Black-Scholes Option Pricing Model Nathan Coelen June 6‚ 2002 1 Introduction Finance is one of the most rapidly changing and fastest growing areas in the corporate business world. Because of this rapid change‚ modern financial instruments have become extremely complex. New mathematical models are essential to implement and price these new financial instruments. The world of corporate finance once managed by business students is now controlled by mathematicians and computer scientists
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Wiener Process Ito ’s Lemma Derivation of Black-Scholes Solving Black-Scholes Introduction to Financial Derivatives Understanding the Stock Pricing Model 22M:303:002 Understanding the Stock Pricing Model 22M:303:002 Wiener Process Ito ’s Lemma Derivation of Black-Scholes Stock Pricing Model Solving Black-Scholes Recall our stochastic dierential equation to model stock prices: dS = σ dX + µ dt S where µ is known as the asset ’s drift ‚ a measure of the average rate
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Black-Scholes Option Pricing Formula In their 1973 paper‚ The Pricing of Options and Corporate Liabilities‚ Fischer Black and Myron Scholes published an option valuation formula that today is known as the Black-Scholes model. It has become the standard method of pricing options. The Black-Scholes model is a tool for equity options pricing. Options traders compare the prevailing option price in the exchange against the theoretical value derived by the Black-Scholes Model in order to determine
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definition‚ the integral evaluates to be 1. Proof of Black Scholes Formula Theorem 2: Assume the stock price following the following PDE Then the option price for a call option with payoff is given by 1 Proof: By Ito’s lemma‚ If form a portfolio P Applying Ito’s lemma Since the portfolio has no risk‚ by no arbitrage‚ it must earn the risk free rate‚ Therefore we have Rearranging the terms we have the Black Scholes PDE With the boundary condition To solve this
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Continuous-Time Models c 2009 by Martin Haugh Fall 2009 Black-Scholes and the Volatility Surface When we studied discrete-time models we used martingale pricing to derive the Black-Scholes formula for European options. It was clear‚ however‚ that we could also have used a replicating strategy argument to derive the formula. In this part of the course‚ we will use the replicating strategy argument in continuous time to derive the Black-Scholes partial differential equation. We will use this PDE and
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All Cash flows are expressed as after tax present values discounted to time zero‚ including capital expenditures At any point "failure‚" investment decision is to stop funding Assuming Standard deviation of 0.5 Using T= 7 years in Black-Scholes Valuation 2 Decision Tree See worksheet "Decision Tree" 3 Detailed description of Real Option Technique "First‚ using a decision tree‚ I came up with a simple expected value of $13‚980‚000 based on the costs
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Laura Martin: Real Options and the Cable Industry Introduction Laura Martin‚ an equity research analyst for cable stocks‚ believes that the best way to value cable stocks is through creative methods such as real options and not through more traditional or typical valuation methods such as EBITDA multiples‚ ROIC analysis and DCF analysis. In 1999 she presented at the Credit Suisse First Boston Broadband conference‚ where she wanted to portray the message that real options is a superior valuation
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used to arrive at the closest estimate of value and yet certain issues will always arise. This case attempts to tackle two approaches in real asset valuation: Discounted Cash Flow (DCF) analysis and the issues surrounding such‚ as well as the Black-Scholes Model for Real Options. Questions to be addressed in the study are: 1. Evaluate Amoco’s and Apache’s corporate objectives and strategies. Is it reasonable to expect that the MW properties are more valuable to Apache than to Amoco? What sources
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Bankok: The University of Thai Chamber ofCommerce. Black‚ F. & Scholes‚ M.‚ 1973. The Pricing of Options and Corporate Liabilities. The Journal of Political Economy‚ 81(3)‚ pp Bowen‚ R. M. & Rajgopal‚ S.‚ 2009. Do Powerful Investors Influence Accounting‚ Governance and Investing Decisions?‚ Washington D.C.: University of Washington. Buffett‚ W. E.‚ 2008. Letter to Shareholders‚ Omaha: Berkshire Hathaway‚ Inc.. Cornell‚ B.‚ 2009. Warren Buffet‚ Black-Scholes and the Valuation of Long-dated Options‚ Pasadena:
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of the option and compare it to the actual price of the option offered by the seller. As mentioned earlier‚ the payoff pattern of the proposed option resembles the payoff of long European Call option from CPL’s point. Therefore‚ CPL should use Black-Scholes option
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