02-BlackScholes(f)

02-BlackScholes(f) - A First Look at the Black-Scholes...

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Primbs, MS&E 345 1 A First Look at the Black-Scholes Equation
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Primbs, MS&E 345 2 Background: Derivative Security: Example: European Call Option. The right, but not the obligation, to purchase a share of stock at a specified price K (the strike price), at a specified date T (the maturity date). A derivative (or derivative security) is a financial instrument whose value depends on the values of other, more basic underlying variables. ([Hull, 1999]). Arbitrage: A riskless profit that involves no investment. (A free lunch)
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Primbs, MS&E 345 3 Assumptions (to be used throughout most of the course) There are no transaction costs (i.e. markets are frictionless) Trading may take place continuously There is no prohibition on short selling The risk free rate is the same for borrowing and lending Assets are perfectly divisible. These are the “standard assumptions”. When I deviate from them, I will mention it specifically, otherwise assume that they are always in force.
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Primbs, MS&E 345 4 The Set-up: Securities: Bond: rBdt dB = Stock: Sdz Sdt dS σ μ + = 0 5 10 15 20 25 30 35 40 45 50 0 2 4 6 8 12 14 Bond: -Deterministic -Exponential Growth -Continuous compounding rt t e B B 0 =
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Primbs, MS&E 345 5 The Set-up: Securities: Bond: rBdt dB = Stock: Sdz Sdt dS σ μ + = 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 Stock: -Geometric Brownian Motion -Log-Normal -Always positive t z t t e S S + - = ) ( 0 2 2 1
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Primbs, MS&E 345 6 The Set-up: Consider a derivative security whose price depends on S t and t. We will call it: ) , ( t S c t Securities: Bond: rBdt dB = Stock: Sdz Sdt dS σ μ + = By Ito’s lemma: dz Sc dt c S Sc c dc S SS S t + + + = ) ( 2 2 2 1
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Primbs, MS&E 345 7 Now we have 3 price processes: Bond: rBdt dB = Stock: Sdz Sdt dS σ μ + = dz Sc dt c S Sc c dc S SS S t + + + = ) ( 2 2 2 1 Derivative:
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02-BlackScholes(f) - A First Look at the Black-Scholes...

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