Featured Post

Physical Aging Ectocranial Suture Closure

Presentation Forensic science reformed wrongdoing examination techniques (White Folkens, 2005). There is no compelling reason to expand on i...

Monday, December 23, 2019

Symptoms And Symptoms Of Lyme Disease - 755 Words

Lyme disease is a bacterial illness that is transfused to humans via a bite from a tick infected with the disease. (Ticks are scientifically classified as Arachnida, a specific classification that also includes spiders.) The most common ticks known to carry Lyme disease are the Deer Tick and the Western Black-Legged Tick. The first manifestation of an infection is typically a rash, which may appear to resemble a bull s eye. The proliferation of the infection progressively brings on symptoms that include headache, fever, muscle and joint pain, fatigue and stiffness of the neck. Lyme disease goes undiagnosed because of the size of these ticks being the size of a peppercorn and unobserved by person that was bitten. In addition, numerous symptoms are like those of the flu and other bacterial infections. Laboratory evaluations may help facilitate at this stage, but may not always give a clear diagnosis. (National Institute of Allergy and Infectious Disease, 2016). Discussion topic s will consist of casual agents including a brief history, characteristics and means of infection from Lyme disease. Epidemiology comprised of the geographical distribution along with persons with prevalent risk factors of contracting Lyme disease. Also included will be how Lyme disease is transmitted and the pathogenesis of the bacterial agent. Signs, symptoms, and complications caused by Lyme disease. What the conclusion looks like for a person whom has been diagnosed along with treatmentShow MoreRelatedSymptoms And Treatment Of Lyme Disease1454 Words   |  6 PagesIntroduction Lyme disease has been the most commonly reported vector-borne illness in the United States since the Centers for Disease Control and Prevention started reporting it in 1991 [1]. It should be noted that Lyme disease does not occur everywhere in the US, and is heavily concentrated in upper Midwest and northeast United States [1]. This report has been compiled to provide recommendations for antimicrobial prophylactic treatment of Lyme disease once a patient encounters a tick bite. Lyme diseaseRead MoreSymptoms And Treatment Of Lyme Disease1485 Words   |  6 Pagesspreading to become the most common vector-borne disease that occurs in the Northern Hemisphere, according to the Center for Disease Control and Prevention. This â€Å"hidden pandemic†, known as Lyme disease, is silently infecting hundreds of thousands of United States citizens each year, and numbers of new and untreated infections continue to climb as traditional treatments fail and doctors misdiagnose the condition. Spread by tick bites, Lyme disease has affected many lives and continues to infect countlessRead MoreSymptoms And Treatment Of Lyme Disease1997 Words   |  8 PagesIntroduction Lyme Disease is the number one tick-borne disease in the United States and in Massachusetts. It is considered a vector disease because it is spread through the bite of a black-legged tick (also known as a deer tick) that carries the bacterium, Borrelia burgdorferi. Lyme was first diagnosed in 1975 in Lyme, CT and the bacterium that causes Lyme was discovered in 1982 by Willy Burgdorfer (Todar, 2012, p. 1). Lyme disease spreads rapidly and can impact many different organ systemsRead MoreSymptoms And Treatment Of Lyme Disease1770 Words   |  8 Pages Lyme disease is the most common vector-borne disease in the United States. Despite this, adequate prevention is lacking and treatment measures are sometimes inadequate. Vaccinations for Lyme disease developed in the late 1990’s have since been withdrawn from the market, and research is currently underway to create a new vaccine. Educational programs have proven to show an increase in protective behaviors to prevent tick bites and tick-borne illnesses through increased knowledge of repellent useRead MoreSymptoms And Treatment Of Lyme Disease1598 Words   |  7 Pagesntroduction Lyme disease has been the most commonly reported vector-borne illness in the United States since the Centers for Disease Control and Prevention started reporting it in 1991 [1]. It should be noted that Lyme disease does not occur everywhere in the US, and is heavily concentrated in upper Midwest and northeast United States [1]. This report has been compiled to provide recommendations for antimicrobial prophylactic treatment of Lyme disease once a patient encounters a tick bite. Lyme diseaseRead MoreSymptoms And Treatment Of Lyme Disease1897 Words   |  8 PagesBackground: Definition of the condition: Lyme disease is named after the town of Old Lyme in Connecticut, where the first cases of Lyme disease (LD) were discovered in 1975. About twelve children, who lived in the same area of this town, were diagnosed with arthritis which was discovered to be caused by Lyme disease (Levi et al., 2012). In 1982, a scientist named Willy Burgdorfer and his coworkers, established the link between ticks and the transmission of Borralis bacteria which is proven to resultRead MoreSymptoms And Treatment Of Lyme Disease2008 Words   |  9 Pagesâ€Å"Lyme disease is caused by the spirochete Borrelia burgdorferi and is transmitted to humans by the bite of infected blacklegged ticks (Ixodes spp.). Early manifestations of infection include fever, headache, fatigue, and a characteristic skin rash called erythema migrans. Left untreated, late manifestations involving the joints, heart, and nervous system can occur. A Healthy People 2010 objective (14-8) is to reduce the annual incidence of Lyme disease to 9.7 new cases per 100,000 population in 10Read MoreSymptoms And Treatment Of Lyme Disease1710 Words   |  7 PagesaIntroduction Lyme disease has been the most commonly reported vector-borne illness in the United States since the Centers for Disease Control and Prevention started reporting it in 19911. It should be noted that Lyme disease does not occur everywhere in the US, and is heavily concentrated in upper Midwest and northeast United States1. This report has been compiled to provide recommendations for antimicrobial prophylactic treatment of Lyme disease once a patient encounters a tick bite. Lyme disease is causedRead MoreLong Term Antibiotic Treatment Of Persistent Symptoms Attributed Lyme Disease2057 Words   |  9 PagesThe research question of this study was to determine if longer-term antibiotic treatment of persistent symptoms attributed to Lyme disease resulted in a better outcome than shorter-term antibiotic treatment1. Evaluate the review of related research. How well did the authors provide a context for the current research in light of previous literature and gaps in current literature and knowledge? In the introduction, the authors indicate that previous randomized, clinical trials have not provided sufficientRead MoreLyme Disease : A Rapidly Spreading Infectious Disease1666 Words   |  7 PagesIntroduction Lyme disease is a rapidly spreading infectious disease in the United States, with over 25,000 confirmed cases in 2013.5 It was first discovered in the early-1970’s in the town of Lyme, Connecticut when a group of children started to present with rheumatoid arthritis symptoms. Some of these children presented with a rash and researchers connected the symptoms to occurring during peak tick season. By the mid-1970’s, the researchers began describing the symptoms and coining the term â€Å"Lyme disease†

Sunday, December 15, 2019

Anomalies in Option Pricing Free Essays

string(185) " shrewd investors will go short the first and long the second; that is, they will write calls and sell bonds \(borrow\), while simultaneously buying both puts and the underlying stock\." Anomalies in option pricing: the Black-Scholes model revisited New England Economic Review, March-April, 1996 by Peter Fortune This study is the third in a series of Federal Reserve Bank of Boston studies contributing to a broader understanding of derivative securities. The first (Fortune 1995) presented the rudiments of option pricing theory and addressed the equivalence between exchange-traded options and portfolios of underlying securities, making the point that plain vanilla options – and many other derivative securities – are really repackages of old instruments, not novel in themselves. That paper used the concept of portfolio insurance as an example of this equivalence. We will write a custom essay sample on Anomalies in Option Pricing or any similar topic only for you Order Now The second (Minehan and Simons 1995) summarized the presentations at â€Å"Managing Risk in the ’90s: What Should You Be Asking about Derivatives? â€Å", an educational forum sponsored by the Boston Fed. Related Results Trust, E-innovation and Leadership in Change Foreign Banks in United States Since World War II: A Useful Fringe Building Your Brand With Brand Line Extensions The Impact of the Structure of Debt on Target Gains Project Management Standard Program. The present paper addresses the question of how well the best-known option pricing model – the Black-Scholes model – works. A full evaluation of the many option pricing models developed since their seminal paper in 1973 is beyond the scope of this paper. Rather, the goal is to acquaint a general audience with the key characteristics of a model that is still widely used, and to indicate the opportunities for improvement which might emerge from current research and which are undoubtedly the basis for the considerable current research on derivative securities. The hope is that this study will be useful to students of financial markets as well as to financial market practitioners, and that it will stimulate them to look into the more recent literature on the subject. The paper is organized as follows. The next section briefly reviews the key features of the Black-Scholes model, identifying some of its most prominent assumptions and laying a foundation for the remainder of the paper. The second section employs recent data on almost one-half million options transactions to evaluate the Black-Scholes model. The third section discusses some of the reasons why the Black-Scholes odel falls short and assesses some recent research designed to improve our ability to explain option prices. The paper ends with a brief summary. Those readers unfamiliar with the basics of stock options might refer to Fortune (1995). Box 1 reviews briefly the fundamental language of options and explains the notation used in the paper. I. The Black-Scholes Model In 1973, Myron Scholes and the late Fischer Black published their seminal paper on option pricing (Black and Scholes 1973). The Black-Scholes model revolutionized financial economics in several ways. First, it contributed to our understanding of a wide range of contracts with option-like features. For example, the call feature in corporate and municipal bonds is clearly an option, as is the refinancing privilege in mortgages. Second, it allowed us to revise our understanding of traditional financial instruments. For example, because shareholders can turn the company over to creditors if it has negative net worth, corporate debt can be viewed as a put option bought by the shareholders from creditors. The Black-Scholes model explains the prices on European options, which cannot be exercised before the expiration date. Box 2 summarizes the Black-Scholes model for pricing a European call option on which dividends are paid continuously at a constant rate. A crucial feature of the model is that the call option is equivalent to a portfolio constructed from the underlying stock and bonds. The â€Å"option-replicating portfolio† consists of a fractional share of the stock combined with borrowing a specific amount at the riskless rate of interest. This equivalence, developed more fully in Fortune (1995), creates price relationships which are maintained by the arbitrage of informed traders. The Black-Scholes option pricing model is derived by identifying an option-replicating portfolio, then equating the option’s premium with the value of that portfolio. An essential assumption of this pricing model is that investors arbitrage away any profits created by gaps in asset pricing. For example, if the call is trading â€Å"rich,† investors will write calls and buy the replicating portfolio, thereby forcing the prices back into line. If the option is trading low, traders will buy the option and short the option-replicating portfolio (that is, sell stocks and buy bonds in the correct proportions). By doing so, traders take advantage of riskless opportunities to make profits, and in so doing they force option, stock, and bond prices to conform to an equilibrium relationship. Arbitrage allows European puts to be priced using put-call parity. Consider purchasing one call that expires at time T and lending the present value of the strike price at the riskless rate of interest. The cost is [C. sub. t] + X[e. sup. -r(T-t)]. (See Box 1 for notation: C is the call premium, X is the call’s strike price, r is the riskless interest rate, T is the call’s expiration date, and t is the current date. At the option’s expiration the position is worth the highest of the stock price ([S. sub. T]) or the strike price, a value denoted as max([S. sub. T], X). Now consider another investment, purchasing one put with the same strike price as the call, plus buying the fraction [e. sup. -q(T-t)] of one share of the stock. Denoting the put premium by P and the stock price by S, then the cost of this is [P. sub. t] + [e. sup. -q(T-t)][S. sub. t], and, at time T, the value at this position is also max([S. sub. T], X). (1) Because both positions have the same terminal value, arbitrage will force them to have the same initial value. Suppose that [C. sub. t] + X[e. sup. -r(T-t)] [greater than] [P. sub. t] + [e. sup. -q(T-t)][S. sub. t], for example. In this case, the cost of the first position exceeds the cost of the second, but both must be worth the same at the option’s expiration. The first position is overpriced relative to the second, and shrewd investors will go short the first and long the second; that is, they will write calls and sell bonds (borrow), while simultaneously buying both puts and the underlying stock. You read "Anomalies in Option Pricing" in category "Papers" The result will be that, in equilibrium, equality will prevail and [C. sub. t] + X[e. sup. r(T-t)] = [P. sub. t] + [e. sup. -q(T-t)][S. sub. t]. Thus, arbitrage will force a parity between premiums of put and call options. Using this put-call parity, it can be shown that the premium for a European put option paying a continuous dividend at q percent of the stock price is: [P. sub. t] = -[e. sup. -q(T-t)][S. sub. t]N(-[d. sub. 1]) + X[e. sup. -r(T-t)]N(-[d. sub. 2]) where [d. sub. 1] and [d. sub. 2] are defined as in Box 2. The importance of arbitrage in the pricing of options is clear. However, many option pricing models can be derived from the assumption of complete arbitrage. Each would differ according to the probability distribution of the price of the underlying asset. What makes the Black-Scholes model unique is that it assumes that stock prices are log-normally distributed, that is, that the logarithm of the stock price is normally distributed. This is often expressed in a â€Å"diffusion model† (see Box 2) in which the (instantaneous) rate of change in the stock price is the sum of two parts, a â€Å"drift,† defined as the difference between the expected rate of change in the stock price and the dividend yield, and â€Å"noise,† defined as a random variable with zero mean and constant variance. The variance of the noise is called the â€Å"volatility† of the stock’s rate of price change. Thus, the rate of change in a stock price vibrates randomly around its expected value in a fashion sometimes called â€Å"white noise. † The Black-Scholes models of put and call option pricing apply directly to European options as long as a continuous dividend is paid at a constant rate. If no dividends are paid, the models also apply to American call options, which can be exercised at any time. In this case, it can be shown that there is no incentive for early exercise, hence the American call option must trade like its European counterpart. However, the Black-Scholes model does not hold for American put options, because these might be exercised early, nor does it apply to any American option (put or call) when a dividend is paid. (2) Our empirical analysis will sidestep those problems by focusing on European-style options, which cannot be exercised early. A call option’s intrinsic value is defined as max(S – X,0), that is, the largest of S – X or zero; a put option’s intrinsic value is max(X – S,0). When the stock price (S) exceeds a call option’s strike price (X), or falls short of a put option’s strike price, the option has a positive intrinsic value because if it could be immediately exercised, the holder would receive a gain of S – X for a call, or X – S for a put. However, if S [less than] X, the holder of a call will not exercise the option and it has no intrinsic value; if X [greater than] S this will be true for a put. The intrinsic value of a call is the kinked line in Figure 1 (a put’s intrinsic value, not shown, would have the opposite kink). When the stock price exceeds the strike price, the call option is said to be in-the-money. It is out-of-the-money when the stock price is below the strike price. Thus, the kinked line, or intrinsic value, is the income from immediately exercising the option: When the option is out-of-the-money, its intrinsic value is zero, and when it is in the money, the intrinsic value is the amount by which S exceeds X. Convexity, the Call Premium, and the Greek Chorus The premium, or price paid for the option, is shown by the curved line in Figure 1. This curvature, or â€Å"convexity,† is a key characteristic of the premium on a call option. Figure 1 shows the relationship between a call option’s premium and the underlying stock price for a hypothetical option having a 60-day term, a strike price of $50, and a volatility of 20 percent. A 5 percent riskless interest rate is assumed. The call premium has an upward-sloping relationship with the stock price, and the slope rises as the stock price rises. This means that the sensitivity of the call premium to changes in the stock price is not constant and that the option-replicating portfolio changes with the stock price. The convexity of option premiums gives rise to a number of technical concepts which describe the response of the premium to changes in the variables and parameters of the model. For example, the relationship between the premium and the stock price is captured by the option’s Delta ([Delta]) and its Gamma ([Gamma]). Defined as the slope of the premium at each stock price, the Delta tells the trader how sensitive the option price is to a change in the stock price. (3) It also tells the trader the value of the hedging ratio. (4) For each share of stock held, a perfect hedge requires writing 1/[[Delta]. ub. c] call options or buying 1/[[Delta]. sub. p] puts. Figure 2 shows the Delta for our hypothetical call option as a function of the stock price. As S increases, the value of Delta rises until it reaches its maximum at a stock price of about $60, or $10 in-the-money. After that point, the option premium and the stock price have a 1:1 relationship. The increasing Delta also means that the hedging ratio falls as the stock price rises. At higher stock prices, fewer call options need to be written to insulate the investor from changes in the stock price. The Gamma is the change in the Delta when the stock price changes. (5) Gamma is positive for calls and negative for puts. The Gamma tells the trader how much the hedging ratio changes if the stock price changes. If Gamma is zero, Delta would be independent of S and changes in S would not require adjustment of the number of calls required to hedge against further changes in S. The greater is Gamma, the more â€Å"out-of-line† a hedge becomes when the stock price changes, and the more frequently the trader must adjust the hedge. Figure 2 shows the value of Gamma as a function of the amount by which our hypothetical call option is in-the-money. (6) Gamma is almost zero for deep-in-the-money and deep-out-of-the-money options, but it reaches a peak for near-the-money options. In short, traders holding near-the-money options will have to adjust their hedges frequently and sizably as the stock price vibrates. If traders want to go on long vacations without changing their hedges, they should focus on far-away-from-the-money options, which have near-zero Gammas. A third member of the Greek chorus is the option’s Lambda, denoted by [Lambda], also called Vega. (7) Vega measures the sensitivity of the call premium to changes in volatility. The Vega is the same for calls and puts having the same strike price and expiration date. As Figure 2 shows, a call option’s Vega conforms closely to the pattern of its Gamma, peaking for near-the-money options and falling to zero for deep-out or deep-in options. Thus, near-the-money options appear to be most sensitive to changes in volatility. Because an option’s premium is directly related to its volatility – the higher the volatility, the greater the chance of it being deep-in-the-money at expiration – any propositions about an option’s price can be translated into statements about the option’s volatility, and vice versa. For example, other things equal, a high volatility is synonymous with a high option premium for both puts and calls. Thus, in many contexts we can use volatility and premium interchangeably. We will use this result below when we address an option’s implied volatility. Other Greeks are present in the Black-Scholes pantheon, though they are lesser gods. The option’s Rho ([Rho]) is the sensitivity of the call premium to changes in the riskless interest rate. (8) Rho is always positive for a call (negative for a put) because a rise in the interest rate reduces the present value of the strike price paid (or received) at expiration if the option is exercised. The option’s Theta ([Theta]) measures the change in the premium as the term shortens by one time unit. (9) Theta is always negative because an option is less valuable the shorter the time remaining. The Black-Scholes Assumptions The assumptions underlying the Black-Scholes model are few, but strong. They are: * Arbitrage: Traders can, and will, eliminate any arbitrage profits by simultaneously buying (or writing) options and writing (or buying) the option-replicating portfolio whenever profitable opportunities appear. * Continuous Trading: Trading in both the option and the underlying security is continuous in time, that is, transactions can occur simultaneously in related markets at any instant. * Leverage: Traders can borrow or lend in unlimited amounts at the riskless rate of interest. Homogeneity: Traders agree on the values of the relevant parameters, for example, on the riskless rate of interest and on the volatility of the returns on the underlying security. * Distribution: The price of the underlying security is log-normally distributed with statistically independent price changes, and with constant mean and constant variance. * Continuous Prices: No discontinuous jumps occur in the price of the underlying security. * Transactions Costs: The cost of engaging in arbitrage is negligibly small. The arbitrage assumption, a fundamental proposition in economics, has been discussed above. The continuous trading assumption ensures that at all times traders can establish hedges by simultaneously trading in options and in the underlying portfolio. This is important because the Black-Scholes model derives its power from the assumption that at any instant, arbitrage will force an option’s premium to be equal to the value of the replicating portfolio. This cannot be done if trading occurs in one market while trading in related markets is barred or delayed. For example, during a halt in trading of the underlying security one would not expect option premiums to conform to the Black-Scholes model. This would also be true if the underlying security were inactively traded, so that the trader had â€Å"stale† information on its price when contemplating an options transaction. The leverage assumption allows the riskless interest rate to be used in options pricing without reference to a trader’s financial position, that is, to whether and how much he is borrowing or lending. Clearly this is an assumption adopted for convenience and is not strictly true. However, it is not clear how one would proceed if the rate on loans was related to traders’ financial choices. This assumption is common to finance theory: For example, it is one of the assumptions of the Capital Asset Pricing Model. Furthermore, while private traders have credit risk, important players in the option markets, such as nonfinancial corporations and major financial institutions, have very low credit risk over the lifetime of most options (a year or less), suggesting that departures from this assumption might not be very important. The homogeneity assumption, that traders share the same probability beliefs and opportunities, flies in the face of common sense. Clearly, traders differ in their judgments of such important things as the volatility of an asset’s future returns, and they also differ in their time horizons, some thinking in hours, others in days, and still others in weeks, months, or years. Indeed, much of the actual trading that occurs must be due to differences in these judgments, for otherwise there would be no disagreements with â€Å"the market† and financial markets would be pretty dull and uninteresting. The distribution assumption is that stock prices are generated by a specific statistical process, called a diffusion process, which leads to a normal distribution of the logarithm of the stock’s price. Furthermore, the continuous price assumption means that any changes in prices that are observed reflect only different draws from the same underlying log-normal distribution, not a change in the underlying probability distribution itself. II. Tests of the Black-Scholes Model. Assessments of a model’s validity can be done in two ways. First, the model’s predictions can be confronted with historical data to determine whether the predictions are accurate, at least within some statistical standard of confidence. Second, the assumptions made in developing the model can be assessed to determine if they are consistent with observed behavior or historical data. A long tradition in economics focuses on the first type of tests, arguing that â€Å"the proof is in the pudding. It is argued that any theory requires assumptions that might be judged â€Å"unrealistic,† and that if we focus on the assumptions, we can end up with no foundations for deriving the generalizations that make theories useful. The only proper test of a theory lies in its predictive ability: The theory that consistently predicts best is the best theory, regardless of the assumptions required to generate the theory. Tests based on assumptions are justified by the principle of â€Å"garbage in-garbage out. † This approach argues that no theory derived from invalid assumptions can be valid. Even if it appears to have predictive abilities, those can slip away quickly when changes in the eThe Data The data used in this study are from the Chicago Board Options Exchange’s Market Data Retrieval System. The MDR reports the number of contracts traded, the time of the transaction, the premium paid, the characteristics of the option (put or call, expiration date, strike price), and the price of the underlying stock at its last trade. This information is available for each option listed on the CBOE, providing as close to a real-time record of transactions as can be found. While our analysis uses only records of actual transactions, the MDR also reports the same information for every request of a quote. Quote records differ from the transaction records only in that they show both the bid and asked premiums and have a zero number of contracts traded. nvironment make the invalid assumptions more pivotal. The data used are for the 1992-94 period. We selected the MDR data for the SP 500-stock index (SPX) for several reasons. First, the SPX options contract is the only European-style stock index option traded on the CBOE. All options on individual stocks and on other indices (for example, the SP 100 index, the Major Market Index, the NASDAQ 100 index) are American options for which the Black-Scholes model would not apply. The ability to focus on a European-style option has several advantages. By allowing us to ignore the potential influence of early exercise, a possibility that significantly affects the premiums on American options on dividend-paying stocks as well as the premiums on deep-in-the-money American put options, we can focus on options for which the Black-Scholes model was designed. In addition, our interest is not in individual stocks and their options, but in the predictive power of the Black-Scholes option pricing model. Thus, an index option allows us to make broader generalizations about model performance than would a select set of equity options. Finally, the SP 500 index options trade in a very active market, while options on many individual stocks and on some other indices are thinly traded. The full MDR data set for the SPX over the roughly 758 trading days in the 1992-94 period consisted of more than 100 million records. In order to bring this down to a manageable size, we eliminated all records that were requests for quotes, selecting only records reflecting actual transactions. Some of these transaction records were cancellations of previous trades, for example, trades made in error. If a trade was canceled, we included the records of the original transaction because they represented market conditions at the time of the trade, and because there is no way to determine precisely which transaction was being canceled. We eliminated cancellations because they record the SP 500 at the time of the cancellation, not the time of the original trade. Thus, cancellation records will contain stale prices. This screening created a data set with over 726,000 records. In order to complete the data required for each transaction, the bond-equivalent yield (average of bid and asked prices) on the Treasury bill with maturity closest to the expiration date of the option was used as a riskless interest rate. These data were available for 180-day terms or less, so we excluded options with a term longer than 180 days, leaving over 486,000 usable records having both CBOE and Treasury bill data. For each of these, we assigned a dividend yield based on the SP 500 dividend yield in the month of the option trade. Because each record shows the actual SP 500 at almost the same time as the option transaction, the MDR provides an excellent basis for estimating the theoretically correct option premium and evaluating its relationship to actual option premiums. There are, however, some minor problems with interpreting the MDR data as providing a trader’s-eye view of option pricing. The transaction data are not entered into the CBOE computer at the exact moment of the trade. Instead, a ticket is filled out and then entered into the computer, and it is only at that time that the actual level of the SP 500 is recorded. In short, the SP 500 entries necessarily lag behind the option premium entries, so if the SP 500 is rising (falling) rapidly, the reported value of the SPX will be above (below) the true value known to traders at the time of the transaction Test 1: An Implied Volatility Test A key variable in the Black-Scholes model is the volatility of returns on the underlying asset, the SPX in our case. Investors are assumed to know the true standard deviation of the rate of return over the term of the option, and this information is embedded in the option premium. While the true volatility is an unobservable variable, the market’s estimate of it can be inferred from option premiums. The Black-Scholes model assumes that this â€Å"implied volatility† is an optimal forecast of the volatility in SPX returns observed over the term of the option. The calculation of an option’s implied volatility is reasonably straightforward. Six variables are needed to compute the predicted premium on a call or put option using the Black-Scholes model. Five of these can be objectively measured within reasonable tolerance levels: the stock price (S), the strike price (X), the remaining life of the option (T – t), the riskless rate of interest over the remaining life of the option (r), typically measured by the rate of interest on U. S. Treasury securities that mature on the option’s expiration date, and the dividend yield (q). The sixth variable, the â€Å"volatility† of the return on the stock price, denoted by [Sigma], is unobservable and must be estimated using numerical methods. Using reasonable values of all the known variables, the implied volatility of an option can be computed as the value of [Sigma] that makes the predicted Black-Scholes premium exactly equal to the actual premium. An example of the computation of the implied volatility on an option is shown in Box 3. The Black-Scholes model assumes that investors know the volatility of the rate of return on the underlying asset, and that this volatility is measured by the (population) standard deviation. If so, an option’s implied volatility should differ from the true volatility only because of random events. While these discrepancies might occur, they should be very short-lived and random: Informed investors will observe the discrepancy and engage in arbitrage, which quickly returns things to their normal relationships. Figure 3 reports two measures of the volatility in the rate of return on the SP 500 index for each trading day in the 1992-94 period. (10) The â€Å"actual† volatility is the ex post standard deviation of the daily change in the logarithm of the SP 500 over a 60-day horizon, converted to a percentage at an annual rate. For example, for January 5, 1993 the standard deviation of the daily change in lnSP500 was computed for the next 60 calendar days; this became the actual volatility for that day. Note that the actual volatility is the realization of one outcome from the entire probability distribution of the standard deviation of the rate of return. While no single realization will be equal to the â€Å"true† volatility, the actual volatility should equal the true volatility, â€Å"on average. † The second measure of volatility is the implied volatility. This was constructed as follows, using the data described above. For each trading day, the implied volatility on call options meeting two criteria was computed. The criteria were that the option had 45 to 75 calendar days to expiration (the average was 61 days) and that it be near the money (defined as a spread between SP 500 and strike price no more than 2. 5 percent of the SP 500). The first criterion was adopted to match the term of the implied volatility with the 60-day term of the actual volatility. The second criterion was chosen because, as we shall see later, near-the-money options are most likely to conform to Black-Scholes predictions. The Black-Scholes model assumes that an option’s implied volatility is an optimal forecast of the volatility in SPX returns observed over the term of the option. Figure 3 does not provide visual support for the idea that implied volatilities deviate randomly from actual volatility, a characteristic of optimal forecasting. While the two volatility measures appear to have roughly the same average, extended periods of significant differences are seen. For example, in the last half of 1992 the implied volatility remained well above the actual volatility, and after the two came together in the first half of 1993, they once again diverged for an extended period. It is clear from this visual record that implied volatility does not track actual volatility well. However, this does not mean that implied volatility provides an inferior forecast of actual volatility: It could be that implied volatility satisfies all the scientific requirements of a good forecast in the sense that no other forecasts of actual volatility are better. In order to pursue the question of the informational content of implied volatility, several simple tests of the hypothesis that implied volatility is an optimal forecast of actual volatility can be applied. One characteristic of an optimal forecast is that the forecast should be unbiased, that is, the forecast error (actual volatility less implied volatility) should have a zero mean. The average forecast error for the data shown in Figure 3 is -0. 7283, with a t-statistic of -8. 22. This indicates that implied volatility is a biased forecast of actual volatility. A second characteristic of an optimal forecast is that the forecast error should not depend on any information available at the time the forecast is made. If information were available that would improve the forecast, the forecaster should have already included it in making his forecast. Any remaining forecasting errors should be random and uncorrelated with information available before the day of the forecast. To implement this â€Å"residual information test,† the forecast error was regressed on the lagged values of the SP 500 in the three days prior to the forecast. 11) The F-statistic for the significance of the regression coefficients was 4. 20, with a significance level of 0. 2 percent. This is strong evidence of a statistically significant violation of the residual information test. The conclusion that implied volatility is a poor forecast of actual volatility has been reached in several other studies using different methods and data. For example, Canina and Figlewski ( 1993), using data for the SP 100 in the years 1983 to 1987, found that implied volatility had almost no informational content as a prediction of actual volatility. However, a recent review of the literature on implied volatility (Mayhew 1995) mentions a number of papers that give more support for the forecasting ability of implied volatility. Test 2: The Smile Test One of the predictions of the Black-Scholes model is that at any moment all SPX options that differ only in the strike price (having the same term to expiration) should have the same implied volatility. For example, suppose that at 10:15 a. m. on November 3, transactions occur in several SPX call options that differ only in the strike price. Because each of the options is for the same interval of time, the value of volatility embedded in the option premiums should be the same. This is a natural consequence of the fact that the variability in the SP 500’s return over any future period is independent of the strike price of an SPX option. One approach to testing this is to calculate the implied volatilities on a set of options identical in all respects except the strike price. If the Black-Scholes model is valid, the implied volatilities should all be the same (with some slippage for sampling errors). Thus, if a group of options all have a â€Å"true† volatility of, say, 12 percent, we should find that the implied volatilities differ from the true level only because of random errors. Possible reasons for these errors are temporary deviations of premiums from equilibrium levels, or a lag in the reporting of the trade so that the value of the SPX at the time stamp is not the value at the time of the trade, or that two options might have the same time stamp but one was delayed more than the other in getting into the computer. This means that a graph of the implied volatilities against any economic variable should show a flat line. In particular, no relationship should exist between the implied volatilities and the strike price or, equivalently, the amount by which each option is â€Å"in-the-money. † However, it is widely believed that a â€Å"smile† is present in option prices, that is, options far out of the money or far in the money have higher implied volatilities than near-the-money options. Stated differently, deep-out and far-in options trade â€Å"rich† (overpriced) relative to near-the-money options. If true, this would make a graph of the implied volatilities against the value by which the option is in-the-money look like a smile: high implied volatilities at the extremes and lower volatilities in the middle. In order to test this hypothesis, our MDR data were screened for each day to identify any options that have the same characteristics but different strike [TABULAR DATA FOR TABLE 1 OMITTED] prices. If 10 or more of these â€Å"identical† options were found, the average implied volatility for the group was computed and the deviation of each option’s implied volatility from its group average, the Volatility Spread, was computed. For each of these options, the amount by which it is in-the-money was computed, creating a variable called ITM (an acronym for in-the-money). ITM is the amount by which an option is in-the-money. It is negative when the option is out-of-the-money. ITM is measured relative to the SP 500 index level, so it is expressed as a percentage of the SP 500. The Volatility Spread was then regressed against a fifth-order polynomial equation in ITM. This allows for a variety of shapes of the relationship between the two variables, ranging from a flat line if Black-Scholes is valid (that is, if all coefficients are zero), through a wavy line with four peaks and troughs. The Black-Scholes prediction that each coefficient in the polynomial regression is zero, leading to a flat line, can be tested by the F-statistic for the regression. The results are reported in Table 1, which shows the F-statistic for the hypothesis that all coefficients of the fifth-degree polynomial are jointly zero. Also reported is the proportion of the variation in the Volatility Spreads, which is explained by variations in ITM ([R. sup. 2]). The results strongly reject the Black-Scholes model. The F-statistics are extremely high, indicating virtually no chance that the value of ITM is irrelevant to the explanation of implied volatilities. The values of [R. sup. 2] are also high, indicating that ITM explains about 40 to 60 percent of the variation in the Volatility Spread. Figure 4 shows, for call options only, the pattern of the relationship between the Volatility Spread and the amount by which an option is in-the-money. The vertical axis, labeled Volatility Spread, is the deviation of the implied volatility predicted by the polynomial regression from the group mean of implied volatilities for all options trading on the same day with the same expiration date. For each year the pattern is shown throughout that year’s range of values for ITM. While the pattern for each year looks more like Charlie Brown’s smile than the standard smile, it is clear that there is a smile in the implied volatilities: Options that are further in or out of the money appear to carry higher volatilities than slightly out-of-the-money options. The pattern for extreme values of ITM is more mixed. Test 3: A Put-Call Parity Test Another prediction of the Black-Scholes model is that put options and call options identical in all other respects should have the same implied volatilities and should trade at the same premium. This is a consequence of the arbitrage that enforces put-call parity. Recall that put-call parity implies [P. sub. t] + [e. sup. -q(T – t)][S. sub. t] = [C. sub. t] + [Xe. sup. -r(T – t)]. A put and a call, having identical strike prices and terms, should have equal premiums if they are just at-the-money in a present value sense. If, as this paper does, we interpret at-the-money in current dollars rather than present value (that is, as S = X rather than S = [Xe. sup. -r(t – q)(T – t)]), at-the-money puts should have a premium slightly below calls. Because an option’s premium is a direct function of its volatility, the requirement that put premiums be no greater than call premiums for equivalent at-the-money options implies that implied volatilities for puts be no greater than for calls. For each trading day in the 1992-94 period, the difference between implied volatilities for at-the-money puts and calls having the same expiration dates was computed, using the [+ or -]2. 5 percent criterion used above. (12) Figure 5 shows this difference. While puts sometimes have implied volatility less than calls, the norm is for higher implied volatilities for puts. Thus, puts tend to trade â€Å"richer† than equivalent calls, and the Black-Scholes model does not pass this put-call parity test. How to cite Anomalies in Option Pricing, Papers

Saturday, December 7, 2019

The One Percent Research Paper free essay sample

The documentary deals with the disparity between the wealthy elite and the citizenry and how they are both so far removed from one another. â€Å"As of 2010, the top 1% of households (the upper class) owned 35. 4% of all privately held wealth. † (Domhoff, 2010, The Wealth Distribution, para. 1). The producer and interviewer presents this film through many wealthy American businessmen, critics, economists and even his own family to explain this major social gap that exists on our home front. When looking at the differences side-by-side, it is hard to grasp that we all live in the same place, the United States of America. The film was created by Jamie Johnson, the heir to one of America’s most affluent families. Being born with a â€Å"silver spoon,† Jamie never really had anything to worry about in life from private schools to private jets, equestrian clubs and charitable dinner parties. But, he always felt something was missing in his life and he couldn’t quite put a finger on it. We will write a custom essay sample on The One Percent Research Paper or any similar topic specifically for you Do Not WasteYour Time HIRE WRITER Only 13.90 / page The fortune that Jamie inherited on his 21st birthday was from one of the wealthiest family-owned companies in the United States, Johnson and Johnson. THE ONE PERCENTPage 3 Ryan M. Kerrick Mr. Richard Cannella English Composition II March 18, 2012 His great grandfather â€Å"started the company in 1886† (Johnson and Johnson, 1997) and it continued to grow well beyond imagination. Jamie would always wonder what made him deserve this prosperous lifestyle. After self-examination, Jamie was determined to investigate some of the questions haunting him in his head about the wealth disparity in America. Attempting to bring his mind to ease, he decided to create this documentary, â€Å"The One Percent. † Within the first few minutes of the film I noticed a sign labeled â€Å"Private Property Members Only. To me the sign is showing how the wealthy pride themselves in being part of such an elite club. Meanwhile, on the other side of the spectrum, the working class feel like they are not truly part of society at all. In the beginning of the documentary you see the Johnson’s having a family meeting. At first it looks normal, but they are not discussing chores around the house. It is a â€Å"family meeting† with their financial wealth and money management advisors. The meeting is centered on managing their wealth and assets and essentially turning their millions in to more millions. The consensus from the advisors is that every year the family fortune tenfold and they continue to become richer and richer. Jamie seems to be upsetting his father with the making of this documentary and the advisors seem hesitant to talk about money and wealth on camera. THE ONE PERCENTPage 4 Ryan M. Kerrick Mr. Richard Cannella English Composition II March 18, 2012 His father’s initial reaction is that his son’s documentary is nonsense, but something that might have a huge ripple effect if taken seriously. Jamie does a great job trying to get answers and asking difficult questions to the wealthy elite of America. The footage he presents is of people giving their most honest views and thoughts and it is evident there is a huge gap between the wealthy and the poor. Jamie Johnson interviewed numerous people within different social classes. They ranged from Milton Freedman and Steve Forbes, who owns his own private cruise ship, to some local residents of the south side of Chicago, who live in poverty without locks on their mailboxes. Jamie is presenting the social gap with visuals broken down buildings compared to mansions, a homeless man asking for money compared to fancy beach resorts and post hurricane Katrina victims with private country clubs. A poignant moment that stood out to me in the documentary was when Jamie interviewed Nicole Buffet, the granddaughter of Warren Buffet through marriage (his son Peter’s ex-wife’s daughter. ) It was comforting to watch and I feel even Jamie felt a sense of self-awareness as he interviewed the young female. She seemed so confident in who she was but most of all peaceful, content and happy with the simple things in life. In this situation, you can see money seems to be the root of all evil. Even to the point of ridding someone of your family that has great memories of you. THE ONE PERCENTPage 5 Ryan M. Kerrick Mr. Richard Cannella English Composition II March 18, 2012 She talked of her â€Å"grandpa† as the loving man she knows him as (not as multimillion dollar business man. ) In response to her participation in the documentary, he wrote to her â€Å"I have not emotionally or legally adopted you as a grandchild, nor have the rest of my family adopted you as a niece of cousin. † (Schroeder, 2008, p. 976) He disclaims her as a granddaughter despite all the good she says about him just because of her role in the film. People argue that Buffet was not out of place because Nicole was adopted or a step child and was not part of his immediate family. I thought the same until I stumbled upon an article written in The Wall Street Journal. The article stated â€Å"Susan Buffett, Warrens first wife, who died in 2004, named Nicole in her will as one of her adored grandchildren and left her $100,000. She added that Nicole shall have the same status and benefits as if they were children of my son, Peter A. Buffett. † Also, â€Å"a source close to the family says Nicole spent very little time with Warren Buffett over the years but that he paid for Nicoles school and living expenses until she was 28. Nicole says that Mr. Buffetts reaction may have reflected his philosophy about wealth. Sharing my experience as a Buffett was stepping outside the box, she says. † (Frank, R. 2008). Another part of the film that stood out to me is when Jamie interviewed the taxi cab driver and I did like what the man had to say. He said, â€Å"My family is one of the richest families in the world, but not with money. With love, kindness, tolerance and patience. Qualities that are worth more than money and you can’t buy that. † THE ONE PERCENTPage 6 Ryan M. Kerrick Mr. Richard Cannella English Composition II March 18, 2012 This showed the much clear distinction in values between the rich and the working class. Comparing what the taxi driver had said to the values of Warren Buffet who wrote his granddaughter disclaiming because she did not support the family lifestyle, which would you prefer? Watching this documentary I came to find that with money also comes a fear of losing that money and becoming consumed by it. Along with money comes the changing of your values and whole aspect on life. It allows families in America to move up in class, often times allowing them to adopt different ideas and different family values. After viewing this documentary my analysis on the disparity of the wealth gap is that it is reality and there isn’t much we can do about it. I am able to see what people have to go through to make it to the top. Business men do not become who they are by being nice to people. They have to be aggressive in the business world, cut throat, sharp and willing to do whatever it takes to achieve their dreams. That might come with risks or even mean walking all over people. But, sometimes to make a difference, you have to ride through hell to make it to heaven. This might be a hard pill for some people to swallow, but, it is reality and it is the truth. People have not become moguls overnight singing KUM-BAH-YA and dancing around a fire. Las Vegas was built on mob money before it was cleaned up and presented with a new face by entrepreneur investing. This is business. You have to be able to stomach it and it is not for the weak hearted. I would therefore have to agree with what I have seen in the documentary regarding Jamie’s father and his behavior. THE ONE PERCENTPage 7 Ryan M. Kerrick Mr. Richard Cannella English Composition II March 18, 2012 He did what he had to do to get to where he is today even though he inherited his thrown. In my personal opinion, if you look hard enough you will always find dirt and the top of the social ladder. You do not only have to be rich for that either, all of humanity is flawed in its own way. I do not believe that everyone was born to be a millionaire. However, I do believe that in our own way, if we preserve and strive to work hard, we are all able to be â€Å"millionaires† in our own eyes and live fulfilling lives and contribute to making our society a better place to live. Being unemployed, uneducated and living off welfare is not fair to the people who work hard to pay taxes to support their fellow citizens. I consider it to be a lazy and irresponsible way of life. However, it is a choice in life you have. The money is out there for the taking so it is also your prerogative whether you choose to go out and get it or not. Make your decision wisely and keep your values in mind while climbing the social ladder if that is the route you decide to take. THE ONE PERCENTPage 8 Ryan M. Kerrick Mr. Richard Cannella English Composition II March 18, 2012 REFERENCES Johnson and Johnson. (1997). History of Johnson and Johnson. Retrieved from http://www. jnj. com/connect/about-jnj/company-history/ Schroeder, A. (2008). The Snowball: Warren Buffet and the Business of Life. Domhoff, G. (2010). Wealth, Income, and Power: The Wealth Distribution. Retrieved from http://whorulesamerica. net/power/wealth. html Frank, R. (2008). The Wall Street Journal: The Rich Man’s Michael Moore. Retrieved from http://online. wsj. com/article/SB120371859381786725. html? mod=fpa_mostpop