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# FESC1 - Financial Engineering with Stochastic Calculus I...

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Financial Engineering with Stochastic Calculus I Johannes Wissel Cornell University Fall 2010

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0. Introduction
Motivation: Some examples of financial derivatives I an airline seeks protection against rising oil prices ( forward contract) I a company wants to hedge the risk in a payment obligation in foreign currency at a future time ( call option) I a fund manager wants to protect a stock position against losses ( put option) Main objectives of financial engineering (FE) I development of quantitative models for financial markets I design, pricing, and hedging of financial derivatives I development of quantitative methods of risk management Stochastic calculus I provides math. framework for continuous time models in FE I asset values are modeled by stochastic processes I trading strategy values are modeled by stochastic integrals

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Course syllabus I: Introduction: financial engineering, binomial model II: Background in probability: information and σ -algebras, independence, general conditional expectations, martingales, fundamental theorem of asset pricing III: Brownian motion (BM): scaled random walks, definition of BM, distribution of BM, filtration for BM, martingale property of BM, quadratic variation IV: Stochastic calculus: stochastic integral, Itˆ o processes, Itˆ o-Doeblin formula, Black-Scholes-Merton equation, multivariable stochastic calculus V: Risk-neutral pricing: Girsanov’s theorem, risk-neutral measure, martingale representation, fundamental theorems of asset pricing VI: Miscellaneous topics (if time permits): dividends, forwards and futures, PDE pricing techniques.
1. The binomial model Main goals I Introduce some fundamental ideas and concepts in FE I Introduce a math. model which later serves as a main building block for Brownian motion and continuous time models

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Motivating problem: Pricing of options Consider a financial asset S , e.g. a stock, which is traded on an exchange. Denote the market price of the asset at time t by S t . I Call option on the asset S : A derivative which gives the holder the right, but not the obligation, to buy the asset S at a future time T for a pre-arranged price K from the issuer (to “exercise the option”). I Terminology: S underlying , K strike price , T maturity of the option. I Value of the option at maturity? I If S T K , option is worthless. I If S T > K , exercise the option (buy S for K ) and sell S on the exchange for S T , making a profit S T - K . Thus, option value at maturity is ( S T - K ) + (option payoff ). I Question: What is the value of the option prior to T ?
Motivating problem: Pricing of options Consider a financial asset S , e.g. a stock, which is traded on an exchange. Denote the market price of the asset at time t by S t . I Put option on the asset S : A derivative which gives the holder the right, but not the obligation, to sell the asset S at a future time T for a pre-arranged price K to the issuer (to “exercise the option”).

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FESC1 - Financial Engineering with Stochastic Calculus I...

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