Intro To Financial Math

Intro To Financial Math - Introduction to Financial...

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Unformatted text preview: Introduction to Financial Mathematics November 19, 2008 2 Contents 1 Introduction to Financial derivatives 5 1.1 Most common financial derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.1 Derivatives and strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.2 Forward contracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.1.3 Call options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.1.4 Put options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.1.5 Who buys call and put contracts? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.1.6 Bank account and zero-coupon bond . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2 Arbitrage and pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2.1 Call-put parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2 Discrete time models 15 2.1 One step binomial tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1.1 Dynamics of the stock, strategies and complete markets . . . . . . . . . . . . . . . . . 15 2.1.2 No arbitrage condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.1.3 Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Two step binomial tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2.1 Model and strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2.2 Pricing of a European call. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2.3 Replicating and pricing of the European option. . . . . . . . . . . . . . . . . . . . . . 22 2.2.4 Risk-neutral measure and martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2.5 Replicable claims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2.6 The lookback option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2.7 A first look at American options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3 General N-step binomial tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.3.1 Pricing of general European option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.3.2 Pricing of path independent European option . . . . . . . . . . . . . . . . . . . . . . . 33 2.3.3 Pricing of the American put . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3 Continuous time models 37 3.1 Continuous time models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.1.1 Continuous trading approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.1.2 Smoothness of the paths of the stock price . . . . . . . . . . . . . . . . . . . . . . . . .Smoothness of the paths of the stock price ....
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Intro To Financial Math - Introduction to Financial...

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