lec09 - Lecture 9: Using the Black-Scholes Formula This...

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Unformatted text preview: Lecture 9: Using the Black-Scholes Formula This lecture extends the Black-Scholes formula to price European style options on dividend paying stocks, on currencies, and on futures. We also derive an approximate pricing formula for American style call options on dividend paying stocks. Finally, we discuss about other, not as popular, extensions of the model. I. Options on Dividend Paying Stocks A. Stock with Known Dividend B. Stock with Known Dividend Yield II. Options Currencies III. Approximate Prices of American Style Options IV. Options on Futures A. Preliminaries B. Black’s Model for Options on Futures V. Some Exotics A. Gap Options B. Exchange Options VI. Other Extensions of the Black-Scholes Formula Using the Black-Scholes Formula I. Options on Dividend Paying Stocks A. Stock with Known Dividend Consider a European style call option that matures at time T on a stock that pays a known dividend of D at time t D , where t < t D < T . SI The option is on the stock after the dividend payment, – So the underlying security is not the stock. – The underlying is the stock less the PV of the dividend. In other words, the current value of the stock S(t) is the sum of two components: SI The PV of the known dividend, PV (D) . SI The PV of the expected stock price at time T , after the dividend payment, denoted v (t) . v (t) D S(t) NUL PV (D) – We did the exact same thing with futures. The call is only on the second component. Bus 35100 Page 2 Robert Novy-Marx Using the Black-Scholes Formula SI We then calculate the value of the call by substituting v (t) for S(t) in the Black-Scholes formula: C D NUL S NUL PV (D) SOH N(d 1 ) NUL K e NUL r(T NUL t) N(d 2 ) where d 1 D ln DLE S NUL PV (D) K DC1 C (r C ESC 2 2 )(T NUL t) ESC p T NUL t d 2 D d 1 NUL ESC p T NUL t Important: Don’t forget to use S NUL PV (D) in d 1 ! SI The dividend reduces the chances of finishing in-the-money. The same argument works for multiple dividends between time t and T , as long as we know the present value of all dividends paid. Bus 35100 Page 3 Robert Novy-Marx Using the Black-Scholes Formula B. Stock with Known Dividend Yield Same reasoning works for pricing European calls on stocks that pay a known dividend yield ı . SI Again, today’s share price reflects two value components: – The PV of the dividends that leave the firm before maturity at T , (1 NUL e NUL ı (T NUL t) )S t . – The PV of the assets that remain in the firm, e NUL ı (T NUL t) S t . SI The call is on one share of stock at time T . – I.e., only on the assets that remain in the firm. – So the underlying security is v (t) D e NUL ı (T NUL t) S(t). Again, we did the exact same thing for futures. SI If you buy a share and reinvest dividends, then after T NUL t has passed you own e ı (T NUL t) shares....
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This note was uploaded on 08/14/2009 for the course BUS 35100 taught by Professor Novy-marx during the Winter '07 term at CHIC.

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lec09 - Lecture 9: Using the Black-Scholes Formula This...

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