lec03

# lec03 - Lecture 3 No-Arbitrage Bounds on Options Deriving...

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Lecture 3: No-Arbitrage Bounds on Options Deriving pricing formulas for options is more difFcult than for forwards/futures. To price an option we have to make assumptions about the behavior of the underlying security’s prices (wait for Lecture 6). We can, however, derive some general restrictions on option prices utilizing the no-arbitrage principle. I. Notation II. “Intuitive” Relations III. No-Arbitrage Bounds A. All Stock Options B. Options on Non Dividend Paying Stocks C. Options on Dividend Paying Stocks IV. Put-Call Parity A. Options on Non Dividend Paying Stocks B. Options on Dividend Paying Stocks

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No-Arbitrage Bounds on Options I. Notation Current date: t Maturity (or expiration) date: T Price of the underlying asset: S(t) D S Strike (or exercise) price: K Price of \$1 bond maturing at T : B(t, T ) D B = PV(\$1) = e N r(t,T ) S (T N t) = N 1 C R(t, T ) S N (T N t) Value of a European call option: c (S, K, t, T ) Value of an American call option: C(S, K, t, T ) Value of a European put option: p (S, K, t, T ) Value of an American put option: P(S, K, t, T ) Bus 35100 Page 2 Robert Novy-Marx
No-Arbitrage Bounds on Options II. Intuitive Relations How does the price of an option change when we increase one of its inputs but keep all others Fxed? Euro. Euro. Amer. Amer. Input Call Put Call Put Stock Price Dividend Yield Strike Price Time to Expiration Risk-±ree Rate Volatility Bus 35100 Page 3 Robert Novy-Marx

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No-Arbitrage Bounds on Options III. No-Arbitrage Bounds A. All Stock Options The following restrictions hold for all stock options, regardless of whether the underlying stock pays dividends. 1. A call option is never worth more than the stock C(S(t),K, t, T ) D S(t) c(S(t),K, t, T ) D S(t) 2. A put option is never worth more than the strike P(S(t), K, t, T ) D K p(S(t), K, t, T ) D K Why? Bus 35100 Page 4 Robert Novy-Marx
No-Arbitrage Bounds on Options 3. Options never have negative value ( why? ) c(S,K, t, T ) N 0 C(S,K, t, T ) N 0 p(S, K, t, T ) N 0 P(S, K, t, T ) N 0 4. A European put option is never worth more than the present value of the strike price ( why? ) p(S, K, t, T ) D K S B(t, T ) 5. American options are at least as valuable as European options ( why? ) C(S, K, t,T ) N c(S,K, t, T ) P(S, K, t,T ) N p(S, K, t, T ) Bus 35100 Page 5 Robert Novy-Marx

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lec03 - Lecture 3 No-Arbitrage Bounds on Options Deriving...

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