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lecture07

# lecture07 - The binomial model The Black-Scholes model...

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The binomial model The Black-Scholes model Lecture 7 - Black-Scholes model BUSI 588, Fall 2011 Diego Garc´ ıa Kenan-Flagler Business School UNC at Chapel Hill September 19th, 2011 c Diego Garc´ ıa, BUSI 588, Kenan-Flagler, Fall 2011 Lecture 7 - Black-Scholes model 1 / 30 The binomial model The Black-Scholes model Outline 1 The binomial model Revisiting Shostakovich - Topalov The binomial formula 2 The Black-Scholes model Preliminaries The formula Volatility Handouts today: Class slides. Case 7 solutions (Topalov). c Diego Garc´ ıa, BUSI 588, Kenan-Flagler, Fall 2011 Lecture 7 - Black-Scholes model 2 / 30

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The binomial model The Black-Scholes model Last class Heavy-duty accounting (Shostakovich) In a binomial world we can price assets recursively using the formula V t = ˆ p u V ut +1 + ˆ p d V dt +1 1 + r f ; take the expectation (using risk-neutral probabilities) of the values next period (as a function of u and d states), and discount at the risk-free rate. This recursive risk-neutral price was the same thing as a dynamic replication argument, in which we try to replicate the replicating portfolio (recursively). c Diego Garc´ ıa, BUSI 588, Kenan-Flagler, Fall 2011 Lecture 7 - Black-Scholes model 3 / 30 The binomial model The Black-Scholes model Calibration It is common to calibrate trees making u and d states equally likely and using u = e σ T / n ; d = 1 / u ; where σ is the volatility of the underlying asset, T is time to expiration and n is the number of steps in the tree. Risk-free rates and dividend yields should be consistent with the tree. Per period one plus risk-free rate: r = (1 + r f ) T / n Topalov is about showing how Black-Scholes is a large tree calibrated as above (let n ↑ ∞ ). c Diego Garc´ ıa, BUSI 588, Kenan-Flagler, Fall 2011 Lecture 7 - Black-Scholes model 4 / 30
The binomial model The Black-Scholes model Topalov 3-step Stock price evolution 25.00 26.49 28.06 29.73 23.60 25.00 26.49 22.27 23.60 21.02 ˆ p u = r - d u - d = 1 . 0041 - 0 . 9439 1 . 0594 - 0 . 9439 = 0 . 5208 ˆ p d = 1 - ˆ p u = 0 . 4792 c Diego Garc´ ıa, BUSI 588, Kenan-Flagler, Fall 2011 Lecture 7 - Black-Scholes model 5 / 30 The binomial model The Black-Scholes model Topalov 3-step What is the value of a security that gives you \$1 in the underlying asset ends up at \$29.73 at date 3? ˆ p 3 u / r 3 = 0 . 13957 What is the value of a security that gives you \$1 in the underlying asset ends up at \$26.49 at date 3? p 2 u ˆ p d / r 3 = 0 . 3852 What is the value of a security that gives you \$1 in the underlying asset ends up at \$23.60 at date 3? p u ˆ p 2 d / r 3 = 0 . 3544 What is the value of a security that gives you \$1 in the underlying asset ends up at \$21.02 at date 3? ˆ p 3 d / r 3 = 0 . 1087 c Diego Garc´ ıa, BUSI 588, Kenan-Flagler, Fall 2011 Lecture 7 - Black-Scholes model 6 / 30

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The binomial model The Black-Scholes model Topalov 12-step What is the price of a security that pays off \$1 in the stock goes up twelve times in the weekly binomial tree?
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