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 Delta Hedging
Issued on September 23 2010 par Strategies Options
Delta Hedging is a very famous strategy to try to smooth the variation of the P&L of an option position.
Delta hedging is a broad concept, widly used with options trading.
The best way to grasp how it works out is to value the expected profit and variance from de delta hedged strategy. I - ATMF Straddle : A Naturally Delta Hedged Strategy We know that we can approximate the p&l of an option by continuously rebalancing an underlying portfolio, keeping the same delta of that option. Looking at that portfolio would have been the same as looking at the option itself. It leads to take a look at a continously delta hedged portfolio is the same as looking at a portfolio made of both options, a call and a put. A straddle is a perfect example. II - Expected profit from an ATMF straddle Given At The Money Forward Relationships 2 we know that the ATMF option value expectation is : E[Value] = ( 1 / ( √ ( 2 п ) ) ) σ . √ (T-t) . S "It's just today's option value !" With, t the starting date T the maturity S the underlying The profit expectation is then: E[Profit] = [ Final Value ] - [ Starting Value ] E[Profit] = [( 1 / ( √ ( 2 п ) ) ) (σ'). √ (T-t) . S] - [( 1 / ( √ ( 2 п ) ) ) (σ). √ (T-t) . S] E[Profit] = ( 1 / ( √ ( 2 п ) ) ) (σ' - σ). √ (T-t) . S "If σ is the implied volatility bought and σ' is the realized volatility". ATMF straddle Value is twice as for a call, because call and put have the same one. It leads to : E[Profit Straddle] = 2 . ( 1 / ( √ ( 2 п ) ) ) (σ' - σ) . √ (T-t) . S E[Profit Straddle] = ( √ [ ( 2 / п ) . (T-t) ] ) (σ' - σ) . S "If σ' > σ there is a gain, if σ' < σ this is a loss." III - Expected profit Variance for an ATMF straddle The variance of this expectation is : E[(Profit Straddle)²] - ( E[Profit Straddle] )² = (1-( 2 / п )) . σ'² . S² . ( T - t ) "The profit expectation variance doesn't depend on implied volatility. Just the realized one." Hedging : Principles Delta Hedging Principles Gamma Hedging Principles Vega Hedging Principles Hedging : A First Attempt Delta Hedging : A First Attempt Gamma Hedging : A First Attempt Hedging Delta Hedging Strategies Options |
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