Suitability Of Black Scholes Model And Pricing Derivatives Finance Essay - Free Essay Example

Sample details Pages: 9 Words: 2713 Downloads: 7 Date added: 2017/06/26 Category Finance Essay Type Research paper Did you like this example? Since the Black-Scholes (B-S) Model was proposed, it became a widely used pricing model in the options market. This paper critically discusses the suitability of using the Black-Scholes model for pricing derivatives from two points: its own accuracy and the accuracy of input data. Finally, it is safety to conclude that the B-S pricing model is only the best tool currently. 1 Introduction of the Black-Scholes (BS) option pricing model Since the option firstly came into the market in1973, it became one of the best choices among derivatives for investors to invest, speculate and hedge. Then with the option being extensively and fruitfully applied, a lot of models for pricing are proposed by many researchers after in-depth study and exploration, such as the Black-Scholes (B-S) Model (Black and Scholes, 1973), the Binomial Pricing Model (Chalasani,1999 and Lee, S. Park, H., and Jeon, 2007), Monte Carlo Simulation (Rubinstein, 1981), Finite Difference Method and so on. And the most influential one is Black-Scholes (B-S) model created by Fisher Black and Myron Scholes (1973). It has already been considered as one of the most successful models in applied economics. Based on the assumptions that stock prices follows a geometric Brownian motion and the logarithm of stock prices obeys normal distribution, a portfolio including a stock and its derivative is constructed. The proceeds of two positions in the portfolio are highly negative correlated and the stock earnings (loss) are always offset by derivative securities losses (gains). As the portfolio is risk-free, the yield is equal to the risk-free interest rate in the case of Risk-free in a small time interval. Therefore, the present value of the portfolio is determined by the risk-free rate and the duration. The BS model is as follows: In this expression, ÃÆ'Ã… ½Ãƒâ€šÃ‚ ¼ is the instantaneous expected return on the stockError: Reference source not found is the instantaneous volatility of the return, and z (t) is a standard Brownian motion or a Wiener process and S is the underlying asset. According to the model, the Black-Scholes equation, was derived by setting an instantaneous riskless portfolio composed by appropriately weighted stocks, options and bonds. The B-S models specific pricing formulas are as follows: Don’t waste time! Our writers will create an original "Suitability Of Black Scholes Model And Pricing Derivatives Finance Essay" essay for you Create order , for the call option, and , for the put option, where Where R is the constant risk-free interest rate and N(x) is the normal cumulative density function, K is exercise price, Error: Reference source not found is the standard deviation of stock returns, T is the time to maturity options. According to Bruno Dupire, implied Black-Scholes volatilities strongly depend on the maturity and the strike of the European option under scrutiny. Then, this problem was solved easily by Merton in 1973. That makes the BS model more applicable. Just like what Black had pointed out during his lifetime, the B-S model for option pricing should really be called Black-Merton-Scholes model. 2 The Suitability of the Black-Scholes (B-S) Model The output accuracy of any theoretical pricing model depends on the exactness of input and the model itself. Therefore, perfect combination of accurate models and accurate input data creates perfect result. Both are indispensable. 2.1 The accuracy of the model itself The accuracy of Model assumptions essentially determined whether the model is a perfect description of the real world. In deriving the B-S model, many important assumes are used: 2.1.1 Market is frictionless market with transaction efficiency (Black and Scholes, 1973) Under this assumption, in essence, it implies that the underlying assets can be freely traded without any restrictions. Everybody is free to borrow funds at the same rate. Undoubtedly, it does not take the impact of taxes into account. However, the markets operation is not without friction in the real world. In some markets, such as Chinas stock market, the underlying asset cannot be traded freely, because there are restrictions on daily change limits. Whats more, there is also a complete ban or limits on short selling in some of the stock market. And sometimes the proceeds from short selling cannot be fully used. Wherefore, if the freedom of short selling is not given, the demand of securing the short equity using the put option will be more intense. This may result in a higher price of the put option than the call option. While options trading are not taxed, almost no one option investment will be better than another under the influence of tax policy issues. Thus, most traders do not usually consider the tax implications. In the real world, traders cannot borrow unlimited funds. If the funds can be borrowed for free, traders can always borrow money, and then put them into the Options Clearing House. Since the lending rates are equal, the Options Clearing House should also pay the interests because of the initial performance bond (margin). Therefore, obtaining the initial performance bond is not a problem. However, traders may be forced to close the contract before expiration of the option because of not making up the maintenance margin. Even if borrowing unlimited funds, traders have to face different interest rates on loans, always higher than lending rates. Finally, if the difference between the rates of borrowing and loading is more obvious, pricing models offer less reliable data. The transaction cost in the options investment is another factor which has to be considered as well. Broker commissions, clearing fees and membership fees and market makers bid-ask spread are all transaction costs. Once considering transaction costs, many investment portfolios theoretically acceptable are often not feasible. 2.1.2 The Stock Pays no Dividends (Black and Scholes, 1973) The B-S model assumes that the stock pays no divides or other distributions. This is clearly an unrealistic assumption. Payment of dividends will increase the intrinsic value of the put option and reduce the value of the call option. As there are protective measures for options paying dividends in the early OTC (Over-the-Counter) market. For example, the adjustment of exercise price is used to eliminate the impact of dividends paid on the option value. But the current contractual terms o f stock options will not be adjusted with cash bonus payment correspondingly in floor trading and the OTC market. Finally, the dividends become an important factor pricing the options. 2.1.3 The Short-Term Interest Rate is Constant through Time (Black and Scholes, 1973) The interest rate is assumed to be risk-free and fixed in the B-S pricing model. In most markets, risk-free interest rate is the interest rates on government securities. However, the actual market risk-free interest rate is usually not fixed but variable. Because the impact of changes in interest rates is a function of the options duration, and most of all listed options contract period of less than 9 months, it is not sufficient to constitute a significant impact on value, unless the interest rate changes to a large extent and the actual value is high. Therefore, the interest rate is not an important factor compared to the underlying asset price or price volatility. But this does not mean that traders can comp letely ignore the possibility of fluctuation in interest rates. Particularly, after the long-term stock options were introduced into the market, the interest rate impact is more and more important. 2.1.4 The variance rate of the return on the stock is constant (Black and Scholes, 1973) The B-S option pricing model assumes that when option expires, the standard deviation of expected returns of stock remain unchanged. That means that the future stock price volatility is constant. But actually, stock price volatility is influenced not only by stock prices, but also by the time the option expires, and other factors. It cannot remain constant in the life of the option. If the stock price volatility associated with stock prices, the stock price may deviate from the lognormal distribution in the B-S pricing model (Corrado, and Miller, 1993). Then pricing error may exist according to the B-S pricing model. If the stock price and volatility are positive correlated, the B-S pricing mode l will tend to underestimate the value of the call option in the virtual the state and overestimate the value of the state of the put option in the virtual the state. When the stock price goes up, the volatility is also rising. That means higher stock price appears in a higher probability in the geometric Brownian motion. Correspondingly, when stock prices fall, the volatility falls down. That means lower stock prices arise in a less probability in the geometric Brownian movement. In contrast, if the stock price and volatility are negative correlated, the B-S pricing model will tend to overestimate the value of call option price in the virtual the state and underestimate the value of the price of a put option in a virtual state for the following reasons. When volatility declined with stock prices rising, a high stock price is difficult to achieve. When volatility increased with stock prices falling, a very low stock price is easy to achieve. 2.1.5 Underlying Asset Price Changes i n a Continuous Manner (Black and Scholes, 1973) The underlying asset price changes have the following three forms (Thomas, Copeland, Fred Weston Kuldeep Shastr. 2010): diffusion form, beating form and diffusion in the form beating. In the diffusion form, the price changes in a continuous smooth manner, such as the changes in temperature, a typical spread in the form. In pure beating form, the prices remained unchanged during a period and then instantly jump to another price, such as changes in interest rates set by Central Bank in China. The same situation continues to happen. The form of beating diffusion is the beat of the combination of the form of diffusion and the forms of beating. In addition to the occasional beating in the price, it changes in a continuous smooth way in general. The BS pricing model assumes that stock price movements are spread in the confusion form, and the transaction will be continuous forever. There is no the price gap existing. Obviously, it is a kind of convenient but not precise form of the hypothesis. The option price of the underlying asset does not render the proliferation of forms in the real world as the Exchange will not open 24 hours a day. There is a closing price at the end of each trading day. The next days opening price is not necessarily equal to the closing price the day before. This is clearly will cause the price spread of the gap and is not allowed by the confusion form. Even in normal trading hours, the proliferation of forms of assumption may not be set up. Once the big news is released in the market, it may result in the price gapped up or down suddenly. 2.1.6 Logarithm of the Stock Prices Obey Normal Distribution at Maturity (Black and Scholes, 1973) The B-S pricing model assumes that stock prices have a lognormal distribution. It means that the logarithm of stock prices conform to normal distribution when the option expires. Actually, the stock price is not strictly logarithmic normal distributio n. The left tail of its distribution curve is a longer length or the right tail rather long and the broad peak is flat or tall. The kurtosis and skewness of its distribution curve is not necessarily zero. If the logarithmic of stock price is not accurate normal distribution at the end of option, it may generate pricing bias. Therefore, it is the BS model that does not depict the world perfectly in some degree. In response to these shortcomings, the options researchers conducted study and exploration of the B-S model in-depth and put forward many amendments to the pricing model. There are a series of amendments (Bakshi. Cao and Chen, 1997), such as the one proposed by Merton in considering of the effects from bonus in 1973, the American call option pricing model made by Roll in 1977, the futures option pricing model established by Black in 1976, the random interest rate pricing model set up by Moton, the stochastic volatility model created by Hull and White in 1987 and the pure ho pping model built by Cox and Ross and so on. Although the extended model is closer to the actual situation in the real world, the increased variables make the complex mathematical structure is more difficult to understand. The accuracy of the input data is also harder to ensure. However, since each of those above models is just an extension of a part of many defects in the model, no model has yet been sent out to overcome all the shortcomings. Even if someone tries to develop such a model, there is nothing but only a bunch of complicated mathematical formula with no practical value will be got. 2.2 The accuracy of model input data Based on the B-S option pricing model, the stock price, exercise price, option period, risk-free interest rate and stock price volatility all are the model input variables. Apart from the volatility, the remaining four variables can be observed directly in the market. Thus, the accuracy of the input data is mainly determined by the volatility (James. Doran. Ehud. Ronn. 2005). Usually, there are two ways for estimating volatility: calculating the standard deviation of returns based on historical stock price data and weighting average on the implicit volatility implied in the market price. In spite of the volatility compared to the data which the model requires may have some bias when the option expires, it is a good approximation. After all it is impossible to get an accurate data on the future. 2.3 The Suitability of the Black-Scholes (B-S) Model Although the B-S pricing model assumptions cannot perfectly describe the real world with many drawbacks, it is still widely used in practice. The reason is that the model is not only easy to understand, but also the model input variables are relatively simple. To some certain extent, this ensures the accuracy of input data. In the practical application process of the B-S pricing model, the actual employees can adopt some simple extension models to overcome its shortcomings. For the random price changes, more trading techniques are utilized to overcome the problem of pricing bias rather than the use of the more complex extended model. Usually, the trading techniques are that taking different pricing volatilities for the different price and different maturity options. There are three specific operating methods as follows: 2.3.1 The Curve of the Exercise Price of Volatility According to the different option exercise prices, actual practitioners calculate the corresponding implicit price volatility and drawn the volatility curves with the changes of the option exercise price. If the curve is concave, it were called the volatility smile curve; If it is convex, it is known as the frown curve. According to the implementation curve of price volatility, it is impossible to estimate the different option exercise prices with different volatilities for the same stock. 2.3.2 The Structure of Volatility Period Actual practitioners can also draw the curve of the structure of volatility period based on the implied volatility. The curve reflects the relationship between the volatility and options expiration time. In the light of the curve, it can price the different option exercise prices with different volatilities for the same stock to show the volatility changes under it duration. 2.3.3 Volatility Matrix A coordinate of the volatility matrix is the strike price and the other coordinate is the time to maturity. The data in matrix are the implied price volatilities calculated from the BS pricing model. If a specific execution price and the option price in expiration date cannot be directly observed from the market, the options implied price volatility can be determined by linear interpolation. When there is a need for a new valuation of options, the corresponding strike price and the implicit pricing volatility in expiration date can be found from the matrix. Actually the relationship between the volatility structure and its changes with the exercise price are taken into account in the volatility matrix. 3 Conclusions Although the B-S pricing model is not very accurate, it is better than other option valuation methods and is still an indispensable trading analysis tool. Most of options traders think that the deficiencies of the BS pricing model should be offset by trading experience rather than more complex models. Owning the B-S pricing model in options market just likes holding a candle into the dark room. Sometimes the flickering candlelight may lead us to judge wrongly. With more and more study and research, there will be more appropriate option pricing model in the future than the B-S pricing model undoubtedly.

The Hippie Subculture of the 1960s Essay - 1356 Words

The 1960’s was a decade to remember. Hippies, The Draft, Civil Rights, and the Vietnam War were all events that took place in this youthful generation. Drugs claimed numerous lives furthermore the birth rate for young adolescents reached sky high. There was a major turning point on the idealistic life in the sixties its effect tarnished many families.  ¶ The corner of Haight and Ashbury marks the spot of where the hippie subculture began. â€Å" Initially it was a youth movement that started during the early 1960’s and spread all over the world† (Stone, Skip). The word hippie derives from â€Å"hipster which is someone who rejects the established culture† (Web Dictionary). Hippies rejected and established institutions, criticized middle-class†¦show more content†¦Velvet, leather, denim, Indian cotton and silk were all popular fabrics. Op art, paisleys and psychedelic designs appeared on clothes. Hippies had no problem with nudity in their eyes it was totally natural. Footwear ranged from the basic sandal, zippered boots, to platforms and bright patent leather shoes. Walking barefoot to hippies put them in direct touch with the world around them. Accessories included love beads, bandannas, granny glasses, Bangles, rings, earrings, nose rings, and ankle bracelets were worn. Necklaces were adorned with peace symbols, raised fists, and yin-yang symbols. Many hippies did not wear deodorant, cosmetics or perfume of any kind, preferring to keep their body natural. â€Å" Long hair was there declaration of independence and rebellion against the clean shaven U.S army† (Huber, Adam).  ¶ â€Å" Many hippies tried and used plenty of illegal drugs in the 60’s mainly because their lives were filled with hard days to get past so most hippies used drugs to get out of their regular state of mind so they were worry free† (Stone, Skip). Hippies used a variety of different drugs the main drugs that were experimented with is Marijuana, Shrooms, Heroine, and LSD. 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Describe Functionalist and Marxist Perspective on Education. Free Essays

Describe functionalist and Marxist perspective on education. Functionalist Education can be described as the method whereby society continually passes on the knowledge it has gain from pass generations to the next or future generation. It is centered around and entails the teaching of the three R’s, which includes: arithmetic’s, reading and writing but also encompass social, spiritual, moral and physical component. We will write a custom essay sample on Describe Functionalist and Marxist Perspective on Education. or any similar topic only for you Order Now It is viewed as a secondary agent of socialization as it socializes individuals into the norms and acceptable behaviors of society. Functionalist view education as having many roles but are interested mainly in how education meets society needs. They believe that one of its major roles being to equip individuals with intellectual knowledge. Functionalist also believes that besides it most obvious role of passing on society’s knowledge, education also has other latent roles that affects the individual on a holistical level and is reflective of society values, beliefs and norms. According to Durkheim individuals learn to socialize with one another within the schools through the integration of people from all diversity of society and this he believes is one of the latent roles of education, which he also call â€Å"moral education†. Besides this, functionalist believes that education have other latent or subtle roles that are used to instill in individuals values that stem from political and economic system and as a result help them build their self-confidence and prepares them for meeting society demands. For example, independence is taught through individualism. Functionalist also, believe that sorting (separating students on the basis of merit) another function of the school is essential in determine from early on who is capable of doing what and what job they will be better suited to in the future. The next function to take place after this is networking, they believe this function facilitates matchmaking and is unavoidable within the schools setting as individual of background, interest and education comes together to learn. Marxist Marxist sees education as a continuation of functionalist ideologies. They believe that education is used by functionalist to maintain the line of segregation that exists between the classes. Hence the reason why the Marxist view education as being used to create an â€Å"obedient working class† continuing the status quo. Acting as a persistent cause of inequality continuing to maintain the power structures and giving them the tool to create a â€Å"docile† work force. However, it is the view of the Marxist that the education system could be used as a system that helps in sorting the individuals of society into what is their calling position in life, it is the way it is done that their exist a problem. The problems they believe exist in the strategy that is used to by the functionalist to achieve this. This strategy they see as based on the ideals of functionalism which is according to class, race and gender. This is achieved in the form of the hidden agenda that exist within the curriculum that is used to educate. This hidden curriculum Marxist believes is present in every classroom and is taught through the use of the curriculum, hence the reason it is call the hidden curriculum. To the Marxist school of taught the hidden curriculum is present in numerous forms, this is evident in the funding that is available to school in affluent areas. These school according to Marxism are funded by â€Å"property taxes† because of there affluent location. Because of this they can afford to pay higher salaries to their teacher and this give them better teachers, available books and even technology. This in turn cause the students that attend these school to have better opportunities than child children who are schooled in rural areas that do not get the amount of funding that the other school get. Some of theses children are not even given the opportunity to go to college or even complete school. This inequality is also present in the exams that are given to children of the two classes. This is seen in the traditional IQ test that is given to student for acceptance into school. This test is Suppose to test the mental knowledge of the child but the formation of the test is not based on the intelligence of the child but on cultural knowledge. This creates a bias because children from a poor social background will not be able to answer question base on the culture of the affluence classes. This is called Cultural bias and is caused by inequality of the classes. They disagree with the idea that education can be used as a tool for any individual to move up within the social classes that exist in society. They believe that these principles are far from the truth. As religion is used to opiate the people against the real issues that affect them and to help in keeping them from solving their problems so to is education used to discourage the people from helping themselves and to remain within their social classes. Marxist see education as an influential tool in maintaining the lines that separate the have from the have not. Another argument of the functionalist school of taught is the use of meritocracy to help individual achieve status in life. However that Marxist sees this as a false perception, and believes that through meritocracy functionalist portrays the ideals of inequality of opportunity. Marxist believed that it is just another tool that is used to create an unequal society. The hidden curriculum is also present in the socialization process that takes place in school. It is of the view of many sociologists regardless of association or theory that the school is a major institution in the socialization process. It is for this reason that the Marxist believes that functionalist used this as the bases for the continuation of their theories. They see the hidden curriculum in this case as use to instill the norm and values of the functionalist. This is used to tell student what is deem as wrong or right, not allowing room for questioning. The question that Marxist asks is who is to tell what is right or wrong? Whether culturally, academically or otherwise. The hidden curriculum does not stop at school level but it is seen as being continued through out an individual lifetime whether it is in the home, work place or society as a whole. 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Pseudo-palladian elements in English Neo-classical architecture Essay The academic architects of the Burlington circle felt themselves to be the custodians of the tradition formed by Palladio and Inigo Jones, in whose works they believed that they had discovered the eternal rules of architecture. In theory and practice they attempted to restore this great art to its former glory, and it is well known how closely they followed the precepts of their models. We tend to look at their works in the light of their endeavours rather than with an eye on their achievements. In reality, their Palladianism is a good deal more English than is generally realized. These men could neither ignore the development of the previous hundred years in English architecture nor their own national tradition, and it can be shown that they gave a new meaning to almost all the elements which they derived from Palladio: to his planning, to the ‘monolithic’ character of his structures as well as to his orders and detail. Moreover, certain recurrent motives of importance in English academic architecture between 1720 and 1760 arc not Palladian at all, and others occur only as ephemeral experiments in his work. It is with two such motives that this paper is concerned, and its aim is to study their translation from Italian into English idiom, and thereby to throw new light on the movement as a whole. The ‘Venetian window is the one motive which everybody associates immediately with English Palladianism. It consists of three lights, the large central one being arched while the two smaller ones are covered by a straight architrave. The Basilica at Vicenza is the famous example of Palladio’s use of the motive on a grand scale, and from there it received its name of â€Å"Palladian Motive.† In the Basilica a continuous sequence of the motive in two stories, each framed by a large order, screens the mediaeval town-hall. By the regular repetition of the monumental motive the wall is reduced to a minimum, and a rhythm based on the approximately equal alternation of arched and straight parts prevails. In English 18th century architecture the motive was hardly ever employed in this way.1 It is, however, common in a less monumental form for windows and in isolation. Could English architects quote Palladio’s authority for such a usage? Palladio was not at all fond of the â€Å"Palladian† motive for isolated windows and only one precedent exists, the Villa Angarano near Bassano. This was never finished and cannot have been known to Palladios admirers except in the small illustration in his Qpattro libri dellarcki Uttura. Moreover, the villa belongs to Palladio’s juvenilia (1548), and he never returned to the simple type of the three-light window in his later periods. In the Basilica, as well as in the Villa Angarano, Palladio was influenced by Serlio who in the fourth book of his Architecture, published in Venice in 1537, prominentlyillustratcs the motive in the form of a gallery and also as a window in a house front (PI. 42a).4 Serlio, on his part, popularized a conception which had a long pedigTee and was almost consistently in use from the and or 3rd century a.d. onwards.6 Reduced to its essentials the motive is concerned with the bridging of voids between columns, and its novelty consists in the reconciliation of the straight architrave of the Greeks with the arch of the Romans. But it was not before the early 16th century that the motive received a definite rhythm. Its author seems to have been Bramante, judging from the fact that his pupil Dolcebuono used it in a monumental sequence for the galleries of S. Maurizio in Milan , and that it appeared in Raphael’s circle after Bramante went to Rome. After that, we   find it isolated as a single window and employed by Raphael himself in S. Eligio degli Orrfiti (designed 1509) and in his fresco of the Rorgo Fire (1514). Even earlier Antonio da Sangallo the elder had used it as the central feature for his facade of S. Maria dcllc Lacrimc at Arezzo, and during the next 20 years it is frequently used as a window, as well as in the monumental sequence, by Peruzzi, Giulio Romano, Antonio da Sangallo the younger, Girolamo Genga, Cristoforo Solari and many others.4 From this list of names it is evident that the motive had been completely incorporated into the repertory of Italian architects when Palladio took it up at the end of the ’forties. Leaving aside the monumental use of the motive as being unimportant from the English point of view it appears that as a window it was often designed for church facades, where it was brought into a functional relationship with the entrance.3 Only at a later dale, towards the middle and in the second half of the century, docs it occur frequently, in palaces by Vignola, Ammanati, Vasari, Cigoli, Giacomo della Porta, and in the north with Giovanni da Udine, San- micheli, Serlio and Scamozzi.4 It is used to emphasize the centre of the facade and, as with Palladio, it is frequently framed by a large order. Even when this is not the case, as in Giovanni da Udine’s Palazzo della Provincia at Udine, entrance door and â€Å"Palladian† window form a compact group firmly tied together, and the window appears immovably fixed in the surface of the wall. When Inigo Jones introduced this type of window into English architecture, he reverted, not to Palladio, but to Scamozzi for whom he seems to have had a dislike mixed with admiration.6 Scamozzis Idea della Architettura uititersale4 contains a selection of palazzi with this centre motive, all of which are translations of Serlios prototypes into a later style (PL 42b). The â€Å"Palladian† motive appears prominently in most of the drawings for Whitehall Palace (PI. 42c, d),7 and the way in which it is firmly bound up with the wall between tle vertical frames of the big order and the horizontals of two tiers corresponds exactly to its use by Scamozzi. But particularly in the schemes which were used by Colin Campbell and by William Kent there are important differences which were to bear fruit. In the block-shaped Italian palazzo the Venetian window, applied in one or two stories above the entrance door, forms with it a middle axis, the climax of the steady rhythm of the side bays. In the long drawn-out fronts of the Campbell plates symmetrically arranged pavilions alternate with lower receding parts, and the pavilions with the Venetian windows and loggias stand out not only by virtue of their greater height but also, and even mainly, through being conceived as isolated decorative accents (PI. 42d). The receding parts are left plain so that they do not in any way prepare for the new motives in the pavilions. That the Venetian window has been chosen for its decorative and festive quality and not for its intrinsic functional value, can be shown by a further and stricter analysis. In the Italian palazzo it bridged the central bay, the great width of which is due to the entrance door; at the same time, by repeating the arch of the entrance in the middle light and the straight lintels of the windows in the side lights,it had become the ideal point of intersection between the horizontal and the vertical tendencies in   the facade. Most of the Whitehall drawings follow this conception, but those used by Kent arc dif ferent (PI. 42c).1 Nothing can be more revealing for the character of the â€Å"Palladian† motive than the fact that here the Venetian window does not result from a particularly wide bay, and that instead of accentuating an entrance-door it stands above the unbroken sequence of the ground-floor windows. Thus, the Venetian group in the first and second floors appears, from a functional point of view,.as a casual element, and not necessitated by the structural logic of the building itself. Campbell plates show this same use of the motive, whereas it docs not appear in the original drawings from which Campbell derives. Theoretically Palladio demands that â€Å"the void must be over the void and the solid upon the solid,†2 and Scamozzi discussing the question of the â€Å"suitability of windows to the quality of the building states explicitly that windows above one another in different storeys should have the same width.3 Tins point docs not occur in English theory. On the other hand, Isaac Ware, in the most comprehensive architectural treatise in the English tongue,Complete Body of ArcMtectun (1756), says about Venetian windows that they are â€Å"a kind calculated for shew, and very pompous in their nature; and, when executed with judgment, of extreme elegance.’ The en thusiasm for the Venetian window displayed l »y Inigo Jones and his school was short lived ; Sir Christopher Wren used it exclusively—and on very few occasions -in the east end of churches. It is an unsettled question who initiated the widespread use of the Venetian window in the early 18th century. Not a single Venetian window Is to be found in Campbell’s first volume of the Vitruvius Britarmicus, published in 1715. But shortly afterwards the motive appears almost simultaneously in the Vanbrugh-Hawksmoor group and in the Burlington circle. Campbell may have preceded Vanbrugh by a slight margin with his Venetian windows in the facade of Burlington House, designed in 1717 (PI. 43a). But as these windows were not executed before 1719,5 it is likely that Vanbrugh, between 1718 and 1720, hit upon the same idea in the towers of Eastbury and Seaton Delava!. The Venetian windows in Burlington House are framed by a double order as in some of Scamozzi’s designs, and they reveal immediately their kinship with the Whitehall Palace types. The ground floor windows under the three-light windows correspond again with the other windows in the same storey and the â€Å"Palladian† motive appears isolated. Moreover, the application of this major feature not in the centre but in the projecting comer bap, centrifugally one might say and not centripetally, is quite foreign to Italian usage. Finally, the windows are set in the wall in such a way that a comparatively large piece of solid surface is shown above them. In other words, the Italian method, which is to reduce this wall as much as possible, in order to tie the arched top of the window to the entablature above, was not followed here. In spite of the completely different   character of Seaton Dclaval, Vanbrughs use of the motive is very much on the same lines. But it is important that here the Venetian windows arc not framed by an order; they appear as relatively small voids in the plain wall of the upper part of the towers. The next step in the development towards a conventional English use of the Venetian window is Campbell’s large house at Wanstcad which was demolished in 1822. Campbell published two designs for it in the first volume of Vitruvius Britannicus (1715) which both show a six-column portico in the centre and six bap in each of the almost bare wings.1 But between 1715 and 1720 there occurs a characteristic development, and we witness the growth of a style. In the third volume of Vitruvius Britannicus Campbell re-published his second design with the â€Å"addition of the new towers† planned in 17203—each tower decorated with a Venetian window on the level of the main storey (PI. 43b). The tripartite windows are used to emphasize the comers as in Burlington House; but they arc loosely placed in a large empty wall without a ‘framing’ order as at Seaton Dclaval. Vanbrugh again follows with very similar Venetian windows in the wings of his last work, Grims- thor pe Castle, designed c. 1723. But from a neo-classical standpoint the final design of Wanstcad is more progressive, for wc find here for the first time equal emphasis placed on the Palladian temple motive of the centre and on the Venetian windows of the corners, a trinity which very soon was to become the accepted pattern of such facades. Two years after Wanstcad it was taken up, and elaborated by Campbell in the design for the west front of Houghton Hall (PI. 43c). The string-course between the Venetian windows and the small windows above them is dropped, with the result that the two windows of each side-pavilion form a decorative configuration on the large plain surface of the wall. This configuration should be seen together with the window of the rusticated ground-Hoor below and that of the tower above the comice,5 and be compared with the central axis of a Scamozzi design. Then it will be realized what a difference separates a conception in which every architectural member is charged with functional energy fr om one which tends towards the arrangement of linear patterns on a surface. The grouping of Wanstead and Houghton was repeated most notably in Kent’s south side of Holkham (1734) and Flitcroft’s Woburn Abbey (1747). Alderman Bcckford’s house at Fonthill, Wiltshire, was an almost exact copy of Houghton, combining features of Campbell’s original design with elements of the structure as carried out by Ripley. The same model was followed by Sanderson Miller in the garden front of Croome Court in Worcestershire (r. 1750); but the proportions were changed by replacing the rustic ground-floor by a basement (PI. 43d).In the Duke of Bedford’s scat, Stratton Park in Hampshire, the type was reproduced without the mez- zanine.10 Gopsal in Leicestershire, on the other hand, has three storeys in its middle block, and only two in the strongly projecting comer-wings with the Venetian windows, the result being a complete lack of relationship between the main block of the building and the wings. These must be   regarded as the frame of a picture (PI. 43c).1 The appeal of this type was so strong that even Robert Adam succumbed to it in the Register House at Edinburgh (1769). In all these cases the variations on the archetype are slight. In other examples, however, the departure from it is more considerable without its essential features being sacrificed. In Isaac Ware’s Wrotham Park (c. 1 754) the Venetian windows frame the tetrastyle feature, and the constituent trinity—portico and Venetian windows is closely joined (PI. 44a). Isaac Ware used the same arrangement in another of his designs,and so did lib greater suc cessor, Robert Adam. Applying it in his designs for Witham Park, Somerset, for the completion of King’s College, Cambridge, and for the main block of Stowe (south front, PI. 45g) he gave it new life.6 These last four projects have a peculiarity with which so far we have not met: the Venetian windows appear under a relieving arch. This conception has its own long and interesting history. The idea of placing a small arch within a large one was an old Italian device which Renais sance architects had found in Roman Thermae and other classical structures. It became an ever recurrent theme from the days of Brunelleschi’s Cappclla Pazzi. Its development and fate cannot here be followed up; it must suffice in this connection to say that the motive is extremely rare in its application to the Venetian window, and it seems probable that Palladio was the first to use it in this combination. The only executed example occurs—with a fascinating simplification—in the entrance door to his Villa Pojana (r. 1560),* and the consequences for later Italian architecture remained negligible. But Lord Burlington possessed a number of Palladian drawings7 which exploited the theme in various directions. Of these drawings, the one which Lord Burlington himself copied in General Wade’s house, is perhaps the mo st important for England (PI. 44b). The relieving arch applied to the Venetian window made it possible for Palladio’s design to attain a measure of concentration and unification which far surpassed Scamozzi’s achievement. The large arch of the window almost corresponds to, and repeats, the arch of the entrance, and the small arch is of identical height and width to the relieving arches of the other windows. This type of Venetian window helped to create a structure of unsurpassed lucidity. On the other hand, the use in both stories of rusticated arches, between which the windows appear set in a smooth wall surface, reveals another tendency. The rustication is attached to the wall like a strengthening scaffolding. The two surfaces which thereby become visible on different levels give this system an ambiguity which occurs only in the Mannerist phase of Palladio’s develop ment, and it is characteristic that the idea had come to him from Giulio Romano. Thus, when this drawing with its combination of rustic and smooth surfaces was used in England, architects were following a Palladio who, from a classical point of view, was least himself. Is drawing exerted an extraordinary influence on English architecture, not only through the copy of it embodied in General Wade’s house which became the prototype of a number of houses on this and the other side of the Atlantic, but also throug h the transformations which it underwent in the course of time. These reveal that neo-classical architects had no eye for the tension of this design. William Kent’s Treasury and Horse Guards provide proof of it. In the top storey of the Treasury project (PI. 44c)2 the old pattern -tetrastyle motive and Venetian windows was combined with the rustication and the relieving arches from Palladios drawing. But without Palladios Doric order the rustication has considerably gained in importance and appears as a continuous wail, and not as applied framework. The decorative character of the rusti cated surface with the hollows for the Venetian windows is still more striking in the Parade side of the Horse Guards (PI. 45c). In both these buildings another idea appears which was not Kent’s own. The Venetian windows are not confined to the comer pavilions but one is also used in the centre. With the abandoning of the tetrastyle feature in the Horse Guards the three Venetian windows remain the chief accents of the facade. Behind this conception lies another Palladian project. Burlingtons collection con- tained a design with three recessed Venetian windows in a plain wall (PI. 45a); this design was quite exceptional for the South, it was probably never meant to be executed, and nothing like it is to be seen in Italy. It is significant for Lord Burlingtons own development and the severe turn in English neo-classicism that a few years after the building of General Wade’s house the Earl applied this arrangement to a building intended for liimsclf. He used it for the garden front of his villa at Chiswick (PI. 45b).4 The relieving arches of the Venetian windows appear here as if cut out of the flat wall with a knife, and the strongly linear character of the design has a rest raint not to be found in Houghton and its derivatives. The triple emphasis on the centre and the sides was now, perhaps for the first time, achieved with the same â€Å"Palladian† motive, and it is this peculiar and essentially un Italian arrangement which had such a strong appeal in this country. Kent applied this system in the north front of Ilolkham and anticipated here in a plain wall the Parade front of the Horse Guards.5 This makes it obvious that the employment of a different surface medium did not necessitate a great change of design. The garden front of Chiswick formed also the basis of Isaac Ware’s design for the Mansion House,5 and derivatives may be found in a long series of buildings such as Vardy’s street front of Spencer House,7 Paines house for Sir Matthew Fcatherstonchaugh in Wrhitchall (PI. 45c)8 or Robert Adam’s Gosford House, East Lothian. But the design for the park front of Sir Matthew’s house which was published by Paine but not executed shows an interesting version of the three-window type. Each window is framed by large Corinthian double columns, and the wall has shrunk considerably, so that a continuous sequence of windows replaces the isolated accents of earlier designs. We are much nearer a genuine Italian conception than in the earlier phase of nco-classicism and, indeed, the design was inspired by another Palladian drawing in Burlingtons collection (PI. 43d)1 which, characteristically, was not used in Burlington’s own generation. Robert Adam relumed to it in the design of the side blocks of Stowe and Kedlcston. In dealing with the use of the Venetian window in two important house types represented by the west from of Houghton and the garden front of Chiswick, this survey leaves out the innumer able buildings in which the Venetian window, more or less loosely connected with these types, plays an important part right down to the end of the 18th century. It appears in the centre alone under a tetrastyle motive and without it, in the wings alone with or without any middle accent, and in both forms, simple and recessed. Yet in almost all these fronts the same law of a decorative relationship between wall and voids remains binding. Quite often such aberrations from the main types are occasioned by the shape and size of the building, and in a small and high front like Isaac Ware’s house in South Audlcy Street, London,4 three variations of Venetian windows one above the other form the only axis. Moreover, this motive was so completely a bsorbed in England that it sank down from the level of ‘high art’ and was widely used as a decorative feature of popular architecture.