Theta is one of the Greek factor sensitivities used by traders to measure exposures in derivatives portfolios. It measures a portfolio's linear exposure to the passage of time. Specifically, it tells you how rapidly a portfolio's market value will change with time, assuming that all market variables—underliers, implied volatilities, interest rates, etc.—do not change. Let t denote time, and let tp denote the portfolio's value at time t (see the notation conventions documentation). Formally, theta is the partial derivative of the portfolio's value with respect to time:
where the derivative is evaluated at the current time, t = 0. This technical definition leads to an approximation for the behavior of a portfolio.
where
Suppose a portfolio has a theta of –1.1MM JPY/year. If all market variables remain constant, over a single week, the portfolio should lose about
Portfolios which have positive gamma or vega usually have negative theta. If other market variables remain constant, they will lose value over time. Portfolios which have negative gamma or vega usually have positive theta. If other market variables remain constant, they will gain value over time.
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is a small interval of time and
is the change in the portfolio's value that will occur during that
interval, assuming all market variables remain the same.
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