Rho

简要解答

Rho measures how much an option's price changes when interest rates move by one percentage point. A call with a rho of 0.08 gains about $0.08 per share if rates rise from 4% to 5%; a put with a rho of −0.06 loses about $0.06 on the same move.

Rho is the quietest of the greeks for most listed options. Interest rates move rarely and gradually compared with stock prices and implied volatility, and short-dated options barely respond when they do.

It matters most for long-dated contracts, where a rate change has years to act on the price. Like the other greeks on Options Band, rho comes from Cboe's delayed feed.

深入了解

Compare two calls on the same $100 stock, both struck at $100. The one-month call might show a rho of 0.03: if rates rise from 4% to 5%, it gains about $0.03 per share — a rounding error next to daily price swings. The one-year call might show a rho of 0.40, so the same rate move adds about $0.40 to an option that might cost $10.00 — a full 4% of its price.

The signs follow from what an option lets its holder defer. A call fixes a future purchase price while the cash sits earning interest in the meantime, so higher rates make calls slightly more valuable: positive rho. A put fixes a future sale price, delaying the receipt of cash, so higher rates weigh on puts: negative rho.

Dividends push in the opposite direction and are a separate input to the model: expected dividends lower call values and raise put values, and a surprise dividend change can move long-dated options in a way none of the displayed greeks predicted.

In practice, rho earns attention in two situations: options with a year or more to expiration, and periods when central banks are moving rates quickly. Elsewhere its effect is usually swamped by theta and vega.

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Rho is the partial derivative of option value with respect to the risk-free interest rate, scaled to a one-percentage-point change.

ρ = ∂V/∂r

Under Black-Scholes, European call rho is positive and put rho negative, with magnitude growing with time to expiration and with depth in the money — a deep in-the-money call behaves like deferred stock ownership and carries nearly the full interest sensitivity of the strike's discounted value.

Limitations: the model assumes one flat rate, while real financing rates vary by term, so a single rho understates the differences across expirations. For American options, early-exercise behavior itself shifts with rates — higher rates make early exercise of puts more attractive — which a simple rho does not capture. Sensitivity to dividend yield is a separate derivative, sometimes labeled epsilon or psi, and is not part of rho.

The common misconception is that rho can be ignored outright. For short-dated options that is roughly true; for multi-year options during a fast rate cycle, rho-sized effects reach the same order of magnitude as vega's.