Lecture10_PricingOptionI - NBA 6730: Derivative Securities...

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1 NBA 6730: Derivative Securities Lecture 10: Introduction to Option Pricing 09/28/2010 George Gao NBA6730-Derivative Securities I 2 Agenda Pricing an option needs assumptions about the behavior of the underlying security’s prices. The basic idea is to construct a replicating portfolio of stock and bond that dynamically replicates the payoff of an option. In this lecture, we learn Binomial option pricing that is used most on the Street. Bounds on options Early exercise of American options Binomial model (plain-vanilla European call and put) Risk-Neutral pricing
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2 NBA6730-Derivative Securities I 3 (1) Bounds on Options Recall our notation: small letters for European options, big letters for American options Strike, maturity, underlying security price ܥ ݐ≜ܥ ܵ ݐ,ܭ ,ܶ,ܲ ݐ≜ܲ ܵ ݐ,ܭ ,ܶ,ܿ ݐ,݌ ݐ Options never have negative value ܥሺݐሻ ൒ 0,ܿሺݐሻ ൒ 0, ܲሺݐሻ ൒ 0,݌ሺݐሻ ൒ 0 A call option is never worth more than the stock ܥ ݐ൑ܵ ݐ,ܿ ݐ൑ܵ ݐ Otherwise, you make a riskless profit by buying a stock and selling a call option immediately NBA6730-Derivative Securities I 4 (1) Bounds on Options A put option is never worth more than the strike ܲ ݐ൑ܭ ݐ൑ܭ Otherwise, you make a riskless profit by selling a put option (you pay the strike price ܭ at most ) A European put option is never worth more than the present value of the strike price ݌ ݐ൑ܲ ܸ ܭൌܭ ݁ ି௥ሺ்ି௧ሻ A European put can only be exercised at maturity ܶ . Thus, you sell a put at ݐ and get ݌ , and meantime you can invest ܸܲሺܭሻ in T-bill at ݐ
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3 NBA6730-Derivative Securities I 5 (1) Bounds on Options American options with more time to maturity are at least as valuable as the same options with less time to maturity. For ܶ ൐ܶ : ܥ ݐ,ܶ ൒ܥ ݐ,ܶ ݐ,ܶ ൒ܲ ݐ,ܶ An American option is at least as valuable as its intrinsic value ܥ ݐ൒m a x 0, ܵ ݐ െ ܭ , ܲሺݐሻ ൒ maxሾ0, ܭ െ ܵ ݐሿ Otherwise, buy a call and execute the option immediately. You make riskless profit by getting S-K and paying C . Similar arguments for a put. The two types of bounds above don’t always hold for European options NBA6730-Derivative Securities I 6 (1) Bounds on Options Options on non dividend paying stocks Call option: ܥ ݐ൒ܿ ݐ൒m a x 0, ܵ ݐെܭ ∙ܤ ݐ,ܶ Put option: ܲሺݐሻ ൒ ݌ሺݐሻ ൒ maxሾ0, ܭ ∙ ܤ ݐ,ܶ െ ܵ ݐሿ Options on dividend paying stocks Call option: ܥ ݐ൒ܿ ݐ൒ܵ ݐെܭ ∙ܤ ݐ,ܶ െ ܸܲሺܦሻ Put option: ܲ ݐ൒݌ ݐ൒ܭ
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Lecture10_PricingOptionI - NBA 6730: Derivative Securities...

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