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# Delta hedge

A **Delta-hedge** or **Δ-hedge** is a self financing portfolio consisting of the underlying asset,S say, and a standard bond, B say, that replicates (or in some cases approximates) the value of a derivative, X say. Let F(t,s) be price at time t
of the derivative X given that S(t)=s. Let V(t) be the value of the self-financing portfolio at time t, i.e. V(t)=a(t)S(t)+b(t)B(t), where a(t) is the number of the underlying asset S and b(t) is the number of bonds B in the portfolio. If the market is complete we should have that a(t)=(∂/∂s)F(t,s)|S(t)=s and b(t)=(F(t,S(t))-S(t)a(t))/B(t) at all times to perfectly replicate the contract X. So the number of the underlying assets S should equal the Δ for the contract X. This is the reason for the name Δ-hedge. A short position
in the derivative X and a long position in the hedging portfolio V would therfore have Δ=0; such a position is called Δ-neutral. In order to keep this position Δ-neutral we must continuously trade on the market. This is not possible in practice so one trade at discrete time points and reset the position to Δ-neutral at theese time points. This will however lead to that the self-financing portfolio will not exactly replicate the value of X, F(t,S(t)) anymore and b(t) will no longer equal (F(t,S(t))-S(t)a(t))/B(t). But for many contracts trading once a day will not lead to large hedging errors. To decrease the hedging errors it is sometimes possible to add more assets to the hedging portfolio such as other derivatives (usually some liguid put or call options). It is the possible to make the portfolio both Δ and Γ neutral,
so called Delta-gamma hedging, or Δ and Ѵ neutral so called d Delta-vega hedging. This makes it possible to trade less frequently and still keeping the hedging errors small.
## References

M. Brode'n and M. Wiktorsson. (2009) Hedging Errors Induced by Discrete Trading under an Adaptive Trading Strategy. Preprint.

M. Brode'n and M. Wiktorsson. (2009) On the Convergence of Higher Order Hedging Schemes. Preprint

E. Gobet and E. Teman (2001). Discrete time hedging errors for options with irregular pay-offs.*Finance and Stochastics*, **5**(3), 357-367.

A. Makhlouf and E Gobet. (2009). The tracking error rate of the Delta-Gamma hedging strategy. Preprint

M. Brode'n and M. Wiktorsson. (2009) On the Convergence of Higher Order Hedging Schemes. Preprint

E. Gobet and E. Teman (2001). Discrete time hedging errors for options with irregular pay-offs.

A. Makhlouf and E Gobet. (2009). The tracking error rate of the Delta-Gamma hedging strategy. Preprint

Questions: Magnus Wiktorsson

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