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# lec08A - Lecture 8 Black-Scholes Option Pricing formula...

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Unformatted text preview: Lecture 8: Black-Scholes Option Pricing formula This lecture presents the famous Black-Scholes option pricing formula. It also shows one way to derive the formula, in a way that is easier and more intuitive than Black and Scholes’ original derivation. I. Assumptions II. Replicating Dynamic Trading Strategy III. Black-Scholes Formula IV. Interpretation V. Derivations of B-S Black-Scholes Option Pricing Formula I. Black-Scholes Assumptions 1. Financial markets are frictionless: SI No taxes or transaction costs. SI Assets are perfectly divisible. SI No restrictions on short sales. 2. The interest rates ( r , c.c.) for borrowing and lending are the same, and constant, from t D to T . 3. The stock pays no dividends from t D to T . 4. Stock prices conform to the log-normal model: SI Stock prices follows a continuous path. SI The log return is normally distributed: ln DC2 S(t 2 ) S(t 1 ) DC3 CAN N STX SYN (t 2 NUL t 1 ), ESC 2 (t 2 NUL t 1 ) ETX – Independent on non-overlapping samples. – SYN and ESC constant. B35100 Page 2 Robert Novy-Marx Black-Scholes Option Pricing Formula II. Replicating Dynamic Trading Strategy With these assumptions, Black and Scholes ( 1973 ) showed that: SI You can replicate the payoff of a European call option at maturity by – buying a certain amount of stock, partially financed with borrowing, at time t D , – dynamically trading this portfolio until time T . The value of the European call option equals the initial cost of this replicating portfolio. SI Just like binomial pricing! The fractional share of stock in the replicating portfolio ( i.e., the hedge ratio or delta ) satisfies DC4 SOH D @ C @ S DC4 1 SI Also, SOH changes constantly. – With the frictionless market assumptions, however, the net cost of adjustments is zero. B35100 Page 3 Robert Novy-Marx Black-Scholes Option Pricing Formula III. Black-Scholes Formula Black and Scholes showed that the value of a European call option on a stock that pays no dividends is: C(S, K, T, r, ESC ) D SN(d 1 ) NUL Ke NUL rT N(d 2 ) where N( SOH ) is the cumulative density function of a standard normal random variable, and: d 1 D ln NUL S K SOH C DLE r C ESC 2 2 DC1 T ESC p T d 2 D d 1 NUL ESC p T Put-call parity implies the value of a European put on the same stock with the same strike is P(S, K,T, r, ESC ) D Ke NUL rT N( NUL d 2 ) NUL SN( NUL d 1 ) B35100 Page 4 Robert Novy-Marx Black-Scholes Option Pricing Formula C(S,K,T, r, ESC ) D SN(d 1 ) NUL Ke NUL rT N(d 1 NUL ESC p T ) d 1 D ln NUL S K SOH C DLE r C ESC 2 2 DC1 T ESC p T Inputs to the Black-Scholes formula: S = current stock price K = exercise price T = time to expiration r = risk-free rate (c. c., annualized ) ESC = S.D. of annual log returns (“volatility”) Note: The expected return SYN is not an input to the formula....
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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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lec08A - Lecture 8 Black-Scholes Option Pricing formula...

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