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

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

Page1 / 47

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

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online