Submitted By hussein99

Words 797

Pages 4

Words 797

Pages 4

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 negatively correlated with the volatility, the Black-Scholes formula overprices out-of-the-money options and underprices in-the-money options.

Although these results are obtained for European call option, they can be directly used for European put options by applying the put-call parity equation. Furthermore, these results can be also applied for American call options on…...

... 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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... 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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...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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...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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...How might you use options to protect yourself against the risk of a price decline? How might you “subsidize” some or all of the cost of protection? Comment on the risks of your chosen “subsidy” strategy. 1) If we are concerned about the possibility of bad news, to protect ourselves we would buy puts to insure our stock. The put price on Feb. 1994 is $1.125 | ST<$55 | ST >$55 | Value of stock | ST-$55 | ST-$55 | Value of buying put | $55- ST – $1.125 | -$1.125 | Total Value | -$1.125 | ST-$56.125 | 2) If I want to subsidize the cost of buying put, I can write some call. The call price on Feb. 1994 is $1.375. If writing $1.125/$1.375=0.818 shares of call, we can subsidize the cost completely. | ST<$55 | ST >$55 | Value of stock | ST-$55 | ST-$55 | Value of buying put | $55- ST – $1.125 | -$1.125 | Value of writing call | 0.818*$1.375 | 0.818($1.375+$55- ST) | Total Value | 0 | 0.182 *(ST-$55) | 3) I sacrifice a part of potential upside return to subsidize the cost of buying puts QUESTION 4 Compare the prices of options on Lotus and AT&T. Why are options with identical strike prices and maturity dates written on stocks with identical prices selling for different prices? Do options on one of these two stocks provide investors with superior investment opportunities? If all else equal, volatility determines price * According to Black Scholes, volatility is the only variable that affects option pricing, given all other......

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...these differences, GV allows customers to self-select the right option for themselves, signaling their willingness to pay. As a result, this strategy not only incentivizes customers who can go to the movie on weekdays to attend lower-demanded shows, but also extracts more revenue from customers who are only able to attend popular timeslots. Example: Ticket of Interest Mon-Thu Fri Weekend and Public Holidays Before 6pm After 6pm 2D Annabelle GV Plaza $8.50 $9.50 $12.50 $12.50 c) Benefit-based Pricing i) Cinema Versioning: Currently GV offers five different versions of cinemas: Standard, GV Gold Class, GVmax, D-box and Gemini . By offering customers different options with different benefits, GV can encourage those customers seeking greater utility to purchase ticket of the higher-priced cinemas like Gold Class, while those with minimal utility requirements and lower WTP can buy ticket of the standard cinema. ii) Movie Versioning: 3D movies give customers a more realistic experience, thus, are priced higher than 2D movies. Customers can self select the type of movie they want to watch depending on their willingness to pay and preference. iii) Add-on: GV adds complementary products such as popcorn, drinks and merchandises to its product offerings together with the movie tickets to serve customers who wish to have snacks during the movie screenings. Similarly, online booking is also a form of add-on pricing where customers who derive extra utility from not queuing......

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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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...Marketing 635 MARKETING ANALYTICS AND PRICING Fall 2015 MW 9:35-10:50 – WCBA 184 Instructor: Office: Office Hours: Office Phone: E-mail: Website: Dr. Yan Liu 220U Wehner Building by appointment 845-2547** yliu@mays.tamu.edu http://elearning.tamu.edu **Outside of the classroom, my preferred method of communication is via e-mail. Please note that I will often use e-mail to communicate with you class information. I will send these messages to your neo email account, so please check this account on a regular basis. Required Materials (1) Text Book (2) Nagle, Thomas T., John E. Hogan and Joseph Zale, the Strategy and Tactics of Pricing, Fifth Edition, Pearson-Prentice Hall, Upper Saddle River, NJ. Packet of Cases and Readings. (http://cb.hbsp.harvard.edu/cbmp/access/38228396) Click the link above and set up a HBP account as a student if you don’t have one.(you can use any email address, not necessarily school email address) Choose coursepack mktg653 and make the purchase (audio version of the cases are optional). You have one-year access to this online course pack. Please save the cases to your computer for future usage. Although you purchase digital version of the cases, you can print them out if you wish. Course Prerequisite Completed MKTG 625 Course Description This course is designed to introduce you to pricing fundamentals and pricing strategies. The course is quantitative in nature and will cover a number...

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...Option Pricing, Interest Rate Risk in U.S Diana PĂUN & Ramona GOGONCEA (2013). Interest Rate Risk Management and the Use of Derivative Securities. Economia Seria Management. Retrieved from: <http://www.management.ase.ro/reveconomia/2013-2/4.pdf> The study by these two authors aims at demonstrating how derivative financial instruments can be utilized to prudently manage interest rate risk majorly faced by numerous banks and financial institutions as well as enable the efficient application of monitoring and control tools. There are a couple of risk management methods at the disposal of banks including both balance sheet and off the balance sheet such as the gap method of managing interest rate risk for purposes of controlling short-term rates exposure, combined with derivatives such as options to manage the residual interest rate exposures. Interest rate risks emanate from interest rates sensitivity differentials of capital outflows and inflows. Due to the common view or misconception that high interest rates are the best way of fighting inflation, banks’’ engaging in monetary policy. Financial institutions play a major role in influencing interest rates since they engage in releasing capita to the public by buying assets in the primary markets and selling securities in the secondary market so as to fund purchase of assets. Furthermore, any interest-bearing asset for instance a loan or bond may face interest rate risk caused by changes in the value of assets......

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...studio is obtained from the popular movies. If Arundel had the option to choose to produce the sequel movie out of all the movies released by a studio they have a contract with, they could decide producing only the one with high return and achieve a positive NPV for this business, provided that the studio and Arundel will agree on a set and appropriate price per movie. In this scenario, it is also essential to buy the rights on the movies before they are even made in order to achieve information symmetry, where all relevant information is known to all parties involved. Otherwise Arundel would find itself in a risky situation where if negotiated per movie, the studio would have access to more information in comparison to Arundel and also negotiating on the price of a movie would be based on perceptions of each party because it would also have to depend on artistic judgements in addition to commercial considerations. Arundel doesn’t want to be in a position, where an artistic judgement was required in lieu of its investors. Valuing the PerFilm Price and Calculation Details The maximum per‐film price for the sequel rights that Arundel Partners should pay is $5.12M. We assume that Arundel Partners will purchase a portfolio of films similar to one used in the analysis. The average net inflow of hypothetical sequels ($21.57M) is used to figure out the value of the state variable for the real options model. The state variable is the average net inf......

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...A Novel Simple but 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......

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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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...stresses on the alignment of marketing strategy with pricing strategy Price In arriving at price, the following factors were considered. * Price sensitivity of target segment: Notably within our target segment, they are not price sensitive given the differentiated product and service. * Price flexibility of target segment: In saying that however, the price sensitivity is wide since they are willing to pay high prices as well as low prices. In that sense the price band width is large within the targeted segment. * Importance of price: Price is relatively unimportant as other attributes such as health concerns and quality were ranked higher than price during the concept testing stage. Cost Price is assumed to be greater than variable cost since we are profit making enterprise * Variable cost: Costs are not that high given that supplies are bought in bulk and the buyer’s purchasing power is not that big given the frequency of deliveries. Brand Positioning: Given that The Bread Break positions itself as a premium product (refer to above) Pricing objective Has to be aligned with company and brand strategy The Bread Break is confident of capturing most of customers’ market value hence premium pricing is adopted Pricing tactics Used to enhance demand * Complimentary Pricing: This was done because since our survey has shown that our target segment would take to this as it would give them more options in addition to gaining their favor and loyalty......

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