Suppose P is the value of a given portfolio This is likely to be a large

Suppose p is the value of a given portfolio this is

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Suppose P is the value of a given portfolio. This is likely to be a large combination of many derivatives positions, but for simplicity, we will assume all are written on the same underlying stock S . Then the Delta (Δ), Gamma (Γ) and Vega ( V ) of the portfolio are defined as Δ( P ) = ∂P ∂S , Γ( P ) = 2 P ∂S 2 , V ( P ) = ∂P ∂σ , that is, the sensitivities of the value of the portfolio and the Delta to a small change in the stock price (Delta and Gamma, respectively), and the sensitivity of the value to a small change in the volatility (Vega). Usually, these “Greeks” are computed using the Black-Scholes formulas for the derivatives in the portfolio. For example, for a call option C with time to maturity T , we have Δ( C ) = N ( d 1 ) , Γ( C ) = N ( d 1 ) T , V ( C ) = S T N ( d 1 ) , where S is the current stock price, d 1 is as in the usual Black-Scholes notation, and N is the standard normal density. Delta hedging (meaning holding Δ( P ) stocks) comes from the Black-Scholes theory: even though the hedge would not be perfect under real market conditions even if carried out continually, it is assumed to be approximately effective. Gamma hedging lessens the risk of Delta hedging infrequently: if the Gamma of a portfolio is lowered, then the hedge slippage 72

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caused by updating the Delta hedge after (say) a week will be lessened. The reason for Vega hedging is to reduce the impact of volatility changes on portfolio value. As an example, suppose we have portfolio that is already Delta-neutral (Δ( P ) = 0), but with Γ( P ) = 5000 , V ( P ) = 8000 . Two derivative securities will be needed to both Gamma- and Vega-neutralize this portfolio. In the market we see two derivatives with values C 1 and C 2 and Greeks Γ( C 1 ) = 0 . 5 , V ( C 1 ) = 2 , Δ( C 1 ) = 0 . 6 and Γ( C 2 ) = 0 . 8 , V ( C 2 ) = 1 . 2 , Δ( C 2 ) = 0 . 5 . We create a new portfolio Π by adding x shares of derivative C 1 and y of C 2 : Π = P + xC 1 + yC 2 . We want the Gamma of the new portfolio to be zero: Γ(Π) = 5000 + 0 . 5 x + 0 . 8 y = 0 , (46) and also the Vega: V (Π) = 8000 + 2 x + 1 . 2 y = 0 . (47) Solving (46) and (47) for x and y gives x = 400 , y = 6000 . However the portfolio Π is no longer Delta-neutral. To achieve this, but without messing up the Gamma- and Vega-neutrality, we use the underlying stock S because Γ( S ) = 0 , V ( S ) = 0 , but Δ( S ) = 1. We have Δ(Π) = 0 . 6 × 400 + 0 . 5 × 6000 = 3240 . Therefore, the last step, in order to Delta hedge the portfolio, is to sell 3240 stocks. 73
• Fall '11
• COULON
• Stochastic volatility, Smile Curve

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