Derivatives09-5 - The Greek Letters

Derivatives09-5- - Session 6 The Greek Letters Example(Page 325 • A bank has sold for $300,000 a European call option on 100,000 shares of a

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Unformatted text preview: Session 6 The Greek Letters Example (Page 325) • A bank has sold for $300,000 a European call option on 100,000 shares of a non-dividend- paying stock • S = 49, K = 50, r = 5%, σ = 20%, T = 20 weeks, μ = 13% • The Black-Scholes value of the option is $240,000 • How does the bank hedge its risk? Naked & Covered Positions Naked position Take no action Covered position Buy 100,000 shares today Both strategies leave the bank exposed to significant risk Stop-Loss Strategy This involves: • Buying 100,000 shares as soon as price reaches $50 • Selling 100,000 shares as soon as price falls below $50 This deceptively simple hedging strategy does not work well Relation between delta, gamma, theta ∂ ∂ ∂ ∂ ∂ ∂ σ f t rS f S f S S rf + + = 1 2 2 2 2 2 • Remember PDE: Theta Delta Gamma Fundamental determinants of option value Call value Put Value Current asset price S Delta C 0 < Delta < 1 C- 1 < Delta < Striking price K K K Interest rate r Rho C C Dividend yield q K K Time-to-maturity T Theta C ? Volatility Vega C C Delta (See Figure 15.2, page 329) • Delta ( ∆ ) is the rate of change of the option price with respect to the underlying Option price A B Slope = ∆ Stock price Delta Hedging • This involves maintaining a delta neutral portfolio • The delta of a European call on a non-dividend- paying stock is N ( d 1 ) • The delta of a European put on the stock is [ N ( d 1 ) – 1] Delta Hedging continued • The hedge position must be frequently rebalanced • Delta hedging a written option involves a “buy high, sell low” trading rule • See Tables 15.2 (page 332) and 15.3 (page 333) for examples of delta hedging Example BLACK-SCHOLES OPTION PRICING FORMULA A.Farber Stock price 100 Call Put Dividend yield 0.00% Decomposition of value Striking price 100 Intrinsic val. 0.00 0.00 Maturity (days) 365 Time value 4.88-4.88 Interest rate 5.00% Insurance 5.57 10.45 Volatility 20.00% BS partial differential equation Theta-6.41-1.66 Call Put (r-q)SDelta 3.18-1.82 Price 10.451 5.574 0.5 σ ²S²Gamma 3.75 3.75 Delta 0.6370....
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This note was uploaded on 02/21/2010 for the course FINA 221 taught by Professor Na during the Spring '09 term at HKUST.

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Derivatives09-5- - Session 6 The Greek Letters Example(Page 325 • A bank has sold for $300,000 a European call option on 100,000 shares of a

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