# LEC11 - ACCG329 Lecture 11: The Greeks &amp; Lecture 11:...

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1 ACCG329 Lecture 11: The Greeks & ACCG329 Lecture 11: The Greeks & Value-at-Risk Semester 1, 2009 Egon Kalotay Delta (See Figure 17.2, page 361) •D e l t a ( Δ ) is the rate of change of the option price with respect to the underlying Option price A B Slope = Δ Stock price

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2 Delta Hedging • This involves maintaining a delta neutral portfolio The delta of a European call on a stock paying The delta of a European call on a stock paying dividends at rate q is N ( d 1 ) e qT • The delta of a European put is e qT [ N ( d 1 ) – 1] • The hedge position must be frequently rebalanced • Delta hedging a written option involves a “buy high, sell low” trading rule sell low trading rule • See Tables 17.2 (page 364) and 17.3 (page 365) for examples of delta hedging Using Futures for Delta Hedging The delta of a futures contract is ( r-q ) T • The delta of a futures contract is e times the delta of a spot contract • The position required in futures for delta hedging is therefore e - ( r-q ) T times the position required in the corresponding spot contract
3 Theta Theta Θ ) of a derivative (or portfolio of • Theta ( ) of a derivative (or portfolio of derivatives) is the rate of change of the value with respect to the passage of time • The theta of a call or put is usually negative. This means that, if time passes with the price of the underlying asset and its volatility of the underlying asset and its volatility remaining the same, the value of the option declines Gamma • Gamma ( Γ ) is the rate of change of delta Δ ) with respect to the price of the delta ( ) with respect to the price of the underlying asset • Gamma is greatest for options that are close to the money (see Figure 17.9, page 372)

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4 Gamma Addresses Delta Hedging Errors Caused By Curvature (Figure 17.7, page 369) Ca Call price C '' C ' S C Stock price S ' Interpretation of Gamma • For a delta neutral portfolio, ΔΠ ≈ Θ Δ t + ½ ΓΔ S 2 ΔΠ Δ S ΔΠ Δ S Negative Gamma Positive Gamma
5 Relationship Between Delta, Gamma, and Theta For a portfolio of derivatives on a stock paying a continuous dividend yield at rate q 1 Θ Δ Γ Π + + = () rq S S r 2 22 σ Vega & Rho • Vega ( ν ) is the rate of change of the value of a derivatives portfolio with respect to volatility – Vega tends to be greatest for options that are close to the money (See Figure 17.11, page 374) • Rho is the rate of change of the value of a derivative with respect to the interest rate – For currency options there are 2 rhos

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6 Managing Delta, Gamma, & Vega Δ
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## This note was uploaded on 08/13/2011 for the course ACCG 329 taught by Professor Egonkalotay during the Three '09 term at Macquarie.

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LEC11 - ACCG329 Lecture 11: The Greeks &amp; Lecture 11:...

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