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A derivative is a financial instrument that has a value determined by the price of something else, such as options. The crucial idea behind the derivation was to hedge perfectly the option by buying and selling the underlying asset in just the right way and consequently "eliminate risk" (Ray, 2012). The derivative asset we will be most interested in is a European call option. A call option gives the holder of the option the right to buy the underlying asset by a certain date for a certain price, but a put option gives the holder the right to sell the underlying asset by a certain date for a certain price. The date in the contract is known as the expiration date or maturity date; the price in the contract is known as the exercise price or strike price. The market price of the underlying asset on the valuation date is spot price or stock price. Intrinsic value is the difference between the current stock market price and the exercise price or simply higher of zero. American options can be exercised at any time up to the expiration date. European options can be exercised only on the expiration date itself. (Hull, 2012). For example, consider a July European call option contract on XYZ with strike price $70. When the contract expires in July, if the price of XYZ stock is $75 the owner will exercise the option and realize a profit of $5. He will buy the stock for $70 from the seller of the option and immediately sell the stock for $75. On the other hand, if a share of XYZ is worth $67 the owner of the option will not exercise the option and it will expire worthless. In this case, the buyer would lose the purchase price of the option. One of the…...

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...Universidad Iberoamericana Cálculo Vectorial Avanzado “Modelo de Black-Scholes y las Opciones Europeas” Mtra. Teresa Martínez Palacios Luisa Adame Elías México, D.F., a 8 de mayo de 2012. Modelo de Black-Scholes y las Opciones Europeas Resumen La finalidad de este trabajo es entender el Modelo de Black-Scholes-Merton. Este método es el que se utiliza con mayor frecuencia para la valuación de opciones europeas en el mercado de derivados. Para poder comprender de mejor manera explicaremos de manera básica el mercado de derivados así como las opciones europeas. Conoceremos parte de la historia del método de Black-Scholes, obtendremos su ecuación diferencial parcial y conoceremos como aplicarla. También veremos un aplicación práctica donde alguna empresa muestra la manera en que llego a utilizar este tipo de opciones. Introducción Un derivado es un instrumento financiero que asegura el precio a futuro de la compra o venta sobre un activo (llamado activo subyacente), para prevenir o adelantarse a las posibles variaciones al alza o a la baja del precio que se generen sobre éste. Su principal característica es que son dependientes al valor del activo subyacente. Por ejemplo, el precio del oro, del petróleo (en el caso de commodities), o de acciones, índices bursátiles, tasa de interés, valores de renta fija, etc. (en el caso de instrumentos financieros). Entre los derivados más utilizados en el mercado se encuentran los......

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... 2 let u=z- , then e 1 z 2 2 dz du 1 and z = u= and z=- u=- - =- + dz Now we make the substitution u z and the integral becomes u 2 1 12 1 z 1 z 2 u z du u 1 1 2 2 dz dz e e e 2 du dz 2 2 u 2 z 12 u 2 but the function f u e is symmetric about zero, i.e. f u f u it follows that 12 12 u u 1 1 2 e du 2 e 2 du N 2 hence we obtain 1 t 2 1 1 z2 t z 2 e dz e 2 N as claimed. e 2 7 Theorem (Key Result for option pricing): Let V ~ log normal m, s 2 so that the standard deviation of the log of V is s var log e V Then the expectation E max V K ,0 is given by the formula E max V K ,0 E V N d1 K N d 2 where 1 log e E V / K s 2 2 d1 s 1 log e E V / K s 2 2 d s d2 1 s 8 proof Since V is lognormal we have E V e 1 m s 2 2 1 It follows that ln E V m s 2 2 1 and that m ln E V s 2 2 the variable z ln V m s has the standard normal distribution because ln V N m, s 2 The variable V can be written as V exp ln V exp sZ m We can express the expectation as an......

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...Individual’s Assignment Atlantic Computer: A Bundle of Pricing Options As Atlantic computer was largest manufacturer of servers and other hi-tech product, Jason Jowers has been assigned the task of developing the pricing structure for the Atlantic Bundle, a unique combination of the TRONN server along with the software tool - Performance Enhancing Server Accelerator – called PESA. The TRONN server has been specifically designed to address the current US market demand. In conjunction with the PESA, the TRONN ‘s performance capacity is four times faster than standard speed. Atlantic has continued to hold a significant portion (20% revenues) of the high-performance sector, but as the continual growth of the internet reached new heights, the demand of a basic server increasing rapidly. Hence, in order to meet the demand of market, the Atlantic company is planning to launch a basic server TRONN with a software tool PESA which will grow up the efficiency of server approximately four times. First of all, apart from choosing the suitable pricing method for Atlantic computers, the broad of this company also need to consider about the lifecycle of their product. Basing on the case of IBM HTTP Server or The server products of Microsoft, it seems that the lifecycle of high tech products is decreasing rapidly nowadays. According to some experts in this field, they believe that the average life expectancy of these products is around 3 or 4 years depending on many factors which......

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...determinants of the option price in the Black-Scholes option pricing model for European options is likely to change the price of a call option. A derivative is a financial instrument that has a value determined by the price of something else, such as options. The crucial idea behind the derivation was to hedge perfectly the option by buying and selling the underlying asset in just the right way and consequently "eliminate risk" (Ray, 2012). The derivative asset we will be most interested in is a European call option. A call option gives the holder of the option the right to buy the underlying asset by a certain date for a certain price, but a put option gives the holder the right to sell the underlying asset by a certain date for a certain price. The date in the contract is known as the expiration date or maturity date; the price in the contract is known as the exercise price or strike price. The market price of the underlying asset on the valuation date is spot price or stock price. Intrinsic value is the difference between the current stock market price and the exercise price or simply higher of zero. American options can be exercised at any time up to the expiration date. European options can be exercised only on the expiration date itself. (Hull, 2012). For example, consider a July European call option contract on XYZ with strike price $70. When the contract expires in July, if the price of XYZ stock is $75 the owner will exercise the option and realize a......

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...Finance Std: xxxxxxxxx Introduction Asset pricing models are very useful tools in calculating the risk and their respected return for the investors and they are being widely used by financial analyst. From different theories we can determine the value of assets into three steps i.e., Expected Cash Flow, number of periods and the expected rate of returns. Investors have several questions before investing his money in any stock or in any other commodity that is what should be the accuracy of prices of selling or buying the stocks, what could be the risk, what are the factors should be considered that ignores uncertainty and the expected returns of the stock. The Capital Asset Pricing Model (CAPM) and Arbitrage Pricing Theory (APT) both are well known pricing model determines the risk factor for analyzing the appropriate returns for the investors in their own unique ways. CAPM model uses the whole market environment as one factor but on the other hand APT uses five different economics factor which is more detailed in describing risk which accelerates for these factors. The adoption of CAPM is in practice but other hand its various criticisms are documented on it as well and academics are working on the new approaches of it such as APT and others is discussed in later paragraphs. In this assignment I will discuss the assumptions of CAPM and APT model and their pros and cons and the limitations of CAPM over APT models. CAPM and its Shortcomings Hary Markowitz......

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... Jon M. Huntsman School of Business Master of Science in Financial Economics August 2013 Pricing and Hedging Asian Options By Vineet B. Lakhlani Pricing and Hedging Asian Options Table of Contents Table of Contents 1. Introduction to Derivatives 2. Exotic Options 2.1. Introduction to Asian Options 3.1. Binomial Option Pricing Model 3.2. Black-Scholes Model 3.2.1. Black-Scholes PDE Derivation 3.2.2. Black-Scholes Formula 1 2 3 4 4 5 6 7 3 3. Option Pricing Methodologies 4. Asian Option Pricing 4.1. 4.2. 4.3. 4.4. Closed Form Solution (Black-Scholes Formula) QuantLib/Boost Monte Carlo Simulations Price Characteristics 8 8 10 11 14 5. Hedging 5.1. Option Greeks 5.2. Characteristics of Option Delta (Δ) 5.3. Delta Hedging 5.3.1. Delta-Hedging for 1 Day 5.4. Hedging Asian Option 5.5. Other Strategies 6. Conclusion 16 17 17 19 20 22 25 26 27 32 34 Appendix i. ii. iii. Tables References Code: Black-Scholes Formula For European & Asian (Geometric) Option 1 Pricing and Hedging Asian Options 1. Introduction to Derivatives: Financial derivatives have been in existence as long as the invention of writing. The first derivative contracts—forward contracts—were written in cuneiform script on clay tablets. The evidence of the first written contract was dates back to in nineteenth century BC......

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...Three different methods of option pricing The three different methods of option pricing are: The Black-Scholes model, binomial trees and Monte Carlo Simulation. The three different methods of option pricing are: The Black-Scholes model, binomial trees and Monte Carlo Simulation. The three different methods of option pricing are: The Black-Scholes model, binomial trees and Monte Carlo Simulation. The three different methods of option pricing are: The Black-Scholes model, binomial trees and Monte Carlo Simulation. The three different methods of option pricing are: The Black-Scholes model, binomial trees and Monte Carlo Simulation. The three different methods of option pricing are: The Black-Scholes model, binomial trees and Monte Carlo Simulation. The three different methods of option pricing are: The Black-Scholes model, binomial trees and Monte Carlo Simulation. The three different methods of option pricing are: The Black-Scholes model, binomial trees and Monte Carlo Simulation. The three different methods of option pricing are: The Black-Scholes model, binomial trees and Monte Carlo Simulation. The three different methods of option pricing are: The Black-Scholes model, binomial trees and Monte Carlo Simulation. The three different methods of option pricing are: The Black-Scholes model, binomial trees and Monte Carlo Simulation. The three different methods of option pricing are: The Black-Scholes model, binomial trees and Monte Carlo Simulation. The......

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...Valuing Stock Options: The Black-Scholes-Merton Model Chapter 13 Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright © John C. Hull 2013 1 The Black-Scholes-Merton Random Walk Assumption Consider a stock whose price is S In a short period of time of length Dt, the return on the stock (DS/S) is assumed to be normal with: mean m Dt standard deviation s Dt m is the annualized expected return and s is the annualized volatility. Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright © John C. Hull 2013 2 Why can we say that? Assume that the Normal(m,s2) annual return is made up of the sum of n returns of shorter horizons (eg. monthly, weekly): m E ( ri ) E (ri ) nE (ri ) i 1 i 1 n n 2 n n thus E (ri ) m / n thus Var (ri ) s 2 / n s Var ( ri ) Var (ri ) nVar (ri ) i 1 i 1 We have n=1/Dt intervals of length Dt in a year (eg. for monthly n=1/(1/12) = 12 intervals of length 1/12 of a year), therefore: E (ri ) m / n m / (1/ Dt ) mDt Var (ri ) s 2 / n s 2 / (1/ Dt ) s 2 Dt Sigma(ri ) s Dt Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright © John C. Hull 2013 3 The Lognormal Property These assumptions imply that ln ST is normally (Gaussian) distributed with mean: ln S 0 (m s 2 / 2)T and standard deviation: s T Because the logarithm of ST is normal, the future value or price (at time T) of the stock ST...

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...Microsoft Stock Options, we started by examining the value by using the Black-Scholes Valuation method. We were given that the time period would be T=6 years, and that both the strike price (K) and the Stock Price (S0) were equal to 66.625. When examining the case provided, it gave the data for both volatility and the risk-free rate in 2000. The volatility for this case in 2000 was 0.33 and the risk-free rate was 6.20%. Next, we calculated d1 and d2 in order to evaluate the price of the call option. D1 and d2 were both calculated at 0.63427 and -0.17406, respectively. This was determined by using the formulas to find d1 and d2 using the Black-Scholes method of valuation. Lastly, using the Black-Scholes method, the price of the call following all of these conditions was determined to by $29.31. So, the value of 70 million options is simply 70 million multiplied by the price of the call, $29.31. This was found to be $2,052,045,496.29. This data can be seen in Table 1 in the appendix. Next, I examined the value of the 70 million options when T followed the distribution of: 10% at 4 years, 20% at 5 years, 40% at 6 years, 20% at 7 years, and 10% at 8 years. Following this distribution set, and calculating the summation of all 70 million options at each different time, T, gave a smaller valuation when compared to the value of all 70 million options at one time, T=6 ($2,040,710,000.00 vs all at T=6, $2,052,045,496.29). The following distribution of the call options at......

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...Atlantic Computer: A Bundle of Pricing Options 1. Determine the price for two Tronn servers plus PESA according to the following pricing methods: * Status-quo pricing * Competition-based pricing * Cost-plus Pricing (Hint: footnote # 5) Note: Jowers makes a conservative estimate that two Tronn servers plus PESA equals the performance of four Ontario Zink servers. To calculate the prices you could use the spreadsheet file included in the course content (Week 3). 2. Determine the Value-based price for two Tronn servers plus PESA. Follow these steps use the spreadsheet file included in the course content (Week 3): 1. Calculate the costs of running for a year 4 Ontario Zink serves, 4 Atlantic Tronn servers, and 2 Atlantic Tronn servers with PESA 2. Calculate the annual savings of owning 2 Atlantic Tronn servers with PESA 3. Determine the Value-based price for two Tronn servers plus PESA assuming that 50% of the savings will be passed to the consumer. At the end of the session each team should raise their card with an answer. 4. How is Cadena’s sales force likely to react to your recommendation? What can Jowers recommend to get Cadena’s hardware-oriented sales force to understand and sell the value of the PESA software effectively? In conclusion, based on our analysis we recommend using Competition Based Pricing because this approach acknowledges competitor prices and gives superior services at a same rate.......

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...financial risk as much as possible when do financial decision. Derivative financial instruments such as options, futures and others have been introduced and more commonly used to manage financial risk for improving decision making in this dynamic competitive environment. Options are defined as securities which one party has the right (no obligation) to buy or sell underlying assets with certain price within a certain/specific period of time (Hull, 2012). The option can be either call (right to buy) or put (right to sell) in the form of American options (exercised any time until expiry date) or European options (exercised on expiry date) as either traded options (standard option contracts) or overt-the-counter options (tailor made options). Due to various choices of options, different option pricing models such as Put-Call Parity, Black-Scholes, Cox-Rubenstein Binominal, Risk-Neutral valuation, the Greeks and others has been developed and applied in current financial market. Black-Scholes Option Pricing Model (BS) BS is designed and introduced by Fisher Black and Myron Scholes in 1973 with the assumptions of the market is efficient, returns are lognormal distributed, no commission or transaction cost is charged, no dividend is paid, no penalties to short selling, terms of European option is used, interest rate is remained constant and known rate (Black & Scholes, 1973). Thereafter, the assumption of no dividends has been relaxed by Robert Merton in the......

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...Empirically Consistent Model for Stock Price and Option Pricing HUADONG(HENRY) PANG∗ Quantitative Research, J.P. Morgan Chase & Co. 277 Park Ave., New York, NY, 10017 Third draft, May 16, 2009 Abstract In this paper, we propose a novel simple but empirically very consistent stochastic model for stock price dynamics and option pricing, which not only has the same analyticity as log-normal and Black-Scholes model, but can also capture and explain all the main puzzles and phenomenons arising from empirical stock and option markets which log-normal and Black-Scholes model fail to explain. In addition, this model and its parameters have clear economic interpretations. Large sample empirical calibration and tests are performed and show strong empirical consistency with our model’s assumption and implication. Immediate applications on risk management, equity and option evaluation and trading, etc are also presented. Keywords: Nonlinear model, Random walk, Stock price, Option pricing, Default risk, Realized volatility, Local volatility, Volatility skew, EGARCH. This paper is self-funded and self-motivated. The author is currently working as a quantitative analyst at J.P. Morgan Chase & Co. All errors belong to the author. Email: henry.na.pang@jpmchase.com or hdpang@gmail.com. ∗ 1 Electronic copy available at: http://ssrn.com/abstract=1374688 2 Huadong(Henry) Pang/J.P. Morgan Chase & Co. 1. Introduction The well-known log-normal model for stock price was......

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...Arbitrage: The Key to Pricing Options by Ed Nosal and Tan Wang A rbitrage is the act of simultaneously buying and selling assets or commodities in an attempt to exploit a profitable opportunity. Although the idea behind arbitrage is fairly simple, it is quite powerful because the ability to exploit such opportunities is needed for markets to operate efficiently. Arbitrage ensures, for example, that buyers and sellers of foreign exchange can be assured that they are getting the “correct” rates for the currencies they are buying and selling independent of the national foreign-exchange markets they happen to be using. When markets are efficient, the prices of the objects being traded reflect their true value. And having prices reflect true values is important in decentralized economies, such as the United States, since it is the relative prices of various goods, services, and assets that determines how many will be produced, how they will be allocated, and how funds will be invested. If prices did not reflect true value, then the resulting allocation of goods, services, and investment would not be, in general, economically efficient. This Commentary focuses on a particular episode in which the recognition of an arbitrage “opportunity” made financial markets more efficient. It wasn’t a chance to make a profit that got noticed, it was the way the principles of arbitrage could be applied to the problem of correctly pricing options. Once......

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...Major Findings: This paper examines the problem of pricing a European call on an asset (Stock) that has a stochastic or variable volatility. Addressing this problem was done by investigating two cases: the first case is to determine the option price when the stochastic volatility is independent of stock price. The second case is to determine the option price when the stochastic volatility is correlated with the stock price. This paper provides a solution in series form for the stochastic volatility option, in addition to a discussion about the numerical methods that are used to examine pricing biases, and an investigation about the occurrence of the biases in the case of stochastic volatility. As for the results obtained, this paper presents interesting results for each of the two cases. When the stochastic volatility is independent of stock price, the results show that the price calculated using Black-Scholes equation is overestimated for at-the-money options and underestimated for deep in-and out-of-the-money options. This overpricing takes place for stock prices within about ten percent of the exercise price. Moreover, it is shown that the degree of the pricing bias can be up to five percent of the Black-Scholes price. For the second case when the stock price is positively correlated with the volatility, the results show that the Black-Scholes formula overprices in-the-money options and underprices out-of-the-money options. On the other hand, when the stock price is......

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