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lecture08post - Option price sensitivities Aureliano Buend...

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Option price sensitivities Aureliano Buend´ ıa Lecture 8 - Black-Scholes’ Greeks BUSI 588, Fall 2011 Diego Garc´ ıa Kenan-Flagler Business School UNC at Chapel Hill September 21st, 2011 c Diego Garc´ ıa, BUSI 588, Kenan-Flagler, Fall 2011 Lecture 8 - Black-Scholes’ Greeks 1 / 27

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Option price sensitivities Aureliano Buend´ ıa Outline 1 Option price sensitivities Replication: Δ and Γ Time decay and options Θ The other Greeks 2 Aureliano Buend´ ıa Hedging Gamma and Theta Handouts today: Class slides. Homework 3 solutions. Case 8 solutions. c Diego Garc´ ıa, BUSI 588, Kenan-Flagler, Fall 2011 Lecture 8 - Black-Scholes’ Greeks 2 / 27
Option price sensitivities Aureliano Buend´ ıa Exotic derivatives Problem 4 in HW3 presented a strange derivative. 1156 2720 6400 170 400 25 It looks cheap, but if we simply buy it, then we risk that the stock goes up twice. Key: buy the replicating portfolio of the derivative (cost 1156). This allows us to be hedged completely and pocket the difference. c Diego Garc´ ıa, BUSI 588, Kenan-Flagler, Fall 2011 Lecture 8 - Black-Scholes’ Greeks 3 / 27

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Option price sensitivities Aureliano Buend´ ıa The main Greeks The Black-Scholes option pricing formula (no dividends today) C t = S t N ( x ) - K (1 + r f ) T - t N ( x - σ T - t ) where x = log( S t ) - log K (1+ r f ) T - t σ T - t + 1 2 σ T - t . The Greeks are a bunch of partial derivatives: Delta, Δ = C t /∂ S t . Gamma, Γ = 2 C t /∂ S 2 t . Theta, Θ = C t /∂ t . Recall: partial derivatives measure how something changes with respect to changes in a variable, keeping everything else constant . c Diego Garc´ ıa, BUSI 588, Kenan-Flagler, Fall 2011 Lecture 8 - Black-Scholes’ Greeks 4 / 27
Option price sensitivities Aureliano Buend´ ıa The main Greeks in English Delta, Δ, change in option price per unit change in the underlying asset price. Gamma, Γ, change in Δ per unit change in the underlying asset price. Theta, Θ, change in option price per unit change in time. Vega, change in option price per unit change in volatility. Rho, ρ , change in option price per unit change in the interest rate. Omega, Ω, percent change in option price per 1% percent change in underlying asset price. Key: partial derivatives means we keep everything else constant. c Diego Garc´ ıa, BUSI 588, Kenan-Flagler, Fall 2011 Lecture 8 - Black-Scholes’ Greeks 5 / 27

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Option price sensitivities Aureliano Buend´ ıa Δ - Delta The change in the option price per unit change in the underlying asset Δ C = C t S t = N ( x ); Δ P = Δ C - 1 . It is the number of units of the underlying asset that we need to hold in order to replicate the option’s payoffs. The Δ of a portfolio is simply the same of the Δ’s of the components of the portfolio, i.e. a portfolio with n 1 units of asset 1 and n 2 units of asset 2 has Δ = n 1 Δ 1 + n 2 Δ 2 To delta hedge a portfolio means that the portfolio’s Δ is zero at each date.
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