actual and theoretical prices for Google, Exxon-Mobil and Apple was much more significant than Sony. As the theoretical prices are mere assumptions, it is important to remember that the binomial option valuation approach uses the notion of replication.

Thus this difference can be attributed to differences in stock rises, as changes in stock prices can cause more significant discrepancy in the calculations of higher stock prices. Additionally larger volatilities will have a tendency to overprice options. Finally it is empirical that actual option prices are driven by the market forces of supply and demand, while theoretical values are implied from historical data. More importantly are the differences found between put and call values. Both actual and theoretical put and call values were dissimilar.

This difference indicates a violation in the parity relationship requiring put and call values o be identical, as put prices can be derived from calls in European put and call options. This parity is known as the put-call parity and describes the relationship between the put and call prices for a stock with the same strike price. Calculating the put-call parity produces a misprinting between all values calculated for each stock and exercise price. This violation in the put-call relationship means that an arbitrage opportunity exists.

However, one must consider the significance or extent of the parity, as not all opportunities are worthwhile to exploit after considering the transaction costs associated with purchasing options. Implied Volatility Implied volatility is the fluctuation caused in the stock price as calculated using the put and call prices. In the first step volatilities were calculated using historical data from the two years chosen earlier. Instead implied volatilities are calculated using a reverse engineering approach using put and call prices.

The question here is What standard deviation would be necessary for the option price that can be observed to be consistent with the Black-Schools formula? Hence the implied volatility is the volatility level for the stock that the option price implies, or the tankard deviation of stock returns that is consistent with an options market value. Even though it’s the same stock you wont receive the same implied volatility. We generally don’t find a constant implied volatility, the higher prices of these options are higher than the actual worth of the option, and vice versa, when the exercise price is above current market price.

Using this figure investors can then Judge whether they think the actual stock standard deviation exceeds the implied volatility. Than the implied volatility, the option’s fair price would exceed the observed price. For the analysis, implied volatilities were calculated for all stocks at the three put and call strike prices for the longer termed expiration date. For every calculation, implied volatility exceeded historical volatility, but in some cases the difference was minimal. For example the implied volatility for a Sony put option expiring January was 29. %, while historical standard deviation was 28%. These values were the closest computed values, indicating that the put option is a relatively good buy. Another variation is to compare two options on the same stock with equal expiration dates but different exercise prices. The option with the higher implied volatility would be considered relatively expensive because a higher standard deviation is required to Justify its price. An analyst therefore might consider buying an option with the lower implied volatility and writing the option with the higher implied volatility.

Using this approach an analyst looking at the table in appendix C can quickly find which put or call options might be worth pursuing for each stock at relative strike prices. For example, using the strike prices of $45. 00, $50. 00 and $40. 00 for Sony put options expiring January 2010, the relevant implied volatilities were 29. %, 33. 41%, and 39. 35% respectively. Therefore an analyst would choose the put option with the lowest implied volatility, I. E. Choose 29. 9% or the $45. 00 strike price.

Furthermore one can see that the difference in the implied and historical volatility estimates are greater using call prices than put prices. Thusly implied volatility is a useful measure when choosing and measuring put and call options between stocks and for each stock. Are not computing right or is the market wrong? , the Black-Schools estimates very good for pricing at the money, however once you move outside you receive errors. It soonest take many things into account. When excursive price are to far away from the actual price, Black-Schools is horrible and the binomial tree should be used.

American Put Options The unique attribute that makes American put options more attractive, compared to European put options, is the fact that they can be exercised at any time before maturity. This is a profitability maximizing feature as it allows an individual to take a short position on a put, and use the proceeds to invest in another put. Hence when studying the binomial tree for an option, it is best to exercise the option at a point where the numbers or nodes turn red. Prior to this point it would be most profitable to hold onto the put.

Once you have reached a step or point in the tree where the node turns red, it is indicative that it is most profitable to exercise the option early in order to reinvest proceeds. If hypothetically interest rates rise by one percent, it would be more profitable to hold onto a put option longer. In this case not exercising the option before maturity will produce higher profits than exercising the option for reasons of reinvestment. Finally it is important to note that the highest option prices are the ones most affected by interest rate increases.

In summary, through this analysis one can formulate a thorough strategy to hedge risk and maximize profits. As a process, this paper proves that stocks can be analyzed and compared using historical data for the purpose of creating an efficiently diversified risk weighted portfolio that efficiently hedges risk. Ultimately the stocks chosen during the selection process can be used to purchase call and put options. Furthermore the stocks prices that can be compared to find opportunities for arbitrage and profit minimization through the purchase of put and call options using a binomial tree model.