...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 in Mesopotamia...
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...Derivative Securities Tutorial Questions Option pricing using Simulation Techniques QUESTION 1 This question refers to the VBA program “Monte Carlo European Option.xlsm” as well as the option discussed in lectures (details repeated as follows) Stock price is currently traded at $1.76. Calculate the price of a European call (strike $1.60, maturity 3 months) on one share. Assume risk free rate of 10% p.a., stock price volatility is 30% p.a. and company not expected to pay any dividends over the next 3 months a) Briefly explain the methodology used to value the European call and put. b) Explain how you would modify the code to price an Asian option. c) Approximate the rho by changing the risk free rate to 9%. Compare this to that from the BSM model where [pic]. d) Now assume a dividend yield of 2% p.a payable over the life of the option. Re-price the option (with risk free rate at 10%). Is the price consistent with your expectations? Why/why not? e) Describe how you could modify the code to allow for leptokurtosis (fat tails). QUESTION 2 Describe how you could use a simulation to price an option using the binomial tree approach. QUESTION 3 Describe the difference between a bootstrap and a monte carlo simulation. Discuss one advantage and disadvantage of the monte carlo approach. QUESTION 4 A Public–private partnership (PPP) describes a government service or private business venture which is funded and operated...
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...I. Overview of the Hong Kong Capital Market Located in the heart of Asia, Hong Kong positioned itself to be a major international financial center of the continent. Its capital market is comprised of integrated network of institutions and markets which provide a wide range of products and services to local and international customers and investors. Hong Kong’s financial markets are characterized by a high degree of liquidity and operate under effective and transparent regulations, which meet international standards. The Government of the Hong Kong Special Administrative Region (HKSAR) abides by the principle of keeping intervention into the way in which the market operates to a minimum and has endeavoured to provide a favorable environment in which business operates. Its policy of low and simple taxation allows maximum room for business initiatives and innovation. There is a strong emphasis on the rule of law and fair market. There are no barriers of access to the market by foreign businesses and no restrictions on capital flows into and out of Hong Kong. Hong Kong’s privileged location in the Northeast Asia, on the other hand, makes it a gateway to China. Moreover, Hong Kong is situated at appropriate time zones that allow 24-hour continuous trading of foreign exchange and gold when the two markets in New York and London are closed. II. Financial Players and Intermediaries in Hong Kong Preview:A closer look at the financial markets As of July 2010, there were...
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...Ramos Bus 138 Questions for Analysis: 1. Define future and options contracts and provide an example of settling accounts of two customers that have taken opposing sides in these contracts. A futures contract is a standardized contract between two parties to buy or sell a specified asset of standardized quantity and quality for a price agreed upon today with delivery and payment occurring at a specified future date, the delivery date. Option contract is a contract that allows the holder to buy or sell an underlying security at a given price, known as the strike price. For example, a trader believes that the price of a stock will rise from its current price of $40 to a level nearing $100. Rather than purchasing the stock itself, she can purchase a call option for a fraction of the price at a strike anywhere between $40 and $100. If the stock does indeed rise to $100, and assuming the call option was bought at a strike of $75, the holder stands to gain $25 per share on the contract, minus any premiums paid for the option itself. 2. Did Mr. Leeson fit well in the Singaporean business culture? Using his example or other examples, comment in general regarding the problems that are faced by "Ex Pat's" and their employers. (Ex Pats are workers that are sent from the home office to work in the foreign branch of a multinational). I believe Mr. Leeson did not fit well in the Singaporean business culture. Asian cultures are punctual, formal, polite, and structured when it comes...
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... . . . . . . . . . . . . . . . 27 4.9 Solutions to Selected Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5 Random Walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5.8 Solutions to Selected Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 6 Interest-Rate-Dependent Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 6.9 Solutions to Selected Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 1 The Binomial No-Arbitrage Pricing Model 1.7 Solutions to Selected Exercises Exercise 1.2. Suppose in the situation of Example 1.1.1 that the option sells for 1.20 at time zero. Consider an agent who begins with wealth X0 = 0 and at time zero buys ∆0 shares of stock and Γ0 options....
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...notes embedded with Asian options in Denmark ---- Using a Monte Carlo Method with stochastic volatility (the Heston Model) Aarhus School of Business and Social Science 2011 2 Acknowledgements My gratitude and appreciation goes to my supervisor Peter Lø chte Jø rgensen, for his kind and insightful discussion and guide through my process of writing. I was always impressed by his wisdom, openness and patience whenever I wrote an email or came by to his office with some confusion and difficulty. Especially on access to the information on certain Danish structured products, I have gained great help and support from him. 3 Abstract My interest came after the reading of the thesis proposal on strucured products written by Henrik, as is pointed out and suggested at the last part of this proposal, one of the main limitations of this thesis may be the choice of model. This intrigues my curiosity on pricing Asian options under assumption of stochstic volatility. At first, after the general introduction of strucutred products, the Black Scholes Model and risk neutral pricing has been explained. The following comes the disadvanges of BS model and then moves to the stochastic volatility model, among which the Heston model is highlighted and elaborated. The next part of this thesis is an emricical studying of two structured products embbeded with Asian options in Danish market and follows with a conclusion. Key words: structured products, Asian options, Black-Scholes model...
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...economics, it measures the excess or shortfall of cash flows, in present value terms, once financing charges are met. OPTION PRICING: The buyer of a call option gets the right to buy the underlying the underlying asset at affixed price, where as the buyer of a put option obtains the right to sell the underlying asset at a fixed price. Alternatives to the binomial model In the binomial option pricing model, the underlying asset and risk free lending or borrowing are combined to create a portfolio that had the same cash flows as the option being valued; we called this portfolio the replicating portfolio. Although the binomial model provides the intuitive feel for the determinants of the option value, it requires a large number of inputs in terms of expected future prices at each node. As we can make time periods shorter in the binomial model, we can make assumptions about asset prices. We can assume that price changes becomes smaller as time periods approaches zero leading to continuous price process. THE BLACK – SCHOLES MODEL: When the price process is continuous, that is price changes become smaller as time period gets shorter, the binomial model for pricing options converges on the Black – Scholes model. The model allows us to estimate the value of any option using a small number of inputs, and it has shown to be remarkably robust in valuing many listed options. The Black-Scholes model is used to calculate a theoretical call price (ignoring...
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...weather, there exists a new solution to the financial effect that weather can have on the incomes of economic agents of developed and emerging economies. There are number of instruments and tools available for management of weather risk. Weather risk management is a definitive guide to the rapidly expanding WRM market. It is the most dynamic sector of the financial arena and is drawing the interest of the companies that are seeking to protect against the financial impact of non catastrophic weather. So for hedging weather risk weather derivatives have been developed. DERIVATIVE TRADING Like any other derivatives weather derivatives are structured as Future, Option or swaps based on different weather indices. Usually most weather derivative transactions are done on over the SInternational Financial Future Option Exchange (LIFFE) offer standardized weather contracts. In India derivative instruments are traded both on OTC market and derivative exchanges. OTC contracts such as forward and swaps are bilaterally negotiated between two parties.OTC terms are flexible and customized. In India OTC derivatives are generally prohibited with some exceptions. An exchange traded contract such as future has a standardized format that specifies the underlying assets to be delivered, size of the contract, time period etc. Margin requirement and...
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...1. Summarize the major features of Texas Instruments’ management systems. To what extent, and how, are these system mutually reinforcing? In 1960, Texas Instruments’ (TI) revenue reached $232 million, which was $32 million over company’s president Pat Haggerty’s announced goal of $200 million. However, the industry was facing a recession at the time, and Patrick Haggerty knew that TI needed to develop a system that would help with innovation and growth of the company. In 1962, Duddy Harris was assigned to find such system, which resulted in development of the OST System: Objectives, Strategies and Tactics. Pat Haggert believed that Texas Instruments must be consumer product centered. Under his leadership TI has organized itself into six major business groups. Each group was subdivided into divisions, which in turn is further divided into product customer centers (PCC), or profit centers. Each Product Customer Center (PCC) manager had responsibilities to Create, Make and Market product in their respective area. The goal of PCC is to allow the business to have a close relationship with customers, and enable an entrepreneurial environment for TI’s middle level managers. The major drawback of Texas Instruments profit center’s decentralized matrix system is the lack of control by top management, possibility of friction and conflict of interest between organization units. With over 80 different PCCs within TI; business cohesion can be a major issue. In order to keep individual...
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.............. 8 2.1 FORWARDS...............................................................................................................................8 Settlement of forward contracts ............................................................................................................9 Default risk in forward contracts .........................................................................................................10 2.1.1 2.1.2 2.2 2.3 FUTURES................................................................................................................................11 O PTIONS................................................................................................................................12 Call option...
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.............. 8 2.1 FORWARDS...............................................................................................................................8 Settlement of forward contracts ............................................................................................................9 Default risk in forward contracts .........................................................................................................10 2.1.1 2.1.2 2.2 2.3 FUTURES................................................................................................................................11 O PTIONS................................................................................................................................12 Call option...
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...2: "Which of the following is an agreement to buy or sell an asset at a fixed date in the future, for a fixed price? 1. Swaps 2. Futures 3. Forwards 4. Warrant 2: "_ Are derived by continuously computing the average of the most recent n day’s closing prices. 1. Moving Average 2. Quotes 3. Commodities plots 4. Plots 2: "What is the conversion factor of a US Treasury bond if securities coupon is 6 percent 1. 1 2. 2 3. 3 4. 6 2: "Which of the following are market instruments that are low risks, highly liquid, short term debt instruments issued by governments, financial institutions and corporations? 1. Derivatives 2. Financial Assets 3. Money Market Securities 4. None of the Above 2: "According to this theory, movement in financial markets depends on attitudes of investors on the business cycle. 1. Dow Theory 2. Cycle Theory 3. Elliot Wave Theory 4. None of the above 2: "Which of the following is a measure of the average change in prices of goods and represent inflation at the consumer level 1. Consumer price index 2. Implicit price deflator 3. Gross domestic product 4. None of the above 2: "What Happens when fiscal policy is paralyzed due to political statement of budget constraints 1. Revert to the economic policy established by the goverment 2. Monetary policy will be relied on 3. Opt to the financial policy 4. None of the above 2:...
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...Packaging | 17% | Electrical | 12% | Machinery & equipment | 10% | Consumer durables | 6% | Other | 6% | Every metal traded on the LME must conform to strict specifications regarding quality, lot size and shape. Each LME tradeable contract is likewise governed by rules covering (but not limited to) prompt dates, settlement terms, traded and cleared currencies and minimum tick size. Contract specifications Law / Regulation English Law, with regulation by the LME and the Financial Conduct Authority (FCA). This is a summary of the contract specifications. LME contracts may only be offered or sold to United States foreign futures and options customers by firms registered with the Commodity Futures Trading Commission (CFTC), or firms who are permitted to solicit and accept money from foreign futures and options customers from trading on the LME. * Physical Quality | Primary aluminium with impurities no greater than...
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...Disclaimer This PDF is a section of the Unilever Annual Report & Accounts and Form 20-F 2003 provided to Unilever's shareholders. It does not contain sufficient information to allow a full understanding of the results of the Unilever Group and the state of affairs of Unilever N.V., Unilever PLC or the Unilever Group. For further information the Unilever Annual Report & Accounts and Form 20-F 2003 should be consulted. Certain sections of the Unilever Annual Report & Accounts and Form 20-F 2003 have been audited. Sections that have been audited are set out on pages 73 to 125, 131 to 147 and 149 to 150. The auditable part of the Directors' Remuneration report as set out on page 68 has also been audited. The maintenance and integrity of the Unilever website is the responsibility of the Directors; the work carried out by the auditors does not involve consideration of these matters. Accordingly, the auditors accept no responsibility for any changes that may have occurred to the financial statements since they were initially placed on the website. Legislation in the United Kingdom and the Netherlands governing the preparation and dissemination of financial statements may differ from legislation in other jurisdictions. Disclaimer Except where you are a shareholder, this material is provided for information purposes only and is not, in particular, intended to confer any legal rights on you. The Annual Report & Accounts and Form 20-F does not constitute an invitation to invest in Unilever...
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...Journal of Banking & Finance 37 (2013) 3388–3400 Contents lists available at SciVerse ScienceDirect Journal of Banking & Finance journal homepage: www.elsevier.com/locate/jbf The cross-sectional relation between conditional heteroskedasticity, the implied volatility smile, and the variance risk premium Louis H. Ederington a,⇑, Wei Guan b a b Finance Division, Michael F. Price College of Business, University of Oklahoma, 205A Adams Hall, Norman, OK 73019, USA College of Business, University of South Florida St. Petersburg, 140 Seventh Avenue South, St. Petersburg, FL 33701, USA a r t i c l e i n f o a b s t r a c t This paper estimates how the shape of the implied volatility smile and the size of the variance risk premium relate to parameters of GARCH-type time-series models measuring how conditional volatility responds to return shocks. Markets in which return shocks lead to large increases in conditional volatility tend to have larger variance risk premia than markets in which the impact on conditional volatility is slight. Markets in which negative (positive) return shocks lead to larger increases in future volatility than positive (negative) return shocks tend to have downward (upward) sloping implied volatility smiles. Also, differences in how volatility responds to return shocks as measured by GARCH-type models explain much, but not all, of the variations in excess kurtosis and multi-period skewness across different markets. Ó 2013 Elsevier B.V. All...
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