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99795 
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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 PutCall Parity
2.4 A Binomial Model
2.5 A ThreePeriod Binomial Model
2.5.1 Sample Spaces, σFields, and Filtrations
2.5.2 Random Variables and Stochastic Processes
2.6 Tower Property 2.7 DiscreteTime Martingales
2.7.1 Using Martingales to Price Options
2.7.2 SelfFinancing 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 BLACKSCHOLES EQUATION
5. FURTHER STUDY
6. SOLUTIONS TO SELECTED EXERCISES
7. REFERENCES
ACKNOWLEDGMENTS
ABOUT THE AUTHORS
This Module serves as an undergraduate level introduction
to options pricing by means of arbitrage and
stochastic calculus.
Section 2 introduces the concept
of arbitrage and shows how options are priced so as
to avoid unfair market advantage (arbitrage). Options
also may be used to hedge claims. A threeperiod discrete
binomial model is then used to introduce the concept
of martingales.
Section 3 uses arbitrage to explain
why stock price cannot be a differentiable function of
time. It then develops a stochastic model for stock
price, as described by stochastic differential equations
and their stochastic integral equivalents. Finally,
Section
4 combines the ideas of arbitrage, hedging claims,
and stochastic differential equations to obtain the celebrated
BlackScholesequation for pricing European call
options.



