Delta hedging is the process of setting or keeping the delta of a portfolio of financial instruments zero, or as close to zero as possible - where delta is the sensitivity of the value of a derivative to changes in the price of its underlying instrument; see Hedge (finance). Mathematically, delta is the partial derivative of the portfolio's fair value with respect to the price of the underlying security; see The Greeks. Being delta neutral (or, instantaneously delta-hedged) means that the instantaneous change in value of the portfolio for an infinitesimal change in the value of the underlying is zero. For other uses, see Delta. ... In finance, a portfolio is a collection of investments held by an institution or a private individual. ... Financial instruments package financial capital in readily tradeable forms - they do not exist outside the context of the financial markets. ... In finance, an underlying is an investment from which a derivative security is derived. ... Other uses of the word hedge. ... In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant. ... Definition Fair value, also called fair price, is a concept used in finance and economics. ... In mathematical finance, the Greeks are the quantities representing the market sensitivities of options or other derivatives, with each measuring a different aspect of the risk in an option position, and corresponding to the set of parameters on which the value of an instrument or portfolio of financial instruments is... Delta neutral refers to a portfolio containing options, that has been hedged so that its overall value will not change for small changes in the underliers price. ... In mathematics, an infinitesimal, or infinitely small number, is a number that is smaller in absolute value than any positive real number. ...

Keeping delta at zero is termed a "static delta hedge"; keeping delta close to zero is a "dynamic delta hedge". Delta constantly changes, thus, once the delta of a portfolio has been made zero by adjusting its holdings (typically in the underlying security for a portfolio of derivatives) it is zero only for that instant; delta neutrality is instantaneous. The term static delta hedge is therefore a misnomer and thus (re)setting delta to zero is often preferred. In dynamic delta hedging, the portfolio is readjusted regularly in order to reset the delta to zero. Between readjustments, the portfolio delta will deviate from zero.

In fact, the amount by which a hedge has to be adjusted to stay delta neutral is related to gamma, the second derivative of the portfolio value with respect to the price of the asset in question. For example, if a position is 'long gamma', i.e., has a positive gamma, an increase in the asset price will lead to a positive delta, and one will need to sell some of the asset to 'flatten' the delta. Similarly, a decrease in asset price will cause one to buy more of the asset. From this it is intuitively clear that a high volatility of the underlying asset will lead to trading profits. In mathematical finance, the Greeks are the quantities representing the market sensitivities of options or other derivatives, with each measuring a different aspect of the risk in an option position, and corresponding to the set of parameters on which the value of an instrument or portfolio of financial instruments is...

As above, a portfolio has to be adjusted continuously (i.e. infinitely often in any time interval) in order to maintain absolute delta neutrality. This idea plays an important part in the Black-Scholes model of option pricing; the present value of the expected cost of keeping a position in one option, the underlying asset (and cash) delta neutral is equal to the initial fair value (Black-Scholes price) of the option. For the underlying logic see the discussion at Rational pricing. The Black-Scholes model, often simply called Black-Scholes, is a model of the varying price over time of financial instruments, and in particular stocks. ... Rational pricing is the assumption in financial economics that asset prices (and hence asset pricing models) will reflect the arbitrage-free price of the asset as any deviation from this price will be arbitraged away. This assumption is useful in pricing fixed income securities, particularly bonds, and is fundamental to...

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Delta neutral Delta neutral refers to a portfolio containing options, that has been hedged so that its overall value will not change for small changes in the underliers price. ...