winter2010_notes_part4

# winter2010_notes_part4 - Black-Scholes Formula Consider an...

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1-1 Black-Scholes Formula Consider an European call (or put) option written on a stock Assume that the stock pays dividend at the continuous rate δ

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1-2 Black-Scholes Formula (cont’d) d ln S / K r T T 1 2 1 2 = +−+ () ( ) δσ σ dd T 21 =− σ CS ,K , ,r ,T , Se Nd Ke -T - rT = ( ) ( ) σδ δ 12 P S , K , , r , T , K eNd S eNd -rT - T = ( ) ( ) δ −− Call Option price Put Option price where and
1-3 Black-Scholes (BS) Assumptions Assumptions about stock return distribution ± Continuously compounded returns on the stock are normally distributed and independent over time (no “jumps”) ± The volatility of continuously compounded returns is known and constant ± Future dividends are known, either as dollar amount or as a fixed dividend yield

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1-4 Black-Scholes (BS) Assumptions (cont’d) Assumptions about the economic environment ± The risk-free rate is known and constant ± There are no transaction costs or taxes ± It is possible to short-sell costlessly and to borrow at the risk-free rate
1-5 Applying BS to Other Assets [] d ln F S / F K T T T P T P 1 00 2 1 2 = + ,, () σ σ dd T 21 =− σ CF S,F K, ,T F S Nd F K Nd T P T P T P T P = ( ) ( ) 0 1 0 2 , , ( ) σ− FK K e T P- r T 0, ()= Call Options where , and

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1-6 Applying BS to Other Assets (cont’d) Options on stocks with discrete dividends ± The prepaid forward price for stock with discrete dividends is ± Examples 12.3 and 12.1 S = \$41, K = \$40, σ = 0.3, r = 8%, t = 0.25, Div = \$3 in one month PV ( Div ) = \$3 e -0.08/12 = \$2.98 Use \$41 \$2.98 = \$38.02 as the stock price in BS formula The BS European call price is \$1.763 Compare this to European call on stock without dividends: \$3.399 FS SP VD i v T P T 00 0 ,, ()= ( )
1-7 Applying BS to Other Assets (cont’d) Options on currencies ± The prepaid forward price for the currency is ± Where x is domestic spot rate and r f is foreign interest rate ± Example 12.4 x = \$0.92/ , K = \$0.9, σ = 0.10, r = 6%, T = 1, and δ = 3.2% The dollar-denominated euro call price is \$0.0606 The dollar-denominated euro put price is \$0.0172 Fx x e T P -r T f 00 , ()=

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1-8 Applying BS to Other Assets (cont’d) Options on futures ± The prepaid forward price for a futures contract is the PV of the futures price. Therefore where and ± Example 12.5: Suppose 1-yr. futures price for natural gas is \$2.10/MMBtu, r = 5.5% Therefore, F =\$2.10, K =\$2.10, and δ = 5.5% If σ = 0.25, T = 1, call price = put price = \$0.197721 CF ,K , ,r , t Fe Nd Ke Nd -rt -rt () = ( )( ) σ 12 [] d ln F / K T T 1 2 1 2 = σ dd T 21 =− σ
1-9 Option Greeks What happens to option price when one input changes? ± Delta ( ): change in option price when stock price increases by \$1 ± Gamma ( Γ ): change in delta when option price increases by \$1 ± V ega: change in option price when v olatility increases by 1% ± T heta ( θ ): change in option price when t ime to maturity decreases by 1 day ± R ho ( ρ ): change in option price when interest r ate increases by 1% Greek measures for portfolios ± The Greek measure of a portfolio is weighted average of Greeks of individual portfolio components ∆∆ portfolio i i i n = = ω 1

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1-10 Option Greeks (cont’d)
1-11 Option Greeks (cont’d)

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1-12 Option Greeks (cont’d)
1-13 Option Greeks (cont’d)

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1-14 Option Greeks (cont’d)
1-15 Option Greeks (cont’d)

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1-16 Option Greeks (cont’d)
1-17 Option Greeks (cont’d)

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winter2010_notes_part4 - Black-Scholes Formula Consider an...

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