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Option Pricing Using Arbitrage and Stochastic Calculus

Paul Isihara with Drew Bixby, John Felker, Michael Kerins, Henry Kuo, Daniel Slye and Michael Izatt

 Mathematics Topic:Calculus, Differential Equations, Geometry Application Areas:Finance Prerequisites:Multivariable calculus, differential equations, and probability theory. The Module assumes only elementary knowledge of math finance, such as compound interest.

| ©2009 by COMAP, Inc. | UMAP Journal 30.1, 2009 | 53 pages |

1. INTRODUCTION

2. ARBITRAGE VALUATION OF OPTIONS
2.1 Present Value
2.2 Arbitrage Pricing of a Call Option
2.2.1 Hedging Claims
2.3 Put-Call Parity
2.4 A Binomial Model
2.5 A Three-Period Binomial Model
2.5.1 Sample Spaces, σ-Fields, and Filtrations
2.5.2 Random Variables and Stochastic Processes
2.6 Tower Property 2.7 Discrete-Time Martingales
2.7.1 Using Martingales to Price Options
2.7.2 Self-Financing Portfolios that Replicate a Claim

3. STOCHASTIC CALCULUS AND INSTANTANEOUS PRICE CHANGES
3.1 Variation of the Stock Price Function
3.2 Understanding the General Form of a Stock Price Function
3.2.1 Step 1: Stock Price as a Function of Time and Risk
3.2.2 Step 2: Construct a RandomWalk
3.2.3 Step 3: Construct a Brownian Motion
3.2.4 Step 4: Stock Price as a Function ofTime and Brownian Motion
3.3 Stochastic Integration 3.4 Continuous Time Martingales andWiener Processes
3.5 Mean Square Limits 3.6 The Itˆo Integral
3.7 Stochastic Differential Equations for Stock Price
3.8 Statement of Itˆo’s Lemma
3.9 Utilizing Itˆo’s Lemma

4. THE BLACK-SCHOLES EQUATION

5. FURTHER STUDY

6. SOLUTIONS TO SELECTED EXERCISES

7. REFERENCES

ACKNOWLEDGMENTS