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# lecture09post - Recap on Black-Scholes Volatility American...

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Recap on Black-Scholes Volatility American options Lecture 9 - American options BUSI 588, Fall 2011 Diego Garc´ ıa Kenan-Flagler Business School UNC at Chapel Hill September 26th, 2011 c ± Diego Garc´ ıa, BUSI 588, Kenan-Flagler, Fall 2011 Lecture 9 - American options 1 / 19

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Recap on Black-Scholes Volatility American options Outline 1 Recap on Black-Scholes Expected returns on options Hedging 2 Volatility Basic estimates ARCH/GARCH 3 American options A simple example Early exercise in Galitzin Handouts today: Class slides. Case 9 solutions. c ± Diego Garc´ ıa, BUSI 588, Kenan-Flagler, Fall 2011 Lecture 9 - American options 2 / 19
Recap on Black-Scholes Volatility American options Risk and return for call options What is the expected return of a call option on an underlying asset with a β S = 1? Assume r f = 0 . 05 and E [ r m ] - r f = 5%, so E [ r s ] = 10%. Further let σ = 0 . 5, T = 1, δ = 0, S = 1300, K = 1350. From Black-Scholes: C = 262 . 4 and Δ = 0 . 6072 Key: a call is just a long position in the underlying asset (Δ shares) ﬁnanced with borrowing. As a portfolio, its beta must satisfy: β C = Δ S Δ S + B β S + B Δ S + B β B = Δ S C β S = Ω β S with Ω 1. Using S = 1300, C = 262 . 4, Δ = 0 . 6072, we have Ω = 3 and E [ r c ] = 20%. c ± Diego Garc´ ıa, BUSI 588, Kenan-Flagler, Fall 2011 Lecture 9 - American options 3 / 19

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Recap on Black-Scholes Volatility American options Risk and return for put options A put is also simply a short position in the underlying asset coupled with lending. As a portfolio, its beta must satisfy: β P = Δ S Δ S + B β S + B Δ S + B β B = Δ S P β S = Ω β S but now Ω 0, since Δ 0. With the parameters as in the previous example we have Ω ≈ - 2, so that E [ r p ] = - 5%. c ± Diego Garc´ ıa, BUSI 588, Kenan-Flagler, Fall 2011 Lecture 9 - American options 4 / 19
Volatility American options One last hedging example Suppose we just sold a straddle with a one-year maturity on an underlying asset trading at \$100 ( σ = 0 . 4 and r f = 0 . 05), with a strike of \$100. We have the following information on the straddle (ﬁrst column) and one set of liquid traded options (shorter maturity). Sold options Hedging instruments Strike 100 100 Time 1 0.5 Call price 17.97 12.36 Put price 13.21 9.95 Delta (call) 0.6263 0.5900 Gamma 0.0095 0.0137 (a) How could one hedge the straddle risk by trading in the short-term calls? (b) Can one create a delta- and gamma-neutral position using the options with K = 100 (a call and a put)? c

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## This note was uploaded on 11/25/2011 for the course BUSI 588 taught by Professor Staff during the Fall '10 term at UNC.

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lecture09post - Recap on Black-Scholes Volatility American...

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