Rho

Explained: rho

Rho is one of the Greek factor sensitivities used by traders to measure market risk exposures in derivatives portfolios. It measures a portfolio's linear exposure to changes in the risk-free interest rate.

Let and be current values for the portfolio and underlier (here superscripts 0 indicate the current time t = 0. See the notation conventions documentation). Formally, rho is the partial derivative of the portfolio's value with respect to the risk-free rate:

[1]

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

[2]

where is a small change in the risk-free rate, and is the corresponding change in the portfolio's value.

Suppose a portfolio has a rho of USD –2.3MM. If the risk free rate rises 5 basis points ( = .0005), the portfolio should lose about

For most portfolios, sensitivity to the risk-free rate is minor compared to possible sensitivities to underliers or implied volatilities. For this reason, rho is a less significant Greek.

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 rho. 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. theta Factor sensitivity measuring a portfolio's first order (linear) sensitivity to the passage of time vega Factor sensitivity measuring a portfolio's first order (linear) sensitivity to the implied volatility of an underlier.
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Related Books
Natenberg (1994) and Taleb (1996) discuss rho 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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