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Chapter 17 Option Pricing

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2 Framework Background Definition and payoff Some features about option strategies One-period analysis Put-call parity, Arbitrage bound, American call option Black-Scholes Formula Price using discount factor Derive Black-Scholes differential equation Asset Pricing

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Background (1) Option; Call/Put; Strike Price Expiration Date Underlying Asset European/ American Option Payoff/Profit Asset Pricing

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Background (2) Asset Pricing

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5 Background (3) Some Interesting Features of Options High Beta (High Leverage) - Trading - Hedging Shaping Distribution of Returns: - OTM Put + Stock But Short OTM Put Option and Long Index Return Distribution Extremely Non-normal The Chance of Beating the Index for one or even five years is extremely high, but face the catastrophe risk So what kind of investments can and cannot be made is written in the portfolio management contracts Asset Pricing

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6 Background (4) Strategies By combining options of various strikes, you can buy and sell any piece of the return distribution. A complete set of option is equivalent to complete markets. Forming payoff that depends on the terminal stock price in any way Asset Pricing

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7 One-period analysis The law of one price existence of a discount factor No arbitrage existence of positive discount factor How to pricing option Put-Call Parity Arbitrage Bounds Discount Factors and Arbitrage Bounds Early Exercise Asset Pricing

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8 Put-call parity Strategies (1) hold a call, write a put,same strike price (2) hold stock, borrow strike price X In the book of John C. Hull, Asset Pricing

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Put-call parity 9 According to the Law of One Price, applying to both sides for any m, We get Asset Pricing

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10 Arbitrage bounds Portfolio A dominates portfolio B Arbitrage portfolio Asset Pricing

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11 C Call value Today Call value in here X/Rf S Stock value today Arbitrage bounds Asset Pricing

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12 Discount factors and arbitrage bounds This presentation of arbitrage bound is unsettling for two reasons, First, you many worry that you will not be clever enough to dream up dominating portfolios in more complex circumstances. Second, you may worry that we have not dream up all of the arbitrage portfolios in this circumstance Asset Pricing

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13 Discount factors and arbitrage bounds This is a linear program. In situations where you do not know the answer, you can calculate arbitrage bounds.(Ritchken(1985)) The discount factor method lets you construct the arbitrage bounds Asset Pricing

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14 Early exercise? By applying the absence of arbitrage, we can never exercise an American call option without dividends before the expiration date. S-X is what you get if you exercise now. the value of the call is greater than this value, because you can delay paying the strike, and exercising early loses the option value payoff price Asset Pricing

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15 Black-Scholes Formula (Standard Approach Review) Portfolio Construction: Asset Pricing

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16 Black-Scholes Formula (Standard Approach Review ) Where: Risk Neutral Pricing: Asset Pricing

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17 Black-Scholes Formula (Discount Factor) Write a process for stock and bond, then use to price the option. the Black-Scholes formula results, (1) solve for the finite-horizon discount factor and find the call option price by taking the expectation (2) find a differential equation for the call option and solve it backward Asset Pricing

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18 Black-Scholes Formula (Discount Factor) The call option payoff is The underlying stock follows The is also a money market security that pays the real interest rate In continuous time, all such discount factors are of the form: Asset Pricing

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19 Method 1: price using discount factor Use the discount factor to price the option directly: Where and are solutions to Asset Pricing

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20 How to find analytical expressions for the solutions of equations of the form (17.2) Asset Pricing

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21 Applying the Solution to (17.2) We get: Ignoring the term of And Proof Later Asset Pricing

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22 Evaluate the call option by doing the integral Asset Pricing

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25 Proof: not Affect Asset Pricing

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26 Where: This is the integral under the normal distribution, with mean of and, standard variance of 1,so the integral is 1.we multiply both sides without any change Asset Pricing

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27 Method 2:derive Black-Scholes Differential Equation Guess that solution for the call option is a function of stock price and time to expiration, C=C(S,t). Use Itos lemma to find derivatives of C(S,t) Asset Pricing

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29 This is the Black-Scholes differential equation for the option price This differential equation has an analytic solution, one standard way to solve differential equation is to guess and check, and by taking derivatives you can check that (17.7) does satisfy (17.8) Asset Pricing

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30 Thanks Your suggestion is welcome! Asset Pricing

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