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Unformatted text preview: Lecture 8: BlackScholes Option Pricing formula This lecture presents the famous BlackScholes 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. BlackScholes Formula IV. Interpretation V. Derivations of BS BlackScholes Option Pricing Formula I. BlackScholes 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 lognormal 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 nonoverlapping samples. SYN and ESC constant. B35100 Page 2 Robert NovyMarx BlackScholes 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 NovyMarx BlackScholes Option Pricing Formula III. BlackScholes 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 Putcall 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 NovyMarx BlackScholes 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 BlackScholes formula: S = current stock price K = exercise price T = time to expiration r = riskfree 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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 Winter '07
 NovyMarx
 Pricing

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