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FBE459_3_5_BlackScholes

# FBE459_3_5_BlackScholes - FBE 459 Financial Derivatives...

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FBE 459 – Financial Derivatives Prof. Pedro Matos Lecture 3.5.: Black-Scholes Option Pricing Model The Black-Scholes formula Interpretation and implementation Readings: HULL chapters 12 and 13

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2 Lecture Outline Black-Scholes option pricing model: . Assumptions . B-S formula . Interpreting the formula . Using the B-S formula . Implied volatility
3 Pricing Options Black-Scholes Model • To find exact pricing formulas, we need to make assumptions on the statistical behavior of the price of the underlying asset.

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4 Black-Scholes The Black-Scholes option pricing formula is a formidable achievement ! – Brown (1827), Wiener (1918) – Einstein (1905) – Bachelier (1900) – Modigliani and Miller (1958) – Samuelson (1965), Merton (1969) – Black and Scholes (1973) => Scholes and Merton get 1997 Nobel Prize in Economics
5 Binomial vs. Black-Scholes

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6 Black-Scholes Assumptions • Stock prices are lognormally distributed • Volatility is constant • Markets are frictionless – No transaction costs, bid-ask spreads nor taxes – Borrowing and shorting are unlimited

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9 Black-Scholes Assumptions Prices, S T , are lognormally distributed. This means that continuously compounded returns, r T = ln( S T /S ), are normally distributed. => ( 29 t t N S S σ μ , , 2 ln ln 2 0 - + T T S N S T

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10 Black-Scholes Assumptions Even though returns are symmetric, prices are skewed to the right because of compounding – Suppose S=100 can either increase by ln(1.1) = 9.5% or fall by ln(1/1.1)=- 9.5% with equal probabilities
11 Black-Scholes Formula Black-Scholes formula for the price of a European call on a non-dividend paying stock where today’s stock price strike price time in years until option expiration annualized riskless interest rate annualized stock volatility

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12 Black-Scholes Formula N ( d ) is the standard cumulative normal probability . use tables at end of HULL book or use function NORMDIST() in Excel
13 Binomial vs. Black-Scholes

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14 Black-Scholes Formula • As S 0 becomes very large: . c tends to S – Ke -rT (and p tends to zero) • As S 0 becomes very small: . c tends to zero (and p tends to Ke -rT –S )
15 Black-Scholes Formula Interpreting the B-S formula: 1. Risk Neutral valuation : - find risk neutral probabilities from underlying - apply probabilities to find option price c = e -rT E*[max(0; S T -K)] = PV { E*[benefit receiving S T ] - E*[cost of paying K] } = SN(d1) – Ke -rT N(d2) N(d2 ) = probability that option matures in-the-money => Ke -rT N(d2) = E*PV(K)

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16 Black-Scholes Formula Interpreting the B-S formula: 2. Replicating portfolio : . Find number of shares and bonds that replicate
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FBE459_3_5_BlackScholes - FBE 459 Financial Derivatives...

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