Vega

Explained: kappa vega

Vega (sometimes called kappa) is one of the Greek factor sensitivities used by traders to measure exposures in derivatives portfolios. Portfolios that hold options—either directly or embedded in instruments held by the portfolio—are sensitive to the implied volatilities of the underliers. In general, a long option position will benefit from rising implied volatilities and suffer from declining implied volatilities. Short option positions display opposite behavior.

Mathematically, vega is defined in much the same way as is delta. Like delta, it is a linear approximation for the sensitivity of the market value of a portfolio. The difference is that delta measures sensitivity to an underlier; whereas vega measures sensitivity to its implied volatility.

Example: Portfolio Value as a Function of Implied Volatility Exhibit 1

The market value in USD MM of a hypothetical portfolio as a function of implied volatility. The portfolio is clearly long options because its value increases with increasing implied volatility. Implied volatility is currently 20%. A tangent line has been fit to the graph at that point. The slope of the tangent line is the portfolio's vega.

Exhibit 1 illustrates how the value of a portfolio holding a long options position might respond to changes in implied volatility. A tangent line has been fit to the curve at the current volatility, which is 20%. The slope of that line is the option's vega.

Fitting tangent lines to functions is the province of calculus, so we turn to calculus for the formal definition of vega. Let 0 be the current time. Let and be current values for the portfolio and underlier (see the notation conventions documentation). Vega is the first partial derivative of a portfolio's value with respect to the value of implied volatility:

[1]

This technical definition leads to an approximation for the behavior of a portfolio.

[2]

Here is a small change in the implied volatility from its current value, and Δ⁰p is the corresponding change in the portfolio's value.

Suppose a portfolio has a vega of EUR 2.4MM. If the implied volatility rises 1% (that is, = 0.01), then the portfolio's value will increase by EUR 24,000.

If a portfolio holds options on different underliers, it will have a different vega for each of the implied volatilities.

Related Internal Links
delta and gamma Factor sensitivities measuring a portfolio's first and second order (linear and quadratic) sensitivity to the value of an underlier. derivative instrument An instrument which derives its value from the value of other financial instruments. Article includes a list of vanilla and exotic derivatives. Greeks A set of factor sensitivities, which includes vega. option pricing theory The body of financial theory used by financial engineers to value options and other derivative instruments. option spreads Positions combining one or more options in a single underlier. put-call parity A formula that relates the price of a put to the price of a corresponding call. rho Factor sensitivity measuring a portfolio's first order (linear) sensitivity to the risk-free rate. theta Factor sensitivity measuring a portfolio's first order (linear) sensitivity to the passage of time
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Related Books
Natenberg (1994) and Taleb (1996) discuss vega in the context of trading. Natenberg is introductory. Taleb is a sophisticated book for professional derivatives traders. Option Volatility & Pricing Sheldon Natenberg quality   technical   1994   Dynamic Hedging Nassim Taleb quality   technical   1996
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